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class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目錄</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移去側欄</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">收埋</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">文頭</div> </a> </li> <li id="toc-基本概念" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#基本概念"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>基本概念</span> </div> </a> <ul id="toc-基本概念-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-系統概念" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#系統概念"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>系統概念</span> </div> </a> <button aria-controls="toc-系統概念-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>切換去系統概念細章節</span> </button> <ul id="toc-系統概念-sublist" class="vector-toc-list"> <li id="toc-吸引子" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#吸引子"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>吸引子</span> </div> </a> <ul id="toc-吸引子-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-複雜度" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#複雜度"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>複雜度</span> </div> </a> <ul id="toc-複雜度-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-混沌系統" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#混沌系統"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>混沌系統</span> </div> </a> <ul id="toc-混沌系統-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-簡史" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#簡史"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>簡史</span> </div> </a> <ul id="toc-簡史-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-相關概念" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#相關概念"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>相關概念</span> </div> </a> <ul id="toc-相關概念-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-睇埋" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#睇埋"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>睇埋</span> </div> </a> <ul id="toc-睇埋-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-文獻" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#文獻"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>文獻</span> </div> </a> <ul id="toc-文獻-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-攷" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#攷"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>攷</span> </div> </a> <ul id="toc-攷-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-拎" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#拎"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>拎</span> </div> </a> <ul id="toc-拎-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目錄" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="開/收內容一覽" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">開/收內容一覽</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">混沌理論</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="去睇另一種語文嘅文章。有67種語言版本。" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-67" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">67種語言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Chaosteorie" title="Chaosteorie – 南非荷蘭文" lang="af" hreflang="af" data-title="Chaosteorie" data-language-autonym="Afrikaans" data-language-local-name="南非荷蘭文" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D9%81%D9%88%D8%B6%D9%89_%D8%A7%D9%84%D9%83%D9%88%D9%86" title="نظرية فوضى الكون – 阿拉伯文" lang="ar" hreflang="ar" data-title="نظرية فوضى الكون" data-language-autonym="العربية" data-language-local-name="阿拉伯文" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%86%D8%B6%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%B1%D9%88%D9%8A%D9%86%D8%A9" title="نضرية الروينة – 摩洛哥阿拉伯文" lang="ary" hreflang="ary" data-title="نضرية الروينة" data-language-autonym="الدارجة" data-language-local-name="摩洛哥阿拉伯文" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_del_caos" title="Teoría del caos – 阿斯圖里亞文" lang="ast" hreflang="ast" data-title="Teoría del caos" data-language-autonym="Asturianu" data-language-local-name="阿斯圖里亞文" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Xaos_n%C9%99z%C9%99riyy%C9%99si" title="Xaos nəzəriyyəsi – 亞塞拜然文" lang="az" hreflang="az" data-title="Xaos nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="亞塞拜然文" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A5%D0%B0%D0%BE%D1%81_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Хаос теорияһы – 巴什客爾文" lang="ba" hreflang="ba" data-title="Хаос теорияһы" data-language-autonym="Башҡортса" data-language-local-name="巴什客爾文" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D1%85%D0%B0%D0%BE%D1%81%D0%B0" title="Теория на хаоса – 保加利亞文" lang="bg" hreflang="bg" data-title="Теория на хаоса" data-language-autonym="Български" data-language-local-name="保加利亞文" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%83%E0%A6%99%E0%A7%8D%E0%A6%96%E0%A6%B2%E0%A6%BE_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="বিশৃঙ্খলা তত্ত্ব – 孟加拉文" lang="bn" hreflang="bn" data-title="বিশৃঙ্খলা তত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="孟加拉文" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_del_caos" title="Teoria del caos – 加泰羅尼亞文" lang="ca" hreflang="ca" data-title="Teoria del caos" data-language-autonym="Català" data-language-local-name="加泰羅尼亞文" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%DB%8C%DB%8C_%D8%B4%DB%8E%D9%88%D8%A7%D9%88%DB%8C" title="تیۆریی شێواوی – 索拉尼庫爾德文" lang="ckb" hreflang="ckb" data-title="تیۆریی شێواوی" data-language-autonym="کوردی" data-language-local-name="索拉尼庫爾德文" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teorie_chaosu" title="Teorie chaosu – 捷克文" lang="cs" hreflang="cs" data-title="Teorie chaosu" data-language-autonym="Čeština" data-language-local-name="捷克文" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D0%B0%D0%BE%D1%81_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Хаос теорийĕ – 楚瓦什文" lang="cv" hreflang="cv" data-title="Хаос теорийĕ" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什文" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Damcaniaeth_anhrefn" title="Damcaniaeth anhrefn – 威爾斯文" lang="cy" hreflang="cy" data-title="Damcaniaeth anhrefn" data-language-autonym="Cymraeg" data-language-local-name="威爾斯文" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kaosteori" title="Kaosteori – 丹麥文" lang="da" hreflang="da" data-title="Kaosteori" data-language-autonym="Dansk" data-language-local-name="丹麥文" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="好文"><a href="https://de.wikipedia.org/wiki/Chaosforschung" title="Chaosforschung – 德文" lang="de" hreflang="de" data-title="Chaosforschung" data-language-autonym="Deutsch" data-language-local-name="德文" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CF%84%CE%BF%CF%85_%CF%87%CE%AC%CE%BF%CF%85%CF%82" title="Θεωρία του χάους – 希臘文" lang="el" hreflang="el" data-title="Θεωρία του χάους" data-language-autonym="Ελληνικά" data-language-local-name="希臘文" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Chaos_theory" title="Chaos theory – 英文" lang="en" hreflang="en" data-title="Chaos theory" data-language-autonym="English" data-language-local-name="英文" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/%C4%A4aosoteorio" title="Ĥaosoteorio – 世界文" lang="eo" hreflang="eo" data-title="Ĥaosoteorio" data-language-autonym="Esperanto" data-language-local-name="世界文" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_del_caos" title="Teoría del caos – 西班牙文" lang="es" hreflang="es" data-title="Teoría del caos" data-language-autonym="Español" data-language-local-name="西班牙文" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kaoseteooria" title="Kaoseteooria – 愛沙尼亞文" lang="et" hreflang="et" data-title="Kaoseteooria" data-language-autonym="Eesti" data-language-local-name="愛沙尼亞文" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kaosaren_teoria" title="Kaosaren teoria – 巴斯克文" lang="eu" hreflang="eu" data-title="Kaosaren teoria" data-language-autonym="Euskara" data-language-local-name="巴斯克文" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%A2%D8%B4%D9%88%D8%A8" title="نظریه آشوب – 波斯文" lang="fa" hreflang="fa" data-title="نظریه آشوب" data-language-autonym="فارسی" data-language-local-name="波斯文" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kaaosteoria" title="Kaaosteoria – 