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class="vector-toc-list"> </ul> </li> <li id="toc-Formal_definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formal_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Formal definition</span> </div> </a> <button aria-controls="toc-Formal_definition-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>Toggle Formal definition subsection</span> </button> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> <li id="toc-Geometrical_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometrical_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Geometrical definition</span> </div> </a> <ul id="toc-Geometrical_definition-sublist" class="vector-toc-list"> <li id="toc-Real_dynamical_system" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Real_dynamical_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Real dynamical system</span> </div> </a> <ul id="toc-Real_dynamical_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discrete_dynamical_system" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Discrete_dynamical_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Discrete dynamical system</span> </div> </a> <ul id="toc-Discrete_dynamical_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cellular_automaton" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cellular_automaton"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.3</span> <span>Cellular automaton</span> </div> </a> <ul id="toc-Cellular_automaton-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multidimensional_generalization" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Multidimensional_generalization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.4</span> <span>Multidimensional generalization</span> </div> </a> <ul id="toc-Multidimensional_generalization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compactification_of_a_dynamical_system" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Compactification_of_a_dynamical_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.5</span> <span>Compactification of a dynamical system</span> </div> </a> <ul id="toc-Compactification_of_a_dynamical_system-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Measure_theoretical_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measure_theoretical_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Measure theoretical definition</span> </div> </a> <ul id="toc-Measure_theoretical_definition-sublist" class="vector-toc-list"> <li id="toc-Relation_to_geometric_definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relation_to_geometric_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Relation to geometric definition</span> </div> </a> <ul id="toc-Relation_to_geometric_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Construction_of_dynamical_systems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Construction_of_dynamical_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Construction of dynamical systems</span> </div> </a> <ul id="toc-Construction_of_dynamical_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_dynamical_systems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Linear_dynamical_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Linear dynamical systems</span> </div> </a> <button aria-controls="toc-Linear_dynamical_systems-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>Toggle Linear dynamical systems subsection</span> </button> <ul id="toc-Linear_dynamical_systems-sublist" class="vector-toc-list"> <li id="toc-Flows" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Flows"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Flows</span> </div> </a> <ul id="toc-Flows-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maps" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maps"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Maps</span> </div> </a> <ul id="toc-Maps-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Local_dynamics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Local_dynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Local dynamics</span> </div> </a> <button aria-controls="toc-Local_dynamics-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>Toggle Local dynamics subsection</span> </button> <ul id="toc-Local_dynamics-sublist" class="vector-toc-list"> <li id="toc-Rectification" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rectification"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Rectification</span> </div> </a> <ul id="toc-Rectification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Near_periodic_orbits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Near_periodic_orbits"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Near periodic orbits</span> </div> </a> <ul id="toc-Near_periodic_orbits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conjugation_results" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conjugation_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Conjugation results</span> </div> </a> <ul id="toc-Conjugation_results-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bifurcation_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bifurcation_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bifurcation theory</span> </div> </a> <ul id="toc-Bifurcation_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ergodic_systems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ergodic_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Ergodic systems</span> </div> </a> <ul id="toc-Ergodic_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonlinear_dynamical_systems_and_chaos" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Nonlinear_dynamical_systems_and_chaos"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Nonlinear dynamical systems and chaos</span> </div> </a> <button aria-controls="toc-Nonlinear_dynamical_systems_and_chaos-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>Toggle Nonlinear dynamical systems and chaos subsection</span> </button> <ul id="toc-Nonlinear_dynamical_systems_and_chaos-sublist" class="vector-toc-list"> <li id="toc-Solutions_of_finite_duration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solutions_of_finite_duration"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Solutions of finite duration</span> </div> </a> <ul id="toc-Solutions_of_finite_duration-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> 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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">Dynamical system</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="Go to an article in another language. Available in 48 languages" > <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-48" 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">48 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%AA%D8%AD%D8%B1%D9%8A%D9%83%D9%8A" title="نظام تحريكي – Arabic" lang="ar" hreflang="ar" data-title="نظام تحريكي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Sistema_din%C3%A1micu" title="Sistema dinámicu – Asturian" lang="ast" hreflang="ast" data-title="Sistema dinámicu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Динамична система – Bulgarian" lang="bg" hreflang="bg" data-title="Динамична система" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_din%C3%A0mic" title="Sistema dinàmic – Catalan" lang="ca" hreflang="ca" data-title="Sistema dinàmic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dynamick%C3%BD_syst%C3%A9m" title="Dynamický systém – Czech" lang="cs" hreflang="cs" data-title="Dynamický systém" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Dynamisk_system" title="Dynamisk system – Danish" lang="da" hreflang="da" data-title="Dynamisk system" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dynamisches_System" title="Dynamisches System – German" lang="de" hreflang="de" data-title="Dynamisches System" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/D%C3%BCnaamiline_s%C3%BCsteem" title="Dünaamiline süsteem – Estonian" lang="et" hreflang="et" data-title="Dünaamiline süsteem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CF%85%CE%BD%CE%B1%CE%BC%CE%B9%CE%BA%CF%8C_%CF%83%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1" title="Δυναμικό σύστημα – Greek" lang="el" hreflang="el" data-title="Δυναμικό σύστημα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_din%C3%A1mico" title="Sistema dinámico – Spanish" lang="es" hreflang="es" data-title="Sistema dinámico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Sistema_dinamiko" title="Sistema dinamiko – Basque" lang="eu" hreflang="eu" data-title="Sistema dinamiko" data-language-autonym="Euskara" data-language-local-name="Basque" 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/%D8%B3%D8%A7%D9%85%D8%A7%D9%86%D9%87_%D9%BE%D9%88%DB%8C%D8%A7" title="سامانه پویا – Persian" lang="fa" hreflang="fa" data-title="سامانه پویا" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_dynamique" title="Système dynamique – French" lang="fr" hreflang="fr" data-title="Système dynamique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Sistema_din%C3%A1mico" title="Sistema dinámico – Galician" lang="gl" hreflang="gl" data-title="Sistema dinámico" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8F%99%EC%97%AD%ED%95%99%EA%B3%84" title="동역학계 – Korean" lang="ko" hreflang="ko" data-title="동역학계" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80%E0%A4%AF_%E0%A4%A4%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="गतिकीय तन्त्र – Hindi" lang="hi" hreflang="hi" data-title="गतिकीय तन्त्र" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Dinamikala_sistemo" title="Dinamikala sistemo – Ido" lang="io" hreflang="io" data-title="Dinamikala sistemo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_dinamis" title="Sistem dinamis – Indonesian" lang="id" hreflang="id" data-title="Sistem dinamis" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" 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/Sistema_dinamico" title="Sistema dinamico – Italian" lang="it" hreflang="it" data-title="Sistema dinamico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%93%D7%99%D7%A0%D7%9E%D7%99%D7%AA" title="מערכת דינמית – Hebrew" lang="he" hreflang="he" data-title="מערכת דינמית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Dinamin%C4%97_sistema" title="Dinaminė sistema – Lithuanian" lang="lt" hreflang="lt" data-title="Dinaminė sistema" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Dinamikai_rendszer" title="Dinamikai rendszer – Hungarian" lang="hu" hreflang="hu" data-title="Dinamikai rendszer" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Динамичен систем – Macedonian" lang="mk" hreflang="mk" data-title="Динамичен систем" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Sistema_dinamika" title="Sistema dinamika – Maltese" lang="mt" hreflang="mt" data-title="Sistema dinamika" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_dinamik" title="Sistem dinamik – Malay" lang="ms" hreflang="ms" data-title="Sistem dinamik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%95%E1%80%BC%E1%80%B1%E1%80%AC%E1%80%84%E1%80%BA%E1%80%B8%E1%80%9C%E1%80%B2%E1%80%9C%E1%80%BE%E1%80%AF%E1%80%95%E1%80%BA%E1%80%9B%E1%80%BE%E1%80%AC%E1%80%B8%E1%80%85%E1%80%94%E1%80%85%E1%80%BA" title="ပြောင်းလဲလှုပ်ရှားစနစ် – Burmese" lang="my" hreflang="my" data-title="ပြောင်းလဲလှုပ်ရှားစနစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Dynamisch_systeem" title="Dynamisch systeem – Dutch" lang="nl" hreflang="nl" data-title="Dynamisch systeem" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8A%9B%E5%AD%A6%E7%B3%BB" title="力学系 – Japanese" lang="ja" hreflang="ja" data-title="力学系" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Uk%C5%82ad_dynamiczny" title="Układ dynamiczny – Polish" lang="pl" hreflang="pl" data-title="Układ dynamiczny" 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class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Dynamical_system" title="Dynamical system – Simple English" lang="en-simple" hreflang="en-simple" data-title="Dynamical system" 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/Dynamick%C3%BD_syst%C3%A9m" title="Dynamický systém – Slovak" lang="sk" hreflang="sk" data-title="Dynamický systém" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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/Dinami%C4%8Dni_sistem" title="Dinamični sistem – Slovenian" lang="sl" hreflang="sl" data-title="Dinamični sistem" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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/Dinami%C4%8Dki_sistem" title="Dinamički sistem – Serbian" lang="sr" hreflang="sr" data-title="Dinamički sistem" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Dinami%C4%8Dki_sistem" title="Dinamički sistem – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Dinamički sistem" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a 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href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D0%BD%D0%B0%D0%BC%D1%96%D1%87%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Динамічна система – Ukrainian" lang="uk" hreflang="uk" data-title="Динамічна система" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_th%E1%BB%91ng_%C4%91%E1%BB%99ng_l%E1%BB%B1c" title="Hệ thống động lực – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ thống động lực" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Sistema_dinamiko" title="Sistema dinamiko – Waray" lang="war" hreflang="war" data-title="Sistema dinamiko" data-language-autonym="Winaray" 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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">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Dynamical_systems&redirect=no" class="mw-redirect" title="Dynamical systems">Dynamical systems</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical model of the time dependence of a point in space</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the general aspects of dynamical systems. For the study field, see <a href="/wiki/Dynamical_systems_theory" title="Dynamical systems theory">Dynamical systems theory</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Dynamical" redirects here. For other uses, see <a href="/wiki/Dynamical_(disambiguation)" class="mw-redirect mw-disambig" title="Dynamical (disambiguation)">Dynamical (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">February 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" 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/220px-Lorenz_attractor_yb.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Lorenz_attractor_yb.svg/330px-Lorenz_attractor_yb.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Lorenz_attractor_yb.svg/440px-Lorenz_attractor_yb.svg.png 2x" data-file-width="750" data-file-height="750" /></a><figcaption>The <a href="/wiki/Lorenz_attractor" class="mw-redirect" title="Lorenz attractor">Lorenz attractor</a> arises in the study of the <a href="/wiki/Lorenz_system" title="Lorenz system">Lorenz oscillator</a>, a dynamical system.