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equations</span> </div> </a> <ul id="toc-Partial_differential_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pendula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pendula"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Pendula</span> </div> </a> <ul id="toc-Pendula-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Types_of_nonlinear_dynamic_behaviors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_nonlinear_dynamic_behaviors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Types of nonlinear dynamic behaviors</span> </div> </a> <ul id="toc-Types_of_nonlinear_dynamic_behaviors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_of_nonlinear_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples_of_nonlinear_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Examples of nonlinear equations</span> </div> </a> <ul id="toc-Examples_of_nonlinear_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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Available in 37 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-37" 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">37 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_%D9%84%D8%A7%D8%AE%D8%B7%D9%8A" title="نظام لاخطي – Arabic" lang="ar" hreflang="ar" data-title="نظام لاخطي" 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_non_llinial" title="Sistema non llinial – Asturian" lang="ast" hreflang="ast" data-title="Sistema non llinial" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BB%D1%96%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B4%D1%8B%D0%BD%D0%B0%D0%BC%D1%96%D0%BA%D0%B0" title="Нелінейная дынаміка – Belarusian" lang="be" hreflang="be" data-title="Нелінейная дынаміка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_no_lineal" title="Sistema no lineal – Catalan" lang="ca" hreflang="ca" data-title="Sistema no lineal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%BC%D0%B0%D1%80_%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0" title="Линилле мар динамика – Chuvash" lang="cv" hreflang="cv" data-title="Линилле мар динамика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Neline%C3%A1rn%C3%AD_syst%C3%A9m" title="Nelineární systém – Czech" lang="cs" hreflang="cs" data-title="Nelineární systém" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Nichtlineares_System" title="Nichtlineares System – German" lang="de" hreflang="de" data-title="Nichtlineares 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/Mittelineaarne_s%C3%BCsteem" title="Mittelineaarne süsteem – Estonian" lang="et" hreflang="et" data-title="Mittelineaarne süsteem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_no_lineal" title="Sistema no lineal – Spanish" lang="es" hreflang="es" data-title="Sistema no lineal" 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_ez_lineal" title="Sistema ez lineal – Basque" lang="eu" hreflang="eu" data-title="Sistema ez lineal" 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_%D8%BA%DB%8C%D8%B1%D8%AE%D8%B7%DB%8C" title="سامانه غیرخطی – Persian" lang="fa" hreflang="fa" data-title="سامانه غیرخطی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr badge-Q70893996 mw-list-item" title=""><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_non_lin%C3%A9aire" title="Système non linéaire – French" lang="fr" hreflang="fr" data-title="Système non linéaire" 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_non_linear" title="Sistema non linear – Galician" lang="gl" hreflang="gl" data-title="Sistema non linear" 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%B9%84%EC%84%A0%ED%98%95%EA%B3%84" title="비선형계 – Korean" lang="ko" hreflang="ko" data-title="비선형계" 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%85%E0%A4%B0%E0%A5%87%E0%A4%96%E0%A5%80%E0%A4%AF_%E0%A4%A4%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="अरेखीय तंत्र – Hindi" lang="hi" hreflang="hi" data-title="अरेखीय तंत्र" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_nonlinier" title="Sistem nonlinier – Indonesian" lang="id" hreflang="id" data-title="Sistem nonlinier" 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_non_lineare" title="Sistema non lineare – Italian" lang="it" hreflang="it" data-title="Sistema non lineare" 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%9C%D7%90_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="מערכת לא ליניארית – Hebrew" lang="he" hreflang="he" data-title="מערכת לא ליניארית" 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/Netiesin%C4%97_sistema" title="Netiesinė sistema – Lithuanian" lang="lt" hreflang="lt" data-title="Netiesinė sistema" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%B0%E0%B5%87%E0%B4%96%E0%B5%80%E0%B4%AF%E0%B4%97%E0%B4%A4%E0%B4%BF%E0%B4%95%E0%B4%82" title="അരേഖീയഗതികം – Malayalam" lang="ml" hreflang="ml" data-title="അരേഖീയഗതികം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Niet-lineair_systeem" title="Niet-lineair systeem – Dutch" lang="nl" hreflang="nl" data-title="Niet-lineair 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/%E9%9D%9E%E7%B7%9A%E5%BD%A2%E3%82%B7%E3%82%B9%E3%83%86%E3%83%A0%E8%AB%96" title="非線形システム論 – Japanese" lang="ja" hreflang="ja" data-title="非線形システム論" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Uline%C3%A6rt_system" title="Ulineært system – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ulineært system" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ikkje-line%C3%A6rt_system" title="Ikkje-lineært system – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ikkje-lineært system" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Drgania_nieliniowe" title="Drgania nieliniowe – Polish" lang="pl" hreflang="pl" data-title="Drgania nieliniowe" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_din%C3%A2mico_n%C3%A3o_linear" title="Sistema dinâmico não linear – Portuguese" lang="pt" hreflang="pt" data-title="Sistema dinâmico não linear" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0" title="Нелинейная динамика – Russian" lang="ru" hreflang="ru" data-title="Нелинейная динамика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Nonlinear_system" title="Nonlinear system – Simple English" lang="en-simple" hreflang="en-simple" data-title="Nonlinear 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-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Nelinearni_sistem" title="Nelinearni sistem – Serbian" lang="sr" hreflang="sr" data-title="Nelinearni 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/Nelinearni_sistem" title="Nelinearni sistem – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Nelinearni 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 href="https://fi.wikipedia.org/wiki/Ep%C3%A4lineaarinen_j%C3%A4rjestelm%C3%A4" title="Epälineaarinen järjestelmä – Finnish" lang="fi" hreflang="fi" data-title="Epälineaarinen järjestelmä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Icke-linj%C3%A4rt_system" title="Icke-linjärt system – Swedish" lang="sv" hreflang="sv" data-title="Icke-linjärt system" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BB%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Нелінійна система – Ukrainian" lang="uk" hreflang="uk" data-title="Нелінійна 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_phi_tuy%E1%BA%BFn" title="Hệ thống phi tuyến – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ thống phi tuyến" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%9D%9E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F" title="非线性系统 – Wu" lang="wuu" hreflang="wuu" data-title="非线性系统" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9D%9E%E7%B7%9A%E6%80%A7%E7%B3%BB%E7%B5%B1" title="非線性系統 – Cantonese" lang="yue" 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">System where changes of output are not proportional to changes of input</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"><span>This article is about "nonlinearity" in mathematics, physics and other sciences. For video and film editing, see <a href="/wiki/Non-linear_editing_system" class="mw-redirect" title="Non-linear editing system">Non-linear editing system</a>. For other uses, see <a href="/wiki/Nonlinearity_(disambiguation)" class="mw-disambig" title="Nonlinearity (disambiguation)">Nonlinearity (disambiguation)</a>.