霍普夫分岔的肿瘤生长的数学模型的作用抑制剂和两个时间延迟</t我tle> <link rel="preload stylesheet" as="style" type="text/css" href="https://cdn.bibblio.org/rcm/4.28.0/bib-related-content.min.css"> <link rel="preload" href="https://static.hindawi.com/new/next_assets/2023-08-03-0017a694320dbe69a559e8120be98c/_next/static/css/b2661f2cd4bd618b.css" as="style"> <link rel="stylesheet" href="https://static.hindawi.com/new/next_assets/2023-08-03-0017a694320dbe69a559e8120be98c/_next/static/css/b2661f2cd4bd618b.css" data-n-g=""> <style id="__jsx-903e4b2e5d3241e4">div#__next{display:-webkit-box;display:-webkit-flex;display:-moz-box;display:-ms-flexbox;display:flex;min-height:100vh;-webkit-box-orient:vertical;-webkit-box-direction:normal;-webkit-flex-direction:column;-moz-box-orient:vertical;-moz-box-direction:normal;-ms-flex-direction:column;flex-direction:column}</style> </head> <body> <div id="__next"> <section class="ant-layout"> <div> <header class="ant-layout-header"> <div class="header__width-wrapper"> <div class="ant-anchor-wrapper ant-anchor-logo" style="max-height:100vh"> <div class="ant-anchor ant-anchor-fixed"> <div class="ant-anchor-ink"> <span class="ant-anchor-ink-ball"></span> </div> <div class="ant-anchor-link ant-link-logo"> <a class="ant-anchor-link-title" href="//www.newsama.com/" title=""> <div class="ant-image"> <img alt="标志" class="ant-image-img ant-image-logo" src="https://static.hindawi.com/new/next_assets/2023-08-03-0017a694320dbe69a559e8120be98c/public/logo.svg"> </div></a> </div> </div> </div> <div class="TopNavigation_topNavigationWrapper__QzFKB"> <div class="TopNavigation_rightSideMenuItems__0bKf1"> <ul class="ant-menu-overflow ant-menu ant-menu-root ant-menu-horizontal ant-menu-light header-ant-menu-horizontal" role="menu" tabindex="0" data-menu-list="true" data-testid="desktop-menu"> <li class="ant-menu-overflow-item ant-menu-item ant-menu-item-only-child" style="opacity:1;order:0" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/journals/" aria-label="Journals">期刊</一个></年代pan></li> <li class="ant-menu-overflow-item ant-menu-item ant-menu-item-only-child" style="opacity:1;order:1" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/publish-research/" aria-label="Publish with us">雷电竞下载苹果</a></span></li> <li class="ant-menu-overflow-item ant-menu-item ant-menu-item-only-child" style="opacity:1;order:2" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/publishing-partnerships/" aria-label="Publishing partnerships">raybet推荐吗</a></span></li> <li class="ant-menu-overflow-item ant-menu-item ant-menu-item-only-child" style="opacity:1;order:3" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/about/" aria-label="About us">关于我们</一个></年代pan></li> <li class="ant-menu-overflow-item ant-menu-item ant-menu-item-only-child" style="opacity:1;order:4" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/blog/" aria-label="Blog">博客</一个></年代pan></li> <li class="ant-menu-overflow-item ant-menu-overflow-item-rest ant-menu-submenu ant-menu-submenu-horizontal" style="opacity:0;height:0;overflow-y:hidden;order:9007199254740991;pointer-events:none;position:absolute" aria-hidden="true" role="none"> <div role="menuitem" class="ant-menu-submenu-title" tabindex="-1" aria-expanded="false" aria-haspopup="true"> <span role="img" aria-label="ellipsis" class="anticon anticon-ellipsis"> <svg viewbox="64 64 896 896" focusable="false" data-icon="ellipsis" width="1em" height="1em" fill="currentColor" aria-hidden="true"> <path d="M176 511a56 56 0 10112 0 56 56 0 10-112 0zm280 0a56 56 0 10112 0 56 56 0 10-112 0zm280 0a56 56 0 10112 0 56 56 0 10-112 0z"></path> </svg></span> <i class="ant-menu-submenu-arrow"></i> </div></li> </ul> <div style="display:none" aria-hidden="true"></div> <button data-testid="search-button-icon" aria-label="Search" type="button" class="ant-btn ant-btn-default SearchToggleButton_btn__kmpSM"><span role="img" class="anticon SearchToggleButton_icon__nurvY"> <svg width="1em" height="1em" viewbox="0 0 26 26" fill="none"> <path fill-rule="evenodd" clip-rule="evenodd" d="M22.1,23.813l-5.478-5.479a10.282,10.282,0,1,1,1.709-1.709L23.813,22.1a1.209,1.209,0,0,1-1.71,1.71ZM2.416,10.27A7.851,7.851,0,0,0,15.78,15.864l.041-.043.043-.041A7.853,7.853,0,1,0,2.416,10.27Z" fill="currentColor"></path> </svg></span></button> <div></div> <button data-testid="menu-button-icon" aria-label="Menu" type="button" class="ant-btn ant-btn-default MenuToggleButton_btn__Go00X"><span role="img" class="anticon MenuToggleButton_icon__5uVd5"> <svg width="1em" height="1em" viewbox="0 -2 28 28" fill="none"> <path fill-rule="evenodd" clip-rule="evenodd" d="M1.833,25.667a1.834,1.834,0,0,1,0-3.668H25.667a1.834,1.834,0,0,1,0,3.668Zm0-11a1.833,1.833,0,0,1,0-3.666H25.667a1.833,1.833,0,0,1,0,3.666Zm0-11A1.833,1.833,0,1,1,1.833,0H25.667a1.833,1.833,0,0,1,0,3.666Z" fill="currentColor"></path> </svg></span></button> <div></div> </div> </div> </div> </header> </div> <main class="ant-layout-content ArticleLayout_articleLayoutWrapper__oukM3"> <div> <div id="journal-header" class=" JournalNavigation_journalHeader__vGN2X" data-istranslated="false"> <a href="//www.newsama.com/journals/aaa" class="JournalNavigation_journalHeader__titleLink__sZyGb"><h1 id="journal-header-title" data-testid="journal-header-title" class="JournalNavigation_small__sXDLc JournalNavigation_journalHeader__title__fsEHD">抽象和应用分析</h1></一个><div class="JournalNavigation_navigationBarHorizontal__YojZL"> <div id="navigation-bar-horizontal" class="navigation-bar-horizontal"> <button id="Journal overview" type="button" class="ant-btn ant-btn-default ant-dropdown-trigger bordered"><span role="img" class="anticon"> <svg width="1em" height="1em" viewbox="0 0 32 32" fill="none"> <path fill-rule="evenodd" clip-rule="evenodd" d="M17.773 23.453l11.2-11.265a1.918 1.918 0 000-2.641 1.75 1.75 0 00-2.542 0l-9.929 10.101-9.933-10.1a1.75 1.75 0 00-2.542 0 1.919 1.919 0 000 2.64L15.23 23.453c.352.365.811.547 1.271.547.46 0 .92-.182 1.271-.547z" fill="currentColor"></path> </svg></span><span>日报》概述</年代pan></button> <a class="bordered" href="//www.newsama.com/journals/aaa/guidelines">对于作者</一个><一个cl一个年代年代="bordered" href="//www.newsama.com/publish-research/reviewers">对于审查员</一个><一个cl一个年代年代="bordered" href="//www.newsama.com/publish-research/editors">的编辑器</一个><一个cl一个年代年代="bordered" href="//www.newsama.com/journals/aaa/contents/">表的内容</一个><button id="Special Issues" type="button" class="ant-btn ant-btn-default ant-dropdown-trigger"><span role="img" class="anticon"> <svg width="1em" height="1em" viewbox="0 0 32 32" fill="none"> <path fill-rule="evenodd" clip-rule="evenodd" d="M17.773 23.453l11.2-11.265a1.918 1.918 0 000-2.641 1.75 1.75 0 00-2.542 0l-9.929 10.101-9.933-10.1a1.75 1.75 0 00-2.542 0 1.919 1.919 0 000 2.64L15.23 23.453c.352.365.811.547 1.271.547.46 0 .92-.182 1.271-.547z" fill="currentColor"></path> </svg></span><span>特殊的问题</年代pan></button> </div> </div> </div> </div> <nav class="ant-breadcrumb ArticleLayout_breadcrumbs__hsc4o"> <ol> <li><span class="ant-breadcrumb-link"><a href="//www.newsama.com/journals/aaa/" aria-label="Abstract and Applied Analysis">抽象和应用分析</一个></年代pan><span class="ant-breadcrumb-separator">/</年代pan></li> <li><span class="ant-breadcrumb-link"><a href="//www.newsama.com/journals/aaa/contents/year/2011/" aria-label="2011">2011年</一个></年代pan><span class="ant-breadcrumb-separator">/</年代pan></li> <li><span class="ant-breadcrumb-link">文章</年代pan><span class="ant-breadcrumb-separator">/</年代pan></li> </ol> </nav> <div class="ant-row ThreeColumnLayout_wrapper__gPK_l"> <div class="ant-col ThreeColumnLayout_leftColumn__yqNgV"> <div> <div> <div id="article-table-of-contents" data-ismobile="false"> <div class="TableOfContents_tableOfContents__0jAuv"> <div class="TableOfContents_tableOfContents__title__CH9Lz"> 在这一页上</div> <a class="TableOfContents_tableOfContents__item__JJil1" href="#abstract">文摘</一个><一个cl一个年代年代="TableOfContents_tableOfContents__item__JJil1" href="#introduction">介绍</一个><一个cl一个年代年代="TableOfContents_tableOfContents__item__JJil1" href="#conclusion">结论</一个><一个cl一个年代年代="TableOfContents_tableOfContents__item__JJil1" href="#acknowledgments">确认</一个><一个cl一个年代年代="TableOfContents_tableOfContents__item__JJil1" href="#references">引用</一个><一个cl一个年代年代="TableOfContents_tableOfContents__item__JJil1" href="#copyright">版权</一个><一个cl一个年代年代="TableOfContents_tableOfContents__item__JJil1" href="#related-articles">相关文章</一个></div> </div> </div> </div> </div> <div class="ant-col ThreeColumnLayout_middleColumn__sbd41"> <div class="ArticleMiddleSection_articleMiddleSection__gAv9z"> <div class="articleHeader"> <strong>研究文章|开放获取</年代trong> <div class="articleHeader__meta"> <span>体积2011年|</年代pan> <span>文章的ID980686年|</年代pan> <span class="articleHeader__meta_doiLink"><a href="https://doi.org/10.1155/2011/980686" aria-label="Doi-link" target="_blank" rel="noreferrer">https://doi.org/10.1155/2011/980686</一个></年代pan> </div> <div class="simpleShowMore"> <button class="simpleShowMore__button">显示引用</button> </div> <h1 class="articleHeader__title">霍普夫分岔的肿瘤生长的数学模型的作用抑制剂和两个时间延迟</h1><div class="articleHeader__authors"> <span class="articleHeader__authors_author">宝施,<年代up>1</年代up></span> <span class="articleHeader__authors_author">Fangwei张,<年代up>1</年代up></span> <span class="articleHeader__authors_author">和<年代trong>Shihe徐</年代trong><a href="//www.newsama.com/cdn-cgi/l/email-protection" aria-label="Mail Option"><span role="img" class="anticon"> <svg version="1.0" xmlns="http://www.w3.org/2000/svg" width="1em" height="1em" viewbox="0 0 1401 1101"> <path fill="#6ba439" d="M132.5 2.1C101.4 6 71.1 20.5 48.3 42.4c-22.7 21.7-37.5 48.3-44.6 80l-2.2 10.1v837l2.2 10.1c6.7 30 19.3 53.9 39.8 75.3 23.3 24.3 56.4 40.9 90 45 12.8 1.6 1122.2 1.6 1135 0 38.9-4.8 75.9-25.6 100-56.4 15-19.3 24.2-39 29.8-63.9l2.2-10.1v-837l-2.2-10.1c-7.1-31.8-21.9-58.3-44.6-80.1-20.2-19.2-43.9-31.7-73.6-38.5l-9.6-2.3L705 1.4c-321.9-.1-568.5.2-572.5.7zm1123.4 99.5c1.7.4 3.1 1 3.1 1.4 0 .5-125.6 87.7-279 194L700.9 490.2 422 297C268.5 190.7 143 103.4 143 103c0-.5 1-1.1 2.3-1.3 3.5-.7 1107.1-.8 1110.6-.1zM401 404c164.4 113.8 299.4 207 300 207 .6 0 135.6-93.2 300.1-207 164.5-113.9 299.3-207 299.5-207 .2 0 .4 170.9.4 379.7 0 417.9.5 386-6.1 398.3-6.2 11.6-19 21.4-31.9 24.4-5.8 1.4-67.4 1.6-562 1.6s-556.2-.2-562-1.6c-12.9-3-25.7-12.8-31.9-24.4-6.6-12.3-6.1 19.6-6.1-398.3 0-208.8.2-379.7.5-379.7S236.6 290.1 401 404z"></path> </svg></span></a><sup>2</年代up></span> </div> <div class="simpleShowMore"> <button class="simpleShowMore__button">显示更多</button> </div> <div class="articleHeader__academicEditor"> <strong>学术编辑器:</年代trong> <span>艾伦·c·彼得森</年代pan> </div> <div class="articleHeader__timeline"> <div class="articleHeader__timeline_item "> <strong>收到了</年代trong> <span>2010年11月03</年代pan> </div> <div class="articleHeader__timeline_item "> <strong>接受</年代trong> <span>2011年1月25日</年代pan> </div> <div class="articleHeader__timeline_item articleHeader__timeline_item_sticky"> <strong>发表</年代trong> <span>2011年3月22日</年代pan> </div> </div> </div> <div class="articleBody"> <div class="xml-content"> <h4 class="header" id="abstract">文摘</h4><p>肿瘤生长的数学模型有两个离散延迟进行了研究。拖延,分别代表了时间的细胞进行有丝分裂和细胞所花费的时间来修改细胞的速度损失由于细胞凋亡和杀死细胞的抑制剂。我们展示霍普夫分岔时间延迟的影响,当延迟作为分岔参数之一。</p> <span class="end-abs"></span> <h4 id="introduction">1。介绍</h4><p>在过去四年内,越来越多的偏微分方程模型肿瘤生长或治疗已经发展;比较(<一个href="#B3">1</一个>- - - - - -<一个href="#B22">9</一个>)和引用引用。大部分的模型形式的自由边界问题。严格的数学分析这样的自由边界问题引起了极大的兴趣,和许多有趣的结果;比较(<一个href="#B2">10</一个>- - - - - -<一个href="#B24">20.</一个>)和引用引用。分析这样的自由边界问题不仅为肿瘤医学提供了一个良好的理论基础,而且还极大地丰富了恭敬的方程的理解。</p> <p>在本文中,我们研究肿瘤生长的数学模型有两个离散延迟。拖延,分别代表了时间的细胞进行有丝分裂和细胞所花费的时间来修改细胞的速度损失由于细胞凋亡和杀死细胞的抑制剂。该模型如下:<年代pan class="equation" id="EEq1.1"> <svg height="206.6125" id="M1" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 206.6125" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,206.6125)"> <g transform="translate(72,93.29)"> <text transform="matrix(1,0,0,-1,113.1,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Δ</t年代pan> </text> <text transform="matrix(1,0,0,-1,122.12,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,126.19,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.802592" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.842981" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,156.38,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,161.26,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.3267579" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="32.457787" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="42.185123" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="54.22551" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="62.80257" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="74.842957" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="86.170677" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="90.334175" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="94.247612" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="100.49911" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="115.66525" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="123.05453" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="135.09491" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="141.34642" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="303.48999" y="0.63"> (</t年代pan> <tspan style="font-size: 12.50px; " x="307.6535" y="0.63"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="313.905" y="0.63"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="317.03076" y="0.63"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="323.28226" y="0.63"> )</t年代pan> <tspan style="font-size: 12.50px; " x="-60.07" y="21.700001"> </t年代pan> <tspan style="font-size: 12.50px; " x="-53.69347" y="21.700001"> </t年代pan> </text> <g> <path d="m 101.19,37.38 13.73,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,102.31,25.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.3765302" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.89" y="-8.7600002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="20.053499" y="-8.7600002"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="26.304998" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="31.506247" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.419685" y="-8.7600002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="43.059017" y="-8.7600002"> =</t年代pan> <tspan style="font-size: 12.50px; " x="55.099407" y="-8.7600002"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="61.350906" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="86.481941" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="95.896698" y="-8.7600002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="100.0602" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="111.40041" y="-8.7600002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="115.56392" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="119.47736" y="-8.7600002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="125.72885" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="130.9301" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="134.84354" y="-8.7600002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="142.48288" y="-8.7600002"> =</t年代pan> <tspan style="font-size: 12.50px; " x="154.52327" y="-8.7600002"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,264.18,31.13)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,272.78,34.25)"> <tspan style="font-size: 12.50px; " x="-160.36" y="26.610001"> Δ</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="15.166139" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.555412" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="34.595802" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="40.847301" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="191.97" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="196.1335" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="202.38499" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="205.51074" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="211.76225" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,121.44,4.52)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,125.51,7.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.452508" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.480394" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,155.34,4.52)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,160.21,7.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.9766741" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="32.107704" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="41.835037" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="53.875427" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="62.439983" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="74.48037" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="85.820595" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="89.984093" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="93.89753" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="100.14903" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="115.31517" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="122.70444" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="134.74483" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="140.99634" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="304.54001" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="308.70349" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="314.95499" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="318.08075" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="324.33224" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="-58.48" y="21.07"> </t年代pan> <tspan style="font-size: 12.50px; " x="-52.090965" y="21.07"> </t年代pan> </text> <g> <path d="m 101.73,-18.6 13.37,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,102.67,-30.48)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.3890328" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.71" y="-8.7600002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="19.873499" y="-8.7600002"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="26.124998" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="31.326246" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.239685" y="-8.7600002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="42.879021" y="-8.7600002"> =</t年代pan> <tspan style="font-size: 12.50px; " x="54.919407" y="-8.7600002"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="61.17091" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="86.301941" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="95.366615" y="-8.7600002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="99.530113" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="110.85783" y="-8.7600002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="115.02133" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="118.93477" y="-8.7600002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="125.18627" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="130.40002" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="134.31346" y="-8.7600002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="141.94029" y="-8.7600002"> =</t年代pan> <tspan style="font-size: 12.50px; " x="153.98068" y="-8.7600002"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,263.64,-24.85)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,272.24,-21.73)"> <tspan style="font-size: 12.50px; " x="-291.85001" y="30.549999"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="15.166139" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.555412" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="34.583298" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="40.834797" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="192.50999" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="196.67349" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="202.925" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="206.05075" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="212.30225" y="0"> )</t年代pan> </text> <g> <path d="m -21.19,-56.4 10.56,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-21.19,-68.28)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.6515961" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.95" y="-16.969999"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="19.2015" y="-16.969999"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.803324" y="-16.969999"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,14.87,-46.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,21.83,-51.3)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.0769377" y="0"> )</t年代pan> </text> <g> <path d="m -8.24,-56.4 42.31,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,9.79,-68.28)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="56.919998" y="-1.05"> </t年代pan> <tspan style="font-size: 12.50px; " x="28.950001" y="-8.75"> =</t年代pan> <tspan style="font-size: 12.50px; " x="40.977886" y="-8.75"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="47.229385" y="-8.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,73.71,-45.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.47648" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="9.3908958" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.130272" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="18.125393" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,96.32,-47.69)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,99.94,-45.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 8.75px; " x="-27.48" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,105.44,-59.53)"> <tspan style="font-size: 12.50px; " x="16.190001" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.764123" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.040001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.141224" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="32.342472" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="39.031578" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="50.3843" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,162.23,-62.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,167.1,-62.04)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,180.14,-54.53)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,185.01,-59.53)"> <tspan style="font-size: 12.50px; " x="41.810001" y="7.6999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.6515961" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.528486" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="25.868708" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="32.120209" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,233.82,-45.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.47648" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="9.3908958" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.130272" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="18.125393" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,256.43,-47.69)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,260.06,-45.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 8.75px; " x="-27.49" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,265.55,-62.04)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.689999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="4.79" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.716662" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.566626" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.66905" y="-2.51"> +</t年代pan> <tspan style="font-size: 12.50px; " x="33.00927" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.623596" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="55.540001" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="60.641224" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="65.842468" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="72.544083" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="83.8843" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,355.84,-62.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,360.71,-62.04)"> <tspan style="font-size: 12.50px; " x="12.73" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,378.54,-54.53)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,383.41,-59.53)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.6515961" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.75282" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="26.918959" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.308231" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="46.348621" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="52.600121" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="81.332016" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="85.495514" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="91.747017" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="94.872765" y="0"> 5</t年代pan> <tspan style="font-size: 12.50px; " x="101.12427" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="-248.11" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="-236.78229" y="29.34"> (</t年代pan> <tspan style="font-size: 12.50px; " x="-232.61879" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="-228.70534" y="29.34"> )</t年代pan> <tspan style="font-size: 12.50px; " x="-221.06601" y="29.34"> =</t年代pan> <tspan style="font-size: 12.50px; " x="-209.02562" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="-198.62312" y="29.34"> (</t年代pan> <tspan style="font-size: 12.50px; " x="-194.45963" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="-190.54619" y="29.34"> )</t年代pan> <tspan style="font-size: 12.50px; " x="-184.29469" y="29.34"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-159.16365" y="29.34"> −</t年代pan> <tspan style="font-size: 12.50px; " x="-150.59911" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="-140.72173" y="29.34"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="-128.68135" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="-121.29207" y="29.34"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="-109.26418" y="29.34"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="-103.01269" y="29.34"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="81.339996" y="29.959999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="85.503502" y="29.959999"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="91.754997" y="29.959999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="94.880753" y="29.959999"> 6</t年代pan> <tspan style="font-size: 12.50px; " x="101.13225" y="29.959999"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代vg height="14.375" id="M7" style="vertical-align:-3.13504pt;width:15.3875px;" version="1.1" viewbox="0 0 15.3875 14.375" width="15.3875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.375)"> <g transform="translate(72,-60.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.375" id="M8" style="vertical-align:-3.13504pt;width:15.3875px;" version="1.1" viewbox="0 0 15.3875 14.375" width="15.3875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.375)"> <g transform="translate(72,-60.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>,<年代vg height="10.8125" id="M9" style="vertical-align:-0.20064pt;width:8.5749998px;" version="1.1" viewbox="0 0 8.5749998 10.8125" width="8.5749998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8125)"> <g transform="translate(72,-63.35)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="9.6750002" id="M10" style="vertical-align:-2.29482pt;width:9.6374998px;" version="1.1" viewbox="0 0 9.6374998 9.6750002" width="9.6374998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.675)"> <g transform="translate(72,-64.26)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="10.725" id="M11" style="vertical-align:-3.13504pt;width:20.049999px;" version="1.1" viewbox="0 0 20.049999 10.725" width="20.049999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.475" id="M12" style="vertical-align:-3.13504pt;width:19.6px;" version="1.1" viewbox="0 0 19.6 14.475" width="19.6" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.475)"> <g transform="translate(72,-60.42)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,60.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> </g> </g> </svg>,<年代vg height="12.0125" id="M13" style="vertical-align:-0.17555pt;width:9.3000002px;" version="1.1" viewbox="0 0 9.3000002 12.0125" width="9.3000002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.0125)"> <g transform="translate(72,-62.39)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="10.725" id="M14" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="10.725" id="M15" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>都是正的常数,<年代vg height="9.875" id="M16" style="vertical-align:-2.29482pt;width:10.5375px;" version="1.1" viewbox="0 0 10.5375 9.875" width="10.5375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是一个给定的积极作用。<年代vg height="17.674999" id="M17" style="vertical-align:-3.13504pt;width:171.91251px;" version="1.1" viewbox="0 0 171.91251 17.674999" width="171.91251" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.675)"> <g transform="translate(72,-57.86)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Δ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.92,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-55.38,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="25.918718" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-24.35,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.48,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="8.3269978" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.716031" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="18.191866" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="24.580898" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.682121" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="33.845619" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="38.009121" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,23.63,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,28.51,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.540029" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="13.665779" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="17.141613" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.530645" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="28.63187" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="32.795368" y="0"> )</t年代pan> </text> </g> </g> </svg>。这个词<年代vg height="14.375" id="M18" style="vertical-align:-3.13504pt;width:24.5625px;" version="1.1" viewbox="0 0 24.5625 14.375" width="24.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.375)"> <g transform="translate(72,-60.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-59.74,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>在(<一个href="#EEq1.1">1.1</一个>)是在单位体积消耗的营养;<年代vg height="14.475" id="M19" style="vertical-align:-3.13504pt;width:24.112499px;" version="1.1" viewbox="0 0 24.112499 14.475" width="24.112499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.475)"> <g transform="translate(72,-60.42)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-59.74,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>在(<一个href="#EEq1.1">1.3</一个>)抑制剂在单位体积的消费率;<年代vg height="10.725" id="M20" style="vertical-align:-3.13504pt;width:20.049999px;" version="1.1" viewbox="0 0 20.049999 10.725" width="20.049999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> </g> </g> </svg>反映了肿瘤得到充足的营养,从其表面接收;<年代vg height="14.475" id="M21" style="vertical-align:-3.13504pt;width:19.6px;" version="1.1" viewbox="0 0 19.6 14.475" width="19.6" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.475)"> <g transform="translate(72,-60.42)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,60.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> </g> </g> </svg>反映了不断供应肿瘤从其表面的抑制剂。<年代vg height="10.725" id="M22" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> </g> </g> </svg>代表细胞进行有丝分裂,所花费的时间<年代vg height="10.725" id="M23" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>代表细胞所花费的时间来修改细胞的速度损失由于细胞凋亡和杀死细胞的抑制剂。右边的两个术语(<一个href="#EEq1.1">1.5</一个>)解释如下:第一项是总量增加单位时间间隔诱导细胞增殖;<年代vg height="10.8125" id="M24" style="vertical-align:-0.20064pt;width:17.75px;" version="1.1" viewbox="0 0 17.75 10.8125" width="17.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8125)"> <g transform="translate(72,-63.35)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.764123" y="0"> </t年代pan> </text> </g> </g> </svg>单位体积的细胞增殖率。第二项是总在单位体积收缩时间间隔由细胞凋亡(细胞死亡由于老化)和杀死细胞的抑制剂;细胞凋亡率是假定为常数;<年代vg height="12.1" id="M25" style="vertical-align:-0.20064pt;width:17.75px;" version="1.1" viewbox="0 0 17.75 12.1" width="17.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.1)"> <g transform="translate(72,-62.32)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.926662" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.764123" y="0"> </t年代pan> </text> </g> </g> </svg>不依赖于<年代vg height="6.9875002" id="M26" style="vertical-align:-0.13794pt;width:9.3000002px;" version="1.1" viewbox="0 0 9.3000002 6.9875002" width="9.3000002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,6.9875)"> <g transform="translate(72,-66.41)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>或<年代vg height="13.425" id="M27" style="vertical-align:-2.29482pt;width:8.8500004px;" version="1.1" viewbox="0 0 8.8500004 13.425" width="8.8500004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。