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>dynamical system</b> is a system in which a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> describes the <a href="/wiki/Time" title="Time">time</a> dependence of a <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> in an <a href="/wiki/Ambient_space" class="mw-redirect" title="Ambient space">ambient space</a>, such as in a <a href="/wiki/Parametric_curve" class="mw-redirect" title="Parametric curve">parametric curve</a>. Examples include the <a href="/wiki/Mathematical_model" title="Mathematical model">mathematical models</a> that describe the swinging of a clock <a href="/wiki/Pendulum" title="Pendulum">pendulum</a>, <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">the flow of water in a pipe</a>, the <a href="/wiki/Brownian_motion" title="Brownian motion">random motion of particles in the air</a>, and <a href="/wiki/Population_dynamics" title="Population dynamics">the number of fish each springtime in a lake</a>. The most general definition unifies several concepts in mathematics such as <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> and <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a> by allowing different choices of the space and how time is measured.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2023)">citation needed</span></a></i>]</sup> Time can be measured by integers, by <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a <a href="/wiki/Manifold" title="Manifold">manifold</a> or simply a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, without the need of a <a href="/wiki/Differentiability" class="mw-redirect" title="Differentiability">smooth</a> space-time structure defined on it. </p><p>At any given time, a dynamical system has a <a href="/wiki/State_(controls)" class="mw-redirect" title="State (controls)">state</a> representing a point in an appropriate <a href="/wiki/State_space_(controls)" class="mw-redirect" title="State space (controls)">state space</a>. This state is often given by a <a href="/wiki/Tuple" title="Tuple">tuple</a> of <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> or by a <a href="/wiki/Vector_space" title="Vector space">vector</a> in a geometrical manifold. The <i>evolution rule</i> of the dynamical system is a function that describes what future states follow from the current state. Often the function is <a href="/wiki/Deterministic_system_(mathematics)" class="mw-redirect" title="Deterministic system (mathematics)">deterministic</a>, that is, for a given time interval only one future state follows from the current state.<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> However, some systems are <a href="/wiki/Stochastic_system" class="mw-redirect" title="Stochastic system">stochastic</a>, in that random events also affect the evolution of the state variables. </p><p>In <a href="/wiki/Physics" title="Physics">physics</a>, a <b>dynamical system</b> is described as a "particle or ensemble of particles whose state varies over time and thus obeys <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> involving time derivatives".<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized. </p><p>The study of dynamical systems is the focus of <a href="/wiki/Dynamical_systems_theory" title="Dynamical systems theory">dynamical systems theory</a>, which has applications to a wide variety of fields such as mathematics, physics,<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><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Biology" title="Biology">biology</a>,<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> <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, <a href="/wiki/Engineering" title="Engineering">engineering</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/Economics" title="Economics">economics</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> <a href="/wiki/Cliodynamics" title="Cliodynamics">history</a>, and <a href="/wiki/Medicine" title="Medicine">medicine</a>. Dynamical systems are a fundamental part of <a href="/wiki/Chaos_theory" title="Chaos theory">chaos theory</a>, <a href="/wiki/Logistic_map" title="Logistic map">logistic map</a> dynamics, <a href="/wiki/Bifurcation_theory" title="Bifurcation theory">bifurcation theory</a>, the <a href="/wiki/Self-assembly" title="Self-assembly">self-assembly</a> and <a href="/wiki/Self-organization" title="Self-organization">self-organization</a> processes, and the <a href="/wiki/Edge_of_chaos" title="Edge of chaos">edge of chaos</a> concept. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of a dynamical system has its origins in <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a>, <a href="/wiki/Recurrence_relation" title="Recurrence relation">difference equation</a> or other <a href="/wiki/Time_scale_calculus" class="mw-redirect" title="Time scale calculus">time scale</a>.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as <i>solving the system</i> or <i>integrating the system</i>. If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a <i><a href="/wiki/Trajectory" title="Trajectory">trajectory</a></i> or <i><a href="/wiki/Orbit_(dynamics)" title="Orbit (dynamics)">orbit</a></i>. </p><p>Before the advent of <a href="/wiki/Computers" class="mw-redirect" title="Computers">computers</a>, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. </p><p>For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because: </p> <ul><li>The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as <a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a> or <a href="/wiki/Structural_stability" title="Structural stability">structural stability</a>. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a> changes with the different notions of stability.</li> <li>The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. <a href="/wiki/Linear_dynamical_system" title="Linear dynamical system">Linear dynamical systems</a> and <a href="/wiki/Poincar%C3%A9%E2%80%93Bendixson_theorem" title="Poincaré–Bendixson theorem">systems that have two numbers describing a state</a> are examples of dynamical systems where the possible classes of orbits are understood.</li> <li>The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have <a href="/wiki/Bifurcation_theory" title="Bifurcation theory">bifurcation points</a> where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the <a href="/wiki/Turbulence" title="Turbulence">transition to turbulence of a fluid</a>.</li> <li>The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic systems</a> and a more detailed understanding has been worked out for <a href="/wiki/Anosov_diffeomorphism" title="Anosov diffeomorphism">hyperbolic systems</a>. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> and of <a href="/wiki/Chaos_theory" title="Chaos theory">chaos</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many people regard French mathematician <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> as the founder of dynamical systems.<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> Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the <a href="/wiki/Poincar%C3%A9_recurrence_theorem" title="Poincaré recurrence theorem">Poincaré recurrence theorem</a>, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. </p><p><a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a> developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system. </p><p>In 1913, <a href="/wiki/George_David_Birkhoff" title="George David Birkhoff">George David Birkhoff</a> proved Poincaré's "<a href="/wiki/Poincar%C3%A9%E2%80%93Birkhoff_theorem" title="Poincaré–Birkhoff theorem">Last Geometric Theorem</a>", a special case of the <a href="/wiki/Three-body_problem" title="Three-body problem">three-body problem</a>, a result that made him world-famous. In 1927, he published his <i><a rel="nofollow" class="external text" href="https://archive.org/details/dynamicalsystems00birk/">Dynamical Systems</a></i>. Birkhoff's most durable result has been his 1931 discovery of what is now called the <a href="/wiki/Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">ergodic theorem</a>. Combining insights from <a href="/wiki/Physics" title="Physics">physics</a> on the <a href="/wiki/Ergodic_hypothesis" title="Ergodic hypothesis">ergodic hypothesis</a> with <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, this theorem solved, at least in principle, a fundamental problem of <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>. The ergodic theorem has also had repercussions for dynamics. </p><p><a href="/wiki/Stephen_Smale" title="Stephen Smale">Stephen Smale</a> made significant advances as well. His first contribution was the <a href="/wiki/Horseshoe_map" title="Horseshoe map">Smale horseshoe</a> that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others. </p><p><a href="/wiki/Oleksandr_Mykolaiovych_Sharkovsky" class="mw-redirect" title="Oleksandr Mykolaiovych Sharkovsky">Oleksandr Mykolaiovych Sharkovsky</a> developed <a href="/wiki/Sharkovsky%27s_theorem" class="mw-redirect" title="Sharkovsky's theorem">Sharkovsky's theorem</a> on the periods of <a href="/wiki/Discrete_dynamical_system" class="mw-redirect" title="Discrete dynamical system">discrete dynamical systems</a> in 1964. One of the implications of the theorem is that if a discrete dynamical system on the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> has a <a href="/wiki/Periodic_point" title="Periodic point">periodic point</a> of period 3, then it must have periodic points of every other period. </p><p>In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer <a href="/wiki/Ali_H._Nayfeh" title="Ali H. Nayfeh">Ali H. Nayfeh</a> applied <a href="/wiki/Nonlinear_dynamics" class="mw-redirect" title="Nonlinear dynamics">nonlinear dynamics</a> in <a href="/wiki/Mechanics" title="Mechanics">mechanical</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a> systems.<sup id="cite_ref-Rega_10-0" class="reference"><a href="#cite_note-Rega-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of <a href="/wiki/Machines" class="mw-redirect" title="Machines">machines</a> and <a href="/wiki/Structures" class="mw-redirect" title="Structures">structures</a> that are common in daily life, such as <a href="/wiki/Ships" class="mw-redirect" title="Ships">ships</a>, <a href="/wiki/Crane_(machine)" title="Crane (machine)">cranes</a>, <a href="/wiki/Bridges" class="mw-redirect" title="Bridges">bridges</a>, <a href="/wiki/Buildings" class="mw-redirect" title="Buildings">buildings</a>, <a href="/wiki/Skyscrapers" class="mw-redirect" title="Skyscrapers">skyscrapers</a>, <a href="/wiki/Jet_engines" class="mw-redirect" title="Jet engines">jet engines</a>, <a href="/wiki/Rocket_engines" class="mw-redirect" title="Rocket engines">rocket engines</a>, <a href="/wiki/Aircraft" title="Aircraft">aircraft</a> and <a href="/wiki/Spacecraft" title="Spacecraft">spacecraft</a>.<sup id="cite_ref-fi_11-0" class="reference"><a href="#cite_note-fi-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=3" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the most general sense,<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><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> a <b>dynamical system</b> is a <a href="/wiki/Tuple" title="Tuple">tuple</a> (<i>T</i>, <i>X</i>, Φ) where <i>T</i> is a <a href="/wiki/Monoid" title="Monoid">monoid</a>, written additively, <i>X</i> is a non-empty <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> and Φ is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> </p> <dl><dd><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 \Phi :U\subseteq (T\times X)\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <mi>U</mi> <mo>⊆</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi :U\subseteq (T\times X)\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/732eb5be439c409f5d0ce5be723066e594884e95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.356ex; height:2.843ex;" alt="{\displaystyle \Phi :U\subseteq (T\times X)\to X}"></span></dd></dl> <p>with </p> <dl><dd><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 \mathrm {proj} _{2}(U)=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {proj} _{2}(U)=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4c1da3d9a1d8f0d65baf05c3f01166011a1680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.803ex; height:2.843ex;" alt="{\displaystyle \mathrm {proj} _{2}(U)=X}"></span> (where <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 \mathrm {proj} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {proj} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2919c1f052c657116192b06356d8f981b37accda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.133ex; height:2.676ex;" alt="{\displaystyle \mathrm {proj} _{2}}"></span> is the 2nd <a href="/wiki/Projection_(set_theory)" title="Projection (set theory)">projection map</a>)</dd></dl> <p>and for any <i>x</i> in <i>X</i>: </p> <dl><dd><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 \Phi (0,x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (0,x)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9e28a8cff06417aaade77a86a849dd787747dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.442ex; height:2.843ex;" alt="{\displaystyle \Phi (0,x)=x}"></span></dd> <dd><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 \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651726633cd281ebfdba0136711c7d17de096d99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.384ex; height:2.843ex;" alt="{\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}"></span></dd></dl> <p>for <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_{1},\,t_{2}+t_{1}\in I(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,t_{1},\,t_{2}+t_{1}\in I(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84acbe7b6ebcce85a48629c1868d996694918766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.481ex; height:2.843ex;" alt="{\displaystyle \,t_{1},\,t_{2}+t_{1}\in I(x)}"></span> and <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_{2}\in I(\Phi (t_{1},x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ t_{2}\in I(\Phi (t_{1},x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17f67a2ef8a2474dddaf35661e8f9e4e7395d09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.041ex; height:2.843ex;" alt="{\displaystyle \ t_{2}\in I(\Phi (t_{1},x))}"></span>, where we have defined the set <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 I(x):=\{t\in T:(t,x)\in U\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mo>∈</mo> <mi>T</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>U</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(x):=\{t\in T:(t,x)\in U\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd29eccb991fe8dda96b609b610c8540bad1410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.27ex; height:2.843ex;" alt="{\displaystyle I(x):=\{t\in T:(t,x)\in U\}}"></span> for any <i>x</i> in <i>X</i>. </p><p>In particular, in the case that <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 U=T\times X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>T</mi> <mo>×</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=T\times X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b54a5da30f132fe5663513da84669f6f5aaade79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.338ex; height:2.176ex;" alt="{\displaystyle U=T\times X}"></span> we have for every <i>x</i> in <i>X</i> that <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 