</span> <span>"Nonlinear dynamics" redirects here. 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title="Homeostasis">Homeostasis</a> <br /> <a href="/wiki/Information_theory" title="Information theory">Information theory</a><br /> <a href="/wiki/Operationalization" title="Operationalization">Operationalization</a><br /> <a href="/wiki/Second-order_cybernetics" title="Second-order cybernetics">Second-order cybernetics</a><br /> <a href="/wiki/Self-reference" title="Self-reference">Self-reference</a><br /> <a href="/wiki/System_dynamics" title="System dynamics">System dynamics</a><br /> <a href="/wiki/Systems_science" title="Systems science">Systems science</a><br /> <a href="/wiki/Systems_thinking" title="Systems thinking">Systems thinking</a><br /> <a href="/wiki/Sensemaking" title="Sensemaking">Sensemaking</a><br /> <a href="/wiki/Variety_(cybernetics)" title="Variety (cybernetics)">Variety</a><br /> </p> <a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a><br /></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a class="mw-selflink selflink">Nonlinear dynamics</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="/wiki/Time_series" title="Time series">Time series analysis</a><br /> <p><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a><br /> <a href="/wiki/Phase_space" title="Phase space">Phase space</a><br /> <a href="/wiki/Attractor" title="Attractor">Attractors</a><br /> <a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a><br /> <a href="/wiki/Chaos_theory" title="Chaos theory">Chaos</a><br /> <a href="/wiki/Multistability" title="Multistability">Multistability</a><br /> <a href="/wiki/Bifurcation_theory" title="Bifurcation theory">Bifurcation</a><br /> </p> <a href="/wiki/Coupled_map_lattice" title="Coupled map lattice">Coupled map lattices</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Game_theory" title="Game theory">Game theory</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a><br /> <p><a href="/wiki/Rational_choice_theory" class="mw-redirect" title="Rational choice theory">Rational choice theory</a><br /> <a href="/wiki/Bounded_rationality" title="Bounded rationality">Bounded rationality</a><br /> </p> <a href="/wiki/Evolutionary_game_theory" title="Evolutionary game theory">Evolutionary game theory</a><br /></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Complex_systems" title="Template:Complex systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Complex_systems" title="Template talk:Complex systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Complex_systems" title="Special:EditPage/Template:Complex systems"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Science" title="Science">science</a>, a <b>nonlinear system</b> (or a <b>non-linear system</b>) is a <a href="/wiki/System" title="System">system</a> in which the change of the output is not <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to the change of the input.<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> Nonlinear problems are of interest to <a href="/wiki/Engineer" title="Engineer">engineers</a>, <a href="/wiki/Biologist" title="Biologist">biologists</a>,<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><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/Physicist" title="Physicist">physicists</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><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/Mathematician" title="Mathematician">mathematicians</a>, and many other <a href="/wiki/Scientist" title="Scientist">scientists</a> since most systems are inherently nonlinear in nature.<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> Nonlinear <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler <a href="/wiki/Linear_system" title="Linear system">linear systems</a>. </p><p>Typically, the behavior of a nonlinear system is described in mathematics by a <b>nonlinear system of equations</b>, which is a set of simultaneous <a href="/wiki/Equation" title="Equation">equations</a> in which the unknowns (or the unknown functions in the case of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>) appear as variables of a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> of degree higher than one or in the argument of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of the unknown <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> or <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is <i>linear</i> if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it. </p><p>As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (<a href="/wiki/Linearization" title="Linearization">linearization</a>). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as <a href="/wiki/Soliton" title="Soliton">solitons</a>, <a href="/wiki/Chaos_theory" title="Chaos theory">chaos</a>,<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> and <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularities</a> are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble <a href="/wiki/Randomness" title="Randomness">random</a> behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. </p><p>Some authors use the term <b>nonlinear science</b> for the study of nonlinear systems. This term is disputed by others: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.</p><div class="templatequotecite">— <cite><a href="/wiki/Stanis%C5%82aw_Ulam" title="Stanisław Ulam">Stanisław Ulam</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></cite></div></blockquote> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nonlinear_system&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <a href="/wiki/Linear_map" title="Linear map">linear map</a> (or <i>linear function</i>) <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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> is one which satisfies both of the following properties: </p> <ul><li>Additivity or <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</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 \textstyle f(x+y)=f(x)+f(y);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle f(x+y)=f(x)+f(y);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b409a92a036eeac3db7977fadfcbdff2b1c6bd8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.66ex; height:2.843ex;" alt="{\displaystyle \textstyle f(x+y)=f(x)+f(y);}"></span></li> <li>Homogeneity: <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 \textstyle f(\alpha x)=\alpha f(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle f(\alpha x)=\alpha f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce95bcbd2e18d9396937be05e9e71d2208123a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.556ex; height:2.843ex;" alt="{\displaystyle \textstyle f(\alpha x)=\alpha f(x).