</p> <p>研究肿瘤生长的时间延迟的影响通过使用数学模型的方法是由伯恩<一个href="#B3">1</一个>]。这项研究最近吸引了其他研究人员的注意;比较·博德纳尔和Foryś[<一个href="#B2">10</一个>),Fory和·博德纳尔<一个href="#B11">15</一个>],Foryś和Kolev [<一个href="#B12">16</一个>Banerjee), Sarkar和(<一个href="#B20">7</一个>有一个延迟;和比较Piotrowska [<一个href="#B19">18</一个>,徐<一个href="#B23">19</一个>有两个延迟。这个数学模型建立了通过修改模型,伯恩(<一个href="#B4">2</一个>)通过考虑两个独立的时间延迟效应(<一个href="#B19">18</一个>,<一个href="#B23">19</一个>]。的修改是基于生物因素,看到崔和徐<一个href="#B9">14</一个>详情)。在[<一个href="#B9">14</一个>),作者研究了问题(<一个href="#EEq1.1">1.1</一个>)- (<一个href="#EEq1.1">1.6</一个>)只有一个延迟扩散,也就是说,<年代vg height="14.6" id="M28" style="vertical-align:-3.13504pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 14.6" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>,显示解决方案的动态行为模型的延迟扩散相似的解决方案对应nondelayed问题。本文的目的是探讨影响霍普夫分岔的时间延迟<年代vg height="10.725" id="M29" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>作为分岔参数。</p> <p>表示<年代vg height="20.4125" id="M30" style="vertical-align:-4.12569pt;width:87.487503px;" version="1.1" viewbox="0 0 87.487503 20.4125" width="87.487503" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.4125)"> <g transform="translate(72,-55.67)"> <text transform="matrix(1,0,0,-1,-71.95,59.83)"> <tspan style="font-size: 12.50px; " x="22.059999" y="0.87"> √</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> =</t年代pan> </text> <g> <path d="m -38.28,70.71 36.23,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-38.28,59.83)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.78,56.71)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-21.91,59.83)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.09,56.71)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-6.22,59.83)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>。通过重新调节空间变量,我们可以假设<年代vg height="14.6" id="M31" style="vertical-align:-3.13504pt;width:42.599998px;" version="1.1" viewbox="0 0 42.599998 14.6" width="42.599998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.26,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 1</t年代pan> </text> </g> </g> </svg>。因此,我们有<年代vg height="20.4125" id="M32" style="vertical-align:-4.21346pt;width:57.474998px;" version="1.1" viewbox="0 0 57.474998 20.4125" width="57.474998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.4125)"> <g transform="translate(72,-55.67)"> <text transform="matrix(1,0,0,-1,-71.95,59.92)"> <tspan style="font-size: 12.50px; " x="22.059999" y="0.95999998"> √</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> =</t年代pan> </text> <g> <path d="m -38.28,70.71 12.21,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-38.28,59.92)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.95,56.8)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>。解决方案(<一个href="#EEq1.1">1.1</一个>)- (<一个href="#EEq1.1">1.4</一个>)是<年代pan class="equation" id="EEq1.7"> <svg height="35.825001" id="M33" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 35.825001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,35.825)"> <g transform="translate(72,-43.34)"> <text transform="matrix(1,0,0,-1,46.5,54.35)"> <tspan style="font-size: 12.50px; " x="53.209999" y="-9.1400003"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.4147587" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.578258" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.679482" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="23.893232" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.806671" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="35.433502" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,107.05,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,115.65,63.49)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.50372" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> )</t年代pan> </text> <g> <path d="m 95.18,57.48 48.59,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,95.18,45.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="4.863667" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="8.3395014" y="0"> n</t年代pan> <tspan style="font-size: 12.50px; " x="14.591001" y="0"> h</t年代pan> <tspan style="font-size: 12.50px; " x="25.006001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.333717" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="40.497215" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.410656" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="50.98" y="-16"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="55.843666" y="-16"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="59.3195" y="-16"> n</t年代pan> <tspan style="font-size: 12.50px; " x="65.570999" y="-16"> h</t年代pan> <tspan style="font-size: 12.50px; " x="75.986" y="-16"> </t年代pan> </text> <g> <path d="m 146.16,57.48 30.11,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,158.66,45.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="105.45" y="-17.889999"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.799999" y="-8.75"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="43.93103" y="-8.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="52.995705" y="-8.75"> (</t年代pan> <tspan style="font-size: 12.50px; " x="57.159203" y="-8.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="62.260429" y="-8.75"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="67.461678" y="-8.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="71.375114" y="-8.75"> )</t年代pan> <tspan style="font-size: 12.50px; " x="79.01445" y="-8.75"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,271.09,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,279.69,63.49)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.327718" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.491217" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.404655" y="0"> )</t年代pan> </text> <g> <path d="m 250.92,57.48 65.55,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,250.92,45.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="4.863667" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="8.3395014" y="0"> n</t年代pan> <tspan style="font-size: 12.50px; " x="14.591001" y="0"> h</t年代pan> <tspan style="font-size: 12.50px; " x="27.081497" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="31.244997" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.79657" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.136791" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="53.300289" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.21373" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="61.377228" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="67.940002" y="-16.969999"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="72.803665" y="-16.969999"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="76.279503" y="-16.969999"> n</t年代pan> <tspan style="font-size: 12.50px; " x="82.530998" y="-16.969999"> h</t年代pan> <tspan style="font-size: 12.50px; " x="95.0215" y="-16.969999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="99.184998" y="-16.969999"> </t年代pan> <tspan style="font-size: 12.50px; " x="105.73657" y="-16.969999"> </t年代pan> <tspan style="font-size: 12.50px; " x="110.83779" y="-16.969999"> )</t年代pan> </text> <g> <path d="m 318.86,57.48 47.07,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,339.84,45.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.290001" y="-8.75"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="124.91" y="-8.9499998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="129.0735" y="-8.9499998"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="135.325" y="-8.9499998"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="138.45074" y="-8.9499998"> 7</t年代pan> <tspan style="font-size: 12.50px; " x="144.70226" y="-8.9499998"> )</t年代pan> </text> </g> </g> </svg></span>用(<一个href="#EEq1.7">1.7</一个>)(<一个href="#EEq1.1">1.6</一个>我们获得<年代pan class="equation" id="EEq1.8"> <svg height="52.0625" id="M34" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 52.0625" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,52.0625)"> <g transform="translate(72,-30.35)"> <text transform="matrix(1,0,0,-1,-45.66,49.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="55.939999" y="-7.7199998"> ⎡</t年代pan> <tspan style="font-size: 12.50px; " x="55.939999" y="1.24"> ⎢</t年代pan> <tspan style="font-size: 12.50px; " x="55.939999" y="10.2"> ⎢</t年代pan> <tspan style="font-size: 12.50px; " x="55.939999" y="19.1"> ⎣</t年代pan> <tspan style="font-size: 12.50px; " x="-8.8000002" y="2.53"> </t年代pan> <tspan style="font-size: 12.50px; " x="2.5277181" y="2.53"> (</t年代pan> <tspan style="font-size: 12.50px; " x="6.6912169" y="2.53"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.604656" y="2.53"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.24399" y="2.53"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.271875" y="2.53"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.612095" y="2.53"> (</t年代pan> <tspan style="font-size: 12.50px; " x="45.775597" y="2.53"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.689034" y="2.53"> )</t年代pan> <tspan style="font-size: 12.50px; " x="61.57" y="2.53"> </t年代pan> <tspan style="font-size: 12.50px; " x="68.334122" y="2.53"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,30.01,43.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,38.62,46.97)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1899996" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.04" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.379999" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.23" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.919106" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="49.259327" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,94.29,43.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,99.16,44.45)"> <tspan style="font-size: 12.50px; " x="13.79" y="7.2199998"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.09" y="-13.09"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.419998" y="-10.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.279999" y="-13.09"> </t年代pan> <tspan style="font-size: 12.50px; " x="48.9566" y="-13.09"> −</t年代pan> <tspan style="font-size: 12.50px; " x="60.309326" y="-13.09"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,165.87,54.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,170.75,55.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <g> <path d="m 124.25,50.09 52.35,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,138.63,38.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="39.16" y="0.98000002"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.34" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.503499" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.416939" y="-0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,187.9,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,195.55,46.97)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.34" y="4.9200001"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.709999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.311825" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,228.85,43.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,237.45,46.97)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1999998" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.05" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.389999" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.24" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.916603" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="49.269325" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,293.12,43.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,298,44.45)"> <tspan style="font-size: 12.50px; " x="14.48" y="-2.52"> +</t年代pan> <tspan style="font-size: 12.50px; " x="27.02" y="-9.7700005"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> </text> <g> <path d="m 325.02,50.09 6.25,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,325.02,38.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="21.549999" y="-3.8399999"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.23" y="0.98000002"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.529999" y="-19.33"> </t年代pan> <tspan style="font-size: 12.50px; " x="53.869999" y="-16.82"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.4400001" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.366662" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.216626" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="59.720001" y="-19.33"> </t年代pan> <tspan style="font-size: 12.50px; " x="66.409103" y="-19.33"> −</t年代pan> <tspan style="font-size: 12.50px; " x="77.749329" y="-19.33"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,409.18,54.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,414.05,55.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <g> <path d="m 367.55,50.09 52.35,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,381.94,38.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="39.16" y="0.98000002"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.33" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.493499" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.406939" y="-0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,431.2,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,436.07,57.22)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ⎤</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="8.96"> ⎥</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="17.92"> ⎥</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="26.82"> ⎦</t年代pan> <tspan style="font-size: 12.50px; " x="7.71" y="10.25"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="28.68" y="9.1700001"> (</t年代pan> <tspan style="font-size: 12.50px; " x="32.843498" y="9.1700001"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="39.094997" y="9.1700001"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="42.220749" y="9.1700001"> 8</t年代pan> <tspan style="font-size: 12.50px; " x="48.472248" y="9.1700001"> )</t年代pan> </text> </g> </g> </svg></span>在这里<年代vg height="16.637501" id="M35" style="vertical-align:-2.29482pt;width:152.9875px;" version="1.1" viewbox="0 0 152.9875 16.637501" width="152.9875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.6375)"> <g transform="translate(72,-58.69)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1139669" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.277466" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.154116" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="24.780947" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="36.821335" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="40.984833" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.949486" y="0"> c</t年代pan> <tspan style="font-size: 12.50px; " x="55.500816" y="0"> o</t年代pan> <tspan style="font-size: 12.50px; " x="61.752316" y="0"> t</t年代pan> <tspan style="font-size: 12.50px; " x="65.228149" y="0"> h</t年代pan> <tspan style="font-size: 12.50px; " x="75.64315" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="85.295464" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="96.635689" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="102.88718" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="107.05069" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="110.53902" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,45.47,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>。集<年代vg height="16.5375" id="M36" style="vertical-align:-2.21957pt;width:78.412498px;" version="1.1" viewbox="0 0 78.412498 16.5375" width="78.412498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5375)"> <g transform="translate(72,-58.77)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.5145426" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.678042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.59148" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="24.218311" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="36.258701" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.43,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,-21.56,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.0769377" y="0"> )</t年代pan> </text> </g> </g> </svg>,我们有<年代pan class="equation" id="EEq1.9"> <svg height="19.475" id="M37" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.475" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.475)"> <g transform="translate(72,-56.42)"> <text transform="matrix(1,0,0,-1,0.61,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="-7.1267099" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="3.463331" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.540269" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="19.167099" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="31.207487" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="37.458988" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.22311" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,52.18,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,60.78,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1999998" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.05" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,83.35,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,97.12,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.539105" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.879326" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,127.4,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,132.28,58.52)"> <tspan style="font-size: 12.50px; " x="13.78" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="24.379999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.23" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.919106" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="48.259327" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,186.95,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,191.83,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="19.969999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.82" y="-2.51"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="32.071499" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="39.685825" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,238.49,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,247.1,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1899996" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.04" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,269.66,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,283.43,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.539105" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.879326" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,313.71,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,318.59,58.52)"> <tspan style="font-size: 12.50px; " x="39.93" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.860001" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.459999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.48" y="-2.51"> +</t年代pan> <tspan style="font-size: 12.50px; " x="25.820221" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.746883" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.596848" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="64.309998" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="70.999107" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="82.339325" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,407.34,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,412.21,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="52.540001" y="-2.5599999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="56.703499" y="-2.5599999"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="62.954998" y="-2.5599999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="66.08075" y="-2.5599999"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="72.332253" y="-2.5599999"> )</t年代pan> </text> </g> </g> </svg></span>采用一步法(见,例如,<一个href="#B16">21</一个>]),我们可以很容易地显示,如果存在一个解决方案<年代vg height="14.75" id="M38" style="vertical-align:-3.25793pt;width:120.925px;" version="1.1" viewbox="0 0 120.925 14.75" width="120.925" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.75)"> <g transform="translate(72,-60.2)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="27.744158" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.733814" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="48.074036" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="54.325535" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="58.489033" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-7.05,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,-2.17,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.415239" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,15.65,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,20.53,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>,那么解决方案<年代vg height="14.75" id="M39" style="vertical-align:-3.25793pt;width:120.925px;" version="1.1" viewbox="0 0 120.925 14.75" width="120.925" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.75)"> <g transform="translate(72,-60.2)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.794649" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-35.75,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.87,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="9.364747" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.354404" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="29.694626" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="35.946125" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="40.109623" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,15.65,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,20.53,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>,在那里<年代vg height="10.75" id="M40" style="vertical-align:-0.33858pt;width:41.987499px;" version="1.1" viewbox="0 0 41.987499 10.75" width="41.987499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.75)"> <g transform="translate(72,-63.4)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6773224" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="21.71771" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="14.8625" id="M41" style="vertical-align:-3.25793pt;width:108.25px;" version="1.1" viewbox="0 0 108.25 14.8625" width="108.25" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.8625)"> <g transform="translate(72,-60.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 米</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="25.231054" y="0"> n</t年代pan> <tspan style="font-size: 12.50px; " x="35.658558" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="39.822056" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-10.98,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-6.1,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,5.51,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,10.38,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>定义的公式<年代pan class="equation" id="eq1"> <svg height="57.950001" id="M42" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 57.950001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,57.95)"> <g transform="translate(72,-25.64)"> <text transform="matrix(1,0,0,-1,-53.13,52.96)"> <tspan style="font-size: 12.50px; " x="48.950001" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.602544" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.766043" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.679482" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.306313" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="38.346703" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="54.799999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="61.013992" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,14.29,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,19.16,50.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> +</t年代pan> <tspan style="font-size: 12.50px; " x="19.98" y="5.1900001"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,46.14,66.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> </t年代pan> <tspan style="font-size: 8.75px; " x="3.0997441" y="25.530001"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,53.72,39.27)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,59.93,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.015623" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,80.28,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,88.88,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1999998" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.05" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,111.45,57.96)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,125.21,50.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8600001" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.124483" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="25.477207" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,157.09,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,161.96,50.45)"> <tspan style="font-size: 12.50px; " x="13.79" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="24.389999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.24" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="38.516987" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="49.857208" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,218.23,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,223.1,50.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="19.969999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.82" y="-2.51"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="32.071499" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="39.685825" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,269.77,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,278.37,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1999998" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.05" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,300.93,57.96)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,314.7,50.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.126986" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="25.467207" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,346.58,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,351.45,50.45)"> <tspan style="font-size: 12.50px; " x="39.93" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.860001" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.459999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.48" y="-2.51"> +</t年代pan> <tspan style="font-size: 12.50px; " x="25.820221" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.746883" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.596848" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="64.309998" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="72.586983" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="83.927208" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,441.79,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,446.66,50.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.591596" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.092916" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="11.83" y="21.01"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.993499" y="21.01"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="22.244999" y="21.01"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="25.370749" y="21.01"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="31.62225" y="21.01"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="37.873749" y="21.01"> )</t年代pan> </text> </g> </g> </svg></span>清晰的步骤方法给出了存在唯一解(<一个href="#EEq1.9">1.9</一个>),因为<年代vg height="14.75" id="M43" style="vertical-align:-3.25793pt;width:205px;" version="1.1" viewbox="0 0 205 14.75" width="205" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.75)"> <g transform="translate(72,-60.2)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.2769861" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="19.617207" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-45.93,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.05,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="9.364747" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.654236" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="28.994457" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-5.65,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,2.7,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="20.354883" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.344542" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="40.697266" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="46.948765" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="51.112263" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,60.21,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,65.09,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.415239" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,82.91,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,87.79,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>。</p> <p>从[使用定理1.2<一个href="#B1">22</一个>),我们可以得到负的初始条件<年代vg height="13.9375" id="M44" style="vertical-align:-0.13794pt;width:16.862499px;" version="1.1" viewbox="0 0 16.862499 13.9375" width="16.862499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.9375)"> <g transform="translate(72,-60.85)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.44,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> </g> </g> </svg>的解决方案(<一个href="#EEq1.9">1.9</一个>)可以成为-在一个有限的时间。因此,通过论文的其余部分,我们认为一个积极的解决方案(<一个href="#EEq1.9">1.9</一个>与初始功能)<年代vg height="13.9375" id="M45" style="vertical-align:-0.13794pt;width:16.862499px;" version="1.1" viewbox="0 0 16.862499 13.9375" width="16.862499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.9375)"> <g transform="translate(72,-60.85)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.44,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> </g> </g> </svg>存在每一个<年代vg height="11.0625" id="M46" style="vertical-align:-0.30096pt;width:32.224998px;" version="1.1" viewbox="0 0 32.224998 11.0625" width="32.224998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> 0</t年代pan> </text> </g> </g> </svg>。</p> <h4 id="stability-of-the-stationary-solutions-and-existence-of-local-hopf-bifurcation">2。稳定的固定解决方案和当地霍普夫分岔的存在</h4><p>在本节中,我们将研究稳定的固定解决方案和当地霍普夫分岔的存在。</p> <p>第一步是找到固定的解决方案。固定的解决方案(<一个href="#EEq1.9">1.9</一个>满足的方程)<年代pan class="equation" id="EEq2.1"> <svg height="19.475" id="M47" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.475" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.475)"> <g transform="translate(72,-56.42)"> <text transform="matrix(1,0,0,-1,93.49,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="12.1015" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.865623" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,119.69,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,128.3,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1999998" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.05" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,149.22,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,160.91,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="19.957718" y="-2.51"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="26.209217" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="33.823544" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,201.72,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,210.33,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1899996" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.04" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.591572" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,237.8,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,249.49,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.07" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="19.957718" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.896883" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.734344" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.009998" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="52.349979" y="-2.51"> =</t年代pan> <tspan style="font-size: 12.50px; " x="64.390373" y="-2.51"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="70.641869" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="215.25999" y="-2.5599999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="219.42349" y="-2.5599999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="225.675" y="-2.5599999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="228.80075" y="-2.5599999"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="235.05225" y="-2.5599999"> )</t年代pan> </text> </g> </g> </svg></span>显然,(<一个href="#EEq2.1">2.1</一个>)简单的解决方案<年代vg height="10.9125" id="M48" style="vertical-align:-0.17555pt;width:35.924999px;" version="1.1" viewbox="0 0 35.924999 10.9125" width="35.924999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> 0</t年代pan> </text> </g> </g> </svg>。