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href="/wiki/File:Shish-kebab-skewer-60458_640.jpg" class="mw-file-description"><img alt="Edge of chaos" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Shish-kebab-skewer-60458_640.jpg/220px-Shish-kebab-skewer-60458_640.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Shish-kebab-skewer-60458_640.jpg/330px-Shish-kebab-skewer-60458_640.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Shish-kebab-skewer-60458_640.jpg/440px-Shish-kebab-skewer-60458_640.jpg 2x" data-file-width="3264" data-file-height="2448" /></a><figcaption></figcaption></figure><div class="infobox-caption"><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>"The truly creative changes and the big shifts occur right at the edge of chaos."<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></p><div class="templatequotecite">— <cite><small>Dr. Robert Bilder, Professor at the UCLA <a href="/wiki/Semel_Institute_for_Neuroscience_and_Human_Behavior" title="Semel Institute for Neuroscience and Human Behavior">Semel Institute for Neuroscience and Human Behavior</a></small></cite></div></blockquote></div></td></tr></tbody></table> <p>The <b>edge of chaos</b> is a transition space between order and <a href="/wiki/Randomness" title="Randomness">disorder</a> that is hypothesized to exist within a wide variety of systems. This transition zone is a region of bounded instability that engenders a constant dynamic interplay between order and disorder.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Even though the idea of the edge of chaos is an abstract one, it has many applications in such fields as ecology,<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> business management,<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> psychology,<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/Political_science" title="Political science">political science</a>, and other domains of the <a href="/wiki/Social_science" title="Social science">social sciences</a>. Physicists have shown that adaptation to the edge of chaos occurs in almost all systems with feedback.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edge_of_chaos&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The phrase <i>edge of chaos</i> was coined in the late 1980s by <a href="/wiki/Chaos_theory" title="Chaos theory">chaos theory</a> physicist <a href="/wiki/Norman_Packard" title="Norman Packard">Norman Packard</a>.<sup id="cite_ref-EOC-T-30_7-0" class="reference"><a href="#cite_note-EOC-T-30-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-EOC-T-20_8-0" class="reference"><a href="#cite_note-EOC-T-20-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In the next decade, Packard and mathematician <a href="/wiki/Doyne_Farmer" class="mw-redirect" title="Doyne Farmer">Doyne Farmer</a> co-authored many papers on understanding how self-organization and order emerges at the edge of chaos.<sup id="cite_ref-EOC-T-30_7-1" class="reference"><a href="#cite_note-EOC-T-30-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> One of the original catalysts that led to the idea of the edge of chaos were the experiments with <a href="/wiki/Cellular_automata" class="mw-redirect" title="Cellular automata">cellular automata</a> done by <a href="/wiki/Computer_scientist" title="Computer scientist">computer scientist</a> <a href="/wiki/Christopher_Langton" title="Christopher Langton">Christopher Langton</a> where a transition phenomenon was discovered.<sup id="cite_ref-EOC-T-19_9-0" class="reference"><a href="#cite_note-EOC-T-19-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-EOC-T-18_10-0" class="reference"><a href="#cite_note-EOC-T-18-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-EOC-T-16_11-0" class="reference"><a href="#cite_note-EOC-T-16-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The phrase refers to an area in the range of a <a href="/wiki/Variable_(programming)" class="mw-redirect" title="Variable (programming)">variable</a>, λ (lambda), which was varied while examining the behaviour of a <a href="/wiki/Cellular_automaton" title="Cellular automaton">cellular automaton</a> (CA). As λ varied, the behaviour of the CA went through a <a href="/wiki/Phase_transition" title="Phase transition">phase transition</a> of behaviours. Langton found a small area conducive to produce CAs capable of <a href="/wiki/Universal_computation" class="mw-redirect" title="Universal computation">universal computation</a>.