I(x)=T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(x)=T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ec9d98eb426eb6deec2b66aeaa8ad32bab1fb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.045ex; height:2.843ex;" alt="{\displaystyle I(x)=T}"></span> and thus that Φ defines a <a href="/wiki/Semigroup_action" title="Semigroup action">monoid action</a> of <i>T</i> on <i>X</i>. </p><p>The function Φ(<i>t</i>,<i>x</i>) is called the <b>evolution function</b> of the dynamical system: it associates to every point <i>x</i> in the set <i>X</i> a unique image, depending on the variable <i>t</i>, called the <b>evolution parameter</b>. <i>X</i> is called <b><a href="/wiki/Phase_space" title="Phase space">phase space</a></b> or <b>state space</b>, while the variable <i>x</i> represents an <b>initial state</b> of the system. </p><p>We often write </p> <dl><dd><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 \Phi _{x}(t)\equiv \Phi (t,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{x}(t)\equiv \Phi (t,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de52b04df90526084d44a8d5e6c83305c5883233" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.289ex; height:2.843ex;" alt="{\displaystyle \Phi _{x}(t)\equiv \Phi (t,x)}"></span></dd> <dd><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 \Phi ^{t}(x)\equiv \Phi (t,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi ^{t}(x)\equiv \Phi (t,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5628d746f45c377395f6e14817e0ecfcd87eb989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.432ex; height:3.009ex;" alt="{\displaystyle \Phi ^{t}(x)\equiv \Phi (t,x)}"></span></dd></dl> <p>if we take one of the variables as constant. The function </p> <dl><dd><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 \Phi _{x}:I(x)\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>:</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{x}:I(x)\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af8bcdb2390e5cb5c35d792eb024edc9889c42fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.693ex; height:2.843ex;" alt="{\displaystyle \Phi _{x}:I(x)\to X}"></span></dd></dl> <p>is called the <b>flow</b> through <i>x</i> and its <a href="/wiki/Graph_(function)" class="mw-redirect" title="Graph (function)">graph</a> is called the <b><a href="/wiki/Trajectory" title="Trajectory">trajectory</a></b> through <i>x</i>. The set </p> <dl><dd><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 \gamma _{x}\equiv \{\Phi (t,x):t\in I(x)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>≡</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>∈</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{x}\equiv \{\Phi (t,x):t\in I(x)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a045112abfddc396ffd69e19d3d6ab6b24e61037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.419ex; height:2.843ex;" alt="{\displaystyle \gamma _{x}\equiv \{\Phi (t,x):t\in I(x)\}}"></span></dd></dl> <p>is called the <b><a href="/wiki/Orbit_(dynamics)" title="Orbit (dynamics)">orbit</a></b> through <i>x</i>. The orbit through <i>x</i> is the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of the flow through <i>x</i>. A subset <i>S</i> of the state space <i>X</i> is called Φ-<b>invariant</b> if for all <i>x</i> in <i>S</i> and all <i>t</i> in <i>T</i> </p> <dl><dd><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 \Phi (t,x)\in S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (t,x)\in S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9d9a81a06e1c9badc03b812ac535ae81d0bc2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.677ex; height:2.843ex;" alt="{\displaystyle \Phi (t,x)\in S.}"></span></dd></dl> <p>Thus, in particular, if <i>S</i> is Φ-<b>invariant</b>, <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 I(x)=T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(x)=T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ec9d98eb426eb6deec2b66aeaa8ad32bab1fb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.045ex; height:2.843ex;" alt="{\displaystyle I(x)=T}"></span> for all <i>x</i> in <i>S</i>. That is, the flow through <i>x</i> must be defined for all time for every element of <i>S</i>. </p><p>More commonly there are two classes of definitions for a dynamical system: one is motivated by <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> and is geometrical in flavor; and the other is motivated by <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a> and is <a href="/wiki/Measure_(mathematics)#Measure_theory" title="Measure (mathematics)">measure theoretical</a> in flavor. </p> <div class="mw-heading mw-heading3"><h3 id="Geometrical_definition">Geometrical definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=4" title="Edit section: Geometrical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the geometrical definition, a dynamical system is the tuple <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 \langle {\mathcal {T}},{\mathcal {M}},f\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\mathcal {T}},{\mathcal {M}},f\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae280ece9a08ab14dcc4c84ce3b4597a6f1c047c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.882ex; height:2.843ex;" alt="{\displaystyle \langle {\mathcal {T}},{\mathcal {M}},f\rangle }"></span>. <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 {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span> is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. <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 {\mathcal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}"></span> is a <a href="/wiki/Manifold" title="Manifold">manifold</a>, i.e. locally a Banach space or Euclidean space, or in the discrete case a <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>. <i>f</i> is an evolution rule <i>t</i> → <i>f</i><sup> <i>t</i></sup> (with <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\in {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b8b08dd997064818370e5546e240e53155d8a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.343ex;" alt="{\displaystyle t\in {\mathcal {T}}}"></span>) such that <i>f<sup> t</sup></i> is a <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> of the manifold to itself. So, f is a "smooth" mapping of the time-domain <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 {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span> into the space of diffeomorphisms of the manifold to itself. In other terms, <i>f</i>(<i>t</i>) is a diffeomorphism, for every time <i>t</i> in the domain <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 {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span> . </p> <div class="mw-heading mw-heading4"><h4 id="Real_dynamical_system">Real dynamical system</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=5" title="Edit section: Real dynamical system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>real dynamical system</i>, <i>real-time dynamical system</i>, <i><a href="/wiki/Continuous_time" class="mw-redirect" title="Continuous time">continuous time</a> dynamical system</i>, or <i><a href="/wiki/Flow_(mathematics)" title="Flow (mathematics)">flow</a></i> is a tuple (<i>T</i>, <i>M</i>, Φ) with <i>T</i> an <a href="/wiki/Open_interval" class="mw-redirect" title="Open interval">open interval</a> in the <a href="/wiki/Real_number" title="Real number">real numbers</a> <b>R</b>, <i>M</i> a <a href="/wiki/Manifold" title="Manifold">manifold</a> locally <a href="/wiki/Diffeomorphic" class="mw-redirect" title="Diffeomorphic">diffeomorphic</a> to a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>, and Φ a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a>. If Φ is <a href="/wiki/Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">continuously differentiable</a> we say the system is a <i>differentiable dynamical system</i>. If the manifold <i>M</i> is locally diffeomorphic to <b>R</b><sup><i>n</i></sup>, the dynamical system is <i>finite-dimensional</i>; if not, the dynamical system is <i>infinite-dimensional</i>. This does not assume a <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic structure</a>. When <i>T</i> is taken to be the reals, the dynamical system is called <i>global</i> or a <i><a href="/wiki/Flow_(mathematics)" title="Flow (mathematics)">flow</a></i>; and if <i>T</i> is restricted to the non-negative reals, then the dynamical system is a <i>semi-flow</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Discrete_dynamical_system">Discrete dynamical system</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=6" title="Edit section: Discrete dynamical system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>discrete dynamical system</i>, <i><a href="/wiki/Discrete-time" class="mw-redirect" title="Discrete-time">discrete-time</a> dynamical system</i> is a tuple (<i>T</i>, <i>M</i>, Φ), where <i>M</i> is a <a href="/wiki/Manifold" title="Manifold">manifold</a> locally diffeomorphic to a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>, and Φ is a function. When <i>T</i> is taken to be the integers, it is a <i>cascade</i> or a <i>map</i>. If <i>T</i> is restricted to the non-negative integers we call the system a <i>semi-cascade</i>.<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-heading4"><h4 id="Cellular_automaton">Cellular automaton</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=7" title="Edit section: Cellular automaton"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>cellular automaton</i> is a tuple (<i>T</i>, <i>M</i>, Φ), with <i>T</i> a <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> such as the <a href="/wiki/Integer" title="Integer">integers</a> or a higher-dimensional <a href="/wiki/Integer_lattice" title="Integer lattice">integer grid</a>, <i>M</i> is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function. As such <a href="/wiki/Cellular_automata" class="mw-redirect" title="Cellular automata">cellular automata</a> are dynamical systems. The lattice in <i>M</i> represents the "space" lattice, while the one in <i>T</i> represents the "time" lattice. </p> <div class="mw-heading mw-heading4"><h4 id="Multidimensional_generalization">Multidimensional generalization</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=8" title="Edit section: Multidimensional generalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called <a href="/wiki/Multidimensional_systems" class="mw-redirect" title="Multidimensional systems">multidimensional systems</a>. Such systems are useful for modeling, for example, <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Compactification_of_a_dynamical_system">Compactification of a dynamical system</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=9" title="Edit section: Compactification of a dynamical system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a global dynamical system (<b>R</b>, <i>X</i>, Φ) on a <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> and <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i>, it is often useful to study the continuous extension Φ* of Φ to the <a href="/wiki/One-point_compactification" class="mw-redirect" title="One-point compactification">one-point compactification</a> <i>X*</i> of <i>X</i>. Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system (<b>R</b>, <i>X*</i>, Φ*). </p><p>In compact dynamical systems the <a href="/wiki/Limit_set" title="Limit set">limit set</a> of any orbit is <a href="/wiki/Non-empty" class="mw-redirect" title="Non-empty">non-empty</a>, <a href="/wiki/Compact_space" title="Compact space">compact</a> and <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Measure_theoretical_definition">Measure theoretical definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=10" title="Edit section: Measure theoretical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Measure-preserving_dynamical_system" title="Measure-preserving dynamical system">Measure-preserving dynamical system</a></div> <p>A dynamical system may be defined formally as a measure-preserving transformation of a <a href="/wiki/Measure_space" title="Measure space">measure space</a>, the triplet (<i>T</i>, (<i>X</i>, Σ, <i>μ</i>), Φ). Here, <i>T</i> is a monoid (usually the non-negative integers), <i>X</i> is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, and (<i>X</i>, Σ, <i>μ</i>) is a <a href="/wiki/Measure_space" title="Measure space">probability space</a>, meaning that Σ is a <a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">sigma-algebra</a> on <i>X</i> and μ is a finite <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a> on (<i>X</i>, Σ). A map Φ: <i>X</i> → <i>X</i> is said to be <a href="/wiki/Measurable_function" title="Measurable function">Σ-measurable</a> if and only if, for every σ in Σ, one has <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 \Phi ^{-1}\sigma \in \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>σ</mi> <mo>∈</mo> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi ^{-1}\sigma \in \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/607d3fe252fc139072a0d91d8dbb09daeaf2eac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.859ex; height:2.676ex;" alt="{\displaystyle \Phi ^{-1}\sigma \in \Sigma }"></span>. A map Φ is said to <b>preserve the measure</b> if and only if, for every <i>σ</i> in Σ, one has <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 \mu (\Phi ^{-1}\sigma )=\mu (\sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (\Phi ^{-1}\sigma )=\mu (\sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f86aabd5c91404b2c1ba4ddbbc0b1e721f6b78a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.191ex; height:3.176ex;" alt="{\displaystyle \mu (\Phi ^{-1}\sigma )=\mu (\sigma )}"></span>. Combining the above, a map Φ is said to be a <b>measure-preserving transformation of <i>X</i> </b>, if it is a map from <i>X</i> to itself, it is Σ-measurable, and is measure-preserving. The triplet (<i>T</i>, (<i>X</i>, Σ, <i>μ</i>), Φ), for such a Φ, is then defined to be a <b>dynamical system</b>. </p><p>The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the <a href="/wiki/Iterated_function" title="Iterated function">iterates</a> <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 \Phi ^{n}=\Phi \circ \Phi \circ \dots \circ \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mo>∘</mo> <mi mathvariant="normal">Φ</mi> <mo>∘</mo> <mo>⋯</mo> <mo>∘</mo> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi ^{n}=\Phi \circ \Phi \circ \dots \circ \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/681995985709ecf80a8dfce5bc91b9bb3be67579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.337ex; height:2.343ex;" alt="{\displaystyle \Phi ^{n}=\Phi \circ \Phi \circ \dots \circ \Phi }"></span> for every integer <i>n</i> are studied. For continuous dynamical systems, the map Φ is understood to be a finite time evolution map and the construction is more complicated. </p> <div class="mw-heading mw-heading4"><h4 id="Relation_to_geometric_definition">Relation to geometric definition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=11" title="Edit section: Relation to geometric definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made. A simple construction (sometimes called the <a href="/wiki/Krylov%E2%80%93Bogolyubov_theorem" title="Krylov–Bogolyubov