}"></span></li></ul> <p>Additivity implies homogeneity for any <a href="/wiki/Rational_number" title="Rational number">rational</a> <i>α</i>, and, for <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a>, for any <a href="/wiki/Real_number" title="Real number">real</a> <i>α</i>. For a <a href="/wiki/Complex_number" title="Complex number">complex</a> <i>α</i>, homogeneity does not follow from additivity. For example, an <a href="/wiki/Antilinear_map" title="Antilinear map">antilinear map</a> is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle </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 f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>x</mi> <mo>+</mo> <mi>β</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>β</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d42f621dff08fef70480e09bb21597e83ab5d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.652ex; height:2.843ex;" alt="{\displaystyle f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)}"></span></dd></dl> <p>An equation written as </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 f(x)=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a25488b81f2af9fd0a2ed4d4530c8c84526b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.282ex; height:2.843ex;" alt="{\displaystyle f(x)=C}"></span></dd></dl> <p>is called <b>linear</b> if <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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> is a linear map (as defined above) and <b>nonlinear</b> otherwise. The equation is called <i>homogeneous</i> if <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 C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f750a7094a396d89a81974cdf35783db2bb287b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C=0}"></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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> is a <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous function</a>. </p><p>The definition <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 f(x)=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a25488b81f2af9fd0a2ed4d4530c8c84526b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.282ex; height:2.843ex;" alt="{\displaystyle f(x)=C}"></span> is very general in 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> can be any sensible mathematical object (number, vector, function, etc.), and the function <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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> can literally be any <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</a>, including integration or differentiation with associated constraints (such as <a href="/wiki/Boundary_values" class="mw-redirect" title="Boundary values">boundary values</a>). If <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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> contains <a href="/wiki/Derivative" title="Derivative">differentiation</a> with respect to <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, the result will be a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Nonlinear_systems_of_equations">Nonlinear systems of equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nonlinear_system&action=edit&section=2" title="Edit section: Nonlinear systems of equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A nonlinear system of equations consists of a set of equations in several variables such that at least one of them is not a <a href="/wiki/Linear_equation" title="Linear equation">linear equation</a>. </p><p>For a single equation of the form <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 f(x)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234abb558dc1c499e7f776c5a3ddd9581152dfda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.325ex; height:2.843ex;" alt="{\displaystyle f(x)=0,}"></span> many methods have been designed; see <a href="/wiki/Root-finding_algorithm" title="Root-finding algorithm">Root-finding algorithm</a>. In the case where <span class="texhtml mvar" style="font-style:italic;">f</span> is a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a>, one has a <i><a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equation</a></i> such as <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^{2}+x-1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+x-1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6465b24065d972602818f67629035203a0af849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.465ex; height:2.843ex;" alt="{\displaystyle x^{2}+x-1=0.}"></span> The general root-finding algorithms apply to polynomial roots, but, generally they do not find all the roots, and when they fail to find a root, this does not imply that there is no roots. Specific methods for polynomials allow finding all roots or the <a href="/wiki/Real_number" title="Real number">real</a> roots; see <a href="/wiki/Real-root_isolation" title="Real-root isolation">real-root isolation</a>. </p><p>Solving <a href="/wiki/Systems_of_polynomial_equations" class="mw-redirect" title="Systems of polynomial equations">systems of polynomial equations</a>, that is finding the common zeros of a set of several polynomials in several variables is a difficult problem for which elaborated algorithms have been designed, such as <a href="/wiki/Gr%C3%B6bner_base" class="mw-redirect" title="Gröbner base">Gröbner base</a> algorithms.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>For the general case of system of equations formed by equating to zero several <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable functions</a>, the main method is <a href="/wiki/Newton%27s_method#Systems_of_equations" title="Newton's method">Newton's method</a> and its variants. Generally they may provide a solution, but do not provide any information on the number of solutions. </p> <div class="mw-heading mw-heading2"><h2 id="Nonlinear_recurrence_relations">Nonlinear recurrence relations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nonlinear_system&action=edit&section=3" title="Edit section: Nonlinear recurrence relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A nonlinear <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a> defines successive terms of a <a href="/wiki/Sequence" title="Sequence">sequence</a> as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the <a href="/wiki/Logistic_map" title="Logistic map">logistic map</a> and the relations that define the various <a href="/wiki/Hofstadter_sequence" title="Hofstadter sequence">Hofstadter sequences</a>. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related <a href="/wiki/Nonlinear_system_identification" title="Nonlinear system identification">nonlinear system identification</a> and analysis procedures.<sup id="cite_ref-SAB1_12-0" class="reference"><a href="#cite_note-SAB1-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains. </p> <div class="mw-heading mw-heading2"><h2 id="Nonlinear_differential_equations">Nonlinear differential equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nonlinear_system&action=edit&section=4" title="Edit section: Nonlinear differential equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Simultaneous_equations" class="mw-redirect" title="Simultaneous equations">system</a> of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> is said to be nonlinear if it is not a <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a>. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a> in fluid dynamics and the <a href="/wiki/Lotka%E2%80%93Volterra_equations" title="Lotka–Volterra equations">Lotka–Volterra equations</a> in biology. </p><p>One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> solutions can be used to construct general solutions through the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</a>. A good example of this is one-dimensional heat transport with <a href="/wiki/Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet boundary conditions</a>, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions. </p> <div class="mw-heading mw-heading3"><h3 id="Ordinary_differential_equations">Ordinary differential equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nonlinear_system&action=edit&section=5" title="Edit section: Ordinary differential equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>First order <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> are often exactly solvable by <a href="/wiki/Separation_of_variables" title="Separation of variables">separation of variables</a>, especially for autonomous equations. For example, the nonlinear 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 {\frac {du}{dx}}=-u^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {du}{dx}}=-u^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6899a5ebd039f236b442d307af38e91c45609d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.672ex; height:5.509ex;" alt="{\displaystyle {\frac {du}{dx}}=-u^{2}}"></span></dd></dl> <p>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 u={\frac {1}{x+C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mi>C</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {1}{x+C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18fa6bc3048f6c78730036605f24912827e0b9b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.201ex; height:5.509ex;" alt="{\displaystyle u={\frac {1}{x+C}}}"></span> as a general solution (and also the special solution <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=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3066ba3977d1575e96f26d821caf3576bd0af4ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle u=0,}"></span> corresponding to the limit of the general solution when <i>C</i> tends to infinity). The equation is nonlinear because it may be written as </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 {\frac {du}{dx}}+u^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {du}{dx}}+u^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d01cb1daffa24c4525aca9919e822898618e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.867ex; height:5.509ex;" alt="{\displaystyle {\frac {du}{dx}}+u^{2}=0}"></span></dd></dl> <p>and the left-hand side of the equation is not a linear function of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> and its derivatives. Note that if the <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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291091343cb7c98ba83d15fdba9fc794a940829f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle u^{2}}"></span> term were replaced 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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span>, the problem would be linear (the <a href="/wiki/Exponential_decay" title="Exponential decay">exponential decay</a> problem). </p><p>Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form</a> solutions, though implicit solutions and solutions involving <a href="/wiki/Nonelementary_integral" title="Nonelementary integral">nonelementary integrals</a> are encountered. </p><p>Common methods for the qualitative analysis of nonlinear ordinary differential equations include: </p> <ul><li>Examination of any <a href="/wiki/Conserved_quantities" class="mw-redirect" title="Conserved quantities">conserved quantities</a>, especially in <a href="/wiki/Hamiltonian_system" title="Hamiltonian system">Hamiltonian systems</a></li> <li>Examination of dissipative quantities (see <a href="/wiki/Lyapunov_function" title="Lyapunov function">Lyapunov function</a>) analogous to conserved quantities</li> <li>Linearization via <a href="/wiki/Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a></li> <li>Change of variables into something easier to study</li> <li><a href="/wiki/Bifurcation_theory" title="Bifurcation theory">Bifurcation theory</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation</a> methods (can be applied to algebraic equations too)</li> <li>Existence of solutions of Finite-Duration,<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> which can happen under specific conditions for some non-linear ordinary differential equations.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Partial_differential_equations">Partial differential equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nonlinear_system&action=edit&section=6" title="Edit section: Partial differential equations"><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/Nonlinear_partial_differential_equation" title="Nonlinear partial differential equation">Nonlinear partial differential equation</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/List_of_nonlinear_partial_differential_equations" title="List of nonlinear partial differential equations">List of nonlinear partial differential equations</a></div> <p>The most common basic approach to studying nonlinear <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly linear). Sometimes, the equation may be transformed into one or more <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a>, as seen in <a href="/wiki/Separation_of_variables" title="Separation of variables">separation of variables</a>, which is always useful whether or not the resulting ordinary differential equation(s) is solvable. </p><p>Another common (though less mathematical) tactic, often exploited in fluid and heat mechanics, is to use <a href="/wiki/Scale_analysis_(mathematics)" title="Scale analysis (mathematics)">scale analysis</a> to simplify a general, natural equation in a certain specific <a href="/wiki/Boundary_value_problem" title="Boundary value problem">boundary value problem</a>. For example, the (very) nonlinear <a href="/wiki/Navier-Stokes_equations" class="mw-redirect" title="Navier-Stokes equations">Navier-Stokes equations</a> can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation. </p><p>Other methods include examining the <a href="/wiki/Method_of_characteristics" title="Method of characteristics">characteristics</a> and using the methods outlined above for ordinary differential equations. </p> <div class="mw-heading mw-heading3"><h3 id="Pendula">Pendula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nonlinear_system&action=edit&section=7" title="Edit section: Pendula"><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/Pendulum_(mathematics)" class="mw-redirect" title="Pendulum (mathematics)">Pendulum (mathematics)</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:PendulumLayout.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/PendulumLayout.svg/200px-PendulumLayout.svg.png" decoding="async" width="200" height="401" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/PendulumLayout.svg/300px-PendulumLayout.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/PendulumLayout.svg/400px-PendulumLayout.svg.png 2x" data-file-width="426" data-file-height="854" /></a><figcaption>Illustration of a pendulum</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:PendulumLinearizations.