<sup id="cite_ref-EOC-T-18_10-1" class="reference"><a href="#cite_note-EOC-T-18-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-EOC-T-19_9-1" class="reference"><a href="#cite_note-EOC-T-19-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-EOC-T-17_12-0" class="reference"><a href="#cite_note-EOC-T-17-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> At around the same time physicist <a href="/wiki/James_P._Crutchfield" title="James P. Crutchfield">James P. Crutchfield</a> and others used the phrase <i>onset of chaos</i> to describe more or less the same concept.<sup id="cite_ref-EOC-T-28_13-0" class="reference"><a href="#cite_note-EOC-T-28-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>In the sciences in general, the phrase has come to refer to a metaphor that some physical, biological, economic and social systems operate in a region between order and either complete <a href="/wiki/Randomness" title="Randomness">randomness</a> or <a href="/wiki/Chaos_theory" title="Chaos theory">chaos</a>, where the <a href="/wiki/Complexity" title="Complexity">complexity</a> is maximal.<sup id="cite_ref-EOC-T-32_14-0" class="reference"><a href="#cite_note-EOC-T-32-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-EOC-T-33_15-0" class="reference"><a href="#cite_note-EOC-T-33-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The generality and significance of the idea, however, has since been called into question by <a href="/wiki/Melanie_Mitchell" title="Melanie Mitchell">Melanie Mitchell</a> and others.<sup id="cite_ref-EOC-T-29_16-0" class="reference"><a href="#cite_note-EOC-T-29-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The phrase has also been borrowed by the business community and is sometimes used inappropriately and in contexts that are far from the original scope of the meaning of the term.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2020)">citation needed</span></a></i>]</sup> </p><p><a href="/wiki/Stuart_Kauffman" title="Stuart Kauffman">Stuart Kauffman</a> has studied <a href="/wiki/Mathematical_model" title="Mathematical model">mathematical models</a> of evolving systems in which the rate of evolution is maximized near the edge of chaos.<sup id="cite_ref-EOC-T-31_17-0" class="reference"><a href="#cite_note-EOC-T-31-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Adaptation">Adaptation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Edge_of_chaos&action=edit&section=2" title="Edit section: Adaptation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Adaptation" title="Adaptation">Adaptation</a> plays a vital role for all living organisms and systems. All of them are constantly changing their inner properties to better fit in the current environment.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The most important instruments for the <a href="/wiki/Adaptation" title="Adaptation">adaptation</a> are the <a href="/wiki/Adaptive_systems" class="mw-redirect" title="Adaptive systems">self-adjusting parameters</a> inherent for many natural systems. The prominent feature of systems with self-adjusting parameters is an ability to avoid <a href="/wiki/Chaos_theory" title="Chaos theory">chaos</a>. The name for this phenomenon is <i>"Adaptation to the edge of chaos"</i>. </p><p>Adaptation to the edge of chaos refers to the idea that many <a href="/wiki/Complex_adaptive_systems" class="mw-redirect" title="Complex adaptive systems">complex adaptive systems</a> (CASs) seem to intuitively evolve toward a regime near the boundary between chaos and order.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Physics has shown that edge of chaos is the optimal settings for control of a system.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> It is also an optional setting that can influence the ability of a physical system to perform primitive functions for computation.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> In CAS, <a href="/wiki/Coevolution" title="Coevolution">coevolution</a> generally occurs near the edge of chaos, and a balance should be maintained between flexibility and stability to avoid structural failure.