接下来,我们考虑积极的解决方案(<一个href="#EEq2.1">2.1</一个>)。从[<一个href="#B13">17</一个>我们知道<年代vg height="13.55" id="M49" style="vertical-align:-2.29482pt;width:26.775px;" version="1.1" viewbox="0 0 26.775 13.55" width="26.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1139669" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.277466" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.154116" y="0"> )</t年代pan> </text> </g> </g> </svg>严格单调递减<年代vg height="11.0625" id="M50" style="vertical-align:-0.30096pt;width:35.924999px;" version="1.1" viewbox="0 0 35.924999 11.0625" width="35.924999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代pan class="equation" id="EEq2.2"> <svg height="33.037498" id="M51" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 33.037498" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,33.0375)"> <g transform="translate(72,-45.57)"> <text transform="matrix(1,0,0,-1,125.49,56.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="6.9516678" y="0"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,122.81,48.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.2664318" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="15.832368" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="20.208368" y="0"> +</t年代pan> </text> <text transform="matrix(1,0,0,-1,149.02,56.25)"> <tspan style="font-size: 12.50px; " x="40.110001" y="-7.25"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1894646" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.352964" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.229614" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.868946" y="0"> =</t年代pan> </text> <g> <path d="m 189.13,59.37 6.25,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,189.13,47.49)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="7.4400001" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="34.121403" y="-8.7600002"> l</t年代pan> <tspan style="font-size: 12.50px; " x="37.597237" y="-8.7600002"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="41.073071" y="-8.7600002"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,221.71,48.74)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.2664318" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="15.832368" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,245.65,56.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1894646" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.352964" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.229614" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.868946" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="38.909336" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="45.160835" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="219.10001" y="0.58999997"> (</t年代pan> <tspan style="font-size: 12.50px; " x="223.2635" y="0.58999997"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="229.515" y="0.58999997"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="232.64075" y="0.58999997"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="238.89224" y="0.58999997"> )</t年代pan> </text> </g> </g> </svg></span>让<年代vg height="17.674999" id="M52" style="vertical-align:-3.13504pt;width:262.83749px;" version="1.1" viewbox="0 0 262.83749 17.674999" width="262.83749" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.675)"> <g transform="translate(72,-57.86)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.191626" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="24.818455" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="36.858845" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="43.110344" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.874466" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-14.73,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-6.13,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1139669" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.277466" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,11.03,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,22.71,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="18.279387" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="24.530886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.145214" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,61.84,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,70.44,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1139669" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.277466" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.829039" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,94.15,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,105.83,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="18.279387" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.218552" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.056011" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="11.0625" id="M53" style="vertical-align:-0.30096pt;width:35.924999px;" version="1.1" viewbox="0 0 35.924999 11.0625" width="35.924999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> 0</t年代pan> </text> </g> </g> </svg>。然后<年代pan class="equation" id="eq2"> <svg height="84.400002" id="M54" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 84.400002" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,84.4)"> <g transform="translate(72,-4.48)"> <text transform="matrix(1,0,0,-1,121.61,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="6.9516678" y="0"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,118.93,54.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.2664318" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="15.823616" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="20.199615" y="0"> +</t年代pan> </text> <text transform="matrix(1,0,0,-1,145.13,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.7270937" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.890593" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.767242" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.406576" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="39.446964" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="46.211086" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,198.69,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,210.07,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.327718" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.942045" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,236,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,247.38,62.57)"> <tspan style="font-size: 12.50px; " x="-291.54001" y="41.939999"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.266884" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.116846" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.443605" y="0"> ,</t年代pan> </text> <text transform="matrix(1,0,0,-1,-37.51,25.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.95,20.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.040149" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.707369" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,5.64,25.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="12.804176" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,25.4,18.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.614123" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,45.36,17.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,53.96,20.63)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,60.07,25.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,65.63,18.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,78.36,25.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,90.04,18.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="19.970221" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.521793" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.13612" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,131.16,17.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,139.76,20.63)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,145.87,25.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,151.43,18.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.401572" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,170.71,25.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,182.39,18.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.18" y="-2.51"> =</t年代pan> <tspan style="font-size: 12.50px; " x="27.207886" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="35.772442" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.324013" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.938339" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,239.32,17.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,247.92,20.63)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,254.79,25.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="12.804176" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,272.47,20.63)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,278.59,25.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,284.15,18.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,296.87,25.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,308.56,18.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9299998" y="7.2199998"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.23" y="-13.09"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,333.9,36.21)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,339.46,28.7)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.401572" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,358.74,36.21)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,370.42,28.7)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <g> <path d="m 327.79,23.76 48.49,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,331.07,10.05)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,337.18,13.49)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,342.74,7.54)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,355.47,13.49)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,367.15,7.54)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.1" y="-13.09"> −</t年代pan> <tspan style="font-size: 12.50px; " x="29.16" y="-22.23"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.924122" y="-22.23"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,410.41,26.64)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 392.79,23.76 29.74,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,392.79,11.88)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.5390692" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.153396" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,413.93,8.75)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,423.73,10.9)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.19" y="-9.7299995"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="41.02" y="-24.219999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="45.183498" y="-24.219999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="51.434998" y="-24.219999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="54.560749" y="-24.219999"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="60.812248" y="-24.219999"> )</t年代pan> </text> </g> </g> </svg></span>由(<一个href="#B7">12</一个>),我们知道<年代vg height="15.4125" id="M55" style="vertical-align:-2.29482pt;width:72.949997px;" version="1.1" viewbox="0 0 72.949997 15.4125" width="72.949997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.4125)"> <g transform="translate(72,-59.67)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.84,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.36,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.702568" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.904057" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="21.067554" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="24.543388" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.69,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-28.22,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.364987" y="0"> )</t年代pan> </text> </g> </g> </svg>严格单调递增(分别地。,减少)<年代vg height="11.0625" id="M56" style="vertical-align:-0.30096pt;width:62.724998px;" version="1.1" viewbox="0 0 62.724998 11.0625" width="62.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.782625" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="43.823013" y="0"> 1</t年代pan> </text> </g> </g> </svg>(职责。<年代vg height="11.0625" id="M57" style="vertical-align:-0.30096pt;width:35.512501px;" version="1.1" viewbox="0 0 35.512501 11.0625" width="35.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.055292" y="0"> 1</t年代pan> </text> </g> </g> </svg>),<年代pan class="equation" id="eq3"> <svg height="37.775002" id="M58" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 37.775002" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,37.775)"> <g transform="translate(72,-41.78)"> <text transform="matrix(1,0,0,-1,111.25,53.7)"> <tspan style="font-size: 12.50px; " x="0" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="6.9516678" y="0"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,108.81,45.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="5.7938242" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="15.351008" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="19.727009" y="0"> +</t年代pan> </text> <text transform="matrix(1,0,0,-1,135.74,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,141.85,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,147.41,62.01)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.702568" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.904057" y="0"> )</t年代pan> </text> <g> <path d="m 135.74,56.83 32.75,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,139.01,44.95)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,145.13,48.39)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,150.68,44.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1700001" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.371488" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="22.48" y="-8.7399998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="34.507885" y="-8.7399998"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.05946" y="-8.7399998"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="67.503304" y="-8.7399998"> l</t年代pan> <tspan style="font-size: 12.50px; " x="70.979134" y="-8.7399998"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="74.454971" y="-8.7399998"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,216.88,46.2)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="5.7938242" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="15.35976" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,241.54,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,247.66,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,253.22,62.01)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.702568" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.904057" y="0"> )</t年代pan> </text> <g> <path d="m 241.54,56.83 32.76,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,244.82,44.95)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,250.93,48.39)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,256.49,44.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="22.469999" y="-8.7399998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="38.290001" y="-15.99"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="4.1700001" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.371488" y="0.0099999998"> )</t年代pan> </text> <g> <path d="m 292.2,56.83 11.42,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,292.2,42.95)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,298.75,48.02)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,304.82,53.7)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="159.92999" y="-0.07"> (</t年代pan> <tspan style="font-size: 12.50px; " x="164.09351" y="-0.07"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="170.345" y="-0.07"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="173.47075" y="-0.07"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="179.72224" y="-0.07"> )</t年代pan> </text> </g> </g> </svg></span>利用这些结果,我们可以很容易地证明以下引理(见[<一个href="#B6">11</一个>]或[<一个href="#B9">14</一个>])。</p> <p><span class="statement" id="lem2.1"><span class="statement-title">引理2.1。</年代pan><i>假设<年代vg height="11.0625" id="M59" style="vertical-align:-0.30096pt;width:62.724998px;" version="1.1" viewbox="0 0 62.724998 11.0625" width="62.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.782625" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="43.823013" y="0"> 1</t年代pan> </text> </g> </g> </svg>。然后下面的断言。</我><年代pan class="list"><span class="list-item"><span class="list-label">(1)</年代pan><span class="list-content"><i>如果<年代vg height="17.762501" id="M60" style="vertical-align:-3.13504pt;width:95.512497px;" version="1.1" viewbox="0 0 95.512497 17.762501" width="95.512497" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.792009" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.76,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.15,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-8.13,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.25,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>然后存在不积极的解决方案(<一个href="#EEq2.1">2.1</一个>),也就是说,问题(<一个href="#EEq1.9">1.9</一个>)没有积极的固定方案。</我></年代pan></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content"><i>如果<年代vg height="17.762501" id="M61" style="vertical-align:-3.13504pt;width:95.512497px;" version="1.1" viewbox="0 0 95.512497 17.762501" width="95.512497" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.792009" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.76,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.15,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-8.13,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.25,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>,那么在这个案子<年代vg height="15.75" id="M62" style="vertical-align:-3.13504pt;width:108.0625px;" version="1.1" viewbox="0 0 108.0625 15.75" width="108.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.76,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.38,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.45,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.16,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>存在不积极的解决方案(<一个href="#EEq2.1">2.1</一个>),也就是说,问题(<一个href="#EEq1.9">1.9</一个>)没有积极的固定方案,相反的情况<年代vg height="15.75" id="M63" style="vertical-align:-3.13504pt;width:108.0625px;" version="1.1" viewbox="0 0 108.0625 15.75" width="108.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.76,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.38,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.45,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.16,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>,存在一个唯一积极的解决方案<年代vg height="11.05" id="M64" style="vertical-align:-3.2316pt;width:44.1875px;" version="1.1" viewbox="0 0 44.1875 11.05" width="44.1875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.05)"> <g transform="translate(72,-63.16)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.05,63.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>(<一个href="#EEq2.1">2.1</一个>),也就是说,问题(<一个href="#EEq1.9">1.9</一个>)有一个独特的静止的正解<年代vg height="11.05" id="M65" style="vertical-align:-3.2316pt;width:44.1875px;" version="1.1" viewbox="0 0 44.1875 11.05" width="44.1875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.05)"> <g transform="translate(72,-63.16)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.05,63.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。此外<年代vg height="16.5875" id="M66" style="vertical-align:-3.2316pt;width:66.474998px;" version="1.1" viewbox="0 0 66.474998 16.5875" width="66.474998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5875)"> <g transform="translate(72,-58.73)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.3,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.82,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.15,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-44.8,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>。</我></年代pan></span></span><i>假设<年代vg height="11.0625" id="M67" style="vertical-align:-0.30096pt;width:35.512501px;" version="1.1" viewbox="0 0 35.512501 11.0625" width="35.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.055292" y="0"> 1</t年代pan> </text> </g> </g> </svg>。然后下面的断言。</我><年代pan class="list"><span class="list-item"><span class="list-label"><i> <svg height="13.45" id="M68" style="vertical-align:-2.21957pt;width:18.35px;" version="1.1" viewbox="0 0 18.35 13.45" width="18.35" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> )</t年代pan> </text> </g> </g> </svg></i></span><span class="list-content"><i>如果<年代vg height="14.6" id="M69" style="vertical-align:-3.13504pt;width:89.425003px;" version="1.1" viewbox="0 0 89.425003 14.6" width="89.425003" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.579458" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.34358" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.2,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.6,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>然后存在不积极的解决方案(<一个href="#EEq2.1">2.1</一个>),也就是说,问题(<一个href="#EEq1.9">1.9</一个>)没有积极的固定方案。</我></年代pan></span><span class="list-item"><span class="list-label"><i> <svg height="13.45" id="M70" style="vertical-align:-2.21957pt;width:18.35px;" version="1.1" viewbox="0 0 18.35 13.45" width="18.35" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> )</t年代pan> </text> </g> </g> </svg></i></span><span class="list-content"><i>如果<年代vg height="17.762501" id="M71" style="vertical-align:-3.13504pt;width:95.512497px;" version="1.1" viewbox="0 0 95.512497 17.762501" width="95.512497" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.792009" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.76,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.15,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-8.13,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.25,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>,那么在这个案子<年代vg height="15.75" id="M72" style="vertical-align:-3.13504pt;width:108.0625px;" version="1.1" viewbox="0 0 108.0625 15.75" width="108.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.76,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.38,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.45,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.16,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>存在不积极的解决方案(<一个href="#EEq2.1">2.1</一个>),也就是说,问题(<一个href="#EEq1.9">1.9</一个>)没有固定的正解,相反的情况<年代vg height="15.75" id="M73" style="vertical-align:-3.13504pt;width:108.0625px;" version="1.1" viewbox="0 0 108.0625 15.75" width="108.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.76,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.38,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.45,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.16,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>,存在一个唯一积极的解决方案<年代vg height="11.05" id="M74" style="vertical-align:-3.2316pt;width:44.1875px;" version="1.1" viewbox="0 0 44.1875 11.05" width="44.1875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.05)"> <g transform="translate(72,-63.16)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.05,63.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>(<一个href="#EEq2.1">2.1</一个>),也就是说,问题(<一个href="#EEq1.9">1.9</一个>)有一个独特的静止的正解。此外<年代vg height="16.5875" id="M75" style="vertical-align:-3.2316pt;width:66.474998px;" version="1.1" viewbox="0 0 66.474998 16.5875" width="66.474998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5875)"> <g transform="translate(72,-58.73)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.3,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.82,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.15,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-44.8,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>。</我></年代pan></span><span class="list-item"><span class="list-label"><i> <svg height="13.6375" id="M76" style="vertical-align:-2.21957pt;width:18.35px;" version="1.1" viewbox="0 0 18.35 13.6375" width="18.35" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6375)"> <g transform="translate(72,-61.09)"> <text transform="matrix(1,0,0,-1,-71.95,63.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 5</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> )</t年代pan> </text> </g> </g> </svg></i></span><span class="list-content"><i>如果<年代vg height="17.762501" id="M77" style="vertical-align:-3.13504pt;width:165.33749px;" version="1.1" viewbox="0 0 165.33749 17.762501" width="165.33749" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.764123" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.85,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.24,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-39.22,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-34.34,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="23.118048" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.24,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,7.84,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.579458" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.34358" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,40.53,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,49.13,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>,那么存在一个唯一<年代vg height="11.9" id="M78" style="vertical-align:-0.30096pt;width:42.262501px;" version="1.1" viewbox="0 0 42.262501 11.9" width="42.262501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.9)"> <g transform="translate(72,-62.48)"> <text transform="matrix(1,0,0,-1,-71.95,62.83)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.07,67.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∗</t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.52,62.83)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样</我></年代pan></span></span><i></i><span class="equation" id="eq4"> <svg height="41.712502" id="M79" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 41.712502" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,41.7125)"> <g transform="translate(72,-38.63)"> <text transform="matrix(1,0,0,-1,162.21,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,168.33,66.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,173.88,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8600001" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.399069" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,192.49,66.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∗</t年代pan> </text> <text transform="matrix(1,0,0,-1,197.56,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <g> <path d="m 162.21,53.68 41.21,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,167.18,41.81)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,173.29,45.24)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,178.85,41.81)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,189.21,45.24)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∗</t年代pan> </text> <text transform="matrix(1,0,0,-1,194.29,41.81)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="13.79" y="-8.75"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.549999" y="-17.889999"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.314121" y="-17.889999"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,238.94,56.57)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 221.32,53.68 29.74,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,221.32,41.81)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.5390692" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.153396" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,242.46,38.68)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,252.26,50.56)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="212.49001" y="-1.63"> (</t年代pan> <tspan style="font-size: 12.50px; " x="216.6535" y="-1.63"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="222.905" y="-1.63"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="226.03075" y="-1.63"> 5</t年代pan> <tspan style="font-size: 12.50px; " x="232.28224" y="-1.63"> )</t年代pan> </text> </g> </g> </svg></span><i>在这里<年代vg height="16.8375" id="M80" style="vertical-align:-2.29482pt;width:73.5625px;" version="1.1" viewbox="0 0 73.5625 16.8375" width="73.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.8375)"> <g transform="translate(72,-58.53)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.75,65.87)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∗</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-34.12,65.87)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∗</t年代pan> </text> <text transform="matrix(1,0,0,-1,-29.05,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,-24.88,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> </g> </g> </svg>,<年代vg height="11.675" id="M81" style="vertical-align:-0.11285pt;width:15.0625px;" version="1.1" viewbox="0 0 15.0625 11.675" width="15.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.675)"> <g transform="translate(72,-62.66)"> <text transform="matrix(1,0,0,-1,-71.95,62.83)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.07,67.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∗</t年代pan> </text> </g> </g> </svg>是最大的<年代vg height="13.6125" id="M82" style="vertical-align:-2.34499pt;width:26.8125px;" version="1.1" viewbox="0 0 26.8125 13.6125" width="26.8125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.191626" y="0"> )</t年代pan> </text> </g> </g> </svg>。表示<年代vg height="14.45" id="M83" style="vertical-align:-2.34499pt;width:69.662498px;" version="1.1" viewbox="0 0 69.662498 14.45" width="69.662498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.45)"> <g transform="translate(72,-60.44)"> <text transform="matrix(1,0,0,-1,-71.95,62.83)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1514759" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.314975" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.76,67.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∗</t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.68,62.83)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> </t年代pan> </text> </g> </g> </svg>。如果<年代vg height="12.225" id="M84" style="vertical-align:-0.30096pt;width:53.612499px;" version="1.1" viewbox="0 0 53.612499 12.225" width="53.612499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.225)"> <g transform="translate(72,-62.22)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="28.919439" y="0"> </t年代pan> </text> </g> </g> </svg>存在不积极的解决方案(<一个href="#EEq2.1">2.1</一个>),也就是说,问题(<一个href="#EEq1.9">1.9</一个>)没有积极的固定方案。如果<年代vg height="15.75" id="M85" style="vertical-align:-3.13504pt;width:135.2625px;" version="1.1" viewbox="0 0 135.2625 15.75" width="135.2625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.930262" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.767723" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.570316" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="44.610703" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-8.62,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,17.32,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,25.92,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>存在一个唯一积极的解决方案<年代vg height="11.05" id="M86" style="vertical-align:-3.2316pt;width:44.1875px;" version="1.1" viewbox="0 0 44.1875 11.05" width="44.1875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.05)"> <g transform="translate(72,-63.16)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.05,63.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>(<一个href="#EEq2.1">2.1</一个>),满足<年代vg height="16.5875" id="M87" style="vertical-align:-3.2316pt;width:66.474998px;" version="1.1" viewbox="0 0 66.474998 16.5875" width="66.474998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5875)"> <g transform="translate(72,-58.73)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.3,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.82,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.15,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-44.8,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>,即问题(<一个href="#EEq1.9">1.9</一个>)有一个独特的积极平稳满意的解决方案<年代vg height="16.5875" id="M88" style="vertical-align:-3.2316pt;width:66.474998px;" version="1.1" viewbox="0 0 66.474998 16.5875" width="66.474998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5875)"> <g transform="translate(72,-58.73)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.3,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.82,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.15,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-44.8,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>。