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/PendulumLinearizations.png/200px-PendulumLinearizations.png" decoding="async" width="200" height="320" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/PendulumLinearizations.png/300px-PendulumLinearizations.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/PendulumLinearizations.png/400px-PendulumLinearizations.png 2x" data-file-width="500" data-file-height="800" /></a><figcaption>Linearizations of a pendulum</figcaption></figure> <p>A classic, extensively studied nonlinear problem is the dynamics of a frictionless <a href="/wiki/Pendulum_(mathematics)" class="mw-redirect" title="Pendulum (mathematics)">pendulum</a> under the influence of <a href="/wiki/Gravity" title="Gravity">gravity</a>. Using <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>, it may be shown<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> that the motion of a pendulum can be described by the <a href="/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> nonlinear 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 {\frac {d^{2}\theta }{dt^{2}}}+\sin(\theta )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\sin(\theta )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/874c2e8a0d854a1d8eb2fc9279b335488564d3b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.056ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\sin(\theta )=0}"></span></dd></dl> <p>where gravity points "downwards" 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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <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 d\theta /dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\theta /dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe083af9808068de003c10018962791d930c010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.524ex; height:2.843ex;" alt="{\displaystyle d\theta /dt}"></span> as an <a href="/wiki/Integrating_factor" title="Integrating factor">integrating factor</a>, which would eventually yield </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 \int {\frac {d\theta }{\sqrt {C_{0}+2\cos(\theta )}}}=t+C_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <msqrt> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>=</mo> <mi>t</mi> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {d\theta }{\sqrt {C_{0}+2\cos(\theta )}}}=t+C_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbacaebec8939b83f97d79268a8e818bf55765cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.352ex; height:6.676ex;" alt="{\displaystyle \int {\frac {d\theta }{\sqrt {C_{0}+2\cos(\theta )}}}=t+C_{1}}"></span></dd></dl> <p>which is an implicit solution involving an <a href="/wiki/Elliptic_integral" title="Elliptic integral">elliptic integral</a>. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the <a href="/wiki/Nonelementary_integral" title="Nonelementary integral">nonelementary integral</a> (nonelementary unless <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 C_{0}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{0}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7e6968c898cf6bc6c1ac6aa3df45a4a0bb71b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.977ex; height:2.509ex;" alt="{\displaystyle C_{0}=2}"></span>). </p><p>Another way to approach the problem is to linearize any nonlinearity (the sine function term in this case) at the various points of interest through <a href="/wiki/Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansions</a>. For example, the linearization at <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 \theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7bc6e34b53e0e8a8815159c356b1acccf7ea24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta =0}"></span>, called the small angle approximation, 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 {\frac {d^{2}\theta }{dt^{2}}}+\theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\theta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce90f07eb2959affb81e22076f8e9224a735108f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.391ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\theta =0}"></span></dd></dl> <p>since <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 \sin(\theta )\approx \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta )\approx \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09cf7dbfccf4569a47c4eeafcfb3f7075d0c4dac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.944ex; height:2.843ex;" alt="{\displaystyle \sin(\theta )\approx \theta }"></span> 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 \theta \approx 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≈</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \approx 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45fa623dea25165982c15695fcc75fe61d9bd404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta \approx 0}"></span>. This is a <a href="/wiki/Simple_harmonic_oscillator" class="mw-redirect" title="Simple harmonic oscillator">simple harmonic oscillator</a> corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <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 \theta =\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4db588619489e27efb50a1d0d5aa016c49ce15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.521ex; height:2.176ex;" alt="{\displaystyle \theta =\pi }"></span>, corresponding to the pendulum being straight up: </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 {\frac {d^{2}\theta }{dt^{2}}}+\pi -\theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\pi -\theta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b0ec2ee1b77e7484cdef080a94a26a0f66807d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.563ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+\pi -\theta =0}"></span></dd></dl> <p>since <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 \sin(\theta )\approx \pi -\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta )\approx \pi -\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d28b5c7c5afa97b37d313a2397240c08ecaf36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.117ex; height:2.843ex;" alt="{\displaystyle \sin(\theta )\approx \pi -\theta }"></span> 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 \theta \approx \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≈</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \approx \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e40af60fdf1675b2e8a1d87584185280d67971c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.521ex; height:2.176ex;" alt="{\displaystyle \theta \approx \pi }"></span>. The solution to this problem involves <a href="/wiki/Hyperbolic_sinusoid" class="mw-redirect" title="Hyperbolic sinusoid">hyperbolic sinusoids</a>, and note that unlike the small angle approximation, this approximation is unstable, meaning 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 |\theta |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\theta |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2880a52da62c0d740bf1f92d9c6ffdbe4772b01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.384ex; height:2.843ex;" alt="{\displaystyle |\theta |}"></span> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state. </p><p>One more interesting linearization is possible around <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 \theta =\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc4628b0f731f81bfabcd8edaca00aa186f03bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.846ex; height:2.843ex;" alt="{\displaystyle \theta =\pi /2}"></span>, around which <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 \sin(\theta )\approx 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta )\approx 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9105777edb335a85f687b7914f6741cfce405c17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.016ex; height:2.843ex;" alt="{\displaystyle \sin(\theta )\approx 1}"></span>: </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 {\frac {d^{2}\theta }{dt^{2}}}+1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccbb5b457c207034fc20a11c2fa3a76d44d959f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.11ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+1=0.