<sup id="cite_ref-CAS-T-29_22-0" class="reference"><a href="#cite_note-CAS-T-29-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CAS-T-30_23-0" class="reference"><a href="#cite_note-CAS-T-30-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CAS-T-35_24-0" class="reference"><a href="#cite_note-CAS-T-35-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CAS-T-37_25-0" class="reference"><a href="#cite_note-CAS-T-37-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> As a response to coping with turbulent environments, CAS bring out <a href="/wiki/Flexibility" class="mw-redirect" title="Flexibility">flexibility</a>, creativity,<sup id="cite_ref-CAS-T-36_26-0" class="reference"><a href="#cite_note-CAS-T-36-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Agility" title="Agility">agility</a>, <a href="/wiki/Anti-fragility" class="mw-redirect" title="Anti-fragility">anti-fragility</a>, and innovation near the edge of chaos, provided these systems are sufficiently decentralized and non-hierarchical.<sup id="cite_ref-CAS-T-35_24-1" class="reference"><a href="#cite_note-CAS-T-35-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CAS-T-30_23-1" class="reference"><a href="#cite_note-CAS-T-30-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CAS-T-29_22-1" class="reference"><a href="#cite_note-CAS-T-29-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>Because of the importance of adaptation in many natural systems, adaptation to the edge of the chaos takes a prominent position in many scientific researches. Physicists demonstrated that adaptation to state at the boundary of chaos and order occurs in population of <a href="/wiki/Cellular_automata" class="mw-redirect" title="Cellular automata">cellular automata</a> rules which optimize the performance evolving with a <a href="/wiki/Genetic_algorithm" title="Genetic algorithm">genetic algorithm</a>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Another example of this phenomenon is the <a href="/wiki/Self-organized_criticality" title="Self-organized criticality">self-organized criticality</a> in <a href="/wiki/Avalanche" title="Avalanche">avalanche</a> and earthquake models.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>The simplest model for chaotic dynamics is the <a href="/wiki/Logistic_map" title="Logistic map">logistic map</a>. Self-adjusting logistic map dynamics exhibit adaptation to the edge of chaos.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> Theoretical analysis allowed prediction of the location of the narrow parameter regime near the boundary to which the system evolves.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </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=Edge_of_chaos&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Self-organized_criticality" title="Self-organized criticality">Self-organized criticality</a></li> <li><a href="/wiki/Complexity_theory_and_organizations" title="Complexity theory and organizations">Complexity theory and organizations</a></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li></ul> <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=Edge_of_chaos&action=edit&section=4" 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"> <div class="mw-references-wrap mw-references-columns"><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 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.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="CITEREFSchwartz2014" class="citation web cs1">Schwartz, Katrina (6 May 2014). <a rel="nofollow" class="external 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(1993). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100608020036/http://web.cecs.pdx.edu/~mm/rev-edge.pdf">"Revisiting the edge of chaos: Evolving cellular automata to perform computations"</a> <span class="cs1-format">(PDF)</span>. <i>Complex Systems</i>. <b>7</b>: 89–130. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/adap-org/9303003">adap-org/9303003</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993adap.org..3003M">1993adap.org..3003M</a>. Archived from <a rel="nofollow" class="external text" href="http://web.cecs.pdx.edu/~mm/rev-edge.