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href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%AC%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%B1_%CE%B4%CE%B9%CE%B1%CE%BA%CE%BB%CE%AC%CE%B4%CF%89%CF%83%CE%B7%CF%82" title="Διάγραμμα διακλάδωσης – Greek" lang="el" hreflang="el" data-title="Διάγραμμα διακλάδωσης" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Diagrama_de_bifurcaci%C3%B3n" title="Diagrama de bifurcación – Spanish" lang="es" hreflang="es" data-title="Diagrama de bifurcación" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Diagramme_de_bifurcation" title="Diagramme de bifurcation – French" lang="fr" hreflang="fr" data-title="Diagramme de bifurcation" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%84%EA%B8%B0_%EB%8B%A4%EC%9D%B4%EC%96%B4%EA%B7%B8%EB%9E%A8" title="분기 다이어그램 – Korean" lang="ko" hreflang="ko" data-title="분기 다이어그램" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Bifurkacijski_dijagram" title="Bifurkacijski dijagram – Croatian" lang="hr" hreflang="hr" data-title="Bifurkacijski dijagram" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%99%D7%90%D7%92%D7%A8%D7%9E%D7%AA_%D7%94%D7%AA%D7%A4%D7%A6%D7%9C%D7%95%D7%AA" title="דיאגרמת התפצלות – Hebrew" lang="he" hreflang="he" data-title="דיאגרמת התפצלות" 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/Bifurkacijos_diagrama" title="Bifurkacijos diagrama – Lithuanian" lang="lt" hreflang="lt" data-title="Bifurkacijos diagrama" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%86%E5%B2%90%E5%9B%B3_(%E5%8A%9B%E5%AD%A6%E7%B3%BB)" title="分岐図 (力学系) – Japanese" lang="ja" hreflang="ja" data-title="分岐図 (力学系)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B8%D1%84%D1%83%D1%80%D0%BA%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%B0%D1%8F_%D0%B4%D0%B8%D0%B0%D0%B3%D1%80%D0%B0%D0%BC%D0%BC%D0%B0" title="Бифуркационная диаграмма – Russian" lang="ru" hreflang="ru" data-title="Бифуркационная диаграмма" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Bifurkacijski_dijagram" title="Bifurkacijski dijagram – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Bifurkacijski dijagram" 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-sv 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Visualization of sudden behavior changes caused by continuous parameter changes</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output 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class="mbox-text-span">This article includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">March 2013</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, particularly in <a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a>, a <b>bifurcation diagram</b> shows the values visited or approached asymptotically (fixed points, <a href="/wiki/Periodic_orbit" class="mw-redirect" title="Periodic orbit">periodic orbits</a>, or <a href="/wiki/Chaos_(mathematics)" class="mw-redirect" title="Chaos (mathematics)">chaotic</a> <a href="/wiki/Attractor" title="Attractor">attractors</a>) of a system as a function of a <a href="/wiki/Bifurcation_theory" title="Bifurcation theory">bifurcation parameter</a> in the system.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Is the parameter of the function to be iterated really called "bifurcation parameter"? What sources substantiate this statement? (February 2024)">citation needed</span></a></i>]</sup> It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of <a href="/wiki/Bifurcation_theory" title="Bifurcation theory">bifurcation theory</a>. In the context of discrete-time dynamical systems, the diagram is also called <b>orbit diagram</b>. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:LogisticMapOrbitDiag_InclNegR.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/LogisticMapOrbitDiag_InclNegR.png/300px-LogisticMapOrbitDiag_InclNegR.png" decoding="async" width="300" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/LogisticMapOrbitDiag_InclNegR.png/450px-LogisticMapOrbitDiag_InclNegR.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/LogisticMapOrbitDiag_InclNegR.png/600px-LogisticMapOrbitDiag_InclNegR.png 2x" data-file-width="9600" data-file-height="5400" /></a><figcaption>A bifurcation diagram of the logistic map</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_map_bifurcation.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Circle_map_bifurcation.jpeg/170px-Circle_map_bifurcation.jpeg" decoding="async" width="170" height="340" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Circle_map_bifurcation.jpeg/255px-Circle_map_bifurcation.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Circle_map_bifurcation.jpeg/340px-Circle_map_bifurcation.jpeg 2x" data-file-width="450" data-file-height="900" /></a><figcaption>Bifurcation diagram of the <a href="/wiki/Circle_map" class="mw-redirect" title="Circle map">circle map</a>. Black regions correspond to <a href="/wiki/Arnold_tongues" class="mw-redirect" title="Arnold tongues">Arnold tongues</a>.</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Logistic_map">Logistic map</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bifurcation_diagram&action=edit&section=1" title="Edit section: Logistic map"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">See also: <a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">Dynamical systems</a> and <a href="/wiki/List_of_chaotic_maps" title="List of chaotic maps">List of chaotic maps</a></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Logistic_Map_Bifurcation_Diagram,_Matplotlib.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Logistic_Map_Bifurcation_Diagram%2C_Matplotlib.svg/300px-Logistic_Map_Bifurcation_Diagram%2C_Matplotlib.