如果<年代vg height="15.75" id="M89" style="vertical-align:-3.13504pt;width:152.375px;" version="1.1" viewbox="0 0 152.375 15.75" width="152.375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.23,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-27.3,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.69,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.703288" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="25.906216" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.75618" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.558773" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="48.586658" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="54.838158" y="0"> </t年代pan> </text> </g> </g> </svg>存在两个正解<年代vg height="12.5" id="M90" style="vertical-align:-3.2316pt;width:130.7625px;" version="1.1" viewbox="0 0 130.7625 12.5" width="130.7625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.5)"> <g transform="translate(72,-62)"> <text transform="matrix(1,0,0,-1,-71.95,65.27)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.07,62.15)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.73,65.27)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-36.17,62.15)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="3.8508799" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.98,65.27)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-5.06,62.15)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,3.29,65.27)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,23.84,62.15)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="3.8508799" y="0"> 2</t年代pan> </text> </g> </g> </svg>(<一个href="#EEq2.1">2.1</一个>),满足<年代vg height="16.5875" id="M91" style="vertical-align:-3.2316pt;width:71.9375px;" version="1.1" viewbox="0 0 71.9375 16.5875" width="71.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5875)"> <g transform="translate(72,-58.73)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.3,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.82,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.15,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="3.8508799" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-40.42,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代vg height="16.5875" id="M92" style="vertical-align:-3.2316pt;width:71.9375px;" version="1.1" viewbox="0 0 71.9375 16.5875" width="71.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5875)"> <g transform="translate(72,-58.73)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.3,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.82,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.15,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="3.8508799" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-40.42,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>,即问题(<一个href="#EEq1.9">1.9</一个>)有两个积极的静止的令人满意的解决方案<年代vg height="16.5875" id="M93" style="vertical-align:-3.2316pt;width:71.9375px;" version="1.1" viewbox="0 0 71.9375 16.5875" width="71.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5875)"> <g transform="translate(72,-58.73)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.3,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.82,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.15,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="3.8508799" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-40.42,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代vg height="16.5875" id="M94" style="vertical-align:-3.2316pt;width:71.9375px;" version="1.1" viewbox="0 0 71.9375 16.5875" width="71.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5875)"> <g transform="translate(72,-58.73)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.3,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.82,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.15,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="3.8508799" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-40.42,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>,分别。</我></年代pan></p> <p>下一步是研究的霍普夫分岔稳定性和(<一个href="#EEq1.9">1.9</一个>)。线性化(<一个href="#EEq1.9">1.9</一个>)积极平稳的解决方案,我们获得<年代pan class="equation" id="EEq2.3"> <svg height="19.35" id="M95" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.35" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.35)"> <g transform="translate(72,-56.52)"> <text transform="matrix(1,0,0,-1,116.93,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="-7.1267099" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="3.47" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="7.6334991" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.546938" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="19.173767" y="-0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="31.214157" y="-0.0099999998"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.778713" y="-0.0099999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,165.68,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,170.56,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.59" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.450001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.126602" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.479324" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,211.44,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,216.32,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="19.957718" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,245.25,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,250.13,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.6" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.450001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.139105" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.479324" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,291.01,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,295.89,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9299998" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="168.86" y="-2.51"> (</t年代pan> <tspan style="font-size: 12.50px; " x="173.0235" y="-2.51"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="179.27499" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="182.40074" y="-2.51"> 6</t年代pan> <tspan style="font-size: 12.50px; " x="188.65225" y="-2.51"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代vg height="17.799999" id="M96" style="vertical-align:-3.2316pt;width:227.96249px;" version="1.1" viewbox="0 0 227.96249 17.799999" width="227.96249" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.8)"> <g transform="translate(72,-57.76)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="20.592442" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.356564" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.93,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.33,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,1.35,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,13.03,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,19.15,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.62,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,35.3,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,46.98,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="18.279387" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="24.530886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.644854" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="34.808353" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,90.31,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,101.99,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> ]</t年代pan> </text> </g> </g> </svg>,<年代vg height="17.799999" id="M97" style="vertical-align:-3.2316pt;width:277.71249px;" version="1.1" viewbox="0 0 277.71249 17.799999" width="277.71249" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.8)"> <g transform="translate(72,-57.76)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.642214" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-28.01,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.41,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.715071" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-0.18,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,11.51,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,17.62,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,21.09,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.715071" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,40.32,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,52.01,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="18.279387" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="24.530886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.644854" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="34.808353" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.359924" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,101.89,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,113.57,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.38205" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.219511" y="0"> </t年代pan> </text> </g> </g> </svg>。</p> <p>类似的线性化(<一个href="#EEq1.9">1.9</一个>)在琐碎的固定解决方案<年代pan class="equation" id="EEq2.4"> <svg height="19.35" id="M98" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.35" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.35)"> <g transform="translate(72,-56.52)"> <text transform="matrix(1,0,0,-1,117.2,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="-7.1267099" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="3.46" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="7.6234989" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.536938" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="19.163769" y="-0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="31.204157" y="-0.0099999998"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.768711" y="-0.0099999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,165.68,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,170.56,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.59" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.450001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.126602" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.479324" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,211.44,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,216.32,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="19.957718" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,244.99,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,249.87,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.59" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.450001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.126602" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.479324" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,290.75,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,295.63,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9299998" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="169.12" y="-2.51"> (</t年代pan> <tspan style="font-size: 12.50px; " x="173.28349" y="-2.51"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="179.535" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="182.66075" y="-2.51"> 7</t年代pan> <tspan style="font-size: 12.50px; " x="188.91225" y="-2.51"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代vg height="14.475" id="M99" style="vertical-align:-3.13504pt;width:75.574997px;" version="1.1" viewbox="0 0 75.574997 14.475" width="75.574997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.475)"> <g transform="translate(72,-60.42)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.9,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="20.592442" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.356564" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.19,60.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> </g> </g> </svg>,<年代vg height="15.75" id="M100" style="vertical-align:-3.13504pt;width:100.7625px;" version="1.1" viewbox="0 0 100.7625 15.75" width="100.7625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.25,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.9,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.642214" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-28.27,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-16.89,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.266884" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.116846" y="0"> </t年代pan> </text> </g> </g> </svg>。</p> <p>我们要求<年代vg height="14.6" id="M101" style="vertical-align:-3.13504pt;width:44.625px;" version="1.1" viewbox="0 0 44.625 14.6" width="44.625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.6" id="M102" style="vertical-align:-3.13504pt;width:44.625px;" version="1.1" viewbox="0 0 44.625 14.6" width="44.625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>。实际上,对于<年代vg height="13.55" id="M103" style="vertical-align:-2.29482pt;width:35.075001px;" version="1.1" viewbox="0 0 35.075001 13.55" width="35.075001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6648188" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.705208" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代vg height="16.637501" id="M104" style="vertical-align:-2.29482pt;width:39.775002px;" version="1.1" viewbox="0 0 39.775002 16.637501" width="39.775002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.6375)"> <g transform="translate(72,-58.69)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.75,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,-60.87,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1139669" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.277466" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.478954" y="0"> )</t年代pan> </text> </g> </g> </svg>严格单调递增在吗<年代vg height="9.8625002" id="M105" style="vertical-align:-2.29482pt;width:7.875px;" version="1.1" viewbox="0 0 7.875 9.8625002" width="7.875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.8625)"> <g transform="translate(72,-64.11)"> <text transform="matrix(1,0,0,-1,-71.95,66.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>(见[<一个href="#B9">14</一个>]),<年代vg height="13.55" id="M106" style="vertical-align:-2.29482pt;width:35.075001px;" version="1.1" viewbox="0 0 35.075001 13.55" width="35.075001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6648188" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.705208" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代pan class="equation" id="eq5"> <svg height="40.799999" id="M107" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 40.799999" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,40.8)"> <g transform="translate(72,-39.36)"> <text transform="matrix(1,0,0,-1,45.38,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.4799995" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.34" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,66.92,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,71.79,61.03)"> <tspan style="font-size: 12.50px; " x="22.73" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1894646" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.352964" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.554453" y="0"> )</t年代pan> </text> <g> <path d="m 45.38,53.58 54.99,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,66.45,41.7)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.6515961" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="38.59" y="-8.75"> ></t年代pan> <tspan style="font-size: 12.50px; " x="50.617886" y="-8.75"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="60.345219" y="-8.75"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="75.398834" y="-8.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,148.05,55.46)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,155.01,50.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.241636" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="10.37749" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.554453" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="22.717951" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="28.919439" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="35.858604" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="47.211327" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="53.46283" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="61.652294" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="65.815788" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="72.017281" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="76.180779" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="83.820114" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="95.860504" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="105.58784" y="0"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="120.62894" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="133.70708" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="126.84293" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="135.03239" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="139.19589" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="145.39738" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="152.33655" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="163.68927" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="169.94078" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="178.13023" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="182.29375" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="188.49522" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="196.13457" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="208.17496" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="214.42645" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="309.73999" y="-2.0999999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="313.9035" y="-2.0999999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="320.155" y="-2.0999999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="323.28076" y="-2.0999999"> 8</t年代pan> <tspan style="font-size: 12.50px; " x="329.53226" y="-2.0999999"> )</t年代pan> </text> </g> </g> </svg></span>这很容易说明<年代vg height="17.075001" id="M108" style="vertical-align:-5.1263pt;width:233.0625px;" version="1.1" viewbox="0 0 233.0625 17.075001" width="233.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.075)"> <g transform="translate(72,-58.34)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="20.592442" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.356564" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.93,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.33,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.241636" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="10.37749" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.491457" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="20.654957" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.856443" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="33.783108" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="45.123325" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="51.374828" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.488792" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="61.652294" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="67.853783" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="72.017281" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="76.180779" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,67.36,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.340992" y="0"> =</t年代pan> <tspan style="font-size: 8.75px; " x="10.336112" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,83.65,63.61)"> <tspan style="font-size: 6.25px; " x="0" y="5.1500001"> </t年代pan> <tspan style="font-size: 6.25px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 6.25px; " x="3.1259999" y="0"> /</t年代pan> <tspan style="font-size: 6.25px; " x="4.8640561" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,96.11,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> 0</t年代pan> </text> </g> </g> </svg>。立即,<年代vg height="18.237499" id="M109" style="vertical-align:-5.1263pt;width:255.52499px;" version="1.1" viewbox="0 0 255.52499 18.237499" width="255.52499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,18.2375)"> <g transform="translate(72,-57.41)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.642214" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-28.01,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.41,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.241636" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="10.37749" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.491457" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="20.654957" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.856443" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="33.783108" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="45.123325" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="51.374828" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.488792" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="61.652294" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="67.853783" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="72.017281" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="76.180779" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,59.28,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.340992" y="0"> =</t年代pan> <tspan style="font-size: 8.75px; " x="10.336112" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="14.92216" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,80.16,62.69)"> <tspan style="font-size: 6.25px; " x="0" y="5.1599998"> </t年代pan> <tspan style="font-size: 6.25px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 6.25px; " x="3.1259999" y="0"> /</t年代pan> <tspan style="font-size: 6.25px; " x="4.8640561" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,91.93,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.352724" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.142813" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="34.183201" y="0"> 0</t年代pan> </text> </g> </g> </svg>。因此,声称是真实的。</p> <p>的特点(<一个href="#EEq2.3">2.6</一个>)如下:<年代pan class="equation" id="eq6"> <svg height="16.65" id="M110" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 16.65" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.65)"> <g transform="translate(72,-58.68)"> <text transform="matrix(1,0,0,-1,146.23,63.01)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="30.307272" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,185.51,59.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,190.38,63.01)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,195.94,68.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.362368" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,210.79,65.82)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,217.69,63.01)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,238,59.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,242.87,63.01)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,248.43,68.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.362368" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,263.28,65.82)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,267.4,63.01)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="197.35001" y="0.52999997"> (</t年代pan> <tspan style="font-size: 12.50px; " x="201.5135" y="0.52999997"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="207.765" y="0.52999997"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="210.89075" y="0.52999997"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="217.14224" y="0.52999997"> )</t年代pan> </text> </g> </g> </svg></span></p> <p>从[<一个href="#B9">14</一个>我们知道,如果<年代vg height="14.6" id="M111" style="vertical-align:-3.13504pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 14.6" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>对于任意的<年代vg height="14.6" id="M112" style="vertical-align:-3.13504pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 14.6" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>解决方案的动态行为的问题(<一个href="#EEq1.9">1.9</一个>)与非负初始功能类似于nondelayed对应问题的解决方案。的连续性,足够小<年代vg height="14.6" id="M113" style="vertical-align:-3.13504pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 14.6" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>解决方案的动态行为的问题(<一个href="#EEq1.9">1.9</一个>)与非负初始功能也相应nonretarded相似解决方案的问题。</p> <p>在下面,我们将研究稳定的固定解决方案和当地霍普夫分岔的存在。从生物学的角度来看是合理的<年代vg height="10.725" id="M114" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>作为分岔参数,详细参见[<一个href="#B19">18</一个>)和引用。</p> <p>时的情况<年代vg height="14.6" id="M115" style="vertical-align:-3.13504pt;width:44.299999px;" version="1.1" viewbox="0 0 44.299999 14.6" width="44.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.6" id="M116" style="vertical-align:-3.13504pt;width:44.299999px;" version="1.1" viewbox="0 0 44.299999 14.6" width="44.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>研究了在<一个href="#B17">23</一个>,<一个href="#B18">24</一个>,可以发现以下引理的证明。</p> <p><span class="statement" id="lem2.2"><span class="statement-title">引理2.2。</年代pan><i>考虑方程</我><年代pan class="equation" id="EEq2.5"> <svg height="19.35" id="M117" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.35" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.35)"> <g transform="translate(72,-56.52)"> <text transform="matrix(1,0,0,-1,129.36,61.13)"> <tspan style="font-size: 12.50px; " x="40.560001" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="46.41" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="55.369999" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="-6.6766019" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="2.28" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="6.4434991" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.356938" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.983768" y="-0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.024157" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="61.220001" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="67.909103" y="7.1054274e-015"> −</t年代pan> <tspan style="font-size: 12.50px; " x="79.249329" y="7.1054274e-015"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,215.02,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,219.89,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.110001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="13.141248" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.959999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.649105" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="45.989326" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,272.28,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,277.16,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.78" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="181.33" y="-2.51"> (</t年代pan> <tspan style="font-size: 12.50px; " x="185.4935" y="-2.51"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="191.745" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="194.87074" y="-2.51"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="201.12225" y="-2.51"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="207.37375" y="-2.51"> )</t年代pan> </text> </g> </g> </svg></span><i>用一个非负最初的连续函数<年代vg height="15.05" id="M118" style="vertical-align:-3.49493pt;width:118.6875px;" version="1.1" viewbox="0 0 118.6875 15.05" width="118.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.05)"> <g transform="translate(72,-59.96)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.790329" y="0"> ∶</t年代pan> <tspan style="font-size: 12.50px; " x="21.655195" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.818695" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.383251" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.784786" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="45.998535" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="52.250038" y="0"> ]</t年代pan> <tspan style="font-size: 12.50px; " x="61.964867" y="0"> →</t年代pan> <tspan style="font-size: 12.50px; " x="79.106483" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,16.41,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> +</t年代pan> </text> </g> </g> </svg>,在那里<年代vg height="10.725" id="M119" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="10.725" id="M120" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>是积极的常数,<年代vg height="14.6" id="M121" style="vertical-align:-3.13504pt;width:104.75px;" version="1.1" viewbox="0 0 104.75 14.6" width="104.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8648672" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.905256" y="0"> 米</t年代pan> <tspan style="font-size: 12.50px; " x="31.632589" y="0"> 一个</t年代pan> <tspan style="font-size: 12.50px; " x="37.183922" y="0"> x</t年代pan> <tspan style="font-size: 12.50px; " x="47.598923" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="51.762421" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-13.77,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-8.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,2.71,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,7.59,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="13.4875" id="M122" style="vertical-align:-2.34499pt;width:10.675px;" version="1.1" viewbox="0 0 10.675 13.4875" width="10.675" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是一个连续可微的非线性函数。假设(<一个href="#EEq2.5">2.10</一个>)简单固定的解决方案,<年代vg height="13.6125" id="M123" style="vertical-align:-2.34499pt;width:68.550003px;" version="1.1" viewbox="0 0 68.550003 13.6125" width="68.550003" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="17.354164" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="22.555412" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="28.806911" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="36.446243" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="48.486633" y="0"> 0</t年代pan> </text> </g> </g> </svg>。让周围的线性化方程的平凡解(<一个href="#EEq2.5">2.10</一个>)如下:</我><年代pan class="equation" id="EEq2.6"> <svg height="19.35" id="M124" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.35" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.35)"> <g transform="translate(72,-56.52)"> <text transform="matrix(1,0,0,-1,119.2,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="-6.6766019" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="2.28" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="6.4434991" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.356938" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.983768" y="-0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.024157" y="-0.0099999998"> −</t年代pan> <tspan style="font-size: 12.50px; " x="38.588711" y="-0.0099999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,166.5,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,171.37,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.96" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.82" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.496601" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="32.849327" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,210.62,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,215.5,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="19.957718" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,244.17,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,249.05,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.96" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.81" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.499105" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="32.839325" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,288.29,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,293.17,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9299998" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="165.32001" y="-2.51"> (</t年代pan> <tspan style="font-size: 12.50px; " x="169.48351" y="-2.51"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="175.735" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="178.86075" y="-2.51"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="185.11224" y="-2.51"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="191.36375" y="-2.51"> )</t年代pan> </text> </g> </g> </svg></span><i>然后</我><年代pan class="list"><span class="list-item"><span class="list-label">(1)</年代pan><span class="list-content"><i>如果<年代vg height="14.6" id="M125" style="vertical-align:-3.13504pt;width:44.299999px;" version="1.1" viewbox="0 0 44.299999 14.6" width="44.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.6" id="M126" style="vertical-align:-3.13504pt;width:59.700001px;" version="1.1" viewbox="0 0 59.700001 14.6" width="59.