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 8 June 2010<span class="reference-accessdate">. Retrieved <span class="nowrap">4 May</span> 2007</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Complex+Systems&rft.atitle=Revisiting+the+edge+of+chaos%3A+Evolving+cellular+automata+to+perform+computations&rft.volume=7&rft.pages=89-130&rft.date=1993&rft_id=info%3Aarxiv%2Fadap-org%2F9303003&rft_id=info%3Abibcode%2F1993adap.org..3003M&rft.aulast=Mitchell&rft.aufirst=Melanie&rft.au=Hraber%2C+Peter+T.&rft.au=Crutchfield%2C+James+P.&rft_id=http%3A%2F%2Fweb.cecs.pdx.edu%2F~mm%2Frev-edge.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEdge+of+chaos" class="Z3988"></span></li> <li>Melanie Mitchell, James P. Crutchfield and Peter T. Hraber. <i><a rel="nofollow" class="external text" href="http://web.cecs.pdx.edu/~mm/dyn-comp-edge.pdf">Dynamics, Computation, and the "Edge of Chaos": A Re-Examination</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100608012919/http://web.cecs.pdx.edu/~mm/dyn-comp-edge.pdf">Archived</a> 8 June 2010 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i></li> <li><i>Origins of Order: Self-Organization and Selection in Evolution</i> by Stuart Kauffman</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoraBialek2010" class="citation journal cs1">Mora, Thierry; Bialek, William (2010). "Are biological systems poised at criticality?". <i>Journal of Statistical Physics</i>. <b>144</b> (2): 268–302. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1012.2242">1012.2242</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011JSP...144..268M">2011JSP...144..268M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10955-011-0229-4">10.1007/s10955-011-0229-4</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Physics&rft.atitle=Are+biological+systems+poised+at+criticality%3F&rft.volume=144&rft.issue=2&rft.pages=268-302&rft.date=2010&rft_id=info%3Aarxiv%2F1012.2242&rft_id=info%3Adoi%2F10.1007%2Fs10955-011-0229-4&rft_id=info%3Abibcode%2F2011JSP...144..268M&rft.aulast=Mora&rft.aufirst=Thierry&rft.au=Bialek%2C+William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEdge+of+chaos" 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=Edge_of_chaos&action=edit&section=5" 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://bactra.org/notebooks/edge-of-chaos.html">"The Edge of Chaos"</a> – a criticism of the idea's prevalence.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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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:Chaos_theory" title="Template:Chaos theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Chaos_theory" title="Template talk:Chaos theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Chaos_theory" title="Special:EditPage/Template:Chaos theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Chaos_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Core" scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Core</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Attractor" title="Attractor">Attractor</a></li> <li><a href="/wiki/Bifurcation_theory" title="Bifurcation theory">Bifurcation</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Limit_set" title="Limit set">Limit set</a></li> <li><a href="/wiki/Lyapunov_exponent" title="Lyapunov exponent">Lyapunov exponent</a></li> <li><a href="/wiki/Orbit_(dynamics)" title="Orbit (dynamics)">Orbit</a></li> <li><a href="/wiki/Periodic_point" title="Periodic point">Periodic point</a></li> <li><a href="/wiki/Phase_space" title="Phase space">Phase space</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anosov_diffeomorphism" title="Anosov diffeomorphism">Anosov diffeomorphism</a></li> <li><a href="/wiki/Arnold_tongue" title="Arnold tongue">Arnold tongue</a></li> <li><a href="/wiki/Axiom_A" title="Axiom A">axiom A dynamical system</a></li> <li><a href="/wiki/Bifurcation_diagram" title="Bifurcation diagram">Bifurcation diagram</a></li> <li><a href="/wiki/Box-counting_dimension" class="mw-redirect" title="Box-counting dimension">Box-counting dimension</a></li> <li><a href="/wiki/Correlation_dimension" title="Correlation dimension">Correlation dimension</a></li> <li><a href="/wiki/Conservative_system" title="Conservative system">Conservative system</a></li> <li><a href="/wiki/Ergodicity" title="Ergodicity">Ergodicity</a></li> <li><a