theorem">Krylov–Bogolyubov theorem</a>) shows that for a large class of systems it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance. </p><p>Some systems have a natural measure, such as the <a href="/wiki/Liouville%27s_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville measure</a> in <a href="/wiki/Hamiltonian_system" title="Hamiltonian system">Hamiltonian systems</a>, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For chaotic <a href="/wiki/Dissipative_system" title="Dissipative system">dissipative systems</a> the choice of invariant measure is technically more challenging. The measure needs to be supported on the <a href="/wiki/Attractor" title="Attractor">attractor</a>, but attractors have zero <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> and the invariant measures must be singular with respect to the Lebesgue measure. A small region of phase space shrinks under time evolution. </p><p>For hyperbolic dynamical systems, the <a href="/wiki/Sinai%E2%80%93Ruelle%E2%80%93Bowen_measure" title="Sinai–Ruelle–Bowen measure">Sinai–Ruelle–Bowen measures</a> appear to be the natural choice. They are constructed on the geometrical structure of <a href="/wiki/Stable_manifold" title="Stable manifold">stable and unstable manifolds</a> of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems. </p> <div class="mw-heading mw-heading2"><h2 id="Construction_of_dynamical_systems">Construction of dynamical systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=12" title="Edit section: Construction of dynamical systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of <i>evolution in time</i> is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanical systems</a>. But a system of <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> must be solved before it becomes a dynamic system. For example, consider an <a href="/wiki/Initial_value_problem" title="Initial value problem">initial value problem</a> such as the following: </p> <dl><dd><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 {\dot {\boldsymbol {x}}}={\boldsymbol {v}}(t,{\boldsymbol {x}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\boldsymbol {x}}}={\boldsymbol {v}}(t,{\boldsymbol {x}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dd5bba9d6211abb566cc94c1578ceac08acc821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.163ex; height:2.843ex;" alt="{\displaystyle {\dot {\boldsymbol {x}}}={\boldsymbol {v}}(t,{\boldsymbol {x}})}"></span></dd> <dd><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 {\boldsymbol {x}}|_{t=0}={\boldsymbol {x}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}|_{t=0}={\boldsymbol {x}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e15680d9f6f907f61cc6eb664d1f3287646120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.79ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {x}}|_{t=0}={\boldsymbol {x}}_{0}}"></span></dd></dl> <p>where </p> <ul><li><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 {\dot {\boldsymbol {x}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">x</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\boldsymbol {x}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29815af430ef19272d18d70c289e0073e50ff381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.532ex; height:2.176ex;" alt="{\displaystyle {\dot {\boldsymbol {x}}}}"></span> represents the <a href="/wiki/Velocity" title="Velocity">velocity</a> of the material point <b>x</b></li> <li><i>M</i> is a finite dimensional manifold</li> <li><b>v</b>: <i>T</i> × <i>M</i> → <i>TM</i> is a <a href="/wiki/Vector_field" title="Vector field">vector field</a> in <b>R</b><sup><i>n</i></sup> or <b>C</b><sup><i>n</i></sup> and represents the change of <a href="/wiki/Velocity" title="Velocity">velocity</a> induced by the known <a href="/wiki/Force" title="Force">forces</a> acting on the given material point in the phase space <i>M</i>. The change is not a vector in the phase space <i>M</i>, but is instead in the <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> <i>TM</i>.</li></ul> <p>There is no need for higher order derivatives in the equation, nor for the parameter <i>t</i> in <i>v</i>(<i>t</i>,<i>x</i>), because these can be eliminated by considering systems of higher dimensions. </p><p>Depending on the properties of this vector field, the mechanical system is called </p> <ul><li><b>autonomous</b>, when <b>v</b>(<i>t</i>, <b>x</b>) = <b>v</b>(<b>x</b>)</li> <li><b>homogeneous</b> when <b>v</b>(<i>t</i>, <b>0</b>) = 0 for all <i>t</i></li></ul> <p>The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above </p> <dl><dd><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 {\boldsymbol {x}}(t)=\Phi (t,{\boldsymbol {x}}_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}(t)=\Phi (t,{\boldsymbol {x}}_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec6f1dc90a51280a25f803eac25bf3cfe23399ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.226ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {x}}(t)=\Phi (t,{\boldsymbol {x}}_{0})}"></span></dd></dl> <p>The dynamical system is then (<i>T</i>, <i>M</i>, Φ). </p><p>Some formal manipulation of the system of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> shown above gives a more general form of equations a dynamical system must satisfy </p> <dl><dd><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 {\dot {\boldsymbol {x}}}-{\boldsymbol {v}}(t,{\boldsymbol {x}})=0\qquad \Leftrightarrow \qquad {\mathfrak {G}}\left(t,\Phi (t,{\boldsymbol {x}}_{0})\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <mo stretchy="false">⇔</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">G</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\boldsymbol {x}}}-{\boldsymbol {v}}(t,{\boldsymbol {x}})=0\qquad \Leftrightarrow \qquad {\mathfrak {G}}\left(t,\Phi (t,{\boldsymbol {x}}_{0})\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7ee0e21b3c136c0871919a453af320d47de7c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.172ex; height:2.843ex;" alt="{\displaystyle {\dot {\boldsymbol {x}}}-{\boldsymbol {v}}(t,{\boldsymbol {x}})=0\qquad \Leftrightarrow \qquad {\mathfrak {G}}\left(t,\Phi (t,{\boldsymbol {x}}_{0})\right)=0}"></span></dd></dl> <p>where <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 {\mathfrak {G}}:{{(T\times M)}^{M}}\to \mathbf {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">G</mi> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>×</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msup> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {G}}:{{(T\times M)}^{M}}\to \mathbf {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48826ea63d11f9d36ae5b1583780ba65329d2df3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.994ex; height:3.343ex;" alt="{\displaystyle {\mathfrak {G}}:{{(T\times M)}^{M}}\to \mathbf {C} }"></span> is a <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functional</a> from the set of evolution functions to the field of the complex numbers. </p><p>This equation is useful when modeling mechanical systems with complicated constraints. </p><p>Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>—in which case the differential equations are <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=13" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 25em;"> <ul><li><a href="/wiki/Arnold%27s_cat_map" title="Arnold's cat map">Arnold's cat map</a></li> <li><a href="/wiki/Baker%27s_map" title="Baker's map">Baker's map</a> is an example of a chaotic <a href="/wiki/Piecewise_linear_function" title="Piecewise linear function">piecewise linear</a> map</li> <li><a href="/wiki/Dynamical_billiards" title="Dynamical billiards">Billiards</a> and <a href="/wiki/Dynamical_outer_billiards" class="mw-redirect" title="Dynamical outer billiards">outer billiards</a></li> <li><a href="/wiki/Bouncing_ball_dynamics" class="mw-redirect" title="Bouncing ball dynamics">Bouncing ball dynamics</a></li> <li><a href="/wiki/Circle_map" class="mw-redirect" title="Circle map">Circle map</a></li> <li><a href="/wiki/Complex_quadratic_polynomial" title="Complex quadratic polynomial">Complex quadratic polynomial</a></li> <li><a href="/wiki/Double_pendulum" title="Double pendulum">Double pendulum</a></li> <li><a href="/wiki/Dyadic_transformation" title="Dyadic transformation">Dyadic transformation</a></li> <li><a href="/wiki/H%C3%A9non_map" title="Hénon map">Hénon map</a></li> <li><a href="/wiki/Irrational_rotation" title="Irrational rotation">Irrational rotation</a></li> <li><a href="/wiki/Kaplan%E2%80%93Yorke_map" title="Kaplan–Yorke map">Kaplan–Yorke map</a></li> <li><a href="/wiki/List_of_chaotic_maps" title="List of chaotic maps">List of chaotic maps</a></li> <li><a href="/wiki/Lorenz_attractor" class="mw-redirect" title="Lorenz attractor">Lorenz system</a></li> <li><a href="/wiki/Complex_quadratic_polynomial#Map" title="Complex quadratic polynomial">Quadratic map simulation system</a></li> <li><a href="/wiki/R%C3%B6ssler_map" class="mw-redirect" title="Rössler map">Rössler map</a></li> <li><a href="/wiki/Swinging_Atwood%27s_machine" title="Swinging Atwood's machine">Swinging Atwood's machine</a></li> <li><a href="/wiki/Tent_map" title="Tent map">Tent map</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Linear_dynamical_systems">Linear dynamical systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=14" title="Edit section: Linear dynamical systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Linear_dynamical_system" title="Linear dynamical system">Linear dynamical system</a></div> <p>Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the <i>N</i>-dimensional Euclidean space, so any point in phase space can be represented by a vector with <i>N</i> numbers. The analysis of linear systems is possible because they satisfy a <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</a>: if <i>u</i>(<i>t</i>) and <i>w</i>(<i>t</i>) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will <i>u</i>(<i>t</i>) + <i>w</i>(<i>t</i>). </p> <div class="mw-heading mw-heading3"><h3 id="Flows">Flows</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=15" title="Edit section: Flows"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Flow_(mathematics)" title="Flow (mathematics)">flow</a>, the vector field v(<i>x</i>) is an <a href="/wiki/Affine_transformation" title="Affine transformation">affine</a> function of the position in the phase space, that is, </p> <dl><dd><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 {\dot {x}}=v(x)=Ax+b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}=v(x)=Ax+b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e50c441245a312d954d597bc01e96d758859bdf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.351ex; height:2.843ex;" alt="{\displaystyle {\dot {x}}=v(x)=Ax+b,}"></span></dd></dl> <p>with <i>A</i> a matrix, <i>b</i> a vector of numbers and <i>x</i> the position vector. The solution to this system can be found by using the superposition principle (linearity). The case <i>b</i> ≠ 0 with <i>A</i> = 0 is just a straight line in the direction of <i>b</i>: </p> <dl><dd><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 \Phi ^{t}(x_{1})=x_{1}+bt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi ^{t}(x_{1})=x_{1}+bt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/486686b3377dc08ad0e8c44b9f214ebdd4dd283d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.504ex; height:3.009ex;" alt="{\displaystyle \Phi ^{t}(x_{1})=x_{1}+bt.}"></span></dd></dl> <p>When <i>b</i> is zero and <i>A</i> ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if <i>x</i><sub>0</sub> = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the <a href="/wiki/Matrix_exponential" title="Matrix exponential">exponential of a matrix</a>: for an initial point <i>x</i><sub>0</sub>, </p> <dl><dd><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 \Phi ^{t}(x_{0})=e^{tA}x_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi ^{t}(x_{0})=e^{tA}x_{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85622e724fee765e7c698b28acb556fba8cad097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.968ex; height:3.176ex;" alt="{\displaystyle \Phi ^{t}(x_{0})=e^{tA}x_{0}.}"></span></dd></dl> <p>When <i>b</i> = 0, the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of <i>A</i> determine the structure of the phase space. From the eigenvalues and the <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> of <i>A</i> it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin. </p><p>The distance between two different initial conditions in the case <i>A</i> ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for <a href="/wiki/Chaos_theory" title="Chaos theory">chaotic behavior</a>. </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:LinearFields.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/LinearFields.png/500px-LinearFields.png" decoding="async" width="500" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/LinearFields.png/750px-LinearFields.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/LinearFields.png/1000px-LinearFields.png 2x" data-file-width="2048" data-file-height="520" /></a><figcaption>Linear vector fields and a few trajectories.</figcaption></figure> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Maps">Maps</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=16" title="Edit section: Maps"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Discrete-time_dynamical_system" class="mw-redirect" title="Discrete-time dynamical system">discrete-time</a>, <a href="/wiki/Affine_transformation" title="Affine transformation">affine</a> dynamical system has the form of a <a href="/wiki/Matrix_difference_equation" title="Matrix difference equation">matrix difference equation</a>: </p> <dl><dd><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 x_{n+1}=Ax_{n}+b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>A</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n+1}=Ax_{n}+b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc592220436380e1bfa99c2d3984a39ad2f1cadb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.523ex; height:2.509ex;" alt="{\displaystyle x_{n+1}=Ax_{n}+b,}"></span></dd></dl> <p>with <i>A</i> a matrix and <i>b</i> a vector. As in the continuous case, the change of coordinates <i>x</i> → <i>x</i> + (1 − <i>A</i>)<sup> –1</sup><i>b</i> removes the term <i>b</i> from the equation. In the new <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a>, the origin is a fixed point of the map and the solutions are of the linear system <i>A</i><sup> <i>n</i></sup><i>x</i><sub>0</sub>. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map. </p><p>As in the continuous case, the eigenvalues and eigenvectors of <i>A</i> determine the structure of phase space. For example, if <i>u</i><sub>1</sub> is an eigenvector of <i>A</i>, with a real eigenvalue smaller than one, then the straight lines given by the points along <i>α</i> <i>u</i><sub>1</sub>, with <i>α</i> ∈ <b>R</b>, is an invariant curve of the map. Points in this straight line run into the fixed point. </p><p>There are also many <a href="/wiki/List_of_chaotic_maps" title="List of chaotic maps">other discrete dynamical systems</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Local_dynamics">Local dynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=17" title="Edit section: Local dynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a <i>singular point</i> of the vector field (a point where <i>v</i>(<i>x</i>) = 0) will remain a singular point under smooth transformations; a <i>periodic orbit</i> is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible. </p> <div class="mw-heading mw-heading3"><h3 id="Rectification">Rectification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=18" title="Edit section: Rectification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A flow in most small patches of the phase space can be made very simple. If <i>y</i> is a point where the vector field <i>v</i>(<i>y</i>) ≠ 0, then there is a change of coordinates for a region around <i>y</i> where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem. </p><p>The <i>rectification theorem</i> says that away from <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singular points</a> the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space <i>M</i> the dynamical system is <i>integrable</i>. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where <i>v</i>(<i>x</i>) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches. </p> <div class="mw-heading mw-heading3"><h3 id="Near_periodic_orbits">Near periodic orbits</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=19" title="Edit section: Near periodic orbits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point <i>x</i><sub>0</sub> in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to <i>v</i>(<i>x</i><sub>0</sub>). These points are a <a href="/wiki/Poincar%C3%A9_section" class="mw-redirect" title="Poincaré section">Poincaré section</a> <i>S</i>(<i>γ</i>, <i>x</i><sub>0</sub>), of the orbit. The flow now defines a map, the <a href="/wiki/Poincar%C3%A9_map" title="Poincaré map">Poincaré map</a> <i>F</i> : <i>S</i> → <i>S</i>, for points starting in <i>S</i> and returning to <i>S</i>. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes <i>x</i><sub>0</sub>. </p><p>The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map <i>F</i>. By a translation, the point can be assumed to be at <i>x</i> = 0. The Taylor series of the map is <i>F</i>(<i>x</i>) = <i>J</i> · <i>x</i> + O(<i>x</i><sup>2</sup>), so a change of coordinates <i>h</i> can only be expected to simplify <i>F</i> to its linear part </p> <dl><dd><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 h^{-1}\circ F\circ h(x)=J\cdot x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∘</mo> <mi>F</mi> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>J</mi> <mo>⋅</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h^{-1}\circ F\circ h(x)=J\cdot x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a412195d022a54f6f29f5e038bb6ee2b1227cf3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.505ex; height:3.176ex;" alt="{\displaystyle h^{-1}\circ F\circ h(x)=J\cdot x.}"></span></dd></dl> <p>This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If <i>λ</i><sub>1</sub>, ..., <i>λ</i><sub><i>ν</i></sub> are the eigenvalues of <i>J</i> they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form <i>λ</i><sub><i>i</i></sub> – Σ (multiples of other eigenvalues) occurs in the denominator of the terms for the function <i>h</i>, the non-resonant condition is also known as the small divisor problem. </p> <div class="mw-heading mw-heading3"><h3 id="Conjugation_results">Conjugation results</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=20" title="Edit section: Conjugation results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The results on the existence of a solution to the conjugation equation depend on the eigenvalues of <i>J</i> and the degree of smoothness required from <i>h</i>. As <i>J</i> does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of <i>J</i> are not in the unit circle, the dynamics near the fixed point <i>x</i><sub>0</sub> of <i>F</i> is called <i><a href="/wiki/Hyperbolic_fixed_point" class="mw-redirect" title="Hyperbolic fixed point">hyperbolic</a></i> and when the eigenvalues are on the unit circle and complex, the dynamics is called <i>elliptic</i>. </p><p>In the hyperbolic case, the <a href="/wiki/Hartman%E2%80%93Grobman_theorem" title="Hartman–Grobman theorem">Hartman–Grobman theorem</a> gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map <i>J</i> · <i>x</i>. The hyperbolic case is also <i>structurally stable</i>. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of <i>J</i> in the complex plane, implying that the map is still hyperbolic. </p><p>The <a href="/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem" title="Kolmogorov–Arnold–Moser theorem">Kolmogorov–Arnold–Moser (KAM)</a> theorem gives the behavior near an elliptic point. </p> <div class="mw-heading mw-heading2"><h2 id="Bifurcation_theory">Bifurcation theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=21" title="Edit section: Bifurcation theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bifurcation_theory" title="Bifurcation theory">Bifurcation theory</a></div> <p>When the evolution map Φ<sup><i>t</i></sup> (or the <a href="/wiki/Vector_field" title="Vector field">vector field</a> it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the <a href="/wiki/Phase_space" title="Phase space">phase space</a> until a special value <i>μ</i><sub>0</sub> is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation. </p><p>Bifurcation theory considers a structure in phase space (typically a <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a>, a periodic orbit, or an invariant <a href="/wiki/Torus" title="Torus">torus</a>) and studies its behavior as a function of the parameter <i>μ</i>. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems. </p><p>The bifurcations of a hyperbolic fixed point <i>x</i><sub>0</sub> of a system family <i>F<sub>μ</sub></i> can be characterized by the <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> of the first derivative of the system <i>DF</i><sub><i>μ</i></sub>(<i>x</i><sub>0</sub>) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of <i>DF<sub>μ</sub></i> on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on <a href="/wiki/Bifurcation_theory" title="Bifurcation theory">Bifurcation theory</a>. </p><p>Some bifurcations can lead to very complicated structures in phase space. For example, the <a href="/w/index.php?title=Ruelle%E2%80%93Takens_scenario&action=edit&redlink=1" class="new" title="Ruelle–Takens scenario (page does not exist)">Ruelle–Takens scenario</a> describes how a periodic orbit bifurcates into a torus and the torus into a <a href="/wiki/Strange_attractor" class="mw-redirect" title="Strange attractor">strange attractor</a>. In another example, <a href="/wiki/Bifurcation_diagram" title="Bifurcation diagram">Feigenbaum period-doubling</a> describes how a stable periodic orbit goes through a series of <a href="/wiki/Period-doubling_bifurcation" title="Period-doubling bifurcation">period-doubling bifurcations</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Ergodic_systems">Ergodic systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=22" title="Edit section: Ergodic systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></div> <p>In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset <i>A</i> into the points Φ<sup> <i>t</i></sup>(<i>A</i>) and invariance of the phase space means that </p> <dl><dd><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 \mathrm {vol} (A)=\mathrm {vol} (\Phi ^{t}(A)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {vol} (A)=\mathrm {vol} (\Phi ^{t}(A)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceec97d7ac3c4fe273369dc1dabd430bb2ee4d3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.237ex; height:3.009ex;" alt="{\displaystyle \mathrm {vol} (A)=\mathrm {vol} (\Phi ^{t}(A)).}"></span></dd></dl> <p>In the <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian formalism</a>, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the <a href="/wiki/Liouville%27s_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville measure</a>. </p><p>In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution. </p><p>For systems where the volume is preserved by the flow, Poincaré discovered the <a href="/wiki/Poincar%C3%A9_recurrence_theorem" title="Poincaré recurrence theorem">recurrence theorem</a>: Assume the phase space has a finite Liouville volume and let <i>F</i> be a phase space volume-preserving map and <i>A</i> a subset of the phase space. Then almost every point of <i>A</i> returns to <i>A</i> infinitely often. The Poincaré recurrence theorem was used by <a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Zermelo</a> to object to <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a>'s derivation of the increase in entropy in a dynamical system of colliding atoms. </p><p>One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the <a href="/wiki/Ergodic_hypothesis" title="Ergodic hypothesis">ergodic hypothesis</a>. The hypothesis states that the length of time a typical trajectory spends in a region <i>A</i> is vol(<i>A</i>)/vol(Ω). </p><p>The ergodic hypothesis turned out not to be the essential property needed for the development of <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. <a href="/wiki/Bernard_Koopman" title="Bernard Koopman">Koopman</a> approached the study of ergodic systems by the use of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. An observable <i>a</i> is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ<sup> t</sup>. This introduces an operator <i>U</i><sup> <i>t</i></sup>, the <a href="/wiki/Transfer_operator" title="Transfer operator">transfer operator</a>, </p> <dl><dd><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 (U^{t}a)(x)=a(\Phi ^{-t}(x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (U^{t}a)(x)=a(\Phi ^{-t}(x)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5b230fec2f0df937c48b7e9c01b1e6cdc3b2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.551ex; height:3.009ex;" alt="{\displaystyle (U^{t}a)(x)=a(\Phi ^{-t}(x)).}"></span></dd></dl> <p>By studying the spectral properties of the linear operator <i>U</i> it becomes possible to classify the ergodic properties of Φ<sup> <i>t</i></sup>. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ<sup> <i>t</i></sup> gets mapped into an infinite-dimensional linear problem involving <i>U</i>. </p><p>The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">equilibrium statistical mechanics</a>. An average in time along a trajectory is equivalent to an average in space computed with the <a href="/wiki/Statistical_mechanics#Canonical_ensemble" title="Statistical mechanics">Boltzmann factor exp(−β<i>H</i>)</a>. This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. <a href="/wiki/SRB_measure" class="mw-redirect" title="SRB measure">SRB measures</a> replace the Boltzmann factor and they are defined on attractors of chaotic systems. </p> <div class="mw-heading mw-heading2"><h2 id="Nonlinear_dynamical_systems_and_chaos">Nonlinear dynamical systems and chaos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=23" title="Edit section: Nonlinear dynamical systems and chaos"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></div> <p>Simple nonlinear dynamical systems, including <a href="/wiki/Piecewise_linear_function" title="Piecewise linear function">piecewise linear</a> systems, can exhibit strongly unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This unpredictable behavior has been called <i><a href="/wiki/Chaos_theory" title="Chaos theory">chaos</a></i>. <a href="/wiki/Anosov_diffeomorphism" title="Anosov diffeomorphism">Hyperbolic systems</a> are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the <a href="/wiki/Tangent_space" title="Tangent space">tangent spaces</a> perpendicular to an orbit can be decomposed into a combination of two parts: one with the points that converge towards the orbit (the <i>stable manifold</i>) and another of the points that diverge from the orbit (the <i>unstable manifold</i>). </p><p>This branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a <a href="/wiki/Steady_state" title="Steady state">steady state</a> in the long term, and if so, what are the possible <a href="/wiki/Attractor" title="Attractor">attractors</a>?" or "Does the long-term behavior of the system depend on its initial condition?" </p><p>The chaotic behavior of complex systems is not the issue. <a href="/wiki/Meteorology" title="Meteorology">Meteorology</a> has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The <a href="/wiki/Pomeau%E2%80%93Manneville_scenario" title="Pomeau–Manneville scenario">Pomeau–Manneville scenario</a> of the <a href="/wiki/Logistic_map" title="Logistic map">logistic map</a> and the <a href="/wiki/Fermi%E2%80%93Pasta%E2%80%93Ulam%E2%80%93Tsingou_problem" title="Fermi–Pasta–Ulam–Tsingou problem">Fermi–Pasta–Ulam–Tsingou problem</a> arose with just second-degree polynomials; the <a href="/wiki/Horseshoe_map" title="Horseshoe map">horseshoe map</a> is piecewise linear. </p> <div class="mw-heading mw-heading3"><h3 id="Solutions_of_finite_duration">Solutions of finite duration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=24" title="Edit section: Solutions of finite duration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> meaning here that in these solutions the system will reach the value zero at some time, called an ending time, and then stay there forever after. This can occur only when system trajectories are not uniquely determined forwards and backwards in time by the dynamics, thus solutions of finite duration imply a form of "backwards-in-time unpredictability" closely related to the forwards-in-time unpredictability of chaos. This behavior cannot happen for <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuous</a> differential equations according to the proof of the <a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard-Lindelof theorem</a>. These solutions are non-Lipschitz functions at their ending times and cannot be analytical functions on the whole real line. </p><p>As example, the equation: </p> <dl><dd><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 y'=-{\text{sgn}}(y){\sqrt {|y|}},\,\,y(0)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>sgn</mtext> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </msqrt> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>y</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'=-{\text{sgn}}(y){\sqrt {|y|}},\,\,y(0)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b45194c3307d9990aad8650196feb49814b370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.057ex; height:4.843ex;" alt="{\displaystyle y'=-{\text{sgn}}(y){\sqrt {|y|}},\,\,y(0)=1}"></span></dd></dl> <p>Admits the finite duration solution: </p> <dl><dd><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 y(t)={\frac {1}{4}}\left(1-{\frac {t}{2}}+\left|1-{\frac {t}{2}}\right|\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)={\frac {1}{4}}\left(1-{\frac {t}{2}}+\left|1-{\frac {t}{2}}\right|\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e2c5e06f31f65e46217f0b5421badfee9db021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.514ex; height:6.509ex;" alt="{\displaystyle y(t)={\frac {1}{4}}\left(1-{\frac {t}{2}}+\left|1-{\frac {t}{2}}\right|\right)^{2}}"></span></dd></dl> <p>that is zero for <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\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd3aebbfd3012049c7953fb3648456a3d84f0463" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.101ex; height:2.343ex;" alt="{\displaystyle t\geq 2}"></span> and is not Lipschitz continuous at its ending time <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=2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921a343329160933f400f5d14201398fa98313d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.747ex; height:2.176ex;" alt="{\displaystyle t=2.