芬蘭文" lang="fi" hreflang="fi" data-title="Kaaosteoria" data-language-autonym="Suomi" data-language-local-name="芬蘭文" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_du_chaos" title="Théorie du chaos – 法文" lang="fr" hreflang="fr" data-title="Théorie du chaos" data-language-autonym="Français" data-language-local-name="法文" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Anord" title="Anord – 愛爾蘭文" lang="ga" hreflang="ga" data-title="Anord" data-language-autonym="Gaeilge" data-language-local-name="愛爾蘭文" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96" title="混沌理論 – 贛語" lang="gan" hreflang="gan" data-title="混沌理論" data-language-autonym="贛語" data-language-local-name="贛語" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_do_caos" title="Teoría do caos – 加里西亞文" lang="gl" hreflang="gl" data-title="Teoría do caos" data-language-autonym="Galego" data-language-local-name="加里西亞文" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9B%D7%90%D7%95%D7%A1" title="תורת הכאוס – 希伯來文" lang="he" hreflang="he" data-title="תורת הכאוס" data-language-autonym="עברית" data-language-local-name="希伯來文" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="अक्रम सिद्धान्त – 北印度文" lang="hi" hreflang="hi" data-title="अक्रम सिद्धान्त" data-language-autonym="हिन्दी" data-language-local-name="北印度文" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Teorija_kaosa" title="Teorija kaosa – 克羅埃西亞文" lang="hr" hreflang="hr" data-title="Teorija kaosa" data-language-autonym="Hrvatski" data-language-local-name="克羅埃西亞文" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/K%C3%A1oszelm%C3%A9let" title="Káoszelmélet – 匈牙利文" lang="hu" hreflang="hu" data-title="Káoszelmélet" data-language-autonym="Magyar" data-language-local-name="匈牙利文" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_kekacauan" title="Teori kekacauan – 印尼文" lang="id" hreflang="id" data-title="Teori kekacauan" data-language-autonym="Bahasa Indonesia" data-language-local-name="印尼文" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_del_caos" title="Teoria del caos – 義大利文" lang="it" hreflang="it" data-title="Teoria del caos" data-language-autonym="Italiano" data-language-local-name="義大利文" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AB%E3%82%AA%E3%82%B9%E7%90%86%E8%AB%96" title="カオス理論 – 日文" lang="ja" hreflang="ja" data-title="カオス理論" data-language-autonym="日本語" data-language-local-name="日文" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Kyeas_tiori" title="Kyeas tiori – 牙買加克裏奧爾英文" lang="jam" hreflang="jam" data-title="Kyeas tiori" data-language-autonym="Patois" data-language-local-name="牙買加克裏奧爾英文" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%98%BC%EB%8F%88_%EC%9D%B4%EB%A1%A0" title="혼돈 이론 – 韓文" lang="ko" hreflang="ko" data-title="혼돈 이론" data-language-autonym="한국어" data-language-local-name="韓文" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Chaoso_teorija" title="Chaoso teorija – 立陶宛文" lang="lt" hreflang="lt" data-title="Chaoso teorija" data-language-autonym="Lietuvių" data-language-local-name="立陶宛文" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B4%AF%E0%B5%8B%E0%B4%B8%E0%B5%8D_%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%82" title="കയോസ് സിദ്ധാന്തം – 馬來亞拉姆文" lang="ml" hreflang="ml" data-title="കയോസ് സിദ്ധാന്തം" data-language-autonym="മലയാളം" data-language-local-name="馬來亞拉姆文" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D0%B0%D0%BE%D1%81%D1%8B%D0%BD_%D0%BE%D0%BD%D0%BE%D0%BB" title="Хаосын онол – 蒙古文" lang="mn" hreflang="mn" data-title="Хаосын онол" data-language-autonym="Монгол" data-language-local-name="蒙古文" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A5%8B%E0%A4%B2%E0%A4%BE%E0%A4%B9%E0%A4%B2" title="कोलाहल – 馬拉地文" lang="mr" hreflang="mr" data-title="कोलाहल" data-language-autonym="मराठी" data-language-local-name="馬拉地文" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_kekacauan" title="Teori kekacauan – 馬來文" lang="ms" hreflang="ms" data-title="Teori kekacauan" data-language-autonym="Bahasa Melayu" data-language-local-name="馬來文" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Chaostheorie" title="Chaostheorie – 荷蘭文" lang="nl" hreflang="nl" data-title="Chaostheorie" data-language-autonym="Nederlands" data-language-local-name="荷蘭文" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kaosteori" title="Kaosteori – 耐諾斯克挪威文" lang="nn" hreflang="nn" data-title="Kaosteori" data-language-autonym="Norsk nynorsk" data-language-local-name="耐諾斯克挪威文" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kaosteori" title="Kaosteori – 巴克摩挪威文" lang="nb" hreflang="nb" data-title="Kaosteori" data-language-autonym="Norsk bokmål" data-language-local-name="巴克摩挪威文" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%98%E0%A9%9C%E0%A8%AE%E0%A9%B1%E0%A8%B8_%E0%A8%B8%E0%A8%BF%E0%A8%A7%E0%A8%BE%E0%A8%82%E0%A8%A4" title="ਘੜਮੱਸ ਸਿਧਾਂਤ – 旁遮普文" lang="pa" hreflang="pa" data-title="ਘੜਮੱਸ ਸਿਧਾਂਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普文" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pam mw-list-item"><a href="https://pam.wikipedia.org/wiki/Teoriang_chaos" title="Teoriang chaos – 潘帕嘉文" lang="pam" hreflang="pam" data-title="Teoriang chaos" data-language-autonym="Kapampangan" data-language-local-name="潘帕嘉文" class="interlanguage-link-target"><span>Kapampangan</span></a></li><li class="interlanguage-link interwiki-pdc mw-list-item"><a href="https://pdc.wikipedia.org/wiki/Chaos_Theory" title="Chaos Theory – 賓夕法尼亞德文" lang="pdc" hreflang="pdc" data-title="Chaos Theory" data-language-autonym="Deitsch" data-language-local-name="賓夕法尼亞德文" class="interlanguage-link-target"><span>Deitsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Chaos_(matematyka)" title="Chaos (matematyka) – 波蘭文" lang="pl" hreflang="pl" data-title="Chaos (matematyka)" data-language-autonym="Polski" data-language-local-name="波蘭文" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%DB%82_%D8%B4%D9%88%D8%A7%D8%B4" title="نظریۂ شواش – Western Punjabi" lang="pnb" hreflang="pnb" data-title="نظریۂ شواش" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_do_caos" title="Teoria do caos – 葡萄牙文" lang="pt" hreflang="pt" data-title="Teoria do caos" data-language-autonym="Português" data-language-local-name="葡萄牙文" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_haosului" title="Teoria haosului – 羅馬尼亞文" lang="ro" hreflang="ro" data-title="Teoria haosului" data-language-autonym="Română" data-language-local-name="羅馬尼亞文" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D1%85%D0%B0%D0%BE%D1%81%D0%B0" title="Теория хаоса – 俄文" lang="ru" hreflang="ru" data-title="Теория хаоса" data-language-autonym="Русский" data-language-local-name="俄文" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tiur%C3%ACa_d%C3%BB_caos" title="Tiurìa dû caos – 西西里文" lang="scn" hreflang="scn" data-title="Tiurìa dû caos" data-language-autonym="Sicilianu" data-language-local-name="西西里文" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Teorija_kaosa" title="Teorija kaosa – 塞爾維亞克羅埃西亞文" lang="sh" hreflang="sh" data-title="Teorija kaosa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞爾維亞克羅埃西亞文" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Chaos_theory" title="Chaos theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Chaos theory" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Te%C3%B3ria_chaosu" title="Teória chaosu – 斯洛伐克文" lang="sk" hreflang="sk" data-title="Teória chaosu" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克文" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Teorija_kaosa" title="Teorija kaosa – 斯洛維尼亞文" lang="sl" hreflang="sl" data-title="Teorija kaosa" data-language-autonym="Slovenščina" data-language-local-name="斯洛維尼亞文" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D1%85%D0%B0%D0%BE%D1%81%D0%B0" title="Теорија хаоса – 塞爾維亞文" lang="sr" hreflang="sr" data-title="Теорија хаоса" data-language-autonym="Српски / srpski" data-language-local-name="塞爾維亞文" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kaosteori" title="Kaosteori – 瑞典文" lang="sv" hreflang="sv" data-title="Kaosteori" data-language-autonym="Svenska" data-language-local-name="瑞典文" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%92%E0%AE%B4%E0%AF%81%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%AE%E0%AF%88_%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="ஒழுங்கின்மை கோட்பாடு – 坦米爾文" lang="ta" hreflang="ta" data-title="ஒழுங்கின்மை