href="/wiki/False_nearest_neighbors" class="mw-redirect" title="False nearest neighbors">False nearest neighbors</a></li> <li><a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a></li> <li><a href="/wiki/Invariant_measure" title="Invariant measure">Invariant measure</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a></li> <li><a href="/wiki/Measure-preserving_dynamical_system" title="Measure-preserving dynamical system">Measure-preserving dynamical system</a></li> <li><a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li> <li><a href="/wiki/Poincar%C3%A9_section" class="mw-redirect" title="Poincaré section">Poincaré section</a></li> <li><a href="/wiki/Recurrence_plot" title="Recurrence plot">Recurrence plot</a></li> <li><a href="/wiki/SRB_measure" class="mw-redirect" title="SRB measure">SRB measure</a></li> <li><a href="/wiki/Stable_manifold" title="Stable manifold">Stable manifold</a></li> <li><a href="/wiki/Topological_conjugacy" title="Topological conjugacy">Topological conjugacy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Theorems</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ergodic_theory#Ergodic_theorems" title="Ergodic theory">Ergodic theorem</a></li> <li><a href="/wiki/Liouville%27s_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville's theorem</a></li> <li><a href="/wiki/Krylov%E2%80%93Bogolyubov_theorem" title="Krylov–Bogolyubov theorem">Krylov–Bogolyubov theorem</a></li> <li><a href="/wiki/Poincar%C3%A9%E2%80%93Bendixson_theorem" title="Poincaré–Bendixson theorem">Poincaré–Bendixson theorem</a></li> <li><a href="/wiki/Poincar%C3%A9_recurrence_theorem" title="Poincaré recurrence theorem">Poincaré recurrence theorem</a></li> <li><a href="/wiki/Stable_manifold_theorem" title="Stable manifold theorem">Stable manifold theorem</a></li> <li><a href="/wiki/Takens%27s_theorem" title="Takens's theorem">Takens's theorem</a></li></ul> </div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:C%C3%B4ne_textileII.png" class="mw-file-description" title="Conus textile shell"><img alt="Conus textile shell" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/C%C3%B4ne_textileII.png/100px-C%C3%B4ne_textileII.png" decoding="async" width="100" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/C%C3%B4ne_textileII.png/150px-C%C3%B4ne_textileII.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/C%C3%B4ne_textileII.png/200px-C%C3%B4ne_textileII.png 2x" data-file-width="2344" data-file-height="4211" /></a></span> <p><br /> </p> <span typeof="mw:File"><a href="/wiki/File:Circle_map_poincare_recurrence.jpeg" class="mw-file-description" title="Circle map with black Arnold tongues"><img alt="Circle map with black Arnold tongues" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Circle_map_poincare_recurrence.jpeg/100px-Circle_map_poincare_recurrence.jpeg" decoding="async" width="100" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Circle_map_poincare_recurrence.jpeg/150px-Circle_map_poincare_recurrence.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Circle_map_poincare_recurrence.jpeg/200px-Circle_map_poincare_recurrence.jpeg 2x" data-file-width="450" data-file-height="900" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theoretical<br />branches</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bifurcation_theory" title="Bifurcation theory">Bifurcation theory</a></li> <li><a href="/wiki/Control_of_chaos" title="Control of chaos">Control of chaos</a></li> <li><a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical system</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Stability_theory" title="Stability theory">Stability theory</a></li> <li><a href="/wiki/Synchronization_of_chaos" title="Synchronization of chaos">Synchronization of chaos</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Chaotic<br />maps (<a href="/wiki/List_of_chaotic_maps" title="List of chaotic maps">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Discrete</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arnold%27s_cat_map" title="Arnold's cat map">Arnold's cat map</a></li> <li><a href="/wiki/Baker%27s_map" title="Baker's map">Baker's map</a></li> <li><a href="/wiki/Complex_quadratic_polynomial" title="Complex quadratic polynomial">Complex quadratic