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.528486" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.66,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.79,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> </text> </g> </g> </svg>,<年代vg height="30.025" id="M127" style="vertical-align:-7.41113pt;width:149.3625px;" version="1.1" viewbox="0 0 149.3625 30.025" width="149.3625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.42)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.7"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 42.27,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,10.21,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.86,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.41,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.28,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>,然后存在<年代vg height="19.200001" id="M128" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M129" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>简单的解决方案(<一个href="#EEq2.5">2.10</一个>)是渐近稳定的<年代vg height="19.200001" id="M130" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔发生;</我></年代pan></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content"><i>如果<年代vg height="14.6" id="M131" style="vertical-align:-3.13504pt;width:44.299999px;" version="1.1" viewbox="0 0 44.299999 14.6" width="44.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.6" id="M132" style="vertical-align:-3.13504pt;width:86.912498px;" version="1.1" viewbox="0 0 86.912498 14.6" width="86.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.49,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-33.14,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.528486" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-9.9,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-5.02,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> </text> </g> </g> </svg>,平凡解(<一个href="#EEq2.5">2.10</一个>)是不稳定的独立价值观的延迟,而且没有霍普夫分岔。</我></年代pan></span></span><i></i></span></p> <p>利用引理<一个href="#lem2.2">2.2</一个>,我们很容易有以下。</p> <p><span class="statement" id="coro2.2"><span class="statement-title">推论2.3。</年代pan><i>考虑方程</我><年代pan class="equation" id="EEq2.7"> <svg height="19.35" id="M133" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.35" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.35)"> <g transform="translate(72,-56.52)"> <text transform="matrix(1,0,0,-1,129.36,61.13)"> <tspan style="font-size: 12.50px; " x="40.560001" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="46.41" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="55.369999" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="-6.6766019" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="2.28" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="6.4434991" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.356938" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.983768" y="-0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.024157" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="61.220001" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="67.909103" y="7.1054274e-015"> −</t年代pan> <tspan style="font-size: 12.50px; " x="79.249329" y="7.1054274e-015"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,215.02,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,219.89,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.110001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="13.141248" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.959999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.649105" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="45.989326" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,272.28,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,277.16,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.78" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="181.33" y="-2.51"> (</t年代pan> <tspan style="font-size: 12.50px; " x="185.4935" y="-2.51"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="191.745" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="194.87074" y="-2.51"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="201.12225" y="-2.51"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="207.37375" y="-2.51"> )</t年代pan> </text> </g> </g> </svg></span><i>用一个非负最初的连续函数<年代vg height="15.05" id="M134" style="vertical-align:-3.49493pt;width:118.6875px;" version="1.1" viewbox="0 0 118.6875 15.05" width="118.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.05)"> <g transform="translate(72,-59.96)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.790329" y="0"> ∶</t年代pan> <tspan style="font-size: 12.50px; " x="21.655195" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.818695" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.383251" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.784786" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="45.998535" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="52.250038" y="0"> ]</t年代pan> <tspan style="font-size: 12.50px; " x="61.964867" y="0"> →</t年代pan> <tspan style="font-size: 12.50px; " x="79.106483" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,16.41,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> +</t年代pan> </text> </g> </g> </svg>,在那里<年代vg height="10.725" id="M135" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="10.725" id="M136" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>是积极的常数,<年代vg height="14.6" id="M137" style="vertical-align:-3.13504pt;width:104.75px;" version="1.1" viewbox="0 0 104.75 14.6" width="104.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8648672" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.905256" y="0"> 米</t年代pan> <tspan style="font-size: 12.50px; " x="31.632589" y="0"> 一个</t年代pan> <tspan style="font-size: 12.50px; " x="37.183922" y="0"> x</t年代pan> <tspan style="font-size: 12.50px; " x="47.598923" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="51.762421" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-13.77,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-8.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,2.71,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,7.59,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="13.4875" id="M138" style="vertical-align:-2.34499pt;width:10.675px;" version="1.1" viewbox="0 0 10.675 13.4875" width="10.675" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是一个连续可微的非线性函数。假设(<一个href="#EEq2.7">2.12</一个>)积极的固定方案<年代vg height="11.05" id="M139" style="vertical-align:-3.2316pt;width:42.137501px;" version="1.1" viewbox="0 0 42.137501 11.05" width="42.137501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.05)"> <g transform="translate(72,-63.16)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-42.69,63.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>,也就是说,<年代vg height="14.7125" id="M140" style="vertical-align:-3.2316pt;width:80.987503px;" version="1.1" viewbox="0 0 80.987503 14.7125" width="80.987503" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.7125)"> <g transform="translate(72,-60.23)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.97,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.62,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-37.54,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-33.19,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6393328" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>。让周围的线性化方程的积极平稳解决方案(<一个href="#EEq2.5">2.10</一个>)如下:</我><年代pan class="equation" id="EEq2.8"> <svg height="19.35" id="M141" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.35" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.35)"> <g transform="translate(72,-56.52)"> <text transform="matrix(1,0,0,-1,118.94,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="-6.6766019" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="2.28" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="6.4434991" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.356938" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.983768" y="-0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.024157" y="-0.0099999998"> −</t年代pan> <tspan style="font-size: 12.50px; " x="38.588711" y="-0.0099999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,166.5,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,171.37,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.96" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.82" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.496601" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="32.849327" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,210.62,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,215.5,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="19.957718" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,244.43,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,249.31,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.96" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.81" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.499105" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="32.839325" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,288.56,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,293.43,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="165.06" y="-2.51"> (</t年代pan> <tspan style="font-size: 12.50px; " x="169.2235" y="-2.51"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="175.47501" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="178.60075" y="-2.51"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="184.85225" y="-2.51"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="191.10374" y="-2.51"> )</t年代pan> </text> </g> </g> </svg></span><i>然后</我><年代pan class="list"><span class="list-item"><span class="list-label">(1)</年代pan><span class="list-content"><i>如果<年代vg height="14.6" id="M142" style="vertical-align:-3.13504pt;width:44.625px;" version="1.1" viewbox="0 0 44.625 14.6" width="44.625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.6" id="M143" style="vertical-align:-3.13504pt;width:60.362499px;" version="1.1" viewbox="0 0 60.362499 14.6" width="60.362499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.528486" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.14,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.26,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> </text> </g> </g> </svg>,<年代vg height="30.025" id="M144" style="vertical-align:-7.23557pt;width:148.77499px;" version="1.1" viewbox="0 0 148.77499 30.025" width="148.77499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.12)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.25)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.53"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 41.8,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,9.97,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.62,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,37.93,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,42.81,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>,然后存在<年代vg height="19.200001" id="M145" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M146" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>积极的固定解决方案(<一个href="#EEq2.5">2.10</一个>)是渐近稳定的<年代vg height="19.200001" id="M147" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔发生;</我></年代pan></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content"><i>如果<年代vg height="14.6" id="M148" style="vertical-align:-3.13504pt;width:44.625px;" version="1.1" viewbox="0 0 44.625 14.6" width="44.625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.6" id="M149" style="vertical-align:-3.13504pt;width:87.5625px;" version="1.1" viewbox="0 0 87.5625 14.6" width="87.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.22,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-32.88,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.540989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-9.37,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.5,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> </text> </g> </g> </svg>积极的静止的解决方案(<一个href="#EEq2.5">2.10</一个>)是不稳定的独立价值观的延迟,而且没有霍普夫分岔。</我></年代pan></span></span><i></i></span></p> <p>注意到<年代vg height="17.799999" id="M150" style="vertical-align:-3.2316pt;width:227.96249px;" version="1.1" viewbox="0 0 227.96249 17.799999" width="227.96249" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.8)"> <g transform="translate(72,-57.76)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="20.592442" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.356564" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.93,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.33,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,1.35,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,13.03,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,19.15,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.62,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,35.3,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,46.98,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="18.279387" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="24.530886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.644854" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="34.808353" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,90.31,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,101.99,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> ]</t年代pan> </text> </g> </g> </svg>,<年代vg height="17.799999" id="M151" style="vertical-align:-3.2316pt;width:269.26251px;" version="1.1" viewbox="0 0 269.26251 17.799999" width="269.26251" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.8)"> <g transform="translate(72,-57.76)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.99,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.64,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.642214" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-28.01,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.41,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.715071" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-0.18,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,11.51,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,17.62,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,21.09,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.715071" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,40.32,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,52.01,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="18.279387" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="24.530886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.644854" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="34.808353" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.359924" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,101.89,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,113.57,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="22.617928" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.455387" y="0"> </t年代pan> </text> </g> </g> </svg>和<年代vg height="14.7125" id="M152" style="vertical-align:-3.2316pt;width:43.412498px;" version="1.1" viewbox="0 0 43.412498 14.7125" width="43.412498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.7125)"> <g transform="translate(72,-60.23)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.44,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-55.61,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>满足(<一个href="#EEq2.1">2.1</一个>),通过直接计算,我们有<年代pan class="equation" id="eq7"> <svg height="19.475" id="M153" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.475" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.475)"> <g transform="translate(72,-56.42)"> <text transform="matrix(1,0,0,-1,61.54,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,70.51,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,78.16,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.34" y="-1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="11.34" y="1.5700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="13.84" y="-7.1054274e-015"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,100.97,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,105.84,62.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="5.98" y="1.59"> =</t年代pan> <tspan style="font-size: 12.50px; " x="18.007887" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,132.82,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,140.47,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,160.78,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,169.13,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="20.592442" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.356564" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,203.83,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,212.44,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,218.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,224.11,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,238.47,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,250.16,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.6300001" y="-2.51"> +</t年代pan> <tspan style="font-size: 12.50px; " x="19.957718" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.509291" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.123615" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,291.27,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,299.87,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,305.99,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,311.55,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.389069" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,332.46,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,344.15,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9299998" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="114.34" y="-2.5599999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="118.5035" y="-2.5599999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="124.755" y="-2.5599999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="127.88075" y="-2.5599999"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="134.13225" y="-2.5599999"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="140.38374" y="-2.5599999"> )</t年代pan> </text> </g> </g> </svg></span>自<年代vg height="13.55" id="M154" style="vertical-align:-2.29482pt;width:26.775px;" version="1.1" viewbox="0 0 26.775 13.55" width="26.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.1139669" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.277466" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.154116" y="0"> )</t年代pan> </text> </g> </g> </svg>严格单调递减<年代vg height="11.0625" id="M155" style="vertical-align:-0.30096pt;width:35.924999px;" version="1.1" viewbox="0 0 35.924999 11.0625" width="35.924999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> 0</t年代pan> </text> </g> </g> </svg>,我们很容易得到<年代pan class="equation" id="eq8"> <svg height="102.925" id="M156" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 102.925" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,102.925)"> <g transform="translate(72,10.34)"> <text transform="matrix(1,0,0,-1,85.89,50.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,94.85,47.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,103.2,50.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.04" y="-1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="12.04" y="1.5700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.54" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,126.7,47.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,131.57,52.04)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="5.98" y="1.59"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="22.219999" y="-8.9899998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,159.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,165.47,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.389069" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,186.38,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="8.1199999"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,198.07,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <g> <path d="m 153.79,53.58 50.13,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,157.07,39.88)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,163.18,43.32)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,168.74,37.36)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.52"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,183.11,44.5)"> <tspan style="font-size: 8.75px; " x="0" y="7.2800002"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,194.79,37.36)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.8" y="-13.09"> <</t年代pan> <tspan style="font-size: 12.50px; " x="30.549999" y="-22.23"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.314121" y="-22.23"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,239.44,56.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 221.82,53.58 29.75,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,221.82,41.7)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.5515718" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.165899" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,242.96,38.58)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,256.23,50.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="15.053612" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,277.94,55.46)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,283.5,47.94)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,297.86,47.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,302.21,47.94)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-216.32001" y="43.09"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3199997" y="-2.51"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.36039" y="-2.51"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="27.611889" y="-2.51"> ,</t年代pan> </text> <text transform="matrix(1,0,0,-1,94.85,1.73)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,103.2,4.85)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.04" y="-1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="12.04" y="1.5700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.54" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,126.7,1.73)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,131.57,6.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="5.98" y="1.59"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="22.219999" y="-8.9899998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,159.91,20.43)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,165.47,12.92)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.389069" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,186.38,20.43)"> <tspan style="font-size: 8.75px; " x="0" y="8.1300001"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,198.07,12.92)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <g> <path d="m 153.79,7.98 50.13,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,157.07,-5.73)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,163.18,-2.29)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,168.74,-8.24)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,183.11,-1.1)"> <tspan style="font-size: 8.75px; " x="0" y="7.29"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> 3</t年代pan> </text> <text transform="matrix(1,0,0,-1,194.79,-8.24)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.8" y="-13.09"> ></t年代pan> <tspan style="font-size: 12.50px; " x="30.549999" y="-22.23"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.314121" y="-22.23"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,239.44,10.87)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 221.82,7.98 29.75,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,221.82,-3.9)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.5515718" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.165899" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,242.96,-7.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,256.23,4.85)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="15.053612" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,277.94,9.85)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,283.5,2.34)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,297.86,1.73)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,302.21,2.34)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3199997" y="-2.51"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.36039" y="-2.51"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="27.611889" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="156.28" y="-25.360001"> (</t年代pan> <tspan style="font-size: 12.50px; " x="160.4435" y="-25.360001"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="166.69499" y="-25.360001"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="169.82076" y="-25.360001"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="176.07225" y="-25.360001"> 5</t年代pan> <tspan style="font-size: 12.50px; " x="182.32375" y="-25.360001"> )</t年代pan> </text> </g> </g> </svg></span>很明显<年代vg height="14.6" id="M157" style="vertical-align:-3.13504pt;width:44.299999px;" version="1.1" viewbox="0 0 44.299999 14.6" width="44.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>通过直接计算得到<年代pan class="equation" id="eq9"> <svg height="77.4375" id="M158" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 77.4375" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,77.4375)"> <g transform="translate(72,-10.05)"> <text transform="matrix(1,0,0,-1,135.51,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,144.21,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,152.56,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.04" y="-1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="12.04" y="1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.54" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,175.8,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,180.68,56.05)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="5.9699998" y="1.59"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="21.186152" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.023611" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.826204" y="1.59"> ></t年代pan> <tspan style="font-size: 12.50px; " x="43.866592" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,231.89,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,243.27,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="12.54" y="-9.1400003"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.141825" y="-9.1400003"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,270.4,60.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 255.81,57.58 23.19,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,264.02,45.71)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.18" y="-8.75"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-128.50999" y="26.66"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,144.21,15.92)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,152.56,19.05)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.04" y="-1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="12.04" y="1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.54" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,175.8,15.92)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,180.68,20.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="5.9699998" y="1.59"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="21.186152" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.023611" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.826204" y="1.59"> <</t年代pan> <tspan style="font-size: 12.50px; " x="43.866592" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,231.89,15.92)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,243.27,19.05)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="12.54" y="-9.1400003"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.141825" y="-9.1400003"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,270.4,25.06)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 255.81,22.17 23.19,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,264.02,10.3)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.18" y="-8.75"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="194.47" y="-27.6"> (</t年代pan> <tspan style="font-size: 12.50px; " x="198.6335" y="-27.6"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="204.88499" y="-27.6"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="208.01074" y="-27.6"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="214.26225" y="-27.6"> 6</t年代pan> <tspan style="font-size: 12.50px; " x="220.51375" y="-27.6"> )</t年代pan> </text> </g> </g> </svg></span></p> <p>注意到<年代pan class="equation" id="EEq2.9"> <svg height="82.662498" id="M159" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 82.662498" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,82.6625)"> <g transform="translate(72,-5.87)"> <text transform="matrix(1,0,0,-1,79.06,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,86.04,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,98.12,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="13.66" y="-7.25"> </t年代pan> </text> <g> <path d="m 111.35,57.58 7.61,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,111.35,45.71)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.89" y="-6.2399998"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.75" y="-8.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,135.43,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,146.82,54.46)"> <tspan style="font-size: 12.50px; " x="18.68" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="28" y="0"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="43.216152" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.066116" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="53.856205" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="65.896591" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,220.06,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,231.44,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="12.54" y="-9.1400003"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.154327" y="-9.1400003"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,258.57,60.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 243.98,57.58 23.2,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,252.2,45.71)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.65" y="-8.75"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="34.691109" y="-8.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,295.6,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,303.95,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.03" y="-1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="12.03" y="1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.53" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,327.19,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,332.06,56.05)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="4.5799999" y="1.59"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-253" y="39.09"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,86.04,13.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,98.12,16.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="13.66" y="-7.25"> </t年代pan> </text> <g> <path d="m 111.35,20.09 7.61,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,111.35,8.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.89" y="-6.2399998"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.75" y="-8.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,135.43,13.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,146.82,16.96)"> <tspan style="font-size: 12.50px; " x="18.68" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="28" y="0"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="43.216152" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.066116" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="53.856205" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="65.896591" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,220.06,13.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,231.44,16.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="12.54" y="-9.1400003"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.154327" y="-9.1400003"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,258.57,22.97)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 243.98,20.09 23.2,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,252.2,8.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.65" y="-8.75"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="34.691109" y="-8.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,295.6,13.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,303.95,16.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.03" y="-1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="12.03" y="1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.53" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,327.19,13.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,332.06,18.