}"></span></dd></dl> <p>This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) <a href="/wiki/Phase_portrait" title="Phase portrait">phase portraits</a> and approximate periods. </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_nonlinear_dynamic_behaviors">Types of nonlinear dynamic behaviors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nonlinear_system&action=edit&section=8" title="Edit section: Types of nonlinear dynamic behaviors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Amplitude_death" title="Amplitude death">Amplitude death</a> – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system</li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos</a> – values of a system cannot be predicted indefinitely far into the future, and fluctuations are <a href="/wiki/Aperiodic" class="mw-redirect" title="Aperiodic">aperiodic</a></li> <li><a href="/wiki/Multistability" title="Multistability">Multistability</a> – the presence of two or more stable states</li> <li><a href="/wiki/Soliton" title="Soliton">Solitons</a> – self-reinforcing solitary waves</li> <li><a href="/wiki/Limit_cycle" title="Limit cycle">Limit cycles</a> – asymptotic periodic orbits to which destabilized fixed points are attracted.</li> <li><a href="/wiki/Self-oscillation" title="Self-oscillation">Self-oscillations</a> – feedback oscillations taking place in open dissipative physical systems.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Examples_of_nonlinear_equations">Examples of nonlinear equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nonlinear_system&action=edit&section=9" title="Edit section: Examples of nonlinear equations"><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/Algebraic_Riccati_equation" title="Algebraic Riccati equation">Algebraic Riccati equation</a></li> <li><a href="/wiki/Ball_and_beam" title="Ball and beam">Ball and beam</a> system</li> <li><a href="/wiki/Bellman_equation" title="Bellman equation">Bellman equation</a> for optimal policy</li> <li><a href="/wiki/Boltzmann_equation" title="Boltzmann equation">Boltzmann equation</a></li> <li><a href="/wiki/Colebrook_equation" class="mw-redirect" title="Colebrook equation">Colebrook equation</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau theory</a></li> <li><a href="/wiki/Ishimori_equation" title="Ishimori equation">Ishimori equation</a></li> <li><a href="/wiki/Kadomtsev%E2%80%93Petviashvili_equation" title="Kadomtsev–Petviashvili equation">Kadomtsev–Petviashvili equation</a></li> <li><a href="/wiki/Korteweg%E2%80%93de_Vries_equation" class="mw-redirect" title="Korteweg–de Vries equation">Korteweg–de Vries equation</a></li> <li><a href="/wiki/Landau%E2%80%93Lifshitz%E2%80%93Gilbert_equation" title="Landau–Lifshitz–Gilbert equation">Landau–Lifshitz–Gilbert equation</a></li> <li><a href="/wiki/Li%C3%A9nard_equation" title="Liénard equation">Liénard equation</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a> of <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a></li> <li><a href="/wiki/Nonlinear_optics" title="Nonlinear optics">Nonlinear optics</a></li> <li><a href="/wiki/Nonlinear_Schr%C3%B6dinger_equation" title="Nonlinear Schrödinger equation">Nonlinear Schrödinger equation</a></li> <li><a href="/wiki/Power-flow_study" title="Power-flow study">Power-flow study</a></li> <li><a href="/wiki/Richards_equation" title="Richards equation">Richards equation</a> for unsaturated water flow</li> <li><a href="/wiki/Self-balancing_unicycle" class="mw-redirect" title="Self-balancing unicycle">Self-balancing unicycle</a></li> <li><a href="/wiki/Sine-Gordon_equation" title="Sine-Gordon equation">Sine-Gordon equation</a></li> <li><a href="/wiki/Van_der_Pol_oscillator" title="Van der Pol oscillator">Van der Pol oscillator</a></li> <li><a href="/wiki/Vlasov_equation" title="Vlasov equation">Vlasov equation</a></li></ul> </div> <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=Nonlinear_system&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col"> <ul><li><a href="/wiki/Aleksandr_Mikhailovich_Lyapunov" class="mw-redirect" title="Aleksandr Mikhailovich Lyapunov">Aleksandr Mikhailovich Lyapunov</a></li> <li><a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical system</a></li> <li><a href="/wiki/Feedback" title="Feedback">Feedback</a></li> <li><a href="/wiki/Initial_condition" title="Initial condition">Initial condition</a></li> <li><a href="/wiki/Linear_system" title="Linear system">Linear system</a></li> <li><a href="/wiki/Mode_coupling" title="Mode coupling">Mode coupling</a></li> <li><a href="/wiki/Vector_soliton" title="Vector soliton">Vector soliton</a></li> <li><a href="/wiki/Volterra_series" title="Volterra series">Volterra series</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nonlinear_system&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 35em;"> <ol 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 a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .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 class="citation news cs1"><a rel="nofollow" class="external text" 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(2009). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jsc.2008.03.004">"Thirty years of Polynomial System Solving, and now?"</a>. <i>Journal of Symbolic Computation</i>. <b>44</b> (3): 222–231. <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/10.1016%2Fj.jsc.2008.03.004">10.1016/j.jsc.2008.03.004</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Computation&rft.atitle=Thirty+years+of+Polynomial+System+Solving%2C+and+now%3F&rft.volume=44&rft.issue=3&rft.pages=222-231&rft.date=2009&rft_id=info%3Adoi%2F10.1016%2Fj.jsc.2008.03.004&rft.aulast=Lazard&rft.aufirst=D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.jsc.2008.03.004&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANonlinear+system" class="Z3988"></span></span> </li> <li id="cite_note-SAB1-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-SAB1_12-0">^</a></b></span> <span class="reference-text">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</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"><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%3ANonlinear+system" class="Z3988"></span></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"><a rel="nofollow" class="external text" href="http://www.damtp.cam.ac.uk/user/tong/dynamics.html">David Tong: Lectures on Classical Dynamics</a></span> </li> </ol></div> <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=Nonlinear_system&action=edit&section=12" 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: 35em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiederich_Hinrichsen_and_Anthony_J._Pritchard2005" class="citation book cs1"><a href="/wiki/Diederich_Hinrichsen" title="Diederich Hinrichsen">Diederich Hinrichsen</a> and Anthony J. Pritchard (2005). <i>Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness</i>. Springer Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540441250" title="Special:BookSources/9783540441250"><bdi>9783540441250</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+Systems+Theory+I+-+Modelling%2C+State+Space+Analysis%2C+Stability+and+Robustness&rft.pub=Springer+Verlag&rft.date=2005&rft.isbn=9783540441250&rft.au=Diederich+Hinrichsen+and+Anthony+J.