கோட்பாடு" data-language-autonym="தமிழ்" data-language-local-name="坦米爾文" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%AD%E0%B8%A5%E0%B8%A7%E0%B8%99" title="ทฤษฎีความอลวน – 泰文" lang="th" hreflang="th" data-title="ทฤษฎีความอลวน" data-language-autonym="ไทย" data-language-local-name="泰文" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kaos_teorisi" title="Kaos teorisi – 土耳其文" lang="tr" hreflang="tr" data-title="Kaos teorisi" data-language-autonym="Türkçe" data-language-local-name="土耳其文" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D1%85%D0%B0%D0%BE%D1%81%D1%83" title="Теорія хаосу – 烏克蘭文" lang="uk" hreflang="uk" data-title="Теорія хаосу" data-language-autonym="Українська" data-language-local-name="烏克蘭文" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%DB%82_%D8%B4%D9%88%D8%A7%D8%B4" title="نظریۂ شواش – 烏都文" lang="ur" hreflang="ur" data-title="نظریۂ شواش" data-language-autonym="اردو" data-language-local-name="烏都文" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_h%E1%BB%97n_lo%E1%BA%A1n" title="Lý thuyết hỗn loạn – 越南文" lang="vi" hreflang="vi" data-title="Lý thuyết hỗn loạn" data-language-autonym="Tiếng Việt" data-language-local-name="越南文" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AE%BA" title="混沌理论 – 吳語" lang="wuu" hreflang="wuu" data-title="混沌理论" data-language-autonym="吴语" data-language-local-name="吳語" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AE%BA" title="混沌理论 – 中文" lang="zh" hreflang="zh" data-title="混沌理论" data-language-autonym="中文" data-language-local-name="中文" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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mw-portlet-electronpdfservice-sidebar-portlet-heading" > <div class="vector-menu-heading"> 打印/匯出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=show-download-screen"><span>下載PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="呢一版印得嘅版本[p]" accesskey="p"><span>印得嘅版本</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 第啲維基項目 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Chaos_theory" hreflang="en"><span>維基同享</span></a></li><li id="t-wikibase" class="wb-otherproject-link 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">閱讀設定</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移去側欄</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">收埋</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">出自維基百科，自由嘅百科全書</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="yue" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/File:Lorenz_attractor_yb.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Lorenz_attractor_yb.svg/300px-Lorenz_attractor_yb.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Lorenz_attractor_yb.svg/450px-Lorenz_attractor_yb.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Lorenz_attractor_yb.svg/600px-Lorenz_attractor_yb.svg.png 2x" data-file-width="750" data-file-height="750" /></a><figcaption><a href="/wiki/%E6%B4%9B%E5%80%AB%E8%8C%B2%E5%90%B8%E5%BC%95%E5%AD%90" class="mw-redirect" title="洛倫茲吸引子">洛倫茲吸引子</a>出嘅一幅圖</figcaption></figure> <p><b>混沌理論</b>（<a href="/wiki/Wikipedia:%E7%B2%B5%E6%96%87%E6%8B%BC%E9%9F%B3" title="Wikipedia:粵文拼音">粵拼</a>：<span class="plainlinks" style="display:inline-block"><a rel="nofollow" class="external text" href="http://yue.forvo.com/word/yue/混沌理論/#yue"><span>wan<span class="Jpingtone">6</span> deon<span class="Jpingtone">6</span> lei<span class="Jpingtone">5</span> leon<span class="Jpingtone">6</span></span></a></span>；<a href="/wiki/%E8%8B%B1%E6%96%87" title="英文">英文</a>：<span lang="en"><b>chaos theory</b></span>）係<a href="/wiki/%E6%95%B8%E5%AD%B8" title="數學">數學</a>嘅一個子領域，研究「望落好似完全<a href="/wiki/%E9%9A%A8%E6%A9%9F" title="隨機">隨機</a>同無法預測嘅<a href="/wiki/%E7%B3%BB%E7%B5%B1" title="系統">系統</a>」。精確啲講，套理論建基於<a href="/wiki/%E8%9D%B4%E8%9D%B6%E6%95%88%E6%87%89" title="蝴蝶效應">蝴蝶效應</a>－呢個諗頭講响<a href="/wiki/%E9%9D%9E%E7%B7%9A%E6%80%A7%E7%B3%BB%E7%B5%B1" title="非線性系統">非線性系統</a>入面，就算<a href="/wiki/%E5%88%9D%E5%A7%8B%E6%A2%9D%E4%BB%B6" title="初始條件">初始狀態</a>變咗少少，都可以引發好大影響，令個系統跟住落嚟嘅狀態完全唔同嗮<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。 </p><p>舉例說明，一座<a href="/wiki/%E5%9F%8E%E5%B8%82" title="城市">城市</a>係個<a href="/wiki/%E8%A4%87%E9%9B%9C%E5%98%85%E7%B3%BB%E7%B5%B1" class="mw-redirect" title="複雜嘅系統">複雜嘅系統</a><sup id="cite_ref-betten2015_3-0" class="reference"><a href="#cite_note-betten2015-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>：座城市由好多件部份組成，包括係<a href="/wiki/%E5%B8%82%E6%B0%91" class="mw-redirect" title="市民">市民</a>、<a href="/wiki/%E9%A6%AC%E8%B7%AF" title="馬路">馬路</a>等嘅<a href="/wiki/%E5%9F%BA%E5%BB%BA" class="mw-redirect" title="基建">基建</a>以及<a href="/wiki/%E5%BB%BA%E7%AF%89%E7%89%A9" class="mw-redirect" title="建築物">建築物</a>等好多樣嘢；想像喺<a href="/wiki/%E5%B8%82%E4%B8%AD%E5%BF%83" title="市中心">市中心</a>有位市民過馬路嗰陣唔小心（細微變化），有位<a href="/wiki/%E5%8F%B8%E6%A9%9F" title="司機">司機</a>為咗唔想撞到佢而撞咗埋第架車度，場<a href="/wiki/%E4%BA%A4%E9%80%9A%E6%84%8F%E5%A4%96" class="mw-redirect" title="交通意外">交通意外</a>搞到條馬路俾人封咗，於是多位市民返工返學有困難，咁啱得咁橋其中一位返唔到工嘅市民係某大<a href="/wiki/%E4%BC%81%E6%A5%AD" title="企業">企業</a>嘅重要<a href="/wiki/%E7%AE%A1%E7%90%86%E5%B1%A4" class="mw-redirect" title="管理層">管理層</a>，連帶搞到間企業當日嘅工作受阻，進而影響埋<a href="/wiki/%E8%82%A1%E5%B8%82" title="股市">股市</a>變化，引發<a href="/wiki/%E7%B6%93%E6%BF%9F" title="經濟">經濟</a>波動（系統跟住落嚟嘅狀態大變）。好似噉嘅連串變化就係「混沌」講緊嘅嘢－喺現實世界，好多時個系統嘅初始狀態係噉咦變咗啲，就會搞到個系統跟住出現意料之外嘅大變動<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>。 </p><p>混沌嘅諗頭引起咗好多領域嘅工作者關注。試想像<sup id="cite_ref-weinberger2019_5-0" class="reference"><a href="#cite_note-weinberger2019-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>： </p> <ul><li><a href="/wiki/%E9%87%91%E8%9E%8D%E5%AD%B8" title="金融學">金融學</a>：<a href="/wiki/%E8%82%A1%E5%B8%82" title="股市">股市</a>嘅上落受好多<a href="/wiki/%E8%AE%8A%E6%95%B8_(%E7%A7%91%E7%A0%94)" title="變數 (科研)">因素</a>影響，例如<a href="/wiki/%E5%88%A9%E7%8E%87" title="利率">利率</a>以及<a href="/wiki/%E4%BE%9B%E6%87%89%E5%90%8C%E9%9C%80%E6%B1%82" title="供應同需求">供應同需求</a>都會影響<a href="/wiki/%E8%82%A1%E5%83%B9" title="股價">股價</a>；家陣金融工作者想預測股市嘅變化，會唔會有可能利率升 0.1% 咁少，就搞到股市由正常狀態變成「引起<a href="/wiki/%E6%81%90%E6%85%8C%E6%80%A7%E6%8B%8B%E5%94%AE" title="恐慌性拋售">恐慌性拋售</a>，股價插水式下滑」呢？</li> <li><a href="/wiki/%E9%86%AB%E5%AD%B8" title="醫學">醫學</a>：<a href="/wiki/%E4%BA%BA%E9%AB%94" title="人體">人體</a>嘅運作受好多因素影響，例如<a href="/wiki/%E8%A1%80" title="血">血</a>入面各<a href="/wiki/%E5%8C%96%E5%AD%B8%E7%89%A9%E8%B3%AA" title="化學物質">化學物質</a>嘅<a href="/wiki/%E6%BF%83%E5%BA%A6" title="濃度">濃度</a>都可以影響人體能唔能夠正常運作；家陣<a href="/wiki/%E9%86%AB%E8%AD%B7%E4%BA%BA%E5%93%A1" title="醫護人員">醫護人員</a>想喺病人身上用<a href="/wiki/%E8%97%A5" title="藥">藥</a>，會唔會隻藥嘅濃度升咗 0.1% 咁少，就搞到病人由「會好返」變成「會<a href="/wiki/%E6%AD%BB" title="死">死</a>」呢？</li> <li><a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="工程學">工程學</a>：一部<a href="/wiki/%E6%A9%9F%E6%A2%B0" title="機械">機械</a>可以有極多件組成部份；想像家陣<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%B8%AB" title="工程師">工程師</a>想設計一部機械，會唔會部機嘅<a href="/wiki/%E6%BA%AB%E5%BA%A6" title="溫度">溫度</a>或者<a href="/wiki/%E5%A3%93%E5%8A%9B" title="壓力">壓力</a>升咗 0.1% 咁少，就搞到部機由「正常噉行到落去」變成「會<a href="/wiki/%E7%88%86%E7%82%B8" title="爆炸">爆炸</a><a href="/wiki/%E8%91%97%E7%81%AB" class="mw-redirect" title="著火">著火</a>」呢？</li></ul> <p>... 呀噉。有好多唔同領域嘅工作者都有興趣<a href="/wiki/%E7%A0%94%E7%A9%B6" title="研究">研究</a>混沌現象，除咗頭先提到嘅領域之外，<a href="/wiki/%E6%B0%A3%E5%80%99%E5%AD%B8" title="氣候學">氣候學</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>、<a href="/wiki/%E7%94%9F%E6%85%8B%E5%AD%B8" title="生態學">生態學</a>、各門嘅<a href="/wiki/%E7%A4%BE%E6%9C%83%E7%A7%91%E5%AD%B8" title="社會科學">社會科學</a><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>以至<a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="電腦科學">電腦科學</a>等嘅多個領域，都有工作者著手研究混沌。呢啲咁多唔同嘅研究就形成咗個<a href="/wiki/%E8%B7%A8%E5%AD%B8%E7%A7%91" class="mw-redirect" title="跨學科">跨學科</a>嘅領域－混沌理論<sup id="cite_ref-weinberger2019_5-1" class="reference"><a href="#cite_note-weinberger2019-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="基本概念"><span id=".E5.9F.BA.E6.9C.AC.E6.A6.82.E5.BF.B5"></span>基本概念</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=1" title="編輯小節: 基本概念"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:SensInitCond.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/SensInitCond.gif/300px-SensInitCond.gif" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/SensInitCond.gif/450px-SensInitCond.