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="4.5799999" y="1.59"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="126.43" y="-17.26"> (</t年代pan> <tspan style="font-size: 12.50px; " x="130.59351" y="-17.26"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="136.845" y="-17.26"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="139.97075" y="-17.26"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="146.22224" y="-17.26"> 7</t年代pan> <tspan style="font-size: 12.50px; " x="152.47375" y="-17.26"> )</t年代pan> </text> </g> </g> </svg></span>由引理<一个href="#lem2.2">2.2</一个>,我们可以得出以下。<年代pan class="list"><span class="list-item"><span class="list-label">(我)</年代pan><span class="list-content">假设<年代vg height="15.75" id="M160" style="vertical-align:-3.13504pt;width:128.875px;" version="1.1" viewbox="0 0 128.875 15.75" width="128.875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.955507" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="26.431341" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.04567" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="38.209167" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="42.372665" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.18,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,8.21,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="30.025" id="M161" style="vertical-align:-7.41113pt;width:149.3625px;" version="1.1" viewbox="0 0 149.3625 30.025" width="149.3625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.42)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.7"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 42.27,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,10.21,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.86,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.41,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.28,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>保持,然后存在<年代vg height="19.200001" id="M162" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M163" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>简单的解决方案(<一个href="#EEq1.9">1.9</一个>)是渐近稳定的<年代vg height="19.200001" id="M164" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔。</年代pan></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content">假设<年代vg height="15.75" id="M165" style="vertical-align:-3.13504pt;width:128.875px;" version="1.1" viewbox="0 0 128.875 15.75" width="128.875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.955507" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="26.431341" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.04567" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="38.209167" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="42.372665" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.18,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,8.21,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> </text> </g> </g> </svg>持有,平凡解(<一个href="#EEq1.9">1.9</一个>)是不稳定的独立价值观的延迟,而且没有霍普夫分岔。</年代pan></span></span></p> <p>通过简单的计算,我们有<年代pan class="equation" id="EEq2.10"> <svg height="87.650002" id="M166" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 87.650002" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,87.65)"> <g transform="translate(72,-1.88)"> <text transform="matrix(1,0,0,-1,-11.02,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.04,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,8.03,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="13.66" y="-7.25"> </t年代pan> </text> <g> <path d="m 21.27,57.58 7.61,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,21.27,45.71)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.89" y="-6.2399998"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.74" y="-8.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,45.35,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,56.73,54.46)"> <tspan style="font-size: 12.50px; " x="18.68" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.352724" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.620001" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="51.75103" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,115.46,51.33)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,127.54,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="13.23" y="-9.1400003"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.994123" y="-9.1400003"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,154.87,60.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 140.77,57.58 22.71,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,142.6,43.71)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.614327" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,156.77,48.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,168.14,54.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="-179.16" y="39.5"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="-169.44518" y="39.5"> <</t年代pan> <tspan style="font-size: 12.50px; " x="-157.40479" y="39.5"> </t年代pan> <tspan style="font-size: 12.50px; " x="-147.37738" y="39.5"> <</t年代pan> <tspan style="font-size: 12.50px; " x="-135.33699" y="39.5"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="-129.08548" y="39.5"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-113.91935" y="39.5"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,61.21,11.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,73.28,14.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="13.23" y="-9.1400003"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.994123" y="-9.1400003"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,100.62,20.98)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 86.51,18.09 22.71,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,88.35,4.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.6018238" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,102.51,9.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,110.42,14.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="15.328678" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.178642" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.968731" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="38.009121" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,155.78,11.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,167.16,14.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="12.54" y="-9.1400003"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.141825" y="-9.1400003"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,194.29,20.98)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 179.7,18.09 23.19,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,187.91,6.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.139999" y="-8.75"> o</t年代pan> <tspan style="font-size: 12.50px; " x="32.391499" y="-8.75"> r</t年代pan> <tspan style="font-size: 12.50px; " x="46.507385" y="-8.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="56.534794" y="-8.75"> ></t年代pan> <tspan style="font-size: 12.50px; " x="68.57518" y="-8.75"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="74.826683" y="-8.75"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="89.992821" y="-8.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,284.89,11.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,296.97,14.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="13.23" y="-9.1400003"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.994123" y="-9.1400003"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,324.3,20.98)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 310.2,18.09 22.71,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,312.03,4.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.614327" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,326.2,9.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,334.1,14.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="15.341181" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.178642" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.981234" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="38.021622" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,379.46,11.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,390.84,14.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="12.54" y="-9.1400003"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.154327" y="-9.1400003"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,417.97,20.98)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <g> <path d="m 403.38,18.09 23.2,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,411.6,6.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="46.889999" y="-27.6"> (</t年代pan> <tspan style="font-size: 12.50px; " x="51.053497" y="-27.6"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="57.305" y="-27.6"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="60.430748" y="-27.6"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="66.682251" y="-27.6"> 8</t年代pan> <tspan style="font-size: 12.50px; " x="72.933746" y="-27.6"> )</t年代pan> </text> </g> </g> </svg></span> <svg height="9.4375" id="M167" style="vertical-align:-0.5643pt;width:18.9375px;" version="1.1" viewbox="0 0 18.9375 9.4375" width="18.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.4375)"> <g transform="translate(72,-64.45)"> <text transform="matrix(1,0,0,-1,-71.95,65.06)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ⇒</t年代pan> </text> </g> </g> </svg>(<一个href="#EEq1.9">1.9</一个>)有一个积极的固定方案<年代vg height="11.05" id="M168" style="vertical-align:-3.2316pt;width:16.200001px;" version="1.1" viewbox="0 0 16.200001 11.05" width="16.200001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.05)"> <g transform="translate(72,-63.16)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.44,63.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>和<年代vg height="16.5875" id="M169" style="vertical-align:-3.2316pt;width:149.8625px;" version="1.1" viewbox="0 0 149.8625 16.5875" width="149.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5875)"> <g transform="translate(72,-58.73)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.3,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.82,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.15,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-44.8,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="29.394552" y="0"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="44.448166" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,8.62,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,16.96,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.540989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,40.47,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,45.34,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> </text> </g> </g> </svg>。然后是必然的结果<一个href="#coro2.2">2.3</一个>,我们有以下。假设<年代vg height="15.75" id="M170" style="vertical-align:-3.13504pt;width:128.875px;" version="1.1" viewbox="0 0 128.875 15.75" width="128.875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.955507" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="26.431341" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.04567" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="38.209167" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="42.372665" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.18,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,8.21,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="17.762501" id="M171" style="vertical-align:-3.13504pt;width:95.512497px;" version="1.1" viewbox="0 0 95.512497 17.762501" width="95.512497" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.792009" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.76,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.15,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-0.51,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>,然后<年代vg height="30.025" id="M172" style="vertical-align:-7.23557pt;width:148.77499px;" version="1.1" viewbox="0 0 148.77499 30.025" width="148.77499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.12)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.25)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.53"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 41.8,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,9.97,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.62,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,37.93,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,42.81,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>存在<年代vg height="19.200001" id="M173" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M174" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>积极的固定解决方案(<一个href="#EEq1.9">1.9</一个>)是渐近稳定的<年代vg height="19.200001" id="M175" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔。</p> <p>自<年代vg height="11.0625" id="M176" style="vertical-align:-0.30096pt;width:62.724998px;" version="1.1" viewbox="0 0 62.724998 11.0625" width="62.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.782625" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="43.823013" y="0"> 1</t年代pan> </text> </g> </g> </svg>和<年代vg height="17.762501" id="M177" style="vertical-align:-3.13504pt;width:323.33749px;" version="1.1" viewbox="0 0 323.33749 17.762501" width="323.33749" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.927622" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="14.403456" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.00528" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.168779" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.332277" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-34.27,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-22.88,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.306313" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="38.346703" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.45,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,34.52,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.804512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,60.67,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,69.27,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,86.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,95.26,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="15.041109" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,117.29,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,129.36,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.792009" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,155.5,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,164.1,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.090161" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,181.75,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>,<年代vg height="15.75" id="M178" style="vertical-align:-3.13504pt;width:190.8px;" version="1.1" viewbox="0 0 190.8 15.75" width="190.8" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.76,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.38,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.45,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.16,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.703288" y="0"> ⇒</t年代pan> <tspan style="font-size: 12.50px; " x="28.756901" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,41.62,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,49.97,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.528486" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,73.21,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,78.09,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> </text> </g> </g> </svg>。从引理<一个href="#lem2.1">2.1</一个>我们知道,如果<年代vg height="11.0625" id="M179" style="vertical-align:-0.30096pt;width:62.724998px;" version="1.1" viewbox="0 0 62.724998 11.0625" width="62.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.782625" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="43.823013" y="0"> 1</t年代pan> </text> </g> </g> </svg>和<年代vg height="17.762501" id="M180" style="vertical-align:-3.13504pt;width:204.78751px;" version="1.1" viewbox="0 0 204.78751 17.762501" width="204.78751" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.927622" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="14.403456" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.00528" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.168779" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.332277" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-34.27,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-22.88,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.306313" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="38.346703" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.45,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,34.52,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.804512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,60.67,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,69.27,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,86.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>持有,那么(<一个href="#EEq1.9">1.9</一个>)没有固定的正解。由引理<一个href="#lem2.2">2.2</一个>,我们很容易有以下。假设<年代vg height="11.0625" id="M181" style="vertical-align:-0.30096pt;width:62.724998px;" version="1.1" viewbox="0 0 62.724998 11.0625" width="62.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.782625" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="43.823013" y="0"> 1</t年代pan> </text> </g> </g> </svg>和<年代vg height="17.762501" id="M182" style="vertical-align:-3.13504pt;width:204.78751px;" version="1.1" viewbox="0 0 204.78751 17.762501" width="204.78751" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.927622" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="14.403456" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.00528" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.168779" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.332277" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-34.27,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-22.88,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.306313" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="38.346703" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.45,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,34.52,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.804512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,60.67,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,69.27,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,86.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>,然后<年代vg height="30.025" id="M183" style="vertical-align:-7.41113pt;width:149.3625px;" version="1.1" viewbox="0 0 149.3625 30.025" width="149.3625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.42)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.7"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 42.27,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,10.21,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.86,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.41,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.28,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>存在<年代vg height="19.200001" id="M184" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M185" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>简单的解决方案(<一个href="#EEq2.5">2.10</一个>)是渐近稳定的<年代vg height="19.200001" id="M186" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔。</p> <p>假设<年代vg height="11.0625" id="M187" style="vertical-align:-0.30096pt;width:35.512501px;" version="1.1" viewbox="0 0 35.512501 11.0625" width="35.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.055292" y="0"> 1</t年代pan> </text> </g> </g> </svg>,我们有以下。</p> <p>如果<年代vg height="17.762501" id="M188" style="vertical-align:-3.13504pt;width:416.46249px;" version="1.1" viewbox="0 0 416.46249 17.762501" width="416.46249" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.742956" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.50708" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-16.04,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-7.44,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.090161" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="15.25366" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="31.445045" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="38.209167" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.12,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,46.72,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.090161" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,64.36,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,69.24,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> ⇒</t年代pan> <tspan style="font-size: 12.50px; " x="22.680443" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,98.9,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,110.98,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.955507" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="26.431341" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.04567" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="38.209167" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="42.372665" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,160.7,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,172.08,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.352724" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.679482" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.318815" y="0"> ⇒</t年代pan> <tspan style="font-size: 12.50px; " x="41.359924" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,222.15,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,230.5,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.528486" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,253.74,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,258.62,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> </text> </g> </g> </svg>;</p> <p>如果<年代vg height="17.762501" id="M189" style="vertical-align:-3.13504pt;width:418.61249px;" version="1.1" viewbox="0 0 418.61249 17.762501" width="418.61249" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.927622" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="14.403456" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.00528" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.168779" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.332277" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-34.27,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-22.88,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.306313" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="38.346703" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.45,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,34.52,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.804512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,60.67,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,69.27,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,86.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,95.26,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ⇒</t年代pan> <tspan style="font-size: 12.50px; " x="15.216151" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.053612" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.856203" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="37.896591" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,140.5,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,151.88,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,177.81,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,186.42,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.703288" y="0"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="28.756901" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,223.88,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,232.23,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.528486" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,255.47,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,260.34,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> </text> </g> </g> </svg>;</p> <p>如果<年代vg height="17.762501" id="M190" style="vertical-align:-3.13504pt;width:165.33749px;" version="1.1" viewbox="0 0 165.33749 17.762501" width="165.33749" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.764123" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.85,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.24,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.6,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.25,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.24,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,7.84,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.579458" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.34358" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,40.53,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,49.13,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>如果<年代vg height="15.75" id="M191" style="vertical-align:-3.13504pt;width:266.03751px;" version="1.1" viewbox="0 0 266.03751 15.75" width="266.03751" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="28.919439" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.960789" y="0"> ⇒</t年代pan> <tspan style="font-size: 12.50px; " x="60.189442" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="60.026901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="70.829498" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="82.869881" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,18.43,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,29.81,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,55.75,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,64.35,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.703288" y="0"> ⇔</t年代pan> <tspan style="font-size: 12.50px; " x="28.756901" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,101.81,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,110.16,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.528486" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,133.4,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,138.28,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> </text> </g> </g> </svg>;如果<年代vg height="15.75" id="M192" style="vertical-align:-3.13504pt;width:190.8px;" version="1.1" viewbox="0 0 190.8 15.75" width="190.8" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.76,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.38,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.45,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.16,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.703288" y="0"> ⇒</t年代pan> <tspan style="font-size: 12.50px; " x="28.756901" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,41.62,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,49.97,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.528486" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,73.21,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,78.09,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> </text> </g> </g> </svg>;如果<年代vg height="16.5875" id="M193" style="vertical-align:-3.2316pt;width:245.9375px;" version="1.1" viewbox="0 0 245.9375 16.5875" width="245.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5875)"> <g transform="translate(72,-58.73)"> <text transform="matrix(1,0,0,-1,-71.95,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.23,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-27.3,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.69,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.703288" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="25.906216" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.75618" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.558773" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="48.586658" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="54.838158" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="70.879509" y="0"> ⇒</t年代pan> <tspan style="font-size: 12.50px; " x="85.933121" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,73.9,67.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,77.37,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,90.05,58.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="3.8508799" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,98.78,62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="19.667219" y="0"> 0</t年代pan> </text> </g> </g> </svg>然后通过引理<一个href="#lem2.2">2.2</一个>和推论<一个href="#coro2.2">2.3</一个>,我们有以下。<年代pan class="list"><span class="list-item"><span class="list-label">(1)</年代pan><span class="list-content">假设<年代vg height="11.0625" id="M194" style="vertical-align:-0.30096pt;width:35.512501px;" version="1.1" viewbox="0 0 35.512501 11.0625" width="35.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.055292" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.6" id="M195" style="vertical-align:-3.13504pt;width:89.425003px;" version="1.1" viewbox="0 0 89.425003 14.6" width="89.425003" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.579458" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.34358" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.2,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.6,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>或<年代vg height="11.0625" id="M196" style="vertical-align:-0.30096pt;width:35.512501px;" version="1.1" viewbox="0 0 35.512501 11.0625" width="35.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.055292" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="17.762501" id="M197" style="vertical-align:-3.13504pt;width:204.78751px;" version="1.1" viewbox="0 0 204.78751 17.762501" width="204.78751" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.927622" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="14.403456" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.00528" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.168779" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.332277" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-34.27,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-22.88,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.306313" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="38.346703" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.45,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,34.52,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.804512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,60.67,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,69.27,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,86.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>成立,那么对<年代vg height="30.025" id="M198" style="vertical-align:-7.41113pt;width:149.3625px;" version="1.1" viewbox="0 0 149.3625 30.025" width="149.3625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.42)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.7"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 42.27,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,10.21,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.86,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.41,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.28,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>存在<年代vg height="19.200001" id="M199" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M200" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>简单的解决方案(<一个href="#EEq2.5">2.10</一个>)是渐近稳定的<年代vg height="19.200001" id="M201" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔。</年代pan></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content">假设<年代vg height="11.0625" id="M202" style="vertical-align:-0.30096pt;width:35.512501px;" version="1.1" viewbox="0 0 35.512501 11.0625" width="35.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.055292" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="17.762501" id="M203" style="vertical-align:-3.13504pt;width:204.78751px;" version="1.1" viewbox="0 0 204.78751 17.762501" width="204.78751" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.927622" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="14.403456" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.00528" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.168779" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.332277" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-34.27,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-22.88,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.306313" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="38.346703" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.45,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,34.52,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.804512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,60.67,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,69.27,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,86.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>持有。