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Logistic_Map_Bifurcation_Diagram%2C_Matplotlib.svg/450px-Logistic_Map_Bifurcation_Diagram%2C_Matplotlib.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Logistic_Map_Bifurcation_Diagram%2C_Matplotlib.svg/600px-Logistic_Map_Bifurcation_Diagram%2C_Matplotlib.svg.png 2x" data-file-width="576" data-file-height="432" /></a><figcaption>Bifurcation diagram of the <a href="/wiki/Logistic_map" title="Logistic map">logistic map</a>. The <a href="/wiki/Attractor" title="Attractor">attractor</a> for any value of the parameter <i>r</i> is shown on the vertical line at that <i>r</i>.</figcaption></figure> <p>An example is the bifurcation diagram of the <a href="/wiki/Logistic_map" title="Logistic map">logistic map</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1}=rx_{n}(1-x_{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>r</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n+1}=rx_{n}(1-x_{n}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dcedf284ca3050f725beeb111b482ba0c8828e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.351ex; height:2.843ex;" alt="{\displaystyle x_{n+1}=rx_{n}(1-x_{n}).}"></span> </p><p>The bifurcation parameter <i>r</i> is shown on the horizontal axis of the plot and the vertical axis shows the set of values of the <a href="/wiki/Logistic_function" title="Logistic function">logistic function</a> visited asymptotically from almost all initial conditions. </p><p>The bifurcation diagram shows the forking of the periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a <a href="/wiki/Period-doubling_bifurcation" title="Period-doubling bifurcation">period-doubling bifurcation</a>. The ratio of the lengths of successive intervals between values of <i>r</i> for which bifurcation occurs <a href="/wiki/Convergent_series" title="Convergent series">converges</a> to the <a href="/wiki/First_Feigenbaum_constant" class="mw-redirect" title="First Feigenbaum constant">first Feigenbaum constant</a>. </p><p>The diagram also shows period doublings from 3 to 6 to 12 etc., from 5 to 10 to 20 etc., and so forth. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Symmetry_breaking_in_bifurcation_sets">Symmetry breaking in bifurcation sets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bifurcation_diagram&action=edit&section=2" title="Edit section: Symmetry breaking in bifurcation sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Asymbif.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Asymbif.gif/300px-Asymbif.gif" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Asymbif.gif/450px-Asymbif.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/0/07/Asymbif.gif 2x" data-file-width="560" data-file-height="420" /></a><figcaption>Symmetry breaking in <a href="/wiki/Pitchfork_bifurcation" title="Pitchfork bifurcation">pitchfork bifurcation</a> as the parameter <i>ε</i> is varied. <i>ε</i> = 0 is the case of symmetric pitchfork bifurcation.</figcaption></figure> <p>In a dynamical system such as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {x}}+f(x;\mu )+\varepsilon g(x)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ε</mi> <mi>g</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 {\ddot {x}}+f(x;\mu )+\varepsilon g(x)=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25bfcaf80d060625205ef332a4a4f1d529cd6cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.11ex; height:2.843ex;" alt="{\displaystyle {\ddot {x}}+f(x;\mu )+\varepsilon g(x)=0,}"></span> which is <a href="/wiki/Structurally_stable" class="mw-redirect" title="Structurally stable">structurally stable</a> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff2b68dc21692b0fe4c1e1220806e6d88b8ace8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu \neq 0}"></span>, if a bifurcation diagram is plotted, treating <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> as the bifurcation parameter, but for different values 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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span>, the case <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 \varepsilon =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbe653d369abd71fc0be98efbf59e9edc1bdfb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon =0}"></span> is the symmetric pitchfork bifurcation. When <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 \varepsilon \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b164106e88d10861e8dbc94194c20a96202efbde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.344ex; height:2.676ex;" alt="{\displaystyle \varepsilon \neq 0}"></span>, we say we have a pitchfork with <i>broken symmetry.</i> This is illustrated in the animation on the right. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bifurcation_diagram&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a system of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> that describes some physical quantity, that for concreteness could represent one of three examples: 1. the position and velocity of an undamped and frictionless pendulum, 2. a neuron's membrane potential over time, and 3. the average concentration of a virus in a patient's bloodstream. The differential equations for these examples include *parameters* that may affect the output of the equations. Changing the pendulum's mass and length will affect its oscillation frequency, changing the magnitude of injected current into a neuron may transition the membrane potential from resting to spiking, and the long-term viral load in the bloodstream may decrease with carefully timed treatments. </p><p>In general, researchers may seek to quantify how the long-term (asymptotic) behavior of a system of differential equations changes if a parameter is changed. In the <a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a> branch of mathematics, a <b>bifurcation diagram</b> quantifies these changes by showing how fixed points, <a href="/wiki/Periodic_orbit" class="mw-redirect" title="Periodic orbit">periodic orbits</a>, or <a href="/wiki/Chaos_(mathematics)" class="mw-redirect" title="Chaos (mathematics)">chaotic</a> <a href="/wiki/Attractor" title="Attractor">attractors</a> of a system change as a function of <a href="/wiki/Bifurcation_theory" title="Bifurcation theory">bifurcation parameter</a>. Bifurcation diagrams are used to visualize these changes. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bifurcation_diagram&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bifurcation_memory" title="Bifurcation memory">Bifurcation memory</a></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Complex_quadratic_polynomial#Critical_curves" title="Complex quadratic polynomial">Skeleton of bifurcation diagram</a></li> <li><a href="/wiki/Feigenbaum_constants" title="Feigenbaum constants">Feigenbaum constants</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bifurcation_diagram&action=edit&section=5" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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 id="CITEREFGlendinning1994" class="citation book cs1">Glendinning, Paul (1994). <i>Stability, Instability and Chaos</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-41553-5" title="Special:BookSources/0-521-41553-5"><bdi>0-521-41553-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stability%2C+Instability+and+Chaos&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=0-521-41553-5&rft.aulast=Glendinning&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABifurcation+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMay1976" class="citation journal cs1">May, Robert M. (1976). "Simple mathematical models with very complicated dynamics". <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>. <b>261</b> (5560): 459–467. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1976Natur.261..459M">1976Natur.261..459M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F261459a0">10.1038/261459a0</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F104555">10338.dmlcz/104555</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/934280">934280</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:2243371">2243371</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Simple+mathematical+models+with+very+complicated+dynamics&rft.volume=261&rft.issue=5560&rft.pages=459-467&rft.date=1976&rft_id=info%3Ahdl%2F10338.dmlcz%2F104555&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2243371%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1976Natur.261..459M&rft_id=info%3Apmid%2F934280&rft_id=info%3Adoi%2F10.1038%2F261459a0&rft.aulast=May&rft.aufirst=Robert+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABifurcation+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrogatz2000" class="citation book cs1">Strogatz, Steven (2000). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/nonlineardynamic00stro"><i>Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering</i></a></span>. <a href="/wiki/Perseus_Books" class="mw-redirect" title="Perseus Books">Perseus Books</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7382-0453-6" title="Special:BookSources/0-7382-0453-6"><bdi>0-7382-0453-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-linear+Dynamics+and+Chaos%3A+With+applications+to+Physics%2C+Biology%2C+Chemistry+and+Engineering&rft.pub=Perseus+Books&rft.date=2000&rft.isbn=0-7382-0453-6&rft.aulast=Strogatz&rft.aufirst=Steven&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnonlineardynamic00stro&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABifurcation+diagram" class="Z3988"></span></li></ul> <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=Bifurcation_diagram&action=edit&section=6" 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.egwald.ca/nonlineardynamics/logisticsmapchaos.php">The Logistic Map and Chaos</a> by Elmer G. Wiens, egwald.ca</li> <li><a href="https://en.wikiversity.org/wiki/Discrete-time_dynamical_system_orbit_diagram" class="extiw" title="wikiversity:Discrete-time dynamical system orbit diagram">Wikiversity: Discrete-time dynamical system orbit diagram</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r1012311289">.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox-body{font-style:italic}.mw-parser-output .asbox-note{font-size:smaller}.mw-parser-output .asbox .navbar{position:absolute;top:-0.75em;right:1em;display:none}</style><div role="note" class="metadata plainlinks asbox stub"><table role="presentation"><tbody><tr class="noresize"><td><span typeof="mw:File"><a href="/wiki/File:Airplane_vortex_edit.jpg" class="mw-file-description"><img alt="Stub icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Airplane_vortex_edit.jpg/37px-Airplane_vortex_edit.jpg" decoding="async" width="37" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Airplane_vortex_edit.jpg/55px-Airplane_vortex_edit.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Airplane_vortex_edit.jpg/74px-Airplane_vortex_edit.jpg 2x" data-file-width="2976" data-file-height="2420" /></a></span></td><td><p class="asbox-body">This <a href="/wiki/Chaos_theory" title="Chaos theory">chaos theory</a>-related article is a <a href="/wiki/Wikipedia:Stub" title="Wikipedia:Stub">stub</a>. 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