+Pritchard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANonlinear+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJordanSmith2007" class="citation book cs1">Jordan, D. W.; Smith, P. (2007). <i>Nonlinear Ordinary Differential Equations</i> (fourth ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-920824-1" title="Special:BookSources/978-0-19-920824-1"><bdi>978-0-19-920824-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+Ordinary+Differential+Equations&rft.edition=fourth&rft.pub=Oxford+University+Press&rft.date=2007&rft.isbn=978-0-19-920824-1&rft.aulast=Jordan&rft.aufirst=D.+W.&rft.au=Smith%2C+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANonlinear+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKhalil2001" class="citation book cs1">Khalil, Hassan K. (2001). <i>Nonlinear Systems</i>. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-067389-3" title="Special:BookSources/978-0-13-067389-3"><bdi>978-0-13-067389-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+Systems&rft.pub=Prentice+Hall&rft.date=2001&rft.isbn=978-0-13-067389-3&rft.aulast=Khalil&rft.aufirst=Hassan+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANonlinear+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKreyszig1998" class="citation book cs1"><a href="/wiki/Erwin_Kreyszig" title="Erwin Kreyszig">Kreyszig, Erwin</a> (1998). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/advancedengineer0008krey"><i>Advanced Engineering Mathematics</i></a></span>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-15496-9" title="Special:BookSources/978-0-471-15496-9"><bdi>978-0-471-15496-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Engineering+Mathematics&rft.pub=Wiley&rft.date=1998&rft.isbn=978-0-471-15496-9&rft.aulast=Kreyszig&rft.aufirst=Erwin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fadvancedengineer0008krey&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANonlinear+system" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSontag1998" class="citation book cs1"><a href="/wiki/Eduardo_D._Sontag" title="Eduardo D. Sontag">Sontag, Eduardo</a> (1998). <i>Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98489-6" title="Special:BookSources/978-0-387-98489-6"><bdi>978-0-387-98489-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Control+Theory%3A+Deterministic+Finite+Dimensional+Systems.+Second+Edition&rft.pub=Springer&rft.date=1998&rft.isbn=978-0-387-98489-6&rft.aulast=Sontag&rft.aufirst=Eduardo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANonlinear+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=Nonlinear_system&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.dodccrp.org/">Command and Control Research Program (CCRP)</a></li> <li><a rel="nofollow" class="external text" href="http://necsi.edu/guide/concepts/linearnonlinear.html">New England Complex Systems Institute: Concepts in Complex Systems</a></li> <li><a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/">Nonlinear Dynamics I: Chaos</a> at <a rel="nofollow" class="external text" href="http://ocw.mit.edu/OcwWeb/index.htm">MIT's OpenCourseWare</a></li> <li><a rel="nofollow" class="external text" href="http://www.hedengren.net/research/models.htm">Nonlinear Model Library</a> –  (in <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a>) a Database of Physical Systems</li> <li><a rel="nofollow" class="external text" href="http://cnls.lanl.gov/">The Center for Nonlinear Studies at Los Alamos National Laboratory</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Differential_equations" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Differential_equations_topics" title="Template:Differential equations topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Differential_equations_topics" title="Template talk:Differential equations topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Differential_equations_topics" title="Special:EditPage/Template:Differential equations topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Differential_equations" style="font-size:114%;margin:0 4em"><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classification</th><td class="navbox-list-with-group navbox-list navbox-odd" 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%">Operations</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/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation for differentiation</a></li> <li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial</a></li> <li><a href="/wiki/Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li> <li><a href="/wiki/Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li> <li><a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear</a></li> <li><a href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li> <li><a href="/wiki/Holonomic_function" title="Holonomic function">Holonomic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Attributes of variables</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/Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li> <li><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a></li> <li><a href="/wiki/Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Coupled</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Decoupled</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Order</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Degree</a></li> <li><a href="/wiki/Autonomous_system_(mathematics)" title="Autonomous system (mathematics)">Autonomous</a></li> <li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact differential equation</a></li> <li><a href="/wiki/Jet_bundle#Partial_differential_equations" title="Jet bundle">On jet bundles</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Relation to processes</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/Difference_equation" class="mw-redirect" title="Difference equation">Difference</a> (discrete analogue)</li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a> <ul><li><a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li> <li><a href="/wiki/Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solutions</th><td class="navbox-list-with-group navbox-list navbox-odd" 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%">Existence/uniqueness</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/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a></li> <li><a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li> <li><a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution topics</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/Wronskian" title="Wronskian">Wronskian</a></li> <li><a href="/wiki/Phase_portrait" title="Phase portrait">Phase portrait</a></li> <li><a href="/wiki/Phase_space" title="Phase space">Phase