如果<年代vg height="12.225" id="M204" style="vertical-align:-0.30096pt;width:53.612499px;" version="1.1" viewbox="0 0 53.612499 12.225" width="53.612499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.225)"> <g transform="translate(72,-62.22)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="28.919439" y="0"> </t年代pan> </text> </g> </g> </svg>,平凡解(<一个href="#EEq1.9">1.9</一个>)是不稳定的独立价值观的延迟,而且没有霍普夫分岔;如果<年代vg height="15.75" id="M205" style="vertical-align:-3.13504pt;width:108.0625px;" version="1.1" viewbox="0 0 108.0625 15.75" width="108.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.76,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.38,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.45,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.16,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>,然后<年代vg height="30.025" id="M206" style="vertical-align:-7.41113pt;width:149.3625px;" version="1.1" viewbox="0 0 149.3625 30.025" width="149.3625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.42)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.7"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 42.27,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,10.21,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.86,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.41,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.28,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>存在<年代vg height="19.200001" id="M207" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M208" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>简单的解决方案(<一个href="#EEq2.5">2.10</一个>)是渐近稳定的<年代vg height="19.200001" id="M209" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔发生;如果<年代vg height="15.75" id="M210" style="vertical-align:-3.13504pt;width:152.375px;" version="1.1" viewbox="0 0 152.375 15.75" width="152.375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.23,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-27.3,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.69,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.703288" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="25.906216" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.75618" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.558773" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="48.586658" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="54.838158" y="0"> </t年代pan> </text> </g> </g> </svg>,然后<年代vg height="30.025" id="M211" style="vertical-align:-7.23557pt;width:148.77499px;" version="1.1" viewbox="0 0 148.77499 30.025" width="148.77499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.12)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.25)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.53"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 41.8,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,9.97,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.62,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,37.93,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,42.81,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>存在<年代vg height="19.200001" id="M212" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M213" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>积极的固定解决方案<年代vg height="11.05" id="M214" style="vertical-align:-3.2316pt;width:21.674999px;" version="1.1" viewbox="0 0 21.674999 11.05" width="21.674999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.05)"> <g transform="translate(72,-63.16)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.44,63.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="3.8508799" y="0"> 2</t年代pan> </text> </g> </g> </svg>(<一个href="#EEq1.9">1.9</一个>)是渐近稳定的<年代vg height="19.200001" id="M215" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔。</年代pan></span></span></p> <p>我们总结如下。</p> <p><span class="statement" id="thm2.4"><span class="statement-title">定理2.4。</年代pan><i>(我)的假设<年代vg height="15.75" id="M216" style="vertical-align:-3.13504pt;width:128.875px;" version="1.1" viewbox="0 0 128.875 15.75" width="128.875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.955507" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="26.431341" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.04567" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="38.209167" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="42.372665" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.18,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,8.21,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="30.025" id="M217" style="vertical-align:-7.41113pt;width:149.3625px;" version="1.1" viewbox="0 0 149.3625 30.025" width="149.3625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.42)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.7"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 42.27,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,10.21,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.86,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.41,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.28,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>保持,然后存在<年代vg height="19.200001" id="M218" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M219" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>简单的解决方案(<一个href="#EEq1.9">1.9</一个>)是渐近稳定的<年代vg height="19.200001" id="M220" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔。</我><br><i>(2)假设<年代vg height="15.75" id="M221" style="vertical-align:-3.13504pt;width:128.875px;" version="1.1" viewbox="0 0 128.875 15.75" width="128.875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.955507" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="26.431341" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.04567" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="38.209167" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="42.372665" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.18,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,8.21,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> </text> </g> </g> </svg>持有,平凡解(<一个href="#EEq1.9">1.9</一个>)是不稳定的独立价值观的延迟,而且没有霍普夫分岔。</我><br><i>(3)假设<年代vg height="15.75" id="M222" style="vertical-align:-3.13504pt;width:128.875px;" version="1.1" viewbox="0 0 128.875 15.75" width="128.875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.955507" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="26.431341" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.04567" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="38.209167" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="42.372665" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-3.18,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,8.21,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="17.762501" id="M223" style="vertical-align:-3.13504pt;width:95.512497px;" version="1.1" viewbox="0 0 95.512497 17.762501" width="95.512497" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.792009" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.76,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.15,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-0.51,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>,然后<年代vg height="30.025" id="M224" style="vertical-align:-7.23557pt;width:148.77499px;" version="1.1" viewbox="0 0 148.77499 30.025" width="148.77499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.12)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.25)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.53"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 41.8,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,9.97,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.62,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,37.93,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,42.81,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>存在<年代vg height="19.200001" id="M225" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M226" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>积极的固定解决方案(<一个href="#EEq1.9">1.9</一个>)是渐近稳定的<年代vg height="19.200001" id="M227" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔。</我><br><i>(四)假设<年代vg height="11.0625" id="M228" style="vertical-align:-0.30096pt;width:62.724998px;" version="1.1" viewbox="0 0 62.724998 11.0625" width="62.724998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.782625" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="43.823013" y="0"> 1</t年代pan> </text> </g> </g> </svg>和<年代vg height="17.762501" id="M229" style="vertical-align:-3.13504pt;width:204.78751px;" version="1.1" viewbox="0 0 204.78751 17.762501" width="204.78751" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.927622" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="14.403456" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.00528" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.168779" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.332277" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-34.27,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-22.88,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.306313" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="38.346703" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.45,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,34.52,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.804512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,60.67,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,69.27,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,86.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>,然后<年代vg height="30.025" id="M230" style="vertical-align:-7.41113pt;width:149.3625px;" version="1.1" viewbox="0 0 149.3625 30.025" width="149.3625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.42)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.7"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 42.27,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,10.21,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.86,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.41,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.28,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>存在<年代vg height="19.200001" id="M231" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M232" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>简单的解决方案(<一个href="#EEq2.5">2.10</一个>)是渐近稳定的<年代vg height="19.200001" id="M233" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔。</我><br><i>(v)假设<年代vg height="11.0625" id="M234" style="vertical-align:-0.30096pt;width:35.512501px;" version="1.1" viewbox="0 0 35.512501 11.0625" width="35.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.055292" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.6" id="M235" style="vertical-align:-3.13504pt;width:89.425003px;" version="1.1" viewbox="0 0 89.425003 14.6" width="89.425003" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.97,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-52.9,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.579458" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.34358" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.2,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.6,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>或<年代vg height="11.0625" id="M236" style="vertical-align:-0.30096pt;width:35.512501px;" version="1.1" viewbox="0 0 35.512501 11.0625" width="35.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.055292" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="17.762501" id="M237" style="vertical-align:-3.13504pt;width:204.78751px;" version="1.1" viewbox="0 0 204.78751 17.762501" width="204.78751" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.927622" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="14.403456" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.00528" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.168779" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.332277" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-34.27,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-22.88,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.666979" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.306313" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="38.346703" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,22.45,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,34.52,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.804512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,60.67,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,69.27,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,86.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>成立,那么对<年代vg height="30.025" id="M238" style="vertical-align:-7.41113pt;width:149.3625px;" version="1.1" viewbox="0 0 149.3625 30.025" width="149.3625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.42)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.7"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 42.27,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,10.21,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.86,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.41,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.28,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>存在<年代vg height="19.200001" id="M239" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M240" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>简单的解决方案(<一个href="#EEq2.5">2.10</一个>)是渐近稳定的<年代vg height="19.200001" id="M241" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔。</我><br><i>(vi)假设<年代vg height="11.0625" id="M242" style="vertical-align:-0.30096pt;width:35.512501px;" version="1.1" viewbox="0 0 35.512501 11.0625" width="35.512501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.014903" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.055292" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="17.762501" id="M243" style="vertical-align:-3.13504pt;width:165.33749px;" version="1.1" viewbox="0 0 165.33749 17.762501" width="165.33749" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7625)"> <g transform="translate(72,-57.79)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.764123" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.85,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-49.24,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.077658" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.6,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.25,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.24,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,7.84,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.579458" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.34358" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,40.53,57.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,49.13,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>持有。如果<年代vg height="12.225" id="M244" style="vertical-align:-0.30096pt;width:53.612499px;" version="1.1" viewbox="0 0 53.612499 12.225" width="53.612499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.225)"> <g transform="translate(72,-62.22)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="28.919439" y="0"> </t年代pan> </text> </g> </g> </svg>,平凡解(<一个href="#EEq1.9">1.9</一个>)是不稳定的独立价值观的延迟,而且没有霍普夫分岔;如果<年代vg height="15.75" id="M245" style="vertical-align:-3.13504pt;width:108.0625px;" version="1.1" viewbox="0 0 108.0625 15.75" width="108.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.78,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-0.16253901" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.640053" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="22.667938" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.76,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.38,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.45,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,4.16,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> </text> </g> </g> </svg>,然后<年代vg height="30.025" id="M246" style="vertical-align:-7.41113pt;width:149.3625px;" version="1.1" viewbox="0 0 149.3625 30.025" width="149.3625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.42)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.7"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 42.27,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,10.21,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.86,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.41,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="-0.5" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.28,55.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>存在<年代vg height="19.200001" id="M247" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M248" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>简单的解决方案(<一个href="#EEq2.5">2.10</一个>)是渐近稳定的<年代vg height="19.200001" id="M249" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔发生;如果<年代vg height="15.75" id="M250" style="vertical-align:-3.13504pt;width:152.375px;" version="1.1" viewbox="0 0 152.375 15.75" width="152.375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.75)"> <g transform="translate(72,-59.4)"> <text transform="matrix(1,0,0,-1,-71.95,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.61,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.23,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.954548" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-27.3,59.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.69,62.57)"> <tspan style="font-size: 12.50px; " x="0" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.703288" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="25.906216" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.75618" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.558773" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="48.586658" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="54.838158" y="0"> </t年代pan> </text> </g> </g> </svg>,然后<年代vg height="30.025" id="M251" style="vertical-align:-7.23557pt;width:148.77499px;" version="1.1" viewbox="0 0 148.77499 30.025" width="148.77499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.025)"> <g transform="translate(72,-47.98)"> <text transform="matrix(1,0,0,-1,-71.95,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,52.12)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,55.25)"> <tspan style="font-size: 12.50px; " x="44.990002" y="3.53"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.258461" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="38.734295" y="0"> 2</t年代pan> </text> <g> <path d="m 1.01,70.71 41.8,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,1.01,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,9.97,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.21"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,17.62,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,37.93,60.3)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.21"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,42.81,55.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ]</t年代pan> </text> </g> </g> </svg>存在<年代vg height="19.200001" id="M252" style="vertical-align:-4.35121pt;width:41.424999px;" version="1.1" viewbox="0 0 41.424999 19.200001" width="41.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样,对于<年代vg height="19.200001" id="M253" style="vertical-align:-4.35121pt;width:72.4375px;" version="1.1" viewbox="0 0 72.4375 19.200001" width="72.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="16.191385" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.656635" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-23.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.26,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>积极的固定解决方案<年代vg height="11.05" id="M254" style="vertical-align:-3.2316pt;width:21.674999px;" version="1.1" viewbox="0 0 21.674999 11.05" width="21.674999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.05)"> <g transform="translate(72,-63.16)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.44,63.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="3.8508799" y="0"> 2</t年代pan> </text> </g> </g> </svg>(<一个href="#EEq1.9">1.9</一个>)是渐近稳定的<年代vg height="19.200001" id="M255" style="vertical-align:-4.35121pt;width:47.700001px;" version="1.1" viewbox="0 0 47.700001 19.200001" width="47.700001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.2,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-38.76,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>霍普夫分岔。</我></年代pan></p> <h4 id="conclusion">3所示。结论</h4><p>在本文中,我们研究肿瘤生长的数学模型有两个离散延迟。拖延,分别代表了时间的细胞进行有丝分裂和细胞所花费的时间来修改细胞的速度损失由于细胞凋亡和杀死细胞的抑制剂。最终模型的数学公式是延迟微分方程的形式<年代pan class="equation" id="EEq3.1"> <svg height="19.35" id="M256" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.35" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.35)"> <g transform="translate(72,-56.52)"> <text transform="matrix(1,0,0,-1,127.36,61.13)"> <tspan style="font-size: 12.50px; " x="41.740002" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.59" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.189999" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="-7.1267099" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="3.46" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="7.6234989" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.536938" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="19.163769" y="-0.0099999998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="31.204157" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="64.040001" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="70.729103" y="7.1054274e-015"> −</t年代pan> <tspan style="font-size: 12.50px; " x="82.069328" y="7.1054274e-015"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,215.84,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,220.71,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.74" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9299998" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="13.143751" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.59" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.279106" y="-2.51"> −</t年代pan> <tspan style="font-size: 12.50px; " x="47.619328" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,274.74,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,279.61,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8514042" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.79" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="185.14" y="-2.51"> (</t年代pan> <tspan style="font-size: 12.50px; " x="189.3035" y="-2.51"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="195.55499" y="-2.51"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="198.68076" y="-2.51"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="204.93225" y="-2.51"> )</t年代pan> </text> </g> </g> </svg></span>用一个非负最初的连续函数<年代vg height="15.05" id="M257" style="vertical-align:-3.49493pt;width:118.6875px;" version="1.1" viewbox="0 0 118.6875 15.05" width="118.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.05)"> <g transform="translate(72,-59.96)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.790329" y="0"> ∶</t年代pan> <tspan style="font-size: 12.50px; " x="21.655195" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.818695" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.383251" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.784786" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="45.998535" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="52.250038" y="0"> ]</t年代pan> <tspan style="font-size: 12.50px; " x="61.964867" y="0"> →</t年代pan> <tspan style="font-size: 12.50px; " x="79.106483" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,16.41,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> +</t年代pan> </text> </g> </g> </svg>,在那里<年代vg height="10.725" id="M258" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代vg height="10.725" id="M259" style="vertical-align:-3.13504pt;width:14.225px;" version="1.1" viewbox="0 0 14.225 10.725" width="14.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.725)"> <g transform="translate(72,-63.42)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.55,63.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.6" id="M260" style="vertical-align:-3.13504pt;width:107.35px;" version="1.1" viewbox="0 0 107.35 14.6" width="107.35" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8648672" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.905256" y="0"> 米</t年代pan> <tspan style="font-size: 12.50px; " x="31.632589" y="0"> 一个</t年代pan> <tspan style="font-size: 12.50px; " x="37.183922" y="0"> x</t年代pan> <tspan style="font-size: 12.50px; " x="49.68692" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="53.850422" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.69,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-6.82,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,4.79,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,9.67,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>是积极的常数,<年代vg height="13.4875" id="M261" style="vertical-align:-2.34499pt;width:10.675px;" version="1.1" viewbox="0 0 10.675 13.4875" width="10.675" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是一个连续可微的函数。结果表明,两个独立的延迟控制的动力学问题的解决方案(<一个href="#EEq3.1">3.1</一个>)和动态行为不同于相应的nonretarded普通方程<年代pan class="equation" id="EEq3.2"> <svg height="16.5375" id="M262" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 16.5375" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5375)"> <g transform="translate(72,-58.77)"> <text transform="matrix(1,0,0,-1,108.97,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="-7.1267099" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="3.463331" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.540269" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="19.167099" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="31.207487" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.735016" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="45.898514" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="56.488552" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="60.652054" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="64.565491" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="70.816994" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="76.030746" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="86.620781" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="90.784286" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="94.697723" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="98.861221" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="105.11272" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="130.24374" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="139.2084" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="143.3719" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="149.6234" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="157.26274" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="169.30312" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,285.15,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,293.5,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="18.279387" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="171.25" y="0.63"> (</t年代pan> <tspan style="font-size: 12.50px; " x="175.4135" y="0.63"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="181.66499" y="0.63"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="184.79076" y="0.63"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="191.04225" y="0.63"> )</t年代pan> </text> </g> </g> </svg></span>或延迟微分方程只有一个延迟的形式<年代pan class="equation" id="EEq3.3"> <svg height="16.5375" id="M263" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 16.5375" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5375)"> <g transform="translate(72,-58.77)"> <text transform="matrix(1,0,0,-1,147.56,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ̇</t年代pan> <tspan style="font-size: 12.50px; " x="-7.1267099" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="3.4508281" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="7.614327" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.527766" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="19.167099" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="31.207487" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.735016" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="45.898514" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="56.488552" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="60.652054" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="67.341156" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="78.693886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="85.095421" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="91.334412" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="96.548164" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="107.15071" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="111.31421" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="115.22765" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="119.39114" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="125.63014" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="317.19" y="0.63"> (</t年代pan> <tspan style="font-size: 12.50px; " x="321.35349" y="0.63"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="327.60501" y="0.63"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="330.73074" y="0.63"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="336.98224" y="0.63"> )</t年代pan> </text> </g> </g> </svg></span>用一个非负最初的连续函数<年代vg height="15.05" id="M264" style="vertical-align:-3.49493pt;width:118.6875px;" version="1.1" viewbox="0 0 118.6875 15.05" width="118.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.05)"> <g transform="translate(72,-59.96)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.790329" y="0"> ∶</t年代pan> <tspan style="font-size: 12.50px; " x="21.655195" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.818695" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.383251" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.784786" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="45.998535" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="52.250038" y="0"> ]</t年代pan> <tspan style="font-size: 12.50px; " x="61.964867" y="0"> →</t年代pan> <tspan style="font-size: 12.50px; " x="79.106483" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,16.41,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> +</t年代pan> </text> </g> </g> </svg>。然而问题的动态行为(<一个href="#EEq3.2">3.2</一个>)和(<一个href="#EEq3.3">3.3</一个>)是类似的,看到<一个href="#B9">14</一个>]。</p> <h4 id="acknowledgments">确认</h4><p>第三部分作者的工作由NNSF (10926128), NSF, YMF广东省(9251064101000015,LYM10133)。</p> </div> </div> <div class="ArticleReferences_xmlContent__p_40j"> <h4 class="ArticleReferences_references__GH2t_" id="references">引用</h4><olcl一个年代年代="ArticleReferences_orderedReferences__mJr9M"> <li class="ArticleReferences_articleReference__ouEuh" id="B3"> <div class="referenceContent"> <p class="referenceText">h . m .伯恩”时间延迟的影响无血管的肿瘤的生长动力学,”<我>数学生物科学</我>,卷144,不。2、83 - 117年,1997页。</p>视图: <a href="https://doi.org/10.1016/S0025-5564(97)00023-0" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=The%20effect%20of%20time%20delays%20on%20the%20dynamics%20of%20avascular%20tumor%20growth&author=H.%20M.%20Byrne&publication_year=1997" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0904.92023" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B4"> <div class="referenceContent"> <p class="referenceText">h . m .伯恩”增长的nonnecrotic肿瘤存在和缺乏抑制剂的情况下,“<我>数学生物科学</我>,卷130,不。2、151 - 181年,1995页。</p>视图: <a href="https://doi.org/10.1016/0025-5564(94)00117-3" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Growth%20of%20nonnecrotic%20tumors%20in%20the%20presence%20and%20absence%20of%20inhibitors&author=H.%20M.%20Byrne&publication_year=1995" target="_blank" rel="noreferrer">谷歌学术搜索</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B5"> <div class="referenceContent"> <p class="referenceText">h . m .伯恩和m·a·j·牧师”增长的坏死的肿瘤存在和缺乏抑制剂的情况下,“<我>数学生物科学</我>,卷135,不。2、187 - 216年,1996页。</p>视图: <a href="https://doi.org/10.1016/0025-5564(96)00023-5" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Growth%20of%20necrotic%20tumors%20in%20the%20presence%20and%20absence%20of%20inhibitors&author=H.%20M.%20Byrne&author=M.%20A.%20J.%20Chaplain&publication_year=1996" target="_blank" rel="noreferrer">谷歌学术搜索</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B10"> <div class="referenceContent"> <p class="referenceText">m . j .失去r·f·卡拉曼·d·f·Rapacchietta et al .,“细胞的迁移和内化和聚苯乙烯微球在肿瘤细胞球状体,“<我>实验细胞研究</我>,卷141,不。1,第209 - 201页,1982。</p>视图: <a href="https://scholar.google.com/scholar_lookup?title=Migration%20and%20internalization%20of%20cells%20and%20polystyrene%20microspheres%20in%20tumor%20cell%20spheroids&author=M.%20J.%20Dorie&author=R.%20F.%20Kallman&author=D.%20F.%20Rapacchietta&publication_year=1982" target="_blank" rel="noreferrer">谷歌学术搜索</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B14"> <div class="referenceContent"> <p class="referenceText">h·格林斯潘”,由扩散模型实体瘤的生长,”<我>在应用数学的研究</我>,51卷,第340 - 317页,1972年。</p>视图: <a href="https://scholar.google.com/scholar_lookup?title=Models%20for%20the%20growth%20of%20solid%20tumor%20by%20diffusion&author=H.