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon 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.citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFStrogatz2001" class="citation book cs1">Strogatz, S. 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Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-34187-5" title="Special:BookSources/978-0-521-34187-5"><bdi>978-0-521-34187-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Modern+Theory+of+Dynamical+Systems&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1995&rft.isbn=978-0-521-34187-5&rft.aulast=Katok&rft.aufirst=A.&rft.au=Hasselblatt%2C+B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontomo0000kato&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.nature.com/subjects/dynamical-systems">"Nature"</a>. 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"Dynamics of Self-Adjusting Systems With Noise". <i>Chaos: An Interdisciplinary Journal of Nonlinear Science</i>. <b>15</b> (3): 033902. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005Chaos..15c3902M">2005Chaos..15c3902M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1953147">10.1063/1.1953147</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16252993">16252993</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chaos%3A+An+Interdisciplinary+Journal+of+Nonlinear+Science&rft.atitle=Dynamics+of+Self-Adjusting+Systems+With+Noise&rft.volume=15&rft.issue=3&rft.pages=033902&rft.date=2005&rft_id=info%3Apmid%2F16252993&rft_id=info%3Adoi%2F10.1063%2F1.1953147&rft_id=info%3Abibcode%2F2005Chaos..15c3902M&rft.aulast=Melby&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGintautas2008" class="citation journal cs1">Gintautas, V.; et al. (2008). "Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics". <i>J. Stat. Phys</i>. <b>130</b> (3): 617. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0705.0311">0705.0311</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008JSP...130..617G">2008JSP...130..617G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10955-007-9444-4">10.1007/s10955-007-9444-4</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8677631">8677631</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Stat.+Phys.&rft.atitle=Resonant+forcing+of+select+degrees+of+freedom+of+multidimensional+chaotic+map+dynamics&rft.volume=130&rft.issue=3&rft.pages=617&rft.date=2008&rft_id=info%3Aarxiv%2F0705.0311&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8677631%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10955-007-9444-4&rft_id=info%3Abibcode%2F2008JSP...130..617G&rft.aulast=Gintautas&rft.aufirst=V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacksonRadunskaya2015" class="citation book cs1">Jackson, T.; Radunskaya, A. (2015). <i>Applications of Dynamical Systems in Biology and Medicine</i>. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+Dynamical+Systems+in+Biology+and+Medicine&rft.pub=Springer&rft.date=2015&rft.aulast=Jackson&rft.aufirst=T.&rft.au=Radunskaya%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKreyszig2011" class="citation book cs1">Kreyszig, Erwin (2011). <i>Advanced Engineering Mathematics</i>. Hoboken: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-64613-7" title="Special:BookSources/978-0-470-64613-7"><bdi>978-0-470-64613-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Engineering+Mathematics&rft.place=Hoboken&rft.pub=Wiley&rft.date=2011&rft.isbn=978-0-470-64613-7&rft.aulast=Kreyszig&rft.aufirst=Erwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGandolfo2009" class="citation book cs1"><a href="/wiki/Giancarlo_Gandolfo" title="Giancarlo Gandolfo">Gandolfo, Giancarlo</a> (2009) [1971]. <i>Economic Dynamics: Methods and Models</i> (Fourth ed.). Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-13503-3" title="Special:BookSources/978-3-642-13503-3"><bdi>978-3-642-13503-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Economic+Dynamics%3A+Methods+and+Models&rft.place=Berlin&rft.edition=Fourth&rft.pub=Springer&rft.date=2009&rft.isbn=978-3-642-13503-3&rft.aulast=Gandolfo&rft.aufirst=Giancarlo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Holmes, Philip. "Poincaré, celestial mechanics, dynamical-systems theory and "chaos"." <i>Physics Reports</i> 193.3 (1990): 137–163.</span> </li> <li id="cite_note-Rega-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rega_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRega2019" class="citation book cs1">Rega, Giuseppe (2019). "Tribute to Ali H. Nayfeh (1933–2017)". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pAilDwAAQBAJ&pg=PA1"><i>IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems</i></a>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. pp. 1–2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783030236922" title="Special:BookSources/9783030236922"><bdi>9783030236922</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tribute+to+Ali+H.+Nayfeh+%281933%E2%80%932017%29&rft.btitle=IUTAM+Symposium+on+Exploiting+Nonlinear+Dynamics+for+Engineering+Systems&rft.pages=1-2&rft.pub=Springer&rft.date=2019&rft.isbn=9783030236922&rft.aulast=Rega&rft.aufirst=Giuseppe&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpAilDwAAQBAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></span> </li> <li id="cite_note-fi-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-fi_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.fi.edu/laureates/ali-hasan-nayfeh">"Ali Hasan Nayfeh"</a>. <i><a href="/wiki/Franklin_Institute_Awards" title="Franklin Institute Awards">Franklin Institute Awards</a></i>. <a href="/wiki/The_Franklin_Institute" class="mw-redirect" title="The Franklin Institute">The Franklin Institute</a>. 4 February 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">25 August</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Franklin+Institute+Awards&rft.atitle=Ali+Hasan+Nayfeh&rft.date=2014-02-04&rft_id=https%3A%2F%2Fwww.fi.edu%2Flaureates%2Fali-hasan-nayfeh&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Giunti M. and Mazzola C. (2012), "<a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/272943599_Dynamical_Systems_on_Monoids_Toward_a_General_Theory_of_Deterministic_Systems_and_Motion">Dynamical systems on monoids: Toward a general theory of deterministic systems and motion</a>". In Minati G., Abram M., Pessa E. (eds.), <i>Methods, models, simulations and approaches towards a general theory of change</i>, pp. 173–185, Singapore: World Scientific. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4383-32-5" title="Special:BookSources/978-981-4383-32-5">978-981-4383-32-5</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Mazzola C. and Giunti M. (2012), "<a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/281244041_Reversible_dynamics_and_the_directionality_of_time">Reversible dynamics and the directionality of time</a>". In Minati G., Abram M., Pessa E. (eds.), <i>Methods, models, simulations and approaches towards a general theory of change</i>, pp. 161–171, Singapore: World Scientific. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4383-32-5" title="Special:BookSources/978-981-4383-32-5">978-981-4383-32-5</a>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalor2010" class="citation book cs1">Galor, Oded (2010). <i>Discrete Dynamical Systems</i>. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Dynamical+Systems&rft.pub=Springer&rft.date=2010&rft.aulast=Galor&rft.aufirst=Oded&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVardia_T._Haimo1985" class="citation book cs1">Vardia T. Haimo (1985). <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/4048613">"Finite Time Differential Equations"</a>. <i>1985 24th IEEE Conference on Decision and Control</i>. pp. 1729–1733. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FCDC.1985.268832">10.1109/CDC.1985.268832</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:45426376">45426376</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Finite+Time+Differential+Equations&rft.btitle=1985+24th+IEEE+Conference+on+Decision+and+Control&rft.pages=1729-1733&rft.date=1985&rft_id=info%3Adoi%2F10.1109%2FCDC.1985.268832&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A45426376%23id-name%3DS2CID&rft.au=Vardia+T.+Haimo&rft_id=https%3A%2F%2Fieeexplore.ieee.org%2Fdocument%2F4048613&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArnold2006" class="citation book cs1"><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Arnold, Vladimir I.</a> (2006). "Fundamental concepts". <i>Ordinary Differential Equations</i>. Berlin: Springer Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-34563-9" title="Special:BookSources/3-540-34563-9"><bdi>3-540-34563-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fundamental+concepts&rft.btitle=Ordinary+Differential+Equations&rft.place=Berlin&rft.pub=Springer+Verlag&rft.date=2006&rft.isbn=3-540-34563-9&rft.aulast=Arnold&rft.aufirst=Vladimir+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChueshov" class="citation book cs1">Chueshov, I. D. <i>Introduction to the Theory of Infinite-Dimensional Dissipative Systems</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+of+Infinite-Dimensional+Dissipative+Systems&rft.aulast=Chueshov&rft.aufirst=I.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span> online version of first edition on the EMIS site <a rel="nofollow" class="external autonumber" href="http://www.emis.de/monographs/Chueshov/">[1]</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTemam1997" class="citation book cs1">Temam, Roger (1997) [1988]. <i>Infinite-Dimensional Dynamical Systems in Mechanics and Physics</i>. Springer Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Infinite-Dimensional+Dynamical+Systems+in+Mechanics+and+Physics&rft.pub=Springer+Verlag&rft.date=1997&rft.aulast=Temam&rft.aufirst=Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=27" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <p>Works providing a broad coverage: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRalph_Abraham_and_Jerrold_E._Marsden1978" class="citation book cs1"><a href="/wiki/Ralph_Abraham_(mathematician)" title="Ralph Abraham (mathematician)">Ralph Abraham</a> and <a href="/wiki/Jerrold_E._Marsden" title="Jerrold E. Marsden">Jerrold E. Marsden</a> (1978). <i>Foundations of mechanics</i>. Benjamin–Cummings. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-0102-1" title="Special:BookSources/978-0-8053-0102-1"><bdi>978-0-8053-0102-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+mechanics&rft.pub=Benjamin%E2%80%93Cummings&rft.date=1978&rft.isbn=978-0-8053-0102-1&rft.au=Ralph+Abraham+and+Jerrold+E.+Marsden&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span> (available as a reprint: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-40840-6" title="Special:BookSources/0-201-40840-6">0-201-40840-6</a>)</li> <li><i>Encyclopaedia of Mathematical Sciences</i> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/search?fq=x0:jrnl&q=n2:0938-0396">0938-0396</a>) has a sub-series on dynamical systems with reviews of current research.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristian_BonattiLorenzo_J._DíazMarcelo_Viana2005" class="citation book cs1">Christian Bonatti; Lorenzo J. Díaz; Marcelo Viana (2005). <i>Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-22066-4" title="Special:BookSources/978-3-540-22066-4"><bdi>978-3-540-22066-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics+Beyond+Uniform+Hyperbolicity%3A+A+Global+Geometric+and+Probabilistic+Perspective&rft.pub=Springer&rft.date=2005&rft.isbn=978-3-540-22066-4&rft.au=Christian+Bonatti&rft.au=Lorenzo+J.+D%C3%ADaz&rft.au=Marcelo+Viana&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen_Smale1967" class="citation journal cs1"><a href="/wiki/Stephen_Smale" title="Stephen Smale">Stephen Smale</a> (1967). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1967-11798-1">"Differentiable dynamical systems"</a>. <i>Bulletin of the American Mathematical Society</i>. <b>73</b> (6): 747–817. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1967-11798-1">10.1090/S0002-9904-1967-11798-1</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Differentiable+dynamical+systems&rft.volume=73&rft.issue=6&rft.pages=747-817&rft.date=1967&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1967-11798-1&rft.au=Stephen+Smale&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9904-1967-11798-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li></ul> <p>Introductory texts with a unique perspective: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFV._I._Arnold1982" class="citation book cs1"><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">V. I. Arnold</a> (1982). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalmeth0000arno"><i>Mathematical methods of classical mechanics</i></a></span>. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96890-2" title="Special:BookSources/978-0-387-96890-2"><bdi>978-0-387-96890-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+methods+of+classical+mechanics&rft.pub=Springer-Verlag&rft.date=1982&rft.isbn=978-0-387-96890-2&rft.au=V.+I.+Arnold&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalmeth0000arno&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacob_Palis_and_Welington_de_Melo1982" class="citation book cs1"><a href="/wiki/Jacob_Palis" title="Jacob Palis">Jacob Palis</a> and <a href="/wiki/Welington_de_Melo" title="Welington de Melo">Welington de Melo</a> (1982). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/geometrictheoryo0000pali"><i>Geometric theory of dynamical systems: an introduction</i></a></span>. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90668-3" title="Special:BookSources/978-0-387-90668-3"><bdi>978-0-387-90668-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+theory+of+dynamical+systems%3A+an+introduction&rft.pub=Springer-Verlag&rft.date=1982&rft.isbn=978-0-387-90668-3&rft.au=Jacob+Palis+and+Welington+de+Melo&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometrictheoryo0000pali&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Ruelle1989" class="citation book cs1"><a href="/wiki/David_Ruelle" title="David Ruelle">David Ruelle</a> (1989). <i>Elements of Differentiable Dynamics and Bifurcation Theory</i>. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-601710-6" title="Special:BookSources/978-0-12-601710-6"><bdi>978-0-12-601710-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Differentiable+Dynamics+and+Bifurcation+Theory&rft.pub=Academic+Press&rft.date=1989&rft.isbn=978-0-12-601710-6&rft.au=David+Ruelle&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTim_Bedford,_Michael_Keane_and_Caroline_Series,_eds.1991" class="citation book cs1">Tim Bedford, Michael Keane and Caroline Series, <i>eds.