map</a></li> <li><a href="/wiki/Coupled_map_lattice" title="Coupled map lattice">Coupled map lattice</a></li> <li><a href="/wiki/Duffing_map" title="Duffing map">Duffing map</a></li> <li><a href="/wiki/Dyadic_transformation" title="Dyadic transformation">Dyadic transformation</a></li> <li><a href="/wiki/Dynamical_billiards" title="Dynamical billiards">Dynamical billiards</a> <ul><li><a href="/wiki/Outer_billiards" title="Outer billiards">outer</a></li></ul></li> <li><a href="/wiki/Exponential_map_(discrete_dynamical_systems)" title="Exponential map (discrete dynamical systems)">Exponential map</a></li> <li><a href="/wiki/Gauss_iterated_map" title="Gauss iterated map">Gauss map</a></li> <li><a href="/wiki/Gingerbreadman_map" title="Gingerbreadman map">Gingerbreadman map</a></li> <li><a href="/wiki/H%C3%A9non_map" title="Hénon map">Hénon map</a></li> <li><a href="/wiki/Horseshoe_map" title="Horseshoe map">Horseshoe map</a></li> <li><a href="/wiki/Ikeda_map" title="Ikeda map">Ikeda map</a></li> <li><a href="/wiki/Interval_exchange_transformation" title="Interval exchange transformation">Interval exchange map</a></li> <li><a href="/wiki/Irrational_rotation" title="Irrational rotation">Irrational rotation</a></li> <li><a href="/wiki/Kaplan%E2%80%93Yorke_map" title="Kaplan–Yorke map">Kaplan–Yorke map</a></li> <li><a href="/wiki/Langton%27s_ant" title="Langton's ant">Langton's ant</a></li> <li><a href="/wiki/Logistic_map" title="Logistic map">Logistic map</a></li> <li><a href="/wiki/Standard_map" title="Standard map">Standard map</a></li> <li><a href="/wiki/Tent_map" title="Tent map">Tent map</a></li> <li><a href="/wiki/Tinkerbell_map" title="Tinkerbell map">Tinkerbell map</a></li> <li><a href="/wiki/Zaslavskii_map" title="Zaslavskii map">Zaslavskii map</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="margin: 10px 0px">Continuous</div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Double_scroll_attractor" class="mw-redirect" title="Double scroll attractor">Double scroll attractor</a></li> <li><a href="/wiki/Duffing_equation" title="Duffing equation">Duffing equation</a></li> <li><a href="/wiki/Lorenz_system" title="Lorenz system">Lorenz system</a></li> <li><a href="/wiki/Lotka%E2%80%93Volterra_equations" title="Lotka–Volterra equations">Lotka–Volterra equations</a></li> <li><a href="/wiki/Mackey%E2%80%93Glass_equations" title="Mackey–Glass equations">Mackey–Glass equations</a></li> <li><a href="/wiki/Rabinovich%E2%80%93Fabrikant_equations" title="Rabinovich–Fabrikant equations">Rabinovich–Fabrikant equations</a></li> <li><a href="/wiki/R%C3%B6ssler_attractor" title="Rössler attractor">Rössler attractor</a></li> <li><a href="/wiki/Three-body_problem" title="Three-body problem">Three-body problem</a></li> <li><a href="/wiki/Van_der_Pol_oscillator" title="Van der Pol oscillator">Van der Pol oscillator</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical<br />systems</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chua%27s_circuit" title="Chua's circuit">Chua's circuit</a></li> <li><a href="/wiki/Rayleigh%E2%80%93B%C3%A9nard_convection" title="Rayleigh–Bénard convection">Convection</a></li> <li><a href="/wiki/Double_pendulum" title="Double pendulum">Double pendulum</a></li> <li><a href="/wiki/Elastic_pendulum" title="Elastic pendulum">Elastic pendulum</a></li> <li><a href="/wiki/Fermi%E2%80%93Pasta%E2%80%93Ulam%E2%80%93Tsingou_problem" title="Fermi–Pasta–Ulam–Tsingou problem">FPUT problem</a></li> <li><a href="/wiki/H%C3%A9non%E2%80%93Heiles_system" title="Hénon–Heiles system">Hénon–Heiles system</a></li> <li><a href="/wiki/Kicked_rotator" title="Kicked rotator">Kicked rotator</a></li> <li><a href="/wiki/Multiscroll_attractor" title="Multiscroll attractor">Multiscroll attractor</a></li> <li><a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a></li> <li><a href="/wiki/Swinging_Atwood%27s_machine" title="Swinging Atwood's machine">Swinging Atwood's machine</a></li> <li><a href="/wiki/Tilt-A-Whirl" title="Tilt-A-Whirl">Tilt-A-Whirl</a></li> <li><a href="/wiki/Weather_forecasting" title="Weather forecasting">Weather</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Chaos<br />theorists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Michael_Berry_(physicist)" title="Michael Berry (physicist)">Michael