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/8/8e/SensInitCond.gif 2x" data-file-width="560" data-file-height="420" /></a><figcaption><a href="/wiki/%E6%B4%9B%E5%80%AB%E8%8C%B2%E7%B3%BB%E7%B5%B1" title="洛倫茲系統">洛倫茲系統</a>嘅<a href="/wiki/%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC" title="電腦模擬">電腦模擬</a>結果；<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{y}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{y}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f346022cb7e87a6db4eb5dc3eb8c8598b85efe34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.282ex; height:2.176ex;" alt="{\displaystyle {\text{y}}_{0}}"></span> 係噉咦變少少，就令最後嘅 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f60f51fe61ff56fbd54ff40986e1cfbb06a5b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.227ex; height:2.009ex;" alt="{\displaystyle {\text{y}}}"></span> 變化規律唔同嗮。</figcaption></figure> <div role="note" class="hatnote">内文：<a href="/wiki/%E8%9D%B4%E8%9D%B6%E6%95%88%E6%87%89" title="蝴蝶效應">蝴蝶效應</a>同<a href="/wiki/%E8%87%A8%E7%95%8C%E9%81%8E%E6%B8%A1" title="臨界過渡">臨界過渡</a></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E5%8B%95%E6%85%8B%E7%B3%BB%E7%B5%B1" title="動態系統">動態系統</a>同<a href="/wiki/%E6%B1%BA%E5%AE%9A%E8%AB%96" title="決定論">決定論</a></div> <p>混沌理論最基本嘅諗頭係所謂嘅<a href="/wiki/%E8%9D%B4%E8%9D%B6%E6%95%88%E6%87%89" title="蝴蝶效應">蝴蝶效應</a>（butterfly effect）：家吓將研究緊嘅<a href="/wiki/%E7%8F%BE%E8%B1%A1" title="現象">現象</a>想像成一個<a href="/wiki/%E7%B3%BB%E7%B5%B1" title="系統">系統</a>（尤其係<a href="/wiki/%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="複雜系統">複雜系統</a>），當中「現象」可以係<a href="/wiki/%E5%A4%A9%E6%B0%A3" title="天氣">天氣</a>、<a href="/wiki/%E7%94%9F%E6%85%8B%E7%B3%BB%E7%B5%B1" title="生態系統">生態系統</a>、<a href="/wiki/%E4%BA%BA%E9%AB%94" title="人體">人體</a>、<a href="/wiki/%E4%BA%BA%E8%85%A6" title="人腦">人腦</a>、<a href="/wiki/%E8%A4%87%E9%9B%9C" title="複雜">複雜</a>嘅<a href="/wiki/%E6%A9%9F%E6%A2%B0" title="機械">機械</a>以至<a href="/wiki/%E8%82%A1%E5%B8%82" title="股市">股市</a>... 呀噉；一個系統梗會有某啲 <a href="/wiki/Input" class="mw-redirect" title="Input">input</a>，而個系統跟住落嚟嘅狀態會取決於 </p> <ol><li>Input 同</li> <li>系統打前嘅狀態；</li></ol> <p>事實表明，好多時 <b>input 或者打前狀態變咗少少，跟住落嚟嘅變化就可以唔同嗮</b>： </p> <ul><li>舉個具體例子，想像兩個<a href="/wiki/%E7%B6%93%E6%BF%9F%E9%AB%94" class="mw-redirect" title="經濟體">經濟體</a>（複雜系統）A 同 B，一開始嗰陣（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}"></span>）兩個經濟體狀態完全一樣咁滯，不過响一開始嗰時 A 嘅<a href="/wiki/%E4%BA%BA%E5%8F%A3" title="人口">人口</a>大過 B 少少，例如 A 人口係 100,002 而 B 人口係 100,000；<a href="/wiki/%E7%9B%B4%E8%A6%BA" title="直覺">直覺</a>上會認為，A 同 B 跟住落嚟嘅變化會係一樣咁滯，但實際嘅研究發現並唔係噉－<a href="/wiki/%E5%AF%A6%E8%AD%89" title="實證">實證</a>嘅研究表明，兩個經濟體有可能會隨時間變到完全唔一樣，例如 A 變成世界最大經濟體，同時 B 進入<a href="/wiki/%E7%B6%93%E6%BF%9F%E8%A1%B0%E9%80%80" title="經濟衰退">經濟衰退</a>，最後 A 嘅 <a href="/wiki/%E6%9C%AC%E5%9C%B0%E7%94%9F%E7%94%A2%E7%B8%BD%E5%80%BC" title="本地生產總值">GDP</a> 係 B 嘅 10 倍<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>。</li> <li>噉嘅現象，喺<a href="/wiki/%E7%B6%93%E6%BF%9F%E5%AD%B8" title="經濟學">經濟學</a>以外嘅領域都可以觀察得到<sup id="cite_ref-scheffer2009_10-0" class="reference"><a href="#cite_note-scheffer2009-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>，例如喺 1969 年，有<a href="/wiki/%E7%A7%91%E5%AD%B8%E5%AE%B6" title="科學家">科學家</a>做過噉嘅<a href="/wiki/%E7%A0%94%E7%A9%B6" title="研究">研究</a>－想像有隻<a href="/wiki/%E8%9D%B4%E8%9D%B6" title="蝴蝶">蝴蝶</a>喺度飛嚟飛去搵<a href="/wiki/%E8%8A%B1%E8%9C%9C" title="花蜜">花蜜</a>食；根據呢啲科學家計嘅數，隻蝴蝶拍翼會造成<a href="/wiki/%E7%A9%BA%E6%B0%A3" title="空氣">空氣</a>嘅微細流動（細變化），而呢股微細變動有可能引起周圍嘅空氣出現一連串變化，最後喺<a href="/wiki/%E5%9C%B0%E7%90%83" title="地球">地球</a>另一邊引起<a href="/wiki/%E9%BE%8D%E6%8D%B2%E9%A2%A8" title="龍捲風">龍捲風</a>（意料之外嘅大變化）<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>。</li></ul> <p>上述嘅現象仲可以用<a href="/wiki/%E6%95%B8%E5%AD%B8%E5%8C%96" class="mw-redirect" title="數學化">數學化</a>嘅<a href="/wiki/%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC" title="電腦模擬">電腦模擬</a>展示：附圖係混沌理論當中嘅<a href="/wiki/%E6%B4%9B%E5%80%AB%E8%8C%B2%E7%B3%BB%E7%B5%B1" title="洛倫茲系統">洛倫茲系統</a>（Lorenz system）嘅電腦模擬結果；成個系統有幾個<a href="/wiki/%E8%AE%8A%E6%95%B8_(%E7%A7%91%E7%A0%94)" title="變數 (科研)">變數</a>同<a href="/wiki/%E5%8F%83%E6%95%B8" title="參數">參數</a>，包括 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f60f51fe61ff56fbd54ff40986e1cfbb06a5b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.227ex; height:2.009ex;" alt="{\displaystyle {\text{y}}}"></span>，幅圖打戙嗰條軸做 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f60f51fe61ff56fbd54ff40986e1cfbb06a5b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.227ex; height:2.009ex;" alt="{\displaystyle {\text{y}}}"></span>，打橫嗰條軸做<a href="/wiki/%E6%99%82%E9%96%93" title="時間">時間</a>，設 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{y}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{y}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f346022cb7e87a6db4eb5dc3eb8c8598b85efe34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.282ex; height:2.176ex;" alt="{\displaystyle {\text{y}}_{0}}"></span> 做「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f60f51fe61ff56fbd54ff40986e1cfbb06a5b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.227ex; height:2.009ex;" alt="{\displaystyle {\text{y}}}"></span> 嘅初始數值」，唔同色嘅線表示喺唔同 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{y}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{y}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f346022cb7e87a6db4eb5dc3eb8c8598b85efe34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.282ex; height:2.176ex;" alt="{\displaystyle {\text{y}}_{0}}"></span> 之下 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f60f51fe61ff56fbd54ff40986e1cfbb06a5b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.227ex; height:2.009ex;" alt="{\displaystyle {\text{y}}}"></span> 隨時間變化嘅規律。由幅圖睇得出，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{y}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{y}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f346022cb7e87a6db4eb5dc3eb8c8598b85efe34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.282ex; height:2.176ex;" alt="{\displaystyle {\text{y}}_{0}}"></span> 係噉咦變咗少少，就會令最後嘅 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f60f51fe61ff56fbd54ff40986e1cfbb06a5b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.227ex; height:2.009ex;" alt="{\displaystyle {\text{y}}}"></span> 變化規律唔同嗮－嗰幾條唔同色嘅線大約去到 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=11}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>11</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=11}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/675df8c7d932a7f6fc2b712c45f2bb703755590c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.263ex; height:2.176ex;" alt="{\displaystyle t=11}"></span> 嗰陣分開。而且<a href="/wiki/%E6%87%89%E7%94%A8%E6%95%B8%E5%AD%B8" title="應用數學">應用數學</a>方面嘅研究仲發現，就算個系統完全冇任何<a href="/wiki/%E9%9A%A8%E6%A9%9F" title="隨機">隨機</a>喺裏面（完全<a href="/wiki/%E6%B1%BA%E5%AE%9A%E8%AB%96" title="決定論">決定性</a>），噉嘅現象依然有可能會發生<sup id="cite_ref-determismandchaos_13-0" class="reference"><a href="#cite_note-determismandchaos-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>。 </p><p>蝴蝶效應引起咗好多<a href="/wiki/%E7%A7%91%E7%A0%94" class="mw-redirect" title="科研">科研</a>工作者嘅關注：<a href="/wiki/%E7%A7%91%E5%AD%B8" title="科學">科學</a>嘅其中一個終極<a href="/wiki/%E7%9B%AE%E7%9A%84" title="目的">目的</a>，就係想靠實證得到<a href="/wiki/%E7%9F%A5%E8%AD%98" title="知識">知識</a>，用知識幫<a href="/wiki/%E4%BA%BA%E9%A1%9E" class="mw-redirect" title="人類">人類</a>預測<a href="/wiki/%E5%AE%87%E5%AE%99" title="宇宙">宇宙</a>嘅各種現象，無論<a href="/wiki/%E8%87%AA%E7%84%B6%E7%A7%91%E5%AD%B8" title="自然科學">自然科學</a>定<a href="/wiki/%E7%A4%BE%E6%9C%83%E7%A7%91%E5%AD%B8" title="社會科學">社會科學</a>都係噉；混沌嘅存在就正正係令到某啲現象難以預料，對人類預測現象嘅能力造成威脅。研究混沌嘅科學家會用數學化嘅方式思考混沌，並且嘗試將得出嘅數學模型應用落去現實嘅現象度，想從而加深人類對混沌現象嘅理解，解答「有冇方法可以預料混沌幾時會出現」等嘅問題<sup id="cite_ref-scheffer2009_10-1" class="reference"><a href="#cite_note-scheffer2009-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>。 </p> <style data-mw-deduplicate="TemplateStyles:r1242160">.mw-parser-output .gallery-mod{background:transparent;margin-top:0.5em}.mw-parser-output .gallery-mod-center{margin-left:auto;margin-right:auto}.mw-parser-output .gallery-mod-title{text-align:center;font-weight:bold}.mw-parser-output .gallery-mod-box{float:left;border-collapse:collapse;margin:3px}.mw-parser-output .gallery-mod-box .thumb{border:1px solid #ccc;background-color:#F8F8F8;padding:0;text-align:center}.mw-parser-output tr.gallery-mod-text{vertical-align:top}.mw-parser-output tr.gallery-mod-text .core{display:block;font-size:small;padding:0}.mw-parser-output .gallery-mod-text .caption{line-height:1.25em;padding:6px 6px 1px 6px;margin:0;border:none;border-width:0}.mw-parser-output .gallery-mod-footer{text-align:right;font-size:80%;line-height:1em}</style><table class="gallery-mod gallery-mod-center"><tbody><tr><td><table class="gallery-mod-box" style="width:380px"><tbody><tr><td class="thumb" style="height:290px"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Butterfly_in_flight_(1497161126).jpg" class="mw-file-description" title="2007 年影到嘅一隻帝王斑蝶；隻蝴蝶飛嗰陣會猛咁拍翼，造成空氣流動嘅微細變化。"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Butterfly_in_flight_%281497161126%29.jpg/360px-Butterfly_in_flight_%281497161126%29.jpg" decoding="async" width="360" height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Butterfly_in_flight_%281497161126%29.jpg/540px-Butterfly_in_flight_%281497161126%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Butterfly_in_flight_%281497161126%29.jpg/720px-Butterfly_in_flight_%281497161126%29.jpg 2x" data-file-width="2339" data-file-height="1605" /></a><figcaption>2007 年影到嘅一隻<a href="/wiki/%E5%B8%9D%E7%8E%8B%E6%96%91%E8%9D%B6" title="帝王斑蝶">帝王斑蝶</a>；隻蝴蝶飛嗰陣會猛咁拍翼，造成<a href="/wiki/%E7%A9%BA%E6%B0%A3" title="空氣">空氣</a>流動嘅微細變化。</figcaption></figure></td></tr><tr class="gallery-mod-text"><td class="core"><div class="caption" style="min-height:3.1em;width:367px">2007 年影到嘅一隻<a href="/wiki/%E5%B8%9D%E7%8E%8B%E6%96%91%E8%9D%B6" title="帝王斑蝶">帝王斑蝶</a>；隻蝴蝶飛嗰陣會猛咁拍翼，造成<a href="/wiki/%E7%A9%BA%E6%B0%A3" title="空氣">空氣</a>流動嘅微細變化。 </div></td></tr></tbody></table><table class="gallery-mod-box" style="width:380px"><tbody><tr><td class="thumb" style="height:290px"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:F5_tornado_Elie_Manitoba_2007.jpg" class="mw-file-description" title="... 一隻蝴蝶拍翼引起嘅變化，會唔會喺地球另一邊引起一股龍捲風呢？有科學家計過條數，答案係會。"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/F5_tornado_Elie_Manitoba_2007.jpg/360px-F5_tornado_Elie_Manitoba_2007.jpg" decoding="async" width="360" height="270" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/F5_tornado_Elie_Manitoba_2007.jpg/540px-F5_tornado_Elie_Manitoba_2007.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/F5_tornado_Elie_Manitoba_2007.jpg/720px-F5_tornado_Elie_Manitoba_2007.jpg 2x" data-file-width="2592" data-file-height="1944" /></a><figcaption>... 一隻蝴蝶拍翼引起嘅變化，會唔會喺地球另一邊引起一股<a href="/wiki/%E9%BE%8D%E6%8D%B2%E9%A2%A8" title="龍捲風">龍捲風</a>呢？有科學家計過條數，答案係<b>會</b>。</figcaption></figure></td></tr><tr class="gallery-mod-text"><td class="core"><div class="caption" style="min-height:3.1em;width:367px">... 一隻蝴蝶拍翼引起嘅變化，會唔會喺地球另一邊引起一股<a href="/wiki/%E9%BE%8D%E6%8D%B2%E9%A2%A8" title="龍捲風">龍捲風</a>呢？有科學家計過條數，答案係<b>會</b>。 </div></td></tr></tbody></table></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="系統概念"><span id=".E7.B3.BB.E7.B5.B1.E6.A6.82.E5.BF.B5"></span>系統概念</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=2" title="編輯小節: 系統概念"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="吸引子"><span id=".E5.90.B8.E5.BC.95.E5.AD.90"></span>吸引子</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=3" title="編輯小節: 吸引子"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E5%90%B8%E5%BC%95%E5%AD%90" title="吸引子">吸引子</a></div> <div class="mw-heading mw-heading3"><h3 id="複雜度"><span id=".E8.A4.87.E9.9B.9C.E5.BA.A6"></span>複雜度</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=4" title="編輯小節: 複雜度"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">内文：<a href="/wiki/%E8%A4%87%E9%9B%9C%E5%BA%A6" class="mw-redirect" title="複雜度">複雜度</a></div> <div class="mw-heading mw-heading2"><h2 id="混沌系統"><span id=".E6.B7.B7.E6.B2.8C.E7.B3.BB.E7.B5.B1"></span>混沌系統</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=5" title="編輯小節: 混沌系統"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="簡史"><span id=".E7.B0.A1.E5.8F.B2"></span>簡史</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=6" title="編輯小節: 簡史"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>喺 19 世紀尾，<a href="/wiki/%E6%B3%95%E5%9C%8B" title="法國">法國</a><a href="/wiki/%E6%95%B8%E5%AD%B8%E5%AE%B6" title="數學家">數學家</a><a href="/wiki/%E9%BE%90%E5%8A%A0%E8%90%8A" title="龐加萊">龐加萊</a>（Henri Poincaré）等嘅研究者喺度諗<a href="/wiki/%E5%A4%AA%E9%99%BD%E7%B3%BB" title="太陽系">太陽系</a>啲<a href="/wiki/%E5%A4%A9%E9%AB%94" title="天體">天體</a>嘅軌跡會點樣隨時間變化，佢哋用<a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>嚟做分析（用到隨時間嘅<a href="/wiki/%E5%B0%8E%E6%95%B8" title="導數">導數</a>），發覺喺分析多過兩個天體嘅軌道嗰陣，啲現象經已<a href="/wiki/%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="複雜系統">複雜</a>得滯，難以用<a href="/wiki/%E5%85%AC%E5%BC%8F" title="公式">公式</a>描述，啲模型嘅可能結果由「啲天體冚唪唥都完美噉跟住現有嘅軌跡行」以至極端嘅「其中有粒<a href="/wiki/%E8%A1%8C%E6%98%9F" title="行星">行星</a>俾<a href="/wiki/%E9%87%8D%E5%8A%9B" title="重力">重力</a>掟出去太陽系外或者撞落<a href="/wiki/%E5%A4%AA%E9%99%BD" title="太陽">太陽</a>嗰度」都有；打後（用微分方程做）嘅研究又發現，有好多動態系統都有啲噉嘅情況－「只要啲<a href="/wiki/%E5%88%9D%E5%A7%8B%E6%A2%9D%E4%BB%B6" title="初始條件">初始條件</a>嘅數值變咗少少，成個系統嘅狀態就會出現極端嘅變化」，而呢點就係混沌理論上所講嘅「<a class="mw-selflink selflink">混沌</a>」。混沌嘅現象喺<a href="/wiki/%E5%A4%A9%E6%B0%A3" title="天氣">天氣</a>、<a href="/wiki/%E7%94%9F%E6%85%8B%E7%B3%BB%E7%B5%B1" title="生態系統">生態系統</a>以至<a href="/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" title="量子力學">量子力學</a>等多種現象嗰度都會見得到，係廿一世紀初數學界相當重視嘅一套理論<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading2"><h2 id="相關概念"><span id=".E7.9B.B8.E9.97.9C.E6.A6.82.E5.BF.B5"></span>相關概念</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=7" title="編輯小節: 相關概念"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="複雜系統">複雜系統</a>同<a href="/wiki/%E6%8E%A7%E5%88%B6%E7%90%86%E8%AB%96" title="控制理論">控制理論</a></div> <div class="mw-heading mw-heading2"><h2 id="睇埋"><span id=".E7.9D.87.E5.9F.8B"></span>睇埋</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=8" title="編輯小節: 睇埋"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%89%B5%E7%99%BC" title="創發">創發</a></li> <li><a href="/wiki/%E7%AE%97%E8%A1%93%E4%B8%8B%E6%BA%A2" title="算術下溢">算術下溢</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="文獻"><span id=".E6.96.87.E7.8D.BB"></span>文獻</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=9" title="編輯小節: 文獻"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="攷"><span id=".E6.94.B7"></span>攷</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=10" title="編輯小節: 攷"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r2020813">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-3"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://fractalfoundation.org/resources/what-is-chaos-theory/">What is Chaos Theory? - Fractal Foundation</a>". Retrieved 2019-11-24.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin; Cui, Jialin; Faghih-Naini, Sara; Paxson, Wei; Atlas, Robert (2022-07-04). "Three Kinds of Butterfly Effects within Lorenz Models". <i>Encyclopedia</i>. 2 (3): 1250-1259.</span> </li> <li id="cite_note-betten2015-3"><span class="mw-cite-backlink"><a href="#cite_ref-betten2015_3-0">↑</a></span> <span class="reference-text">Bettencourt, L. M. (2015). Cities as complex systems. <i>Modeling complex systems for public policies</i>, 217-236.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Lorenz, Edward (1993). <i>The Essence of Chaos</i>. University of Washington Press. pp. 181-206.</span> </li> <li id="cite_note-weinberger2019-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-weinberger2019_5-0">5.0</a></sup> <sup><a href="#cite_ref-weinberger2019_5-1">5.1</a></sup></span> <span class="reference-text">Weinberger, David (2019). <i>Everyday Chaos - Technology, Complexity, and How We're Thriving in a New World of Possibility</i>. Harvard Business Review Press.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Ivancevic, Vladimir G.; Tijana T. Ivancevic (2008). <i>Complex nonlinearity: chaos, phase transitions, topology change, and path integrals</i>. Springer.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Lorenz, Edward N. (1963). "<a rel="nofollow" class="external text" href="https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml">Deterministic non-periodic flow</a>". <i>Journal of the Atmospheric Sciences</i>. 20 (2): 130-141.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">Mosko M.S., Damon F.H. (Eds.) (2005). <i>On the order of chaos. Social anthropology and the science of chaos</i>. Oxford: Berghahn Books.