</i> (1991). <i>Ergodic theory, symbolic dynamics and hyperbolic spaces</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853390-0" title="Special:BookSources/978-0-19-853390-0"><bdi>978-0-19-853390-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ergodic+theory%2C+symbolic+dynamics+and+hyperbolic+spaces&rft.pub=Oxford+University+Press&rft.date=1991&rft.isbn=978-0-19-853390-0&rft.au=Tim+Bedford%2C+Michael+Keane+and+Caroline+Series%2C+%27%27eds.%27%27&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRalph_H._Abraham_and_Christopher_D._Shaw1992" class="citation book cs1"><a href="/wiki/Ralph_Abraham_(mathematician)" title="Ralph Abraham (mathematician)">Ralph H. Abraham</a> and <a href="/wiki/Robert_Shaw_(Physicist)#Illustrations" class="mw-redirect" title="Robert Shaw (Physicist)">Christopher D. Shaw</a> (1992). <i>Dynamics—the geometry of behavior, 2nd edition</i>. Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-56716-8" title="Special:BookSources/978-0-201-56716-8"><bdi>978-0-201-56716-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics%E2%80%94the+geometry+of+behavior%2C+2nd+edition&rft.pub=Addison-Wesley&rft.date=1992&rft.isbn=978-0-201-56716-8&rft.au=Ralph+H.+Abraham+and+Christopher+D.+Shaw&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li></ul> <p>Textbooks </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKathleen_T._Alligood,_Tim_D._Sauer_and_James_A._Yorke2000" class="citation book cs1">Kathleen T. Alligood, Tim D. Sauer and <a href="/wiki/James_A._Yorke" title="James A. Yorke">James A. Yorke</a> (2000). <i>Chaos. An introduction to dynamical systems</i>. Springer Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94677-1" title="Special:BookSources/978-0-387-94677-1"><bdi>978-0-387-94677-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chaos.+An+introduction+to+dynamical+systems&rft.pub=Springer+Verlag&rft.date=2000&rft.isbn=978-0-387-94677-1&rft.au=Kathleen+T.+Alligood%2C+Tim+D.+Sauer+and+James+A.+Yorke&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOded_Galor2011" class="citation book cs1">Oded Galor (2011). <i><span></span></i>Discrete Dynamical Systems<i><span></span></i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-07185-0" title="Special:BookSources/978-3-642-07185-0"><bdi>978-3-642-07185-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Dynamical+Systems&rft.pub=Springer&rft.date=2011&rft.isbn=978-3-642-07185-0&rft.au=Oded+Galor&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorris_W._Hirsch,_Stephen_Smale_and_Robert_L._Devaney2003" class="citation book cs1"><a href="/wiki/Morris_Hirsch" title="Morris Hirsch">Morris W. Hirsch</a>, <a href="/wiki/Stephen_Smale" title="Stephen Smale">Stephen Smale</a> and <a href="/wiki/Robert_L._Devaney" title="Robert L. Devaney">Robert L. Devaney</a> (2003). <i>Differential Equations, dynamical systems, and an introduction to chaos</i>. Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-349703-1" title="Special:BookSources/978-0-12-349703-1"><bdi>978-0-12-349703-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Equations%2C+dynamical+systems%2C+and+an+introduction+to+chaos&rft.pub=Academic+Press&rft.date=2003&rft.isbn=978-0-12-349703-1&rft.au=Morris+W.+Hirsch%2C+Stephen+Smale+and+Robert+L.+Devaney&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnatole_KatokBoris_Hasselblatt1996" class="citation book cs1">Anatole Katok; Boris Hasselblatt (1996). <i>Introduction to the modern theory of dynamical systems</i>. Cambridge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-57557-7" title="Special:BookSources/978-0-521-57557-7"><bdi>978-0-521-57557-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+modern+theory+of+dynamical+systems&rft.pub=Cambridge&rft.date=1996&rft.isbn=978-0-521-57557-7&rft.au=Anatole+Katok&rft.au=Boris+Hasselblatt&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen_Lynch2010" class="citation book cs1">Stephen Lynch (2010). <i>Dynamical Systems with Applications using Maple 2nd Ed</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4389-8" title="Special:BookSources/978-0-8176-4389-8"><bdi>978-0-8176-4389-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamical+Systems+with+Applications+using+Maple+2nd+Ed.&rft.pub=Springer&rft.date=2010&rft.isbn=978-0-8176-4389-8&rft.au=Stephen+Lynch&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen_Lynch2014" class="citation book cs1">Stephen Lynch (2014). <i>Dynamical Systems with Applications using MATLAB 2nd Edition</i>. Springer International Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3319068190" title="Special:BookSources/978-3319068190"><bdi>978-3319068190</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamical+Systems+with+Applications+using+MATLAB+2nd+Edition&rft.pub=Springer+International+Publishing&rft.date=2014&rft.isbn=978-3319068190&rft.au=Stephen+Lynch&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen_Lynch2017" class="citation book cs1">Stephen Lynch (2017). <i>Dynamical Systems with Applications using Mathematica 2nd Ed</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-61485-4" title="Special:BookSources/978-3-319-61485-4"><bdi>978-3-319-61485-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamical+Systems+with+Applications+using+Mathematica+2nd+Ed.&rft.pub=Springer&rft.date=2017&rft.isbn=978-3-319-61485-4&rft.au=Stephen+Lynch&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen_Lynch2018" class="citation book cs1">Stephen Lynch (2018). <i>Dynamical Systems with Applications using Python</i>. Springer International Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-78145-7" title="Special:BookSources/978-3-319-78145-7"><bdi>978-3-319-78145-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamical+Systems+with+Applications+using+Python&rft.pub=Springer+International+Publishing&rft.date=2018&rft.isbn=978-3-319-78145-7&rft.au=Stephen+Lynch&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_Meiss2007" class="citation book cs1">James Meiss (2007). <i>Differential Dynamical Systems</i>. SIAM. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-635-1" title="Special:BookSources/978-0-89871-635-1"><bdi>978-0-89871-635-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Dynamical+Systems&rft.pub=SIAM&rft.date=2007&rft.isbn=978-0-89871-635-1&rft.au=James+Meiss&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_D._Nolte2015" class="citation book cs1">David D. Nolte (2015). <i>Introduction to Modern Dynamics: Chaos, Networks, Space and Time</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0199657032" title="Special:BookSources/978-0199657032"><bdi>978-0199657032</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Modern+Dynamics%3A+Chaos%2C+Networks%2C+Space+and+Time&rft.pub=Oxford+University+Press&rft.date=2015&rft.isbn=978-0199657032&rft.au=David+D.+Nolte&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJulien_Clinton_Sprott2003" class="citation book cs1">Julien Clinton Sprott (2003). <i><span></span></i>Chaos and time-series analysis<i><span></span></i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850839-7" title="Special:BookSources/978-0-19-850839-7"><bdi>978-0-19-850839-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chaos+and+time-series+analysis&rft.pub=Oxford+University+Press&rft.date=2003&rft.isbn=978-0-19-850839-7&rft.au=Julien+Clinton+Sprott&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteven_H._Strogatz1994" class="citation book cs1"><a href="/wiki/Steven_Strogatz" title="Steven Strogatz">Steven H. Strogatz</a> (1994). <i>Nonlinear dynamics and chaos: with applications to physics, biology chemistry and engineering</i>. Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-54344-5" title="Special:BookSources/978-0-201-54344-5"><bdi>978-0-201-54344-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+dynamics+and+chaos%3A+with+applications+to+physics%2C+biology+chemistry+and+engineering&rft.pub=Addison+Wesley&rft.date=1994&rft.isbn=978-0-201-54344-5&rft.au=Steven+H.+Strogatz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTeschl2012" class="citation book cs1"><a href="/wiki/Gerald_Teschl" title="Gerald Teschl">Teschl, Gerald</a> (2012). <a rel="nofollow" class="external text" href="https://www.mat.univie.ac.at/~gerald/ftp/book-ode/"><i>Ordinary Differential Equations and Dynamical Systems</i></a>. <a href="/wiki/Providence,_Rhode_Island" title="Providence, Rhode Island">Providence</a>: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-8328-0" title="Special:BookSources/978-0-8218-8328-0"><bdi>978-0-8218-8328-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ordinary+Differential+Equations+and+Dynamical+Systems&rft.place=Providence&rft.pub=American+Mathematical+Society&rft.date=2012&rft.isbn=978-0-8218-8328-0&rft.aulast=Teschl&rft.aufirst=Gerald&rft_id=https%3A%2F%2Fwww.mat.univie.ac.at%2F~gerald%2Fftp%2Fbook-ode%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen_Wiggins2003" class="citation book cs1">Stephen Wiggins (2003). <i>Introduction to Applied Dynamical Systems and Chaos</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-00177-7" title="Special:BookSources/978-0-387-00177-7"><bdi>978-0-387-00177-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Applied+Dynamical+Systems+and+Chaos&rft.pub=Springer&rft.date=2003&rft.isbn=978-0-387-00177-7&rft.au=Stephen+Wiggins&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li></ul> <p>Popularizations: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlorin_Diacu_and_Philip_Holmes1996" class="citation book cs1"><a href="/wiki/Florin_Diacu" title="Florin Diacu">Florin Diacu</a> and <a href="/wiki/Philip_Holmes" title="Philip Holmes">Philip Holmes</a> (1996). <i>Celestial Encounters</i>. Princeton. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-02743-2" title="Special:BookSources/978-0-691-02743-2"><bdi>978-0-691-02743-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Celestial+Encounters&rft.pub=Princeton&rft.date=1996&rft.isbn=978-0-691-02743-2&rft.au=Florin+Diacu+and+Philip+Holmes&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_Gleick1988" class="citation book cs1"><a href="/wiki/James_Gleick" title="James Gleick">James Gleick</a> (1988). <a href="/wiki/Chaos:_Making_a_New_Science" title="Chaos: Making a New Science"><i>Chaos: Making a New Science</i></a>. Penguin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-14-009250-9" title="Special:BookSources/978-0-14-009250-9"><bdi>978-0-14-009250-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chaos%3A+Making+a+New+Science&rft.pub=Penguin&rft.date=1988&rft.isbn=978-0-14-009250-9&rft.au=James+Gleick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIvar_Ekeland1990" class="citation book cs1"><a href="/wiki/Ivar_Ekeland" title="Ivar Ekeland">Ivar Ekeland</a> (1990). <i>Mathematics and the Unexpected (Paperback)</i>. University Of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-19990-0" title="Special:BookSources/978-0-226-19990-0"><bdi>978-0-226-19990-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+the+Unexpected+%28Paperback%29&rft.pub=University+Of+Chicago+Press&rft.date=1990&rft.isbn=978-0-226-19990-0&rft.au=Ivar+Ekeland&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIan_Stewart1997" class="citation book cs1">Ian Stewart (1997). <i>Does God Play Dice? The New Mathematics of Chaos</i>. Penguin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-14-025602-4" title="Special:BookSources/978-0-14-025602-4"><bdi>978-0-14-025602-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Does+God+Play+Dice%3F+The+New+Mathematics+of+Chaos&rft.pub=Penguin&rft.date=1997&rft.isbn=978-0-14-025602-4&rft.au=Ian+Stewart&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADynamical+system" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dynamical_system&action=edit&section=28" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.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:var(--background-color-interactive-subtle,#f8f9fa);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;min-width:0}}@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><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.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/en/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/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/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">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Dynamical_systems" class="extiw" title="commons:Category:Dynamical systems">Dynamical systems</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.arxiv.org/list/math.DS/recent">Arxiv preprint server</a> has daily submissions of (non-refereed) manuscripts in dynamical systems.</li> <li><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/Encyclopedia_of_Dynamical_Systems">Encyclopedia of dynamical systems</a> A part of <a href="/wiki/Scholarpedia" title="Scholarpedia">Scholarpedia</a> — peer-reviewed and written by invited experts.</li> <li><a rel="nofollow" class="external text" href="http://www.egwald.ca/nonlineardynamics/index.php">Nonlinear Dynamics</a>. Models of bifurcation and chaos by Elmer G. Wiens</li> <li><a rel="nofollow" class="external text" href="http://amath.colorado.edu/faculty/jdm/faq-Contents.html">Sci.Nonlinear FAQ 2.0 (Sept 2003)</a> provides definitions, explanations and resources related to nonlinear science</li></ul> <dl><dt>Online books or lecture notes</dt></dl> <ul><li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.HO/0111177">Geometrical theory of dynamical systems</a>. Nils Berglund's lecture notes for a course at <a href="/wiki/ETH" class="mw-redirect" title="ETH">ETH</a> at the advanced undergraduate level.</li> <li><a rel="nofollow" class="external text" href="https://archive.org/details/dynamicalsystems00birk">Dynamical systems</a>. George D. Birkhoff's 1927 book already takes a modern approach to dynamical systems.</li> <li><a rel="nofollow" class="external text" href="http://chaosbook.org/">Chaos: classical and quantum</a>. An introduction to dynamical systems from the periodic orbit point of view.</li> <li><a rel="nofollow" class="external text" href="http://www.cs.brown.edu/research/ai/dynamics/tutorial/home.html">Learning Dynamical Systems</a>. Tutorial on learning dynamical systems.</li> <li><a rel="nofollow" class="external text" href="https://www.mat.univie.ac.at/~gerald/ftp/book-ode/">Ordinary Differential Equations and Dynamical Systems</a>. Lecture notes by <a href="/wiki/Gerald_Teschl" title="Gerald Teschl">Gerald Teschl</a></li></ul> <dl><dt>Research groups</dt></dl> <ul><li><a rel="nofollow" class="external text" href="http://www.math.rug.nl/~broer/">Dynamical Systems Group Groningen</a>, IWI, University of Groningen.</li> <li><a rel="nofollow" class="external text" href="http://www-chaos.umd.edu/">Chaos @ UMD</a>. Concentrates on the applications of dynamical systems.</li> <li><a rel="nofollow" class="external autonumber" href="http://www.math.stonybrook.edu/dynamical-systems">[2]</a>, SUNY Stony Brook. Lists of conferences, researchers, and some open problems.</li> <li><a rel="nofollow" class="external text" href="http://www.math.psu.edu/dynsys/">Center for Dynamics and Geometry</a>, Penn State.