Berry</a></li> <li><a href="/wiki/Rufus_Bowen" title="Rufus Bowen">Rufus Bowen</a></li> <li><a href="/wiki/Mary_Cartwright" title="Mary Cartwright">Mary Cartwright</a></li> <li><a href="/wiki/Chen_Guanrong" title="Chen Guanrong">Chen Guanrong</a></li> <li><a href="/wiki/Leon_O._Chua" title="Leon O. Chua">Leon O. Chua</a></li> <li><a href="/wiki/Mitchell_Feigenbaum" title="Mitchell Feigenbaum">Mitchell Feigenbaum</a></li> <li><a href="/wiki/Peter_Grassberger" title="Peter Grassberger">Peter Grassberger</a></li> <li><a href="/wiki/Celso_Grebogi" title="Celso Grebogi">Celso Grebogi</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Martin Gutzwiller</a></li> <li><a href="/wiki/Brosl_Hasslacher" title="Brosl Hasslacher">Brosl Hasslacher</a></li> <li><a href="/wiki/Michel_H%C3%A9non" title="Michel Hénon">Michel Hénon</a></li> <li><a href="/wiki/Svetlana_Jitomirskaya" title="Svetlana Jitomirskaya">Svetlana Jitomirskaya</a></li> <li><a href="/wiki/Bryna_Kra" title="Bryna Kra">Bryna Kra</a></li> <li><a href="/wiki/Edward_Norton_Lorenz" title="Edward Norton Lorenz">Edward Norton Lorenz</a></li> <li><a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a></li> <li><a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoît Mandelbrot</a></li> <li><a href="/wiki/Hee_Oh" title="Hee Oh">Hee Oh</a></li> <li><a href="/wiki/Edward_Ott" title="Edward Ott">Edward Ott</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></li> <li><a href="/wiki/Itamar_Procaccia" title="Itamar Procaccia">Itamar Procaccia</a></li> <li><a href="/wiki/Mary_Rees" title="Mary Rees">Mary Rees</a></li> <li><a href="/wiki/Otto_R%C3%B6ssler" title="Otto Rössler">Otto Rössler</a></li> <li><a href="/wiki/David_Ruelle" title="David Ruelle">David Ruelle</a></li> <li><a href="/wiki/Caroline_Series" title="Caroline Series">Caroline Series</a></li> <li><a href="/wiki/Yakov_Sinai" title="Yakov Sinai">Yakov Sinai</a></li> <li><a href="/wiki/Oleksandr_Mykolayovych_Sharkovsky" class="mw-redirect" title="Oleksandr Mykolayovych Sharkovsky">Oleksandr Mykolayovych Sharkovsky</a></li> <li><a href="/wiki/Nina_Snaith" title="Nina Snaith">Nina Snaith</a></li> <li><a href="/wiki/Floris_Takens" title="Floris Takens">Floris Takens</a></li> <li><a href="/wiki/Audrey_Terras" title="Audrey Terras">Audrey Terras</a></li> <li><a href="/wiki/Mary_Tsingou" title="Mary Tsingou">Mary Tsingou</a></li> <li><a href="/wiki/Marcelo_Viana" title="Marcelo Viana">Marcelo Viana</a></li> <li><a href="/wiki/Amie_Wilkinson" title="Amie Wilkinson">Amie Wilkinson</a></li> <li><a href="/wiki/James_A._Yorke" title="James A. Yorke">James A. Yorke</a></li> <li><a href="/wiki/Lai-Sang_Young" title="Lai-Sang Young">Lai-Sang Young</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />articles</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Butterfly_effect" title="Butterfly effect">Butterfly effect</a></li> <li><a href="/wiki/Complexity" title="Complexity">Complexity</a></li> <li><a class="mw-selflink selflink">Edge of chaos</a></li> <li><a href="/wiki/Predictability" title="Predictability">Predictability</a></li> <li><a href="/wiki/Santa_Fe_Institute" title="Santa Fe Institute">Santa Fe Institute</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Edge_of_chaos&oldid=1254123348">https://en.wikipedia.org/w/index.php?title=Edge_of_chaos&oldid=1254123348</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Chaos_theory" title="Category:Chaos theory">Chaos theory</a></li><li><a href="/wiki/Category:Self-organization" title="Category:Self-organization">Self-organization</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_September_2024" title="Category:Use dmy dates from September 2024">Use dmy dates from September 2024</a></li><li><a href="/wiki/Category:Articles_using_infobox_templates_with_no_data_rows" title="Category:Articles using infobox templates with no data rows">Articles using infobox templates with no data rows</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_November_2020" title="Category:Articles with unsourced statements from November 2020">Articles with unsourced statements from November 2020</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 29 October 2024, at 14:41<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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