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text">Gaspard, P. (2005). <i>Chaos, scattering and statistical mechanics</i> (No. 9). Cambridge University Press.</span> </li> <li id="cite_note-scheffer2009-10"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-scheffer2009_10-0">10.0</a></sup> <sup><a href="#cite_ref-scheffer2009_10-1">10.1</a></sup></span> <span class="reference-text">Scheffer, Marten; Bascompte, Jordi; Brock, William A.; Brovkin, Victor; Carpenter, Stephen R.; Dakos, Vasilis; Held, Hermann; van Nes, Egbert H.; Rietkerk, Max; Sugihara, George (September 2009). "<a rel="nofollow" class="external text" href="https://www.nature.com/articles/nature08227">Early-warning signals for critical transitions</a>". <i>Nature</i>. 461 (7260): 53-59.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text">Edward N. Lorenz (1969). "Atmospheric predictability as revealed by naturally occurring analogues". <i>Journal of the Atmospheric Sciences</i>. 26 (4): 636-646.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text">Hasselblatt, Boris; Anatole Katok (2003). <i>A First Course in Dynamics: With a Panorama of Recent Developments</i>. Cambridge University Press.</span> </li> <li id="cite_note-determismandchaos-13"><span class="mw-cite-backlink"><a href="#cite_ref-determismandchaos_13-0">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://pi.math.cornell.edu/~lipa/mec/lesson1.html">Determinism and Chaos</a>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text">Gaspard, P. (2005). <i>Chaos, scattering and statistical mechanics</i> (No. 9). Cambridge University Press.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="拎"><span id=".E6.8B.8E"></span>拎</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AB%96&action=edit&section=11" title="編輯小節: 拎"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r2018312">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r2018311">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist"><a href="https://commons.wikimedia.org/wiki/%E9%A0%AD%E7%89%88" class="extiw" title="commons:頭版">維基同享</a>有多媒體嘅嘢：<br /><b><a href="https://commons.wikimedia.org/wiki/Category:Chaos_theory" class="extiw" title="commons:Category:Chaos theory"><span style="">混沌理論</span></a></b></div></div> </div> <ul><li><span style="font-family: sans-serif; cursor: default; color: var(--color-subtle, #555); font-size: 0.8em; position: relative; bottom: 0.1em; font-weight: bold;" title="連到英文網頁">（英文）</span> <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/chaos/">混沌</a>，<a href="/wiki/%E5%8F%B2%E4%B8%B9%E7%A6%8F%E5%93%B2%E5%AD%B8%E7%99%BE%E7%A7%91%E5%85%A8%E6%9B%B8" title="史丹福哲學百科全書">史丹福哲學百科全書</a></li> <li><span style="font-family: sans-serif; cursor: default; color: var(--color-subtle, #555); font-size: 0.8em; position: relative; bottom: 0.1em; font-weight: bold;" title="連到英文網頁">（英文）</span> <a rel="nofollow" class="external text" href="https://www.britannica.com/science/chaos-theory">混沌理論</a>，<a href="/wiki/%E4%B8%8D%E5%88%97%E9%A1%9B%E7%99%BE%E7%A7%91%E5%85%A8%E6%9B%B8" title="不列顛百科全書">不列顛百科全書</a></li></ul> <div role="navigation" class="navbox" aria-labelledby="複雜系統" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><a href="/wiki/Template:%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="Template:複雜系統"><abbr title="去睇呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">睇</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1&action=edit&redlink=1" class="new" title="Template talk:複雜系統 (無呢版)"><abbr title="傾呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">傾</abbr></a></li><li class="nv-edit"><a class="external text" href="https://zh-yue.wikipedia.org/w/index.php?title=Template:%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1&action=edit"><abbr title="編輯呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">改</abbr></a></li></ul></div><div id="複雜系統" style="font-size:114%;margin:0 4em"><a href="/wiki/%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="複雜系統">複雜系統</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div id="顧名思義係指複雜嘅系統，變化難以預測">顧名思義係指<a href="/wiki/%E8%A4%87%E9%9B%9C" title="複雜">複雜</a>嘅<a href="/wiki/%E7%B3%BB%E7%B5%B1" title="系統">系統</a>，變化難以預測</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">重要概念</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%A4%87%E9%9B%9C" title="複雜">複雜</a>同<a href="/wiki/%E7%B0%A1%E5%96%AE" title="簡單">簡單</a></li> <li><a href="/wiki/%E7%B5%84%E5%90%88%E7%88%86%E7%99%BC" title="組合爆發">組合爆發</a></li> <li><a href="/wiki/%E5%89%B5%E7%99%BC" title="創發">創發</a>同<a href="/wiki/%E8%87%AA%E6%88%91%E7%B5%84%E7%B9%94" title="自我組織">自我組織</a></li> <li><a href="/wiki/%E5%8B%95%E6%85%8B%E7%B3%BB%E7%B5%B1" title="動態系統">動態系統</a></li> <li><a href="/wiki/%E8%A4%87%E9%9B%9C%E9%81%A9%E6%87%89%E7%B3%BB%E7%B5%B1" title="複雜適應系統">適應系統</a></li> <li><a href="/wiki/%E5%8F%AF%E9%80%B2%E5%8C%96%E5%BA%A6" title="可進化度">可進化</a></li> <li><a href="/wiki/%E6%95%B4%E9%AB%94%E4%B8%BB%E7%BE%A9" title="整體主義">整體主義</a>同<a href="/wiki/%E9%82%84%E5%8E%9F%E8%AB%96" title="還原論">還原論</a></li> <li><a href="/wiki/%E8%87%AA%E7%84%B6%E5%BD%A2%E6%85%8B%E8%A6%8F%E5%BE%8B" title="自然形態規律">自然形態規律</a></li></ul> </div></td><td class="navbox-image" rowspan="8" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/File:Lorenz_attractor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lorenz_attractor.svg/150px-Lorenz_attractor.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lorenz_attractor.svg/225px-Lorenz_attractor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lorenz_attractor.svg/300px-Lorenz_attractor.svg.png 2x" data-file-width="750" data-file-height="750" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">重要例子</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%94%9F%E5%91%BD" title="生命">生命</a>同<a href="/wiki/%E7%94%9F%E7%89%A9" title="生物">生物</a></li> <li><a href="/wiki/%E7%B4%B0%E8%83%9E" title="細胞">細胞</a></li> <li>神經<a href="/wiki/%E7%A5%9E%E7%B6%93%E7%B4%B0%E8%83%9E" title="神經細胞">細胞</a>同<a href="/wiki/%E7%A5%9E%E7%B6%93%E7%B3%BB%E7%B5%B1" title="神經系統">系統</a>－包括<a href="/wiki/%E8%85%A6" title="腦">腦</a>（尤其係<a href="/wiki/%E4%BA%BA%E8%85%A6" title="人腦">人腦</a>）</li> <li><a href="/wiki/%E7%A8%AE%E7%BE%A4" title="種群">種群</a></li> <li><a href="/wiki/%E7%94%9F%E6%85%8B%E7%B3%BB%E7%B5%B1" title="生態系統">生態系統</a>（<a href="/wiki/%E9%97%9C%E9%8D%B5%E7%89%A9%E7%A8%AE" title="關鍵物種">關鍵物種</a>）</li> <li><a href="/wiki/%E8%87%AA%E7%84%B6" title="自然">自然</a></li> <li><a href="/wiki/%E6%B0%A3%E5%80%99" title="氣候">氣候</a></li></ul> <ul><li><a href="/wiki/%E7%B5%84%E7%B9%94" title="組織">組織</a></li> <li><a href="/wiki/%E7%A4%BE%E6%9C%83" title="社會">社會</a></li> <li><a href="/wiki/%E7%B6%93%E6%BF%9F%E9%AB%94" class="mw-redirect" title="經濟體">經濟體</a>（<a href="/wiki/%E9%87%91%E8%9E%8D%E5%B8%82%E5%A0%B4" title="金融市場">金融市場</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%9C%8B%E9%9A%9B%E7%B6%93%E6%BF%9F" class="mw-redirect" title="國際經濟">國際經濟</a>）</li> <li><a href="/wiki/%E5%9F%8E%E5%B8%82" title="城市">城市</a></li> <li>現代<a href="/wiki/%E5%9F%BA%E7%A4%8E%E5%BB%BA%E8%A8%AD" class="mw-redirect" title="基礎建設">基礎建設</a></li> <li>進階<a href="/wiki/%E9%9B%BB%E8%85%A6%E8%BB%9F%E4%BB%B6" title="電腦軟件">電腦軟件</a></li> <li><a href="/wiki/%E5%9C%B0%E7%90%83" title="地球">地球</a></li> <li><a href="/wiki/%E6%98%9F%E7%B3%BB" title="星系">星系</a></li> <li><a href="/wiki/%E5%AE%87%E5%AE%99" title="宇宙">宇宙</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9D%9E%E7%B7%9A%E6%80%A7%E7%B3%BB%E7%B5%B1" title="非線性系統">非線系統</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>（<a href="/wiki/%E5%B8%B8%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="常微分方程">常</a>同<a href="/wiki/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="偏微分方程">偏</a>）</li> <li><a href="/wiki/%E6%99%82%E9%96%93%E5%BA%8F%E5%88%97" title="時間序列">時間序列</a></li> <li><a class="mw-selflink selflink">混沌理論</a>（<a href="/wiki/%E8%87%A8%E7%95%8C%E9%81%8E%E6%B8%A1" title="臨界過渡">臨界過渡</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%90%B8%E5%BC%95%E5%AD%90" title="吸引子">吸引子</a>）</li> <li><a href="/wiki/%E5%88%86%E5%B2%94%E7%90%86%E8%AB%96" title="分岔理論">分岔理論</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%B6%B2%E7%B5%A1%E7%90%86%E8%AB%96" title="網絡理論">網絡理論</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%B6%B2%E7%B5%A1%E7%A7%91%E5%AD%B8" title="網絡科學">網絡科學</a></li> <li><a