</li> <li><a rel="nofollow" class="external text" href="http://www.cds.caltech.edu/">Control and Dynamical Systems</a>, Caltech.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20061018031023/http://lanoswww.epfl.ch/">Laboratory of Nonlinear Systems</a>, Ecole Polytechnique Fédérale de Lausanne (EPFL).</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070208153906/http://www.math.uni-bremen.de/ids.html">Center for Dynamical Systems</a>, University of Bremen</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070406053155/http://www.eng.ox.ac.uk/samp/">Systems Analysis, Modelling and Prediction Group</a>, University of Oxford</li> <li><a rel="nofollow" class="external text" href="http://sd.ist.utl.pt/">Non-Linear Dynamics Group</a>, Instituto Superior Técnico, Technical University of Lisbon</li> <li><a rel="nofollow" class="external text" href="http://www.impa.br/">Dynamical Systems</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170602221933/http://www.impa.br/">Archived</a> 2017-06-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, IMPA, Instituto Nacional de Matemática Pura e Applicada.</li> <li><a rel="nofollow" class="external text" href="http://ndw.cs.cas.cz/">Nonlinear Dynamics Workgroup</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150121174532/http://ndw.cs.cas.cz/">Archived</a> 2015-01-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Institute of Computer Science, Czech Academy of Sciences.</li> <li><a rel="nofollow" class="external text" href="https://dynamicalsystems.upc.edu/">UPC Dynamical Systems Group Barcelona</a>, Polytechnical University of Catalonia.</li> <li><a rel="nofollow" class="external text" href="https://www.ccdc.ucsb.edu/">Center for Control, Dynamical Systems, and Computation</a>, University of California, Santa Barbara.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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4em"><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Core" scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Core</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Attractor" title="Attractor">Attractor</a></li> <li><a href="/wiki/Bifurcation_theory" title="Bifurcation theory">Bifurcation</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Limit_set" title="Limit set">Limit set</a></li> <li><a href="/wiki/Lyapunov_exponent" title="Lyapunov exponent">Lyapunov exponent</a></li> <li><a href="/wiki/Orbit_(dynamics)" title="Orbit (dynamics)">Orbit</a></li> <li><a href="/wiki/Periodic_point" title="Periodic point">Periodic point</a></li> <li><a href="/wiki/Phase_space" title="Phase space">Phase space</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anosov_diffeomorphism" title="Anosov diffeomorphism">Anosov diffeomorphism</a></li> <li><a href="/wiki/Arnold_tongue" title="Arnold tongue">Arnold tongue</a></li> <li><a href="/wiki/Axiom_A" title="Axiom A">axiom A dynamical system</a></li> <li><a href="/wiki/Bifurcation_diagram" title="Bifurcation diagram">Bifurcation diagram</a></li> <li><a href="/wiki/Box-counting_dimension" class="mw-redirect" title="Box-counting dimension">Box-counting dimension</a></li> <li><a href="/wiki/Correlation_dimension" title="Correlation dimension">Correlation dimension</a></li> <li><a href="/wiki/Conservative_system" title="Conservative system">Conservative system</a></li> <li><a href="/wiki/Ergodicity" title="Ergodicity">Ergodicity</a></li> <li><a href="/wiki/False_nearest_neighbors" class="mw-redirect" title="False nearest neighbors">False nearest neighbors</a></li> <li><a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a></li> <li><a href="/wiki/Invariant_measure" title="Invariant measure">Invariant measure</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a></li> <li><a href="/wiki/Measure-preserving_dynamical_system" title="Measure-preserving dynamical system">Measure-preserving dynamical system</a></li> <li><a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li> <li><a href="/wiki/Poincar%C3%A9_section" class="mw-redirect" title="Poincaré section">Poincaré section</a></li> <li><a href="/wiki/Recurrence_plot" title="Recurrence plot">Recurrence plot</a></li> <li><a href="/wiki/SRB_measure" class="mw-redirect" title="SRB measure">SRB measure</a></li> <li><a href="/wiki/Stable_manifold" title="Stable manifold">Stable manifold</a></li> <li><a href="/wiki/Topological_conjugacy" title="Topological conjugacy">Topological conjugacy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Theorems</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ergodic_theory#Ergodic_theorems" title="Ergodic theory">Ergodic theorem</a></li> <li><a href="/wiki/Liouville%27s_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville's theorem</a></li> <li><a href="/wiki/Krylov%E2%80%93Bogolyubov_theorem" title="Krylov–Bogolyubov theorem">Krylov–Bogolyubov theorem</a></li> <li><a href="/wiki/Poincar%C3%A9%E2%80%93Bendixson_theorem" title="Poincaré–Bendixson theorem">Poincaré–Bendixson theorem</a></li> <li><a href="/wiki/Poincar%C3%A9_recurrence_theorem" title="Poincaré recurrence theorem">Poincaré recurrence theorem</a></li> <li><a href="/wiki/Stable_manifold_theorem" title="Stable manifold theorem">Stable manifold theorem</a></li> <li><a href="/wiki/Takens%27s_theorem" title="Takens's theorem">Takens's theorem</a></li></ul> </div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:C%C3%B4ne_textileII.png" class="mw-file-description" title="Conus textile shell"><img alt="Conus textile shell" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/C%C3%B4ne_textileII.png/100px-C%C3%B4ne_textileII.png" decoding="async" width="100" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/C%C3%B4ne_textileII.png/150px-C%C3%B4ne_textileII.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/C%C3%B4ne_textileII.png/200px-C%C3%B4ne_textileII.png 2x" data-file-width="2344" data-file-height="4211" /></a></span> <p><br /> </p> <span typeof="mw:File"><a href="/wiki/File:Circle_map_poincare_recurrence.jpeg" class="mw-file-description" title="Circle map with black Arnold tongues"><img alt="Circle map with black Arnold tongues" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Circle_map_poincare_recurrence.jpeg/100px-Circle_map_poincare_recurrence.jpeg" decoding="async" width="100" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Circle_map_poincare_recurrence.jpeg/150px-Circle_map_poincare_recurrence.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Circle_map_poincare_recurrence.jpeg/200px-Circle_map_poincare_recurrence.jpeg 2x" data-file-width="450" data-file-height="900" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theoretical<br />branches</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bifurcation_theory" title="Bifurcation theory">Bifurcation theory</a></li> <li><a href="/wiki/Control_of_chaos" title="Control of chaos">Control of chaos</a></li> <li><a class="mw-selflink selflink">Dynamical system</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Stability_theory" title="Stability theory">Stability theory</a></li> <li><a href="/wiki/Synchronization_of_chaos" title="Synchronization of chaos">Synchronization of chaos</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Chaotic<br />maps (<a href="/wiki/List_of_chaotic_maps" title="List of chaotic maps">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Discrete</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arnold%27s_cat_map" title="Arnold's cat map">Arnold's cat map</a></li> <li><a href="/wiki/Baker%27s_map" title="Baker's map">Baker's map</a></li> <li><a href="/wiki/Complex_quadratic_polynomial" title="Complex quadratic polynomial">Complex quadratic map</a></li> <li><a href="/wiki/Coupled_map_lattice" title="Coupled map lattice">Coupled map lattice</a></li> <li><a href="/wiki/Duffing_map" title="Duffing map">Duffing map</a></li> <li><a href="/wiki/Dyadic_transformation" title="Dyadic transformation">Dyadic transformation</a></li> <li><a href="/wiki/Dynamical_billiards" title="Dynamical billiards">Dynamical billiards</a> <ul><li><a href="/wiki/Outer_billiards" title="Outer billiards">outer</a></li></ul></li> <li><a href="/wiki/Exponential_map_(discrete_dynamical_systems)" title="Exponential map (discrete dynamical systems)">Exponential map</a></li> <li><a href="/wiki/Gauss_iterated_map" title="Gauss iterated map">Gauss map</a></li> <li><a href="/wiki/Gingerbreadman_map" title="Gingerbreadman map">Gingerbreadman map</a></li> <li><a href="/wiki/H%C3%A9non_map" title="Hénon map">Hénon map</a></li> <li><a href="/wiki/Horseshoe_map" title="Horseshoe map">Horseshoe map</a></li> <li><a href="/wiki/Ikeda_map" title="Ikeda map">Ikeda map</a></li> <li><a href="/wiki/Interval_exchange_transformation" title="Interval exchange transformation">Interval exchange map</a></li> <li><a href="/wiki/Irrational_rotation" title="Irrational rotation">Irrational rotation</a></li> <li><a href="/wiki/Kaplan%E2%80%93Yorke_map" title="Kaplan–Yorke map">Kaplan–Yorke map</a></li> <li><a href="/wiki/Langton%27s_ant" title="Langton's ant">Langton's ant</a></li> <li><a href="/wiki/Logistic_map" title="Logistic map">Logistic map</a></li> <li><a href="/wiki/Standard_map" title="Standard map">Standard map</a></li> <li><a href="/wiki/Tent_map" title="Tent map">Tent map</a></li> <li><a href="/wiki/Tinkerbell_map" title="Tinkerbell map">Tinkerbell map</a></li> <li><a href="/wiki/Zaslavskii_map" title="Zaslavskii map">Zaslavskii map</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Continuous</div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Double_scroll_attractor" class="mw-redirect" title="Double scroll attractor">Double scroll attractor</a></li> <li><a href="/wiki/Duffing_equation" title="Duffing equation">Duffing equation</a></li> <li><a href="/wiki/Lorenz_system" title="Lorenz system">Lorenz system</a></li> <li><a href="/wiki/Lotka%E2%80%93Volterra_equations" title="Lotka–Volterra equations">Lotka–Volterra equations</a></li> <li><a href="/wiki/Mackey%E2%80%93Glass_equations" title="Mackey–Glass equations">Mackey–Glass equations</a></li> <li><a href="/wiki/Rabinovich%E2%80%93Fabrikant_equations" title="Rabinovich–Fabrikant equations">Rabinovich–Fabrikant equations</a></li> <li><a href="/wiki/R%C3%B6ssler_attractor" title="Rössler attractor">Rössler attractor</a></li> <li><a href="/wiki/Three-body_problem" title="Three-body problem">Three-body problem</a></li> <li><a href="/wiki/Van_der_Pol_oscillator" title="Van der Pol oscillator">Van der Pol oscillator</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical<br />systems</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chua%27s_circuit" title="Chua's circuit">Chua's circuit</a></li> <li><a href="/wiki/Rayleigh%E2%80%93B%C3%A9nard_convection" title="Rayleigh–Bénard convection">Convection</a></li> <li><a href="/wiki/Double_pendulum" title="Double pendulum">Double pendulum</a></li> <li><a href="/wiki/Elastic_pendulum" title="Elastic pendulum">Elastic pendulum</a></li> <li><a href="/wiki/Fermi%E2%80%93Pasta%E2%80%93Ulam%E2%80%93Tsingou_problem" title="Fermi–Pasta–Ulam–Tsingou problem">FPUT problem</a></li> <li><a href="/wiki/H%C3%A9non%E2%80%93Heiles_system" title="Hénon–Heiles system">Hénon–Heiles system</a></li> <li><a href="/wiki/Kicked_rotator" title="Kicked rotator">Kicked rotator</a></li> <li><a href="/wiki/Multiscroll_attractor" title="Multiscroll attractor">Multiscroll attractor</a></li> <li><a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a></li> <li><a href="/wiki/Swinging_Atwood%27s_machine" title="Swinging Atwood's machine">Swinging Atwood's machine</a></li> <li><a href="/wiki/Tilt-A-Whirl" title="Tilt-A-Whirl">Tilt-A-Whirl</a></li> <li><a href="/wiki/Weather_forecasting" title="Weather forecasting">Weather</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Chaos<br />theorists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Michael_Berry_(physicist)" title="Michael Berry (physicist)">Michael Berry</a></li> <li><a href="/wiki/Rufus_Bowen" title="Rufus Bowen">Rufus Bowen</a></li> <li><a href="/wiki/Mary_Cartwright" title="Mary Cartwright">Mary Cartwright</a></li> <li><a href="/wiki/Chen_Guanrong" title="Chen Guanrong">Chen Guanrong</a></li> <li><a href="/wiki/Leon_O._Chua" title="Leon O. Chua">Leon O. Chua</a></li> <li><a href="/wiki/Mitchell_Feigenbaum" title="Mitchell Feigenbaum">Mitchell Feigenbaum</a></li> <li><a href="/wiki/Peter_Grassberger" title="Peter Grassberger">Peter Grassberger</a></li> <li><a href="/wiki/Celso_Grebogi" title="Celso Grebogi">Celso Grebogi</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Martin Gutzwiller</a></li> <li><a href="/wiki/Brosl_Hasslacher" title="Brosl Hasslacher">Brosl Hasslacher</a></li> <li><a href="/wiki/Michel_H%C3%A9non" title="Michel Hénon">Michel Hénon</a></li> <li><a href="/wiki/Svetlana_Jitomirskaya" title="Svetlana Jitomirskaya">Svetlana Jitomirskaya</a></li> <li><a href="/wiki/Bryna_Kra" title="Bryna Kra">Bryna Kra</a></li> <li><a href="/wiki/Edward_Norton_Lorenz" title="Edward Norton Lorenz">Edward Norton Lorenz</a></li> <li><a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a></li> <li><a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoît Mandelbrot</a></li> <li><a href="/wiki/Hee_Oh" title="Hee Oh">Hee Oh</a></li> <li><a href="/wiki/Edward_Ott" title="Edward Ott">Edward Ott</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></li> <li><a href="/wiki/Itamar_Procaccia" title="Itamar Procaccia">Itamar Procaccia</a></li> <li><a href="/wiki/Mary_Rees" title="Mary Rees">Mary Rees</a></li> <li><a href="/wiki/Otto_R%C3%B6ssler" title="Otto Rössler">Otto Rössler</a></li> <li><a href="/wiki/David_Ruelle" title="David Ruelle">David Ruelle</a></li> <li><a href="/wiki/Caroline_Series" title="Caroline Series">Caroline Series</a></li> <li><a href="/wiki/Yakov_Sinai" title="Yakov Sinai">Yakov Sinai</a></li> <li><a href="/wiki/Oleksandr_Mykolayovych_Sharkovsky" class="mw-redirect" title="Oleksandr Mykolayovych Sharkovsky">Oleksandr Mykolayovych Sharkovsky</a></li> <li><a href="/wiki/Nina_Snaith" title="Nina Snaith">Nina Snaith</a></li> <li><a href="/wiki/Floris_Takens" title="Floris Takens">Floris Takens</a></li> <li><a href="/wiki/Audrey_Terras" title="Audrey Terras">Audrey Terras</a></li> <li><a href="/wiki/Mary_Tsingou" title="Mary Tsingou">Mary Tsingou</a></li> <li><a href="/wiki/Marcelo_Viana" title="Marcelo Viana">Marcelo Viana</a></li> <li><a href="/wiki/Amie_Wilkinson" title="Amie Wilkinson">Amie Wilkinson</a></li> <li><a href="/wiki/James_A._Yorke" title="James A. Yorke">James A. Yorke</a></li> <li><a href="/wiki/Lai-Sang_Young" title="Lai-Sang Young">Lai-Sang Young</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />articles</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Butterfly_effect" title="Butterfly effect">Butterfly effect</a></li> <li><a href="/wiki/Complexity" title="Complexity">Complexity</a></li> <li><a href="/wiki/Edge_of_chaos" title="Edge of chaos">Edge of chaos</a></li> <li><a href="/wiki/Predictability" title="Predictability">Predictability</a></li> <li><a href="/wiki/Santa_Fe_Institute" title="Santa Fe Institute">Santa Fe Institute</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q638328#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00576625">Japan</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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