%20Greenspan&publication_year=1972" target="_blank" rel="noreferrer">谷歌学术搜索</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B15"> <div class="referenceContent"> <p class="referenceText">惠普格林斯潘:“增长和稳定的细胞培养和固体肿瘤,”<我>理论生物学杂志》上</我>卷,56号1,第242 - 229页,1976。</p>视图: <a href="https://doi.org/10.1016/S0022-5193(76)80054-9" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=On%20the%20growth%20and%20stability%20of%20cell%20cultures%20and%20solid%20tumors&author=H.%20P.%20Greenspan&publication_year=1976" target="_blank" rel="noreferrer">谷歌学术搜索</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B20"> <div class="referenceContent"> <p class="referenceText">r . r . Sarkar Banerjee和美国,“时间延迟模型控制恶性肿瘤的增长,”<我>国家会议上非线性系统和动力学</我>,2006年,页1 - 4。</p>视图: <a href="https://scholar.google.com/scholar_lookup?title=A%20time%20delay%20model%20for%20control%20of%20malignant%20tumor%20growth&author=R.%20R.%20Sarkar&author=S.%20Banerjee" target="_blank" rel="noreferrer">谷歌学术搜索</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B21"> <div class="referenceContent"> <p class="referenceText">k·汤普森和h·伯恩,“造型掩饰的标记细胞肿瘤球状体,“<我>通讯的数学生物学</我>卷,61年,第623 - 601页,1999年。</p>视图: <a href="https://scholar.google.com/scholar_lookup?title=Modelling%20the%20internalisation%20of%20labelled%20cells%20in%20tumor%20spheroids&author=K.%20Thompson&author=H.%20Byrne&publication_year=1999" target="_blank" rel="noreferrer">谷歌学术搜索</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B22"> <div class="referenceContent"> <p class="referenceText">j·沃德和j .王”avascular-tumor增长的数学模型II:造型grwoth饱和,”<我>IMA的数学应用在医学和生物学》杂志上</我>15卷,1-42,1998页。</p>视图: <a href="https://scholar.google.com/scholar_lookup?title=Mathematical%20modelling%20of%20avascular-tumor%20growth%20II%3A%20modelling%20grwoth%0Asaturation&author=J.%20Ward&author=J.%20King&publication_year=1998" target="_blank" rel="noreferrer">谷歌学术搜索</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B2"> <div class="referenceContent"> <p class="referenceText">m·博德纳尔和美国Foryś坏死核心形成时间延迟,”<我>数学生物科学与工程</我>,卷2,不。3、461 - 472年,2005页。</p>视图: <a href="https://scholar.google.com/scholar_lookup?title=Time%20delay%20in%20necrotic%20core%20formation&author=M.%20Bodnar&author=U.%20Fory%C5%9B&publication_year=2005" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1115.34078" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B6"> <div class="referenceContent"> <p class="referenceText">s .崔”分析数学模型的肿瘤的生长外部抑制剂的作用下,“<我>数学生物学》杂志上</我>,44卷,不。5,395 - 426年,2002页。</p>视图: <a href="https://doi.org/10.1007/s002850100130" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Analysis%20of%20a%20mathematical%20model%20for%20the%20growth%20of%20tumors%20under%20the%20action%20of%20external%20inhibitors&author=S.%20Cui&publication_year=2002" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1019.92017" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B7"> <div class="referenceContent"> <p class="referenceText">崔和a·弗里德曼”分析数学模型的抑制剂在肿瘤的生长的影响,“<我>数学生物科学</我>,卷164,不。2、103 - 137年,2000页。</p>视图: <a href="https://doi.org/10.1016/S0025-5564(99)00063-2" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Analysis%20of%20a%20mathematical%20model%20of%20the%20effect%20of%20inhibitors%20on%20the%20growth%20of%20tumors&author=S.%20Cui&author=A.%20Friedman&publication_year=2000" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0998.92022" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B8"> <div class="referenceContent"> <p class="referenceText">崔和a·弗里德曼”分析的数学模型,坏死肿瘤的增长,”<我>《数学分析和应用程序</我>,卷255,不。2、636 - 677年,2001页。</p>视图: <a href="https://doi.org/10.1006/jmaa.2000.7306" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Analysis%20of%20a%20mathematical%20model%20of%20the%20growth%20of%20necrotic%20tumors&author=S.%20Cui&author=A.%20Friedman&publication_year=2001" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0984.35169" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B9"> <div class="referenceContent"> <p class="referenceText">崔和s .徐”分析的数学模型与时间延迟肿瘤细胞的生长增殖,”<我>《数学分析和应用程序</我>,卷336,不。1,第541 - 523页,2007。</p>视图: <a href="https://doi.org/10.1016/j.jmaa.2007.02.047" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Analysis%20of%20mathematical%20models%20for%20the%20growth%20of%20tumors%20with%20time%20delays%20in%20cell%20proliferation&author=S.%20Cui&author=S.%20Xu&publication_year=2007" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1116.92032" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B11"> <div class="referenceContent"> <p class="referenceText">Fory和m·博德纳尔,“时间延迟扩散过程固体无血管的肿瘤,”<我>数学和计算机模拟</我>,37卷,不。11日,第1209 - 1201页,2003年。</p>视图: <a href="https://doi.org/10.1016/S0895-7177(03)80019-5" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Time%20delays%20in%20proliferation%20process%20for%20solid%20avascular%20tumour&author=U.%20Fory&author=M.%20Bodnar&publication_year=2003" target="_blank" rel="noreferrer">谷歌学术搜索</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B12"> <div class="referenceContent"> <p class="referenceText">Foryś和m . Kolev”时间延迟在增殖和凋亡固体无血管的肿瘤,”<我>种群动态的数学模型</我>卷,63<我>巴拿赫中心出版物</我>,页187 - 196年,波兰科学院,华沙,波兰,2004年。</p>视图: <a href="https://scholar.google.com/scholar_lookup?title=Time%20delays%20in%20proliferation%20and%20apoptosis%20for%20solid%20avascular%20tumour&author=U.%20Fory%C5%9B&author=M.%20Kolev&publication_year=2004" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1058.35107" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B13"> <div class="referenceContent"> <p class="referenceText">答:弗里德曼和f . Reitich”分析的数学模型、肿瘤的生长”<我>数学生物学》杂志上</我>,38卷,不。3、262 - 284年,1999页。</p>视图: <a href="https://doi.org/10.1007/s002850050149" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Analysis%20of%20a%20mathematical%20model%20for%20the%20growth%20of%20tumors&author=A.%20Friedman&author=F.%20Reitich&publication_year=1999" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0944.92018" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B19"> <div class="referenceContent"> <p class="referenceText">m . j . Piotrowska”霍普夫分岔固体无血管的肿瘤生长模型和两个离散延迟,”<我>数学和计算机模拟</我>卷,47号5 - 6,597 - 603年,2008页。</p>视图: <a href="https://doi.org/10.1016/j.mcm.2007.02.030" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Hopf%20bifurcation%20in%20a%20solid%20avascular%20tumour%20growth%20model%20with%20two%20discrete%20delays&author=M.%20J.%20Piotrowska&publication_year=2008" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1149.92319" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B23"> <div class="referenceContent"> <p class="referenceText">许,“霍普夫分岔的自由边界问题建模肿瘤生长有两个时间延迟,”<我>混乱,孤波和分形</我>第41卷。。5,2491 - 2494年,2009页。</p>视图: <a href="https://doi.org/10.1016/j.chaos.2008.09.029" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Hopf%20bifurcation%20of%20a%20free%20boundary%20problem%20modeling%20tumor%20growth%20with%20two%20time%20delays&author=S.%20Xu&publication_year=2009" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1198.35277" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B24"> <div class="referenceContent"> <p class="referenceText">美国许”,分析直接影响下的肿瘤生长增殖抑制剂和时间延迟的”<我>非线性分析:现实世界应用程序B</我>,11卷,不。1,第406 - 401页,2010。</p>视图: <a href="https://doi.org/10.1016/j.nonrwa.2008.11.002" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Analysis%20of%20tumor%20growth%20under%20direct%20effect%20of%20inhibitors%20with%20time%20delays%20in%20proliferation&author=S.%20Xu&publication_year=2010" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1186.34067" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B16"> <div class="referenceContent"> <p class="referenceText">j·黑尔<我>泛函微分方程的理论</我>施普林格,纽约,纽约,美国,第二版,1977年,应用数学科学。</p> </div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B1"> <div class="referenceContent"> <p class="referenceText">m·博德纳尔,“nonnegativity延迟微分方程的解决方案,“<我>应用数学的信</我>,13卷,不。6,91 - 95年,2000页。</p>视图: <a href="https://doi.org/10.1016/S0893-9659(00)00061-6" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=The%20nonnegativity%20of%20solutions%20of%20delay%20differential%20equations&author=M.%20Bodnar&publication_year=2000" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0958.34049" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B17"> <div class="referenceContent"> <p class="referenceText">和j·魏,阮x,美国“在时滞微分方程的稳定性和分岔两个延迟,”<我>《数学分析和应用程序</我>,卷236,不。2、254 - 280年,1999页。</p>视图: <a href="https://doi.org/10.1006/jmaa.1999.6418" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=Stability%20and%20bifurcation%20in%20delay-differential%20equations%20with%20two%20delays&author=X.%20Li&author=S.%20Ruan&author=J.%20Wei&publication_year=1999" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0946.34066" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> <li class="ArticleReferences_articleReference__ouEuh" id="B18"> <div class="referenceContent"> <p class="referenceText">m . j . Piotrowska“评论ODE两个离散延迟,”<我>《数学分析和应用程序</我>,卷329,不。1,第676 - 664页,2007。</p>视图: <a href="https://doi.org/10.1016/j.jmaa.2006.06.078" target="_blank" rel="noreferrer">出版商的网站</一个><年代pan class="sep">|</年代pan> <a href="https://scholar.google.com/scholar_lookup?title=A%20remark%20on%20the%20ODE%20with%20two%20discrete%20delays&author=M.%20J.%20Piotrowska&publication_year=2007" target="_blank" rel="noreferrer">谷歌学术搜索</一个><年代pan class="sep">|</年代pan> <a href="https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1123.34058" target="_blank" rel="noreferrer">Zentralblatt数学</一个></div></li> </ol> </div> <h4 class="copyright-title" id="copyright">版权</h4><div class="copyright-body"> 版权©2011鲍施等。这是一个开放分布式下文章<一个rel="license" href="http://creativecommons.org/licenses/by/3.0/">知识共享归属许可</一个>,它允许无限制的使用、分配和复制在任何媒介,提供最初的工作是正确引用。</div> <div id="related-articles"></div> </div> </div> <div class="ant-col ThreeColumnLayout_rightColumn__b_MuL"> <div> <div> <div class="ArticleRightSection_articleRightSection__rG8Zh"> <div class="ArticleToolbar_articleToolbar__8fsAH"> <a href="https://downloads.hindawi.com/journals/aaa/2011/980686.pdf" class="ButtonGroup_buttonGroupWrapper__Rqhy0" role="button" type="primary" aria-label="PDF" tabindex="0" unselectable="on" target="_blank" data-testid="download-pdf"><span class="ButtonGroup_leftGroup__FJAmS"><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg xmlns="http://www.w3.org/2000/svg" width="1em" height="1em" viewbox="0 0 19 24"> <path d="M16.772,24H2.227A2.163,2.163,0,0,1,0,21.746V2.255A2.159,2.159,0,0,1,2.227,0H12.415L12.9.494h.03V.523L19,6.669V21.746A2.161,2.161,0,0,1,16.772,24ZM2.334,1A1.276,1.276,0,0,0,1,2.358V21.642A1.283,1.283,0,0,0,2.334,23H16.665A1.276,1.276,0,0,0,18,21.642V7.087l-.032-.033H14.215A2.131,2.131,0,0,1,12.02,4.832V1.005L12.015,1ZM13,2V4.745A1.192,1.192,0,0,0,14.261,6H17ZM4.012,20A1.032,1.032,0,0,1,3,18.923a2.505,2.505,0,0,1,.967-1.539,8.15,8.15,0,0,1,2.377-1.418c.344-.779.65-1.6.92-2.327l.008-.022.017-.046.009-.024c.2-.539.453-1.209.71-1.827a4.545,4.545,0,0,1-.372-3.862c.267-.569.6-.858.976-.858.556,0,.964.621,1.015,1.546a7.125,7.125,0,0,1-.681,3.143c.158.305.343.624.55.949a7.127,7.127,0,0,0,1.806,1.9,8.681,8.681,0,0,1,2.023-.279,4.23,4.23,0,0,1,1.468.238c.118.047,1.156.478,1.2,1.143a.688.688,0,0,1-.412.67,2.509,2.509,0,0,1-1.164.242,4.757,4.757,0,0,1-.671-.052,6.631,6.631,0,0,1-2.608-1.025c-.163.042-.323.086-.514.138l-.1.026-.15.041c-.235.067-.464.13-.679.185l-.292.076-.025.006-.019,0c-.794.205-1.615.418-2.4.705A14.033,14.033,0,0,1,6,18.437C5.832,18.7,4.941,20,4.012,20ZM6,17a6.865,6.865,0,0,0-1.287.871A1.927,1.927,0,0,0,4,18.831a.12.12,0,0,0,.041.1A.165.165,0,0,0,4.17,19c.327,0,.965-.631,1.356-1.219C5.671,17.564,5.83,17.3,6,17Zm7.044-2A4.632,4.632,0,0,0,12,15.131a4.76,4.76,0,0,0,1.675.792A2.906,2.906,0,0,0,14.32,16a1.629,1.629,0,0,0,.68-.13,2.672,2.672,0,0,0-.584-.528A2.752,2.752,0,0,0,13.045,15ZM8.166,13c-.177.449-.343.884-.481,1.25l-.01.025-.009.024-.007.02-.035.092-.032.085c-.156.409-.37.97-.59,1.5.6-.181,1.229-.341,1.733-.469l.031-.008.055-.014.028-.007.028-.007.307-.078c.191-.048.4-.105.674-.179l.03-.008.005,0,.036-.01.036-.01.01,0,.01,0,.016,0a8.291,8.291,0,0,1-1.566-1.767C8.341,13.285,8.264,13.16,8.166,13Zm.516-5a1.64,1.64,0,0,0-.34.365A2.425,2.425,0,0,0,8.437,11a4.012,4.012,0,0,0,.553-2.22A1.309,1.309,0,0,0,8.681,8Z" fill="currentColor"></path> </svg></span><span>PDF</年代pan></span><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg width="1em" height="1em" viewbox="0 0 20 22"> <defs> <clippath> <path fill-rule="evenodd" clip-rule="evenodd" d="M3,22a3,3,0,0,1-3-3V16a1,1,0,0,1,2,0v3a1,1,0,0,0,1,1H17a1,1,0,0,0,1-1V16a1,1,0,1,1,2,0v3a3,3,0,0,1-3,3Zm6.288-6.3-4-4a1,1,0,0,1,1.414-1.414L9,12.586V1a1,1,0,0,1,2,0V12.586l2.293-2.293a1,1,0,0,1,1.414,1.414l-4,4a1,1,0,0,1-1.423,0Z" fill="currentColor"></path> </clippath> </defs> <g> <path fill-rule="evenodd" clip-rule="evenodd" d="M3,22a3,3,0,0,1-3-3V16a1,1,0,0,1,2,0v3a1,1,0,0,0,1,1H17a1,1,0,0,0,1-1V16a1,1,0,1,1,2,0v3a3,3,0,0,1-3,3Zm6.288-6.3-4-4a1,1,0,0,1,1.414-1.414L9,12.586V1a1,1,0,0,1,2,0V12.586l2.293-2.293a1,1,0,0,1,1.414,1.414l-4,4a1,1,0,0,1-1.423,0Z" fill="currentColor"></path> </g> </svg></span></a> <a href="//www.newsama.com/journals/aaa/2011/980686.ris" class="ButtonGroup_buttonGroupWrapper__Rqhy0" role="button" type="ghost" aria-label="Download Citation" tabindex="0" unselectable="on" data-testid="download-citation"><span class="ButtonGroup_leftGroup__FJAmS"><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg width="1em" height="1em" viewbox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"> <path d="M16 12H8C7.44772 12 7 12.4477 7 13C7 13.5523 7.44772 14 8 14H16C16.5523 14 17 13.5523 17 13C17 12.4477 16.5523 12 16 12Z" fill="currentColor"></path> <path d="M16 16H8C7.44772 16 7 16.4477 7 17C7 17.5523 7.44772 18 8 18H16C16.5523 18 17 17.5523 17 17C17 16.4477 16.5523 16 16 16Z" fill="currentColor"></path> <path d="M10 8H8C7.44772 8 7 8.44772 7 9C7 9.55228 7.44772 10 8 10H10C10.5523 10 11 9.55228 11 9C11 8.44772 10.5523 8 10 8Z" fill="currentColor"></path> <path d="M15 2C15 1.44772 14.5523 1 14 1C13.4477 1 13 1.44772 13 2V8C13 8.55228 13.4477 9 14 9H20C20.5523 9 21 8.55228 21 8C21 7.44772 20.5523 7 20 7H15V2Z" fill="currentColor"></path> <path fill-rule="evenodd" clip-rule="evenodd" d="M14.7071 1.29289C14.5196 1.10536 14.2652 1 14 1H6C4.34315 1 3 2.34315 3 4V20C3 21.6569 4.34315 23 6 23H18C19.6569 23 21 21.6569 21 20V8C21 7.73478 20.8946 7.48043 20.7071 7.29289L14.7071 1.29289ZM6 3H13.5858L19 8.41421V20C19 20.5523 18.5523 21 18 21H6C5.44772 21 5 20.5523 5 20V4C5 3.44772 5.44772 3 6 3Z" fill="currentColor"></path> </svg></span><span>下载引用</年代pan></span><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg width="1em" height="1em" viewbox="0 0 20 22"> <defs> <clippath> <path fill-rule="evenodd" clip-rule="evenodd" d="M3,22a3,3,0,0,1-3-3V16a1,1,0,0,1,2,0v3a1,1,0,0,0,1,1H17a1,1,0,0,0,1-1V16a1,1,0,1,1,2,0v3a3,3,0,0,1-3,3Zm6.288-6.3-4-4a1,1,0,0,1,1.414-1.414L9,12.586V1a1,1,0,0,1,2,0V12.586l2.293-2.293a1,1,0,0,1,1.414,1.414l-4,4a1,1,0,0,1-1.423,0Z" fill="currentColor"></path> </clippath> </defs> <g> <path fill-rule="evenodd" clip-rule="evenodd" d="M3,22a3,3,0,0,1-3-3V16a1,1,0,0,1,2,0v3a1,1,0,0,0,1,1H17a1,1,0,0,0,1-1V16a1,1,0,1,1,2,0v3a3,3,0,0,1-3,3Zm6.288-6.3-4-4a1,1,0,0,1,1.414-1.414L9,12.586V1a1,1,0,0,1,2,0V12.586l2.293-2.293a1,1,0,0,1,1.414,1.414l-4,4a1,1,0,0,1-1.423,0Z" fill="currentColor"></path> </g> </svg></span></a> <div class="ant-collapse ant-collapse-icon-position-end articleToolbarOtherFormats_otherFormatsCollapse__VHaOL"> <div class="ant-collapse-item"> <div class="ant-collapse-header" aria-expanded="false" aria-disabled="false" role="button" tabindex="0"> <div class="ant-collapse-expand-icon"> <span role="img" class="anticon ant-collapse-arrow"> <svg width="1em" height="1em" viewbox="0 0 32 32" fill="none"> <path fill-rule="evenodd" clip-rule="evenodd" d="M17.773 23.453l11.2-11.265a1.918 1.918 0 000-2.641 1.75 1.75 0 00-2.542 0l-9.929 10.101-9.933-10.1a1.75 1.75 0 00-2.542 0 1.919 1.919 0 000 2.64L15.23 23.453c.352.365.811.547 1.271.547.46 0 .92-.182 1.271-.547z" fill="currentColor"></path> </svg></span> </div> <span class="ant-collapse-header-text"><span><span role="img" class="anticon articleToolbarOtherFormats_atofDownloadIcon__YRM5T"> <svg width="1em" height="1em" viewbox="0 0 20 22"> <defs> <clippath> <path fill-rule="evenodd" clip-rule="evenodd" d="M3,22a3,3,0,0,1-3-3V16a1,1,0,0,1,2,0v3a1,1,0,0,0,1,1H17a1,1,0,0,0,1-1V16a1,1,0,1,1,2,0v3a3,3,0,0,1-3,3Zm6.288-6.3-4-4a1,1,0,0,1,1.414-1.414L9,12.586V1a1,1,0,0,1,2,0V12.586l2.293-2.293a1,1,0,0,1,1.414,1.414l-4,4a1,1,0,0,1-1.423,0Z" fill="currentColor"></path> </clippath> </defs> <g> <path fill-rule="evenodd" clip-rule="evenodd" d="M3,22a3,3,0,0,1-3-3V16a1,1,0,0,1,2,0v3a1,1,0,0,0,1,1H17a1,1,0,0,0,1-1V16a1,1,0,1,1,2,0v3a3,3,0,0,1-3,3Zm6.288-6.3-4-4a1,1,0,0,1,1.414-1.414L9,12.586V1a1,1,0,0,1,2,0V12.586l2.293-2.293a1,1,0,0,1,1.414,1.414l-4,4a1,1,0,0,1-1.423,0Z" fill="currentColor"></path> </g> </svg></span><span>下载其他格式</年代pan></span></span> </div> </div> </div> <a href="//www.newsama.com/reprints/980686/confirm" class="ButtonGroup_buttonGroupWrapper__Rqhy0" role="button" type="ghost" aria-label="Order printed copies" tabindex="0" unselectable="on" target="_blank"><span class="ButtonGroup_leftGroup__FJAmS"><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg width="1em" height="1em" viewbox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"> <path fill-rule="evenodd" clip-rule="evenodd" d="M7.41667 1H18.6667C19.1269 1 19.5 1.44772 19.5 2V22C19.5 22.5523 19.1269 23 18.6667 23H7.41667C5.80584 23 4.5 21.433 4.5 19.5V4.5C4.5 2.567 5.80584 1 7.41667 1ZM6.16667 16.3368V4.5C6.16667 3.67157 6.72631 3 7.41667 3H17.8333V16H7.41667C6.96933 16 6.54552 16.1208 6.16667 16.3368ZM17.8333 18H7.41667C6.72631 18 6.16667 18.6716 6.16667 19.5C6.16667 20.3284 6.72631 21 7.41667 21H17.8333V18Z" fill="currentColor"></path> </svg></span><span>订单打印副本</年代pan></span><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg width="1em" height="1em" viewbox="0 0 18 14"> <defs> <clippath id="a"> <path d="M10.293,13.707a1,1,0,0,1,0-1.414L14.586,8H1A1,1,0,0,1,1,6H14.586L10.293,1.707A1,1,0,0,1,11.707.293l6,6a1,1,0,0,1,.172.231h0l.01.019,0,.005.007.014.006.012,0,.008.008.018v0a1,1,0,0,1-.009.817l0,0L17.9,7.44l0,.009-.005.011-.007.013,0,.008-.008.015,0,.005-.01.016,0,0-.011.017,0,0-.012.018,0,0-.012.017,0,0L17.8,7.6l0,0-.011.014,0,.006-.01.012-.009.011-.006.007-.015.017h0l-.04.042-6,6a1,1,0,0,1-1.414,0Z" transform="translate(3 5)"></path> </clippath> </defs> <g transform="translate(-3 -5)"> <path d="M10.293,13.707a1,1,0,0,1,0-1.414L14.586,8H1A1,1,0,0,1,1,6H14.586L10.293,1.707A1,1,0,0,1,11.707.293l6,6a1,1,0,0,1,.172.231h0l.01.019,0,.005.007.014.006.012,0,.008.008.018v0a1,1,0,0,1-.009.817l0,0L17.9,7.44l0,.009-.005.011-.007.013,0,.008-.008.015,0,.005-.01.016,0,0-.011.017,0,0-.012.018,0,0-.012.017,0,0L17.8,7.6l0,0-.011.014,0,.006-.01.012-.009.011-.006.007-.015.017h0l-.04.042-6,6a1,1,0,0,1-1.414,0Z" transform="translate(3 5)" fill="currentColor"></path> </g> </svg></span></a> <div class="articleToolbarStatistics_articleToolbarStatistics__8_woN"> <div class="articleToolbarStatistics_column__R2GqD"> <div> <div> <span>的观点</年代pan> <div class="articleToolbarStatistics_amount__vJIDp"> 820年</div> </div> <div> <span>下载</年代pan> <div class="articleToolbarStatistics_amount__vJIDp"> 832年</div> </div> </div> </div> <div class="articleToolbarStatistics_column__R2GqD articleToolbarStatistics_dimensions__pe7pO"> <div> 引用</div> <span class="__dimensions_badge_embed__ articleToolbarStatistics_circle__p2dSc" data-doi="10.1155/2011/980686" data-style="small_circle"></span> </div> </div> <ul class="articleToolbarSocials_articleToolbarSocials__g02HH"> <li><a href="https://twitter.com/share?url=https://doi.org/10.1155/2011/980686&via=Hindawi" unselectable="on" title="" rel="noopener noreferrer" target="_blank"><span role="img" class="anticon"> <svg xmlns="http://www.w3.org/2000/svg" width="1em" height="1em" viewbox="126.444 2.281 589 589"> <circle cx="420.944" cy="296.781" r="294.5" fill="#2daae1"></circle> <path d="M609.773 179.634c-13.891 6.164-28.811 10.331-44.498 12.204 16.01-9.587 28.275-24.779 34.066-42.86a154.78 154.78 0 0 1-49.209 18.801c-14.125-15.056-34.267-24.456-56.551-24.456-42.773 0-77.462 34.675-77.462 77.473 0 6.064.683 11.98 1.996 17.66-64.389-3.236-121.474-34.079-159.684-80.945-6.672 11.446-10.491 24.754-10.491 38.953 0 26.875 13.679 50.587 34.464 64.477a77.122 77.122 0 0 1-35.097-9.686v.979c0 37.54 26.701 68.842 62.145 75.961-6.511 1.784-13.344 2.716-20.413 2.716-4.998 0-9.847-.473-14.584-1.364 9.859 30.769 38.471 53.166 72.363 53.799-26.515 20.785-59.925 33.175-96.212 33.175-6.25 0-12.427-.373-18.491-1.104 34.291 21.988 75.006 34.824 118.759 34.824 142.496 0 220.428-118.052 220.428-220.428 0-3.361-.074-6.697-.236-10.021a157.855 157.855 0 0 0 38.707-40.158z" fill="#fff"></path> </svg></span></a></li> <li><a href="https://www.linkedin.com/shareArticle/?url=https://doi.org/10.1155/2011/980686" unselectable="on" title="" rel="noopener noreferrer" target="_blank"><span role="img" class="anticon"> <svg xmlns="http://www.w3.org/2000/svg" width="1em" height="1em" viewbox="0 0 256 256" xml:space="preserve"> <g transform="translate(-373.642 -318.344)"> <path fill="#1D8CB5" d="M583.659 328.849H419.625c-19.599 0-35.487 15.888-35.487 35.487v164.015c0 19.599 15.888 35.486 35.487 35.486h164.033c19.599 0 35.487-15.888 35.487-35.486V364.337c.001-19.599-15.888-35.488-35.486-35.488z"></path> <lineargradient id="a" gradientunits="userSpaceOnUse" x1="-808.873" y1="199.892" x2="-860.569" y2="49.203" gradienttransform="rotate(180 -17.27 206.273) scale(.5915)"> <stop offset="0" style="stop-color:#fff"></stop> <stop offset="1" style="stop-color:#fff;stop-opacity:0"></stop> </lineargradient> <path opacity=".781" fill="url(#a)" d="M557.056 338.895H446.227c-29.337 0-52.955 23.81-52.955 53.385v108.127c.95 23.087 4.603 8.491 11.555-17.075 8.081-29.713 34.396-55.682 66.445-75.145 24.461-14.855 51.84-24.341 101.679-25.244 28.265-.511 25.77-36.683-15.895-44.048z"></path> </g> <g fill="#FFF"> <path d="M81.604 207.76V101.49h-35.32v106.27h35.32zM63.951 86.98c12.308 0 19.981-8.161 19.981-18.36-.228-10.426-7.673-18.36-19.75-18.36-12.087 0-19.984 7.934-19.984 18.36 0 10.199 7.662 18.36 19.52 18.36h.233zM101.151 207.76h35.326v-59.345c0-3.178.23-6.352 1.165-8.621 2.547-6.35 8.363-12.92 18.118-12.92 12.784 0 17.893 9.748 17.893 24.032v56.855h35.319v-60.936c0-32.643-17.422-47.83-40.665-47.83-19.053 0-27.421 10.647-32.068 17.902h.238V101.49h-35.326c.469 9.97 0 106.27 0 106.27z"></path> </g> </svg></span></a></li> <li><a href="https://www.facebook.com/sharer/sharer.php?u=https://doi.org/10.1155/2011/980686" unselectable="on" title="" rel="noopener noreferrer" target="_blank"><span role="img" class="anticon"> <svg xmlns="http://www.w3.org/2000/svg" width="1em" height="1em" viewbox="0 0 14222 14222"> <circle cx="7111" cy="7112" r="7111" fill="#1977f3"></circle> <path d="M9879 9168l315-2056H8222V5778c0-562 275-1111 1159-1111h897V2917s-814-139-1592-139c-1624 0-2686 984-2686 2767v1567H4194v2056h1806v4969c362 57 733 86 1111 86s749-30 1111-86V9168z" fill="#fff"></path> </svg></span></a></li> <li><a href="//www.newsama.com/cdn-cgi/l/email-protection" unselectable="on" title="" rel="noopener noreferrer" target="_blank"><span role="img" class="anticon"> <svg version="1.0" xmlns="http://www.w3.org/2000/svg" width="1em" height="1em" viewbox="0 0 1401 1101"> <path fill="#6ba439" d="M132.5 2.1C101.4 6 71.1 20.5 48.3 42.4c-22.7 21.7-37.5 48.3-44.6 80l-2.2 10.1v837l2.2 10.1c6.7 30 19.3 53.9 39.8 75.3 23.3 24.3 56.4 40.9 90 45 12.8 1.6 1122.2 1.6 1135 0 38.9-4.8 75.9-25.6 100-56.4 15-19.3 24.2-39 29.8-63.9l2.2-10.1v-837l-2.2-10.1c-7.1-31.8-21.9-58.3-44.6-80.1-20.2-19.2-43.9-31.7-73.6-38.5l-9.6-2.3L705 1.4c-321.9-.1-568.5.2-572.5.7zm1123.4 99.5c1.7.4 3.1 1 3.1 1.4 0 .5-125.6 87.7-279 194L700.9 490.2 422 297C268.5 190.7 143 103.4 143 103c0-.5 1-1.1 2.3-1.3 3.5-.7 1107.1-.8 1110.6-.1zM401 404c164.4 113.8 299.4 207 300 207 .6 0 135.6-93.2 300.1-207 164.5-113.9 299.3-207 299.5-207 .2 0 .4 170.9.4 379.7 0 417.9.5 386-6.1 398.3-6.2 11.6-19 21.4-31.9 24.4-5.8 1.4-67.4 1.6-562 1.6s-556.2-.2-562-1.6c-12.9-3-25.7-12.8-31.9-24.4-6.6-12.3-6.1 19.6-6.1-398.3 0-208.8.2-379.7.5-379.7S236.6 290.1 401 404z"></path> </svg></span></a></li> </ul> </div> </div> </div> </div> </div> </div> </main> <div> <footer class="ant-layout-footer"> <div class="footer-main-nav-section"> <div class="footer-main-nav"> <div class="ant-col footer-main-nav__column"> <ul class="ant-menu ant-menu-root ant-menu-vertical ant-menu-light footer-main-nav__menu" role="menu" tabindex="0" data-menu-list="true"> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/about/" aria-label="About Us">关于我们</一个></年代pan></li> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/about/contact/" aria-label="Contact us">联系我们</一个></年代pan></li> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/publishing-partnerships/" aria-label="Partnerships">合作伙伴关系</一个></年代pan></li> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/blog/" aria-label="Blog">博客</一个></年代pan></li> </ul> <div style="display:none" aria-hidden="true"></div> </div> <div class="ant-col footer-main-nav__column"> <ul class="ant-menu ant-menu-root ant-menu-vertical ant-menu-light footer-main-nav__menu" role="menu" tabindex="0" data-menu-list="true"> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/journals/" aria-label="Journals">期刊</一个></年代pan></li> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/publish-research/authors/article-processing-charges/" aria-label="Article Processing Charges">文章处理费用</一个></年代pan></li> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/print-editions/" aria-label="Print editions">印刷版</一个></年代pan></li> </ul> <div style="display:none" aria-hidden="true"></div> </div> <div class="ant-col footer-main-nav__column"> <ul class="ant-menu ant-menu-root ant-menu-vertical ant-menu-light footer-main-nav__menu" role="menu" tabindex="0" data-menu-list="true"> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/publish-research/authors/" aria-label="Authors">作者</一个></年代pan></li> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/publish-research/editors/" aria-label="Editors">编辑器</一个></年代pan></li> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/publish-research/reviewers/" aria-label="Reviewers">评论家</一个></年代pan></li> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/publishing-partnerships/" aria-label="Partnerships">合作伙伴关系</一个></年代pan></li> </ul> <div style="display:none" aria-hidden="true"></div> </div> <div class="ant-col footer-main-nav__column"> <ul class="ant-menu ant-menu-root ant-menu-vertical ant-menu-light footer-main-nav__menu" role="menu" tabindex="0" data-menu-list="true"> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/hindawi-xml-corpus/" aria-label="Hindawi XML Corpus">雷电竞APP</a></span></li> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/open-archives-initiative/" aria-label="Open Archives Initiative">开放档案计划</一个></年代pan></li> <li class="ant-menu-item ant-menu-item-only-child footer-main-nav__item" role="menuitem" tabindex="-1"><span class="ant-menu-title-content"><a href="//www.newsama.com/preventing-and-protecting-against-fraud/" aria-label="Fraud prevention">欺诈防范</一个></年代pan></li> </ul> <div style="display:none" aria-hidden="true"></div> </div> </div> <div class="social-links"> 遵循我们:<div class="social-links__icons"> <a aria-label="Twitter" rel="noopener noreferrer" href="https://twitter.com/hindawi" target="_blank" class="social-links__icon"><span role="img" class="anticon"> <svg xmlns="http://www.w3.org/2000/svg" width="40" height="40" viewbox="0 0 40 40" fill="none"> <path fill-rule="evenodd" clip-rule="evenodd" d="M0 20C0 8.95599 8.95599 0 20 0C31.044 0 40 8.95599 40 20C40 31.044 31.044 40 20 40C8.95599 40 0 31.044 0 20ZM29.1449 16.1871C29.1449 15.9885 29.1406 15.7907 29.1318 15.5939C30.0281 14.946 30.8072 14.1373 31.4215 13.2166C30.5991 13.5822 29.7141 13.8284 28.7854 13.9395C29.7333 13.371 30.4608 12.4719 30.8041 11.4005C29.917 11.9263 28.9349 12.3083 27.8891 12.5146C27.0514 11.6223 25.8588 11.0641 24.5383 11.0641C22.0032 11.0641 19.9472 13.1201 19.9472 15.6543C19.9472 16.0144 19.9878 16.3647 20.0665 16.701C16.2512 16.5091 12.868 14.6823 10.6042 11.9046C10.21 12.583 9.9826 13.371 9.9826 14.212C9.9826 15.8047 10.7928 17.2107 12.0255 18.0331C11.2723 18.0099 10.5649 17.803 9.94629 17.4591C9.94568 17.4783 9.94568 17.4969 9.94568 17.5177C9.94568 19.7406 11.528 21.597 13.6285 22.0175C13.2428 22.1228 12.8369 22.1793 12.4179 22.1793C12.1228 22.1793 11.835 22.15 11.5552 22.0963C12.1399 23.9203 13.8342 25.2475 15.8438 25.2847C14.2725 26.5161 12.2931 27.2498 10.1422 27.2498C9.77203 27.2498 9.40643 27.2287 9.04724 27.1863C11.0788 28.4885 13.4912 29.2484 16.0837 29.2484C24.5276 29.248 29.1449 22.2531 29.1449 16.1871Z" fill="#F2F2F2"></path> </svg></span></a> <a aria-label="Linkedin" rel="noopener noreferrer" href="https://www.linkedin.com/company/hindawilimited/" target="_blank" class="social-links__icon"><span role="img" class="anticon"> <svg width="40" height="40" viewbox="0 0 40 40" fill="none" xmlns="http://www.w3.org/2000/svg"> <path fill-rule="evenodd" clip-rule="evenodd" d="M0 20C0 8.95599 8.95599 0 20 0C31.044 0 40 8.95599 40 20C40 31.044 31.044 40 20 40C8.95599 40 0 31.044 0 20ZM9.31732 30.2344H14.1882V15.5801H9.31732V30.2344ZM11.7529 13.5791H11.7212C10.0867 13.5791 9.02954 12.4539 9.02954 11.0477C9.02954 9.60968 10.119 8.51562 11.7853 8.51562C13.4515 8.51562 14.4769 9.60968 14.5087 11.0477C14.5087 12.4539 13.4515 13.5791 11.7529 13.5791ZM26.8814 30.2344H31.7517V21.8317C31.7517 17.3306 29.3488 15.2362 26.1441 15.2362C23.5599 15.2362 22.4017 16.6565 21.7545 17.655V15.5801H16.8839C16.9476 16.955 16.8839 30.2344 16.8839 30.2344H21.7545V22.0508C21.7545 21.6129 21.7859 21.1752 21.9147 20.8621C22.2668 19.9872 23.0682 19.0808 24.4138 19.0808C26.1761 19.0808 26.8814 20.4245 26.8814 22.3947V30.2344Z" fill="#F2F2F2"></path> </svg></span></a> <a aria-label="Youtube" rel="noopener noreferrer" href="https://www.youtube.com/channel/UCDIK_O7FATPEkhad_sQVaSg" target="_blank" class="social-links__icon"><span role="img" class="anticon"> <svg width="40" height="40" viewbox="0 0 40 40" fill="none" xmlns="http://www.w3.org/2000/svg"> <path d="M17.2973 23.7838L23.7838 20L17.2973 16.2162V23.7838Z" fill="#F2F2F2"></path> <path fill-rule="evenodd" clip-rule="evenodd" d="M0 20C0 8.95599 8.95599 0 20 0C31.044 0 40 8.95599 40 20C40 31.044 31.044 40 20 40C8.95599 40 0 31.044 0 20ZM31.9824 26.0324C32.4969 24.0765 32.4969 20.0204 32.4969 20.0204C32.4969 20.0204 32.5177 15.9442 31.9824 13.9883C31.694 12.9178 30.8499 12.0734 29.7794 11.7853C27.8235 11.25 20 11.25 20 11.25C20 11.25 12.1765 11.25 10.2206 11.7648C9.17053 12.0529 8.30566 12.9178 8.01758 13.9883C7.50275 15.9442 7.50275 20 7.50275 20C7.50275 20 7.50275 24.0765 8.01758 26.0117C8.30597 27.0822 9.15009 27.9266 10.2206 28.2147C12.197 28.75 20 28.75 20 28.75C20 28.75 27.8235 28.75 29.7794 28.2352C30.8499 27.9471 31.694 27.103 31.9824 26.0324Z" fill="#F2F2F2"></path> </svg></span></a> <a aria-label="Facebook" rel="noopener noreferrer" href="https://www.facebook.com/HindawiLimited/" target="_blank" class="social-links__icon"><span role="img" class="anticon"> <svg width="40" height="39" viewbox="0 0 40 39" fill="none" xmlns="http://www.w3.org/2000/svg"> <path d="M40 19.4595C40 8.71394 31.044 0 20 0C8.95599 0 0 8.71394 0 19.4595C0 30.205 8.95599 38.9189 20 38.9189C20.1175 38.9189 20.2344 38.9165 20.3516 38.9145V23.7661H16.0547V18.8938H20.3516V15.3084C20.3516 11.1502 22.9605 8.88705 26.7728 8.88705C28.5983 8.88705 30.1675 9.01948 30.625 9.07857V13.4244H27.9962C25.9222 13.4244 25.5206 14.3835 25.5206 15.7906V18.8938H30.4797L29.8334 23.7661H25.5206V38.1671C33.8803 35.8353 40 28.3418 40 19.4595Z" fill="#F2F2F2"></path> </svg></span></a> </div> </div> </div> <div class="footer-minor-nav"> <span><a href="//www.newsama.com/privacy/" class="footer-minor-nav__link" aria-label="Privacy Policy">隐私政策</一个><div class="ant-divider ant-divider-vertical" role="separator"></div></span> <span><a href="//www.newsama.com/terms/" class="footer-minor-nav__link" aria-label="Terms of Service">服务条款</一个><div class="ant-divider ant-divider-vertical" role="separator"></div></span> <span><a href="//www.newsama.com/responsible-disclosure-policy/" class="footer-minor-nav__link" aria-label="Responsible Disclosure Policy">负责信息披露政策</一个><div class="ant-divider ant-divider-vertical" role="separator"></div></span> <span><a href="//www.newsama.com/cookie-policy/" class="footer-minor-nav__link" aria-label="Cookie Policy">饼干的政策</一个><div class="ant-divider ant-divider-vertical" role="separator"></div></span> <span><a href="//www.newsama.com/copyright/" class="footer-minor-nav__link" aria-label="Copyright">版权</一个><div class="ant-divider ant-divider-vertical" role="separator"></div></span> <span><a href="//www.newsama.com/modern-slavery-statement/" class="footer-minor-nav__link" aria-label="Modern slavery statement">现代奴隶制的声明</一个><div class="ant-divider ant-divider-vertical" role="separator"></div></span> <span><a class="footer-minor-nav__link" aria-label="Cookie Preferences">饼干的偏好</一个></年代pan> </div> </footer> </div> </section> <noscript> <iframe src="https://www.newsama.com/ns.html?id=GTM-MQ4MGW" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript> </div> </body> </html>