href="/wiki/%E7%B6%B2%E7%B5%A1%E4%B8%AD%E5%BF%83%E5%BA%A6" title="網絡中心度">中心度</a></li> <li><a href="/wiki/%E7%B6%B2%E7%B5%A1%E8%A6%8F%E5%BE%8B" title="網絡規律">網絡規律</a></li> <li><a href="/wiki/%E7%B4%B0%E4%B8%96%E7%95%8C%E7%B6%B2%E7%B5%A1" title="細世界網絡">細世界網絡</a></li> <li><a href="/wiki/%E7%A5%9E%E7%B6%93%E7%B6%B2%E7%B5%A1" title="神經網絡">神經網絡</a>、<a href="/wiki/%E7%A4%BE%E6%9C%83%E7%B6%B2%E7%B5%A1" title="社會網絡">社會網絡</a>同<a href="/wiki/%E9%9B%BB%E8%85%A6%E7%B6%B2%E7%B5%A1" title="電腦網絡">電腦網絡</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%86%E9%AB%94%E8%A1%8C%E7%82%BA" title="集體行為">集體行為</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%A4%BE%E6%9C%83%E5%8B%95%E5%8A%9B%E5%AD%B8" title="社會動力學">社會動力學</a></li> <li><a href="/wiki/%E8%B6%85%E7%94%9F%E7%89%A9" title="超生物">超生物</a></li> <li><a href="/wiki/%E5%A4%A7%E7%BE%A4%E8%A1%8C%E7%82%BA" title="大群行為">大群行為</a></li> <li><a href="/wiki/%E9%9B%86%E9%AB%94%E6%99%BA%E8%83%BD" title="集體智能">集體智能</a></li> <li><a href="/wiki/%E9%9B%86%E9%AB%94%E8%A1%8C%E5%8B%95" title="集體行動">集體行動</a></li> <li><a href="/wiki/%E9%9B%86%E9%AB%94%E6%84%8F%E8%AD%98" title="集體意識">集體意識</a></li> <li><a href="/wiki/%E7%BE%8A%E7%BE%A4%E5%BF%83%E7%90%86" class="mw-redirect" title="羊群心理">羊群心理</a></li> <li><a href="/wiki/%E5%90%8C%E6%AD%A5" title="同步">同步</a></li> <li><a href="/wiki/%E9%9B%86%E9%AB%94%E6%84%8F%E6%9C%83" title="集體意會">集體意會</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%B3%BB%E7%B5%B1%E7%90%86%E8%AB%96" title="系統理論">系統理論</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Feedback" title="Feedback">Feedback</a></li> <li><a href="/wiki/%E8%87%AA%E6%8C%87" title="自指">自指</a></li> <li><a href="/wiki/%E9%A0%91%E5%81%A5%E5%BA%A6" title="頑健度">頑健</a>同<a href="/wiki/%E5%AE%B9%E9%8C%AF%E8%83%BD%E5%8A%9B" title="容錯能力">容錯</a></li> <li><a href="/wiki/%E5%8F%AF%E6%93%B4%E7%B8%AE%E5%BA%A6" title="可擴縮度">可擴縮度</a></li> <li><a href="/wiki/%E9%AB%94%E5%86%85%E5%B9%B3%E8%A1%A1" title="體内平衡">體内平衡</a></li> <li><a href="/wiki/%E8%87%AA%E7%94%9F%E7%B3%BB%E7%B5%B1" title="自生系統">自生系統</a>同<a href="/wiki/%E8%87%AA%E6%88%91%E8%A4%87%E8%A3%BD" title="自我複製">自我複製</a></li> <li><a href="/wiki/%E5%A4%9A%E7%A9%A9%E6%85%8B" title="多穩態">多穩態</a></li> <li><a href="/wiki/%E7%B3%BB%E7%B5%B1%E5%8B%95%E5%8A%9B%E5%AD%B8" title="系統動力學">系統動力學</a></li> <li><a href="/wiki/%E6%A8%A1%E6%8E%A7%E5%AD%B8" title="模控學">模控學</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%81%8B%E7%AE%97" title="運算">運算</a>相關</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%81%8B%E7%AE%97%E8%A4%87%E9%9B%9C%E5%BA%A6" class="mw-redirect" title="運算複雜度">運算複雜度</a></li> <li><a href="/wiki/%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC" title="電腦模擬">電腦模擬</a>（<a href="/wiki/%E5%80%8B%E9%AB%94%E7%82%BA%E6%9C%AC%E6%A8%A1%E5%9E%8B" title="個體為本模型">個體為本模型</a>）</li> <li><a href="/wiki/%E6%A9%9F%E6%A2%B0%E5%AD%B8%E7%BF%92" title="機械學習">機械學習</a>（<a href="/wiki/%E4%BA%BA%E5%B7%A5%E7%A5%9E%E7%B6%93%E7%B6%B2%E7%B5%A1" title="人工神經網絡">人工神經網絡</a>）</li> <li><a href="/wiki/%E9%80%B2%E5%8C%96%E9%81%8B%E7%AE%97" title="進化運算">進化運算</a>（<a href="/wiki/%E9%81%BA%E5%82%B3%E6%BC%94%E7%AE%97%E6%B3%95" title="遺傳演算法">遺傳演算法</a>）</li></ul> <ul><li><a href="/wiki/%E6%9C%80%E4%BD%B3%E5%8C%96" title="最佳化">最佳化</a>（<a href="/wiki/%E5%85%A8%E5%B1%80%E6%9C%80%E4%BD%B3%E5%8C%96" title="全局最佳化">全局最佳化</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B2%92%E5%AD%90%E7%BE%A4%E6%9C%80%E4%BD%B3%E5%8C%96" title="粒子群最佳化">粒子群最佳化</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%9F%BB%E7%BE%A4%E6%9C%80%E4%BD%B3%E5%8C%96%E6%BC%94%E7%AE%97%E6%B3%95" title="蟻群最佳化演算法">蟻群最佳化演算法</a>）</li> <li><a href="/wiki/%E6%A0%BC%E4%BB%94%E8%87%AA%E5%8B%95%E6%A9%9F" title="格仔自動機">格仔自動機</a>（<a href="/wiki/%E7%94%9F%E5%91%BD%E6%A3%8B" title="生命棋">生命棋</a>）</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">相關領域</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%B3%BB%E7%B5%B1%E7%A7%91%E5%AD%B8" title="系統科學">系統科學</a></li> <li><a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a></li> <li><a href="/wiki/%E9%81%8B%E7%AE%97%E7%90%86%E8%AB%96" title="運算理論">運算理論</a></li> <li><a href="/wiki/%E5%8D%9A%E5%BC%88%E8%AB%96" title="博弈論">博弈論</a></li> <li><a href="/wiki/%E5%9C%B0%E7%90%83%E7%A7%91%E5%AD%B8" title="地球科學">地球科學</a>（<a href="/wiki/%E5%9C%B0%E5%BD%A2%E5%AD%B8" title="地形學">地形學</a>）</li> <li><a href="/wiki/%E7%94%9F%E7%89%A9%E5%AD%B8" title="生物學">生物學</a>（<a href="/wiki/%E7%B3%BB%E7%B5%B1%E7%94%9F%E7%89%A9%E5%AD%B8" title="系統生物學">系統生物學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%80%B2%E5%8C%96%E7%99%BC%E8%82%B2%E7%94%9F%E7%89%A9%E5%AD%B8" title="進化發育生物學">進化發育生物學</a>）</li></ul> <ul><li><a href="/wiki/%E9%80%B2%E5%8C%96%E5%8D%9A%E5%BC%88%E8%AB%96" title="進化博弈論">進化博弈論</a></li> <li><a href="/wiki/%E7%A4%BE%E6%9C%83%E7%A7%91%E5%AD%B8" title="社會科學">社會科學</a>（<a href="/wiki/%E7%90%86%E6%80%A7%E9%81%B8%E6%93%87%E7%90%86%E8%AB%96" title="理性選擇理論">理性選擇理論</a>）</li> <li><a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%B8" title="工程學">工程學</a></li> <li><a href="/wiki/%E6%8E%A7%E5%88%B6%E7%90%86%E8%AB%96" title="控制理論">控制理論</a></li> <li><a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">人工智能</a></li> <li><a href="/wiki/%E9%80%B2%E5%8C%96%E6%A9%9F%E6%A2%B0%E4%BA%BA%E5%AD%B8" title="進化機械人學">進化機械人學</a></li> <li><a href="/wiki/%E4%BA%BA%E5%B7%A5%E7%94%9F%E5%91%BD" title="人工生命">人工生命</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><b>拉雜相關</b><br /> <a href="/wiki/%E5%A5%A7%E5%9D%8E%E5%89%83%E5%88%80" title="奧坎剃刀">奧坎剃刀</a>、<a href="/wiki/%E8%AA%BF%E7%AF%80%E6%95%88%E6%87%89" class="mw-redirect" title="調節效應">調節效應</a>同<a href="/wiki/%E4%B8%8D%E5%85%85%E5%88%86%E6%B1%BA%E5%AE%9A%E8%AB%96" title="不充分決定論">不充分決定論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%A2%8E%E5%BD%A2" title="碎形">碎形</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F" title="隨機">隨機</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%86%B5" title="熵">熵</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%80%97%E6%95%A3%E7%B3%BB%E7%B5%B1" title="耗散系統">耗散系統</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%BB%B2%E6%B5%81" title="滲流">滲流</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%9C%89%E9%99%90%E7%90%86%E6%80%A7" title="有限理性">有限理性</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%AA%AE%E8%88%89%E6%90%9C%E5%B0%8B" title="窮舉搜尋">窮舉搜尋</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%94%9F%E6%88%90%E8%97%9D%E8%A1%93" title="生成藝術">生成藝術</a></div></td></tr></tbody></table></div> <div role="navigation" class="navbox" 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href="/wiki/%E9%9B%86%E5%90%88%E8%AB%96" title="集合論">集合論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BB%A3%E6%95%B8%E5%AD%B8" title="代數學">代數學</a>（<a href="/wiki/%E5%A0%B4_(%E6%95%B8%E5%AD%B8)" class="mw-redirect mw-disambig" title="場 (數學)">場</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B5%84%E5%90%88%E6%95%B8%E5%AD%B8" title="組合數學">組合數學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%9C%96%E8%AB%96" title="圖論">圖論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%A2%E6%95%A3%E5%B9%BE%E4%BD%95%E5%AD%B8" class="mw-redirect" title="離散幾何學">離散幾何學</a></div></td><td class="navbox-image" rowspan="8" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/File:Aizerman2d.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Aizerman2d.jpg/150px-Aizerman2d.jpg" decoding="async" width="150" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Aizerman2d.jpg/225px-Aizerman2d.