space</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a></li> <li><a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic stability</a></li> <li><a href="/wiki/Exponential_stability" title="Exponential stability">Exponential stability</a></li> <li><a href="/wiki/Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li> <li><a href="/wiki/Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series solutions</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> solutions</li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></li> <li><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution methods</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/List_of_mathematical_jargon#Proof_techniques" class="mw-redirect" title="List of mathematical jargon">Inspection</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a></li> <li><a href="/wiki/Separation_of_variables" title="Separation of variables">Separation of variables</a></li> <li><a href="/wiki/Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Method of undetermined coefficients</a></li> <li><a href="/wiki/Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li> <li><a href="/wiki/Integrating_factor" title="Integrating factor">Integrating factor</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transforms</a></li> <li><a href="/wiki/Euler_method" title="Euler method">Euler method</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference method</a></li> <li><a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson method</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta methods</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume method</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</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/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></li> <li><a href="/wiki/List_of_linear_ordinary_differential_equations" title="List of linear ordinary differential equations">List of linear ordinary differential equations</a></li> <li><a href="/wiki/List_of_nonlinear_ordinary_differential_equations" title="List of nonlinear ordinary differential equations">List of nonlinear ordinary differential equations</a></li> <li><a href="/wiki/List_of_nonlinear_partial_differential_equations" title="List of nonlinear partial differential equations">List of nonlinear partial differential equations</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematicians</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/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li> <li><a href="/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li> <li><a href="/wiki/Ernst_Leonard_Lindel%C3%B6f" title="Ernst Leonard Lindelöf">Ernst Lindelöf</a></li> <li><a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/John_Crank" title="John Crank">John Crank</a></li> <li><a href="/wiki/Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li> <li><a href="/wiki/Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li> <li><a href="/wiki/Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li> <li><a href="/wiki/Sofya_Kovalevskaya" title="Sofya Kovalevskaya">Sofya Kovalevskaya</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" aria-labelledby="Complex_systems" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Complex_systems_topics" title="Template:Complex 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class="navbox-group" style="width:1%"><a href="/wiki/Collective_behavior" title="Collective behavior">Collective <br />behavior</a></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/Social_dynamics" title="Social dynamics">Social dynamics</a></li> <li><a href="/wiki/Collective_intelligence" title="Collective intelligence">Collective intelligence</a></li> <li><a href="/wiki/Collective_action" title="Collective action">Collective action</a></li> <li><a href="/wiki/Collective_consciousness" title="Collective consciousness">Collective consciousness</a></li> <li><a href="/wiki/Self-organized_criticality" title="Self-organized criticality">Self-organized criticality</a></li> <li><a href="/wiki/Herd_mentality" title="Herd mentality">Herd mentality</a></li> <li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li> <li><a href="/wiki/Agent-based_modelling" 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href="/wiki/Game_theory" title="Game theory">Game theory</a></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/Prisoner%27s_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a></li> <li><a href="/wiki/Rational_choice_theory" class="mw-redirect" title="Rational choice theory">Rational choice theory</a></li> <li><a href="/wiki/Bounded_rationality" title="Bounded rationality">Bounded rationality</a></li> <li><a href="/wiki/Evolutionary_game_theory" title="Evolutionary game theory">Evolutionary game theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Network_science" title="Network science">Networks</a></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/Social_network_analysis" title="Social network analysis">Social network analysis</a></li> <li><a href="/wiki/Small-world_networks" class="mw-redirect" title="Small-world networks">Small-world networks</a></li> <li><a href="/wiki/Centrality" title="Centrality">Centrality</a></li> <li><a href="/wiki/Network_motif" title="Network motif">Motifs</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Scalability" title="Scalability">Scaling</a></li> <li><a href="/wiki/Robustness_(computer_science)" title="Robustness (computer science)">Robustness</a></li> <li><a href="/wiki/Systems_biology" title="Systems biology">Systems biology</a></li> <li><a href="/wiki/Dynamic_network_analysis" title="Dynamic network analysis">Dynamic networks</a></li> <li><a href="/wiki/Complex_adaptive_system" title="Complex adaptive system">Adaptive networks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Nonlinear_dynamics" class="mw-redirect" title="Nonlinear dynamics">Nonlinear dynamics</a></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/Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li> <li><a href="/wiki/Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">Ordinary differential equations</a></li> <li><a href="/wiki/Phase_space" title="Phase space">Phase space</a></li> <li><a href="/wiki/Attractor" title="Attractor">Attractor</a></li> <li><a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos</a></li> <li><a href="/wiki/Multistability" title="Multistability">Multistability</a></li> <li><a href="/wiki/Bifurcation_theory" title="Bifurcation theory">Bifurcation</a></li> <li><a href="/wiki/Coupled_map_lattices" class="mw-redirect" title="Coupled map lattices">Coupled map lattices</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pattern_formation" title="Pattern formation">Pattern formation</a></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/Reaction-diffusion_systems" class="mw-redirect" title="Reaction-diffusion systems">Reaction-diffusion systems</a></li> <li><a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">Partial differential equations</a></li> <li><a href="/wiki/Dissipative_structures" class="mw-redirect" title="Dissipative structures">Dissipative structures</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation</a></li> <li><a href="/wiki/Cellular_automata" class="mw-redirect" title="Cellular automata">Cellular automata</a></li> <li><a href="/wiki/Spatial_ecology" title="Spatial ecology">Spatial 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