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Aizerman2d.jpg/300px-Aizerman2d.jpg 2x" data-file-width="591" data-file-height="569" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B8%E5%AD%B8%E5%88%86%E6%9E%90" title="數學分析">數<br />學<br />分<br />析</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;">分析基礎</div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%AF%A6%E6%95%B8" title="實數">實數</a>同<a href="/wiki/%E8%99%9B%E6%95%B8" title="虛數">虛數</a>（<a href="/wiki/%E5%AF%A6%E5%88%86%E6%9E%90" title="實分析">實分析</a>同<a href="/wiki/%E8%A4%87%E5%88%86%E6%9E%90" title="複分析">複分析</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%96%93%E8%B7%9D" title="間距">間距</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%84%A1%E7%AA%AE%E7%9B%A1" title="無窮盡">無窮盡</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%84%A1%E7%AA%AE%E5%B0%8F%E9%87%8F" title="無窮小量">無窮小量</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E6%95%B8%E5%88%97" class="mw-redirect" title="數列">數列</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%AD%90%E6%95%B8%E5%88%97" title="子數列">子數列</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%A2%9E%E6%B8%9B%E6%95%B8%E5%88%97" title="增減數列">增減數列</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%83%9D%E6%B0%8F%E6%95%B8%E5%88%97" title="郝氏數列">郝氏數列</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BF%9D%E8%A5%BF%E5%A5%B4-%E8%8F%AF%E5%AF%A6%E6%96%AF%E5%AE%9A%E7%90%86" title="保西奴-華實斯定理">保西奴-華實斯定理</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E5%87%BD%E6%95%B8" title="函數">函數</a></div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%80%A3%E7%BA%8C%E5%87%BD%E6%95%B8" title="連續函數">連續函數</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A5%B5%E9%99%90_(%E5%87%BD%E6%95%B8)" title="極限 (函數)">極限</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%80%BC%E8%BF%91%E7%90%86%E8%AB%96" title="逼近理論">逼近理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90" title="泛函分析">泛函分析</a>（<a href="/wiki/%E8%AE%8A%E5%88%86%E6%B3%95" title="變分法">變分法</a>）</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E5%BE%AE%E7%A9%8D%E5%88%86" title="微積分">微積分</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="微分" scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E5%BE%AE%E5%88%86" title="微分">微分</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>（<a href="/wiki/%E5%B8%B8%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="常微分方程">常</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="偏微分方程">偏</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="隨機微分方程">隨機</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%A4%87%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="複微分方程">複</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%BE%AE%E5%88%86%E5%B9%BE%E4%BD%95" title="微分幾何">微分幾何</a></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E7%A9%8D%E5%88%86" title="積分">積分</a>（<a href="/wiki/%E9%BB%8E%E6%9B%BC%E7%A9%8D%E5%88%86" title="黎曼積分">黎曼積分</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%A9%8D%E5%88%86%E6%96%B9%E7%A8%8B" title="積分方程">積分方程</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%A4%9A%E5%85%83%E5%BE%AE%E7%A9%8D%E5%88%86" title="多元微積分">多元微積分</a>（<a href="/wiki/%E5%90%91%E9%87%8F%E5%BE%AE%E7%A9%8D%E5%88%86" title="向量微積分">向量</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%BC%B5%E9%87%8F%E5%BE%AE%E7%A9%8D%E5%88%86" title="張量微積分">張量</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%A4%9A%E9%87%8D%E7%A9%8D%E5%88%86" title="多重積分">多重積分</a>）</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;">第啲分析</div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%A4%9A%E9%87%8D%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" title="多重線性代數">多線代數</a>（<a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" title="向量空間">向量空間</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%BC%B5%E9%87%8F" title="張量">張量</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%8B%95%E6%85%8B%E7%B3%BB%E7%B5%B1" title="動態系統">動態系統</a>（<a class="mw-selflink selflink">混沌理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%8E%A7%E5%88%B6%E7%90%86%E8%AB%96" title="控制理論">控制理論</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%AB%A7%E6%B3%A2%E5%88%86%E6%9E%90" title="諧波分析">諧波分析</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E5%80%BC%E5%88%86%E6%9E%90" title="數值分析">數值分析</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%81%8B%E7%AE%97%E6%95%B8%E5%AD%B8" title="運算數學">運算數學</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%81%8B%E7%AE%97%E7%90%86%E8%AB%96" title="運算理論">運算理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%81%8B%E7%AE%97%E9%82%8F%E8%BC%AF" title="運算邏輯">運算邏輯</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%87%AA%E5%8B%95%E6%A9%9F%E7%90%86%E8%AB%96" title="自動機理論">自動機理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95" title="演算法">演算法</a>（<a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95%E5%88%86%E6%9E%90" title="演算法分析">分析</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%AF%86%E7%A2%BC%E5%AD%B8" title="密碼學">密碼學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%81%8B%E7%AE%97%E6%95%B8%E8%AB%96" title="運算數論">運算數論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%A7%91%E5%AD%B8%E9%81%8B%E7%AE%97" title="科學運算">科學運算</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6%E4%BB%A3%E6%95%B8" title="電腦代數">電腦代數</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%A6%82%E7%8E%87%E8%AB%96" title="概率論">概率論</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9A%A8%E6%A9%9F%E8%AE%8A%E6%95%B8" title="隨機變數">隨機變數</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E9%81%8E%E7%A8%8B" title="隨機過程">隨機過程</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%A6%AC%E5%8F%AF%E5%A4%AB%E9%8F%88" title="馬可夫鏈">馬可夫鏈</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="概率分佈">概率分佈</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9A%A8%E6%A9%9F%E5%BE%AE%E7%A9%8D%E5%88%86" title="隨機微積分">隨機微積分</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%B1%BA%E7%AD%96%E8%AB%96" title="決策論">決策論</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%8D%9A%E5%BC%88%E8%AB%96" title="博弈論">博弈論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B5%B1%E8%A8%88%E5%AD%B8" title="統計學">統計學</a>（<a href="/wiki/%E6%8F%8F%E8%BF%B0%E7%B5%B1%E8%A8%88%E5%AD%B8" title="描述統計學">描述</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%8E%A8%E8%AB%96%E7%B5%B1%E8%A8%88%E5%AD%B8" title="推論統計學">推論</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%9C%80%E4%BD%B3%E5%8C%96" title="最佳化">最佳化</a>（<a href="/wiki/%E7%B7%9A%E6%80%A7%E8%A6%8F%E5%8A%83" title="線性規劃">線性</a>同<a href="/wiki/%E9%9D%9E%E7%B7%9A%E6%80%A7%E8%A6%8F%E5%8A%83" title="非線性規劃">非線性</a>規劃）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E7%90%86%E7%B6%93%E6%BF%9F%E5%AD%B8" title="數理經濟學">數理經濟學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E7%90%86%E9%87%91%E8%9E%8D%E5%AD%B8" title="數理金融學">數理金融學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%81%8B%E7%B1%8C%E5%AD%B8" title="運籌學">運籌學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B2%BE%E7%AE%97%E5%AD%B8" title="精算學">精算學</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B8%E7%90%86%E7%89%A9%E7%90%86%E5%AD%B8" title="數理物理學">數理物理</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E5%88%86%E6%9E%90%E5%8A%9B%E5%AD%B8" title="分析力學">分析力學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%A0%B4_(%E7%89%A9%E7%90%86)" title="場 (物理)">場</a>（<a href="/wiki/%E8%A6%8F%E7%AF%84%E5%A0%B4%E8%AB%96" title="規範場論">規範場論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%87%8F%E5%AD%90%E5%A0%B4%E8%AB%96" title="量子場論">量子場論</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%87%8F%E5%AD%90%E7%BE%A3" title="量子羣">量子羣</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">拉雜應用</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E6%95%B8%E7%90%86%E5%8C%96%E5%AD%B8" title="數理化學">數理化學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E7%90%86%E7%94%9F%E7%89%A9%E5%AD%B8" title="數理生物學">數理生物學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E7%90%86%E5%BF%83%E7%90%86%E5%AD%B8" title="數理心理學">數理心理學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E7%90%86%E7%A4%BE%E6%9C%83%E5%AD%B8" title="數理社會學">數理社會學</a>（<a href="/wiki/%E7%A4%BE%E6%9C%83%E9%81%B8%E6%93%87%E7%90%86%E8%AB%96" title="社會選擇理論">社會選擇理論</a>）</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">拉雜相關</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E6%95%B8%E5%AD%B8%E6%A8%A1%E5%9E%8B" title="數學模型">數學模型</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%B7%A5%E7%A8%8B%E6%95%B8%E5%AD%B8" title="工程數學">工程數學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%87%89%E7%94%A8%E7%A7%91%E5%AD%B8" title="應用科學">應用科學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A8%A1%E6%8E%A7%E5%AD%B8" title="模控學">模控學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B3%BB%E7%B5%B1" title="系統">系統</a>（<a href="/wiki/%E8%A4%87%E9%9B%9C%E7%B3%BB%E7%B5%B1" title="複雜系統">複雜系統</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E5%AD%B8%E5%96%BA%E8%87%AA%E7%84%B6%E7%A7%91%E5%AD%B8%E4%B8%8A%E5%98%85%E8%B6%85%E5%B8%B8%E6%95%88%E7%94%A8" title="數學喺自然科學上嘅超常效用">數學喺自然科學上嘅超常效用</a></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><b><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" 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