Editing Chaos theory (section)

{{Short description|Field of mathematics and science based on non-linear systems and initial conditions}}
{{Other uses}}
[[File:Lorenz attractor yb.svg|thumb|right|A plot of the [[Lorenz attractor]] for values {{nowrap|''r'' {{=}} 28}}, {{nowrap|σ {{=}} 10}}, {{nowrap|''b'' {{=}} {{sfrac|8|3}}}}]]
[[File:Double-compound-pendulum.gif|thumb|An animation of a [[double pendulum|double-rod pendulum]] at an intermediate energy showing chaotic behavior. Starting the pendulum from a slightly different [[initial condition]] would result in a vastly different [[trajectory]]. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions.]]

'''Chaos theory''' is an [[interdisciplinary]] area of [[Scientific method|scientific study]] and branch of [[mathematics]]. It focuses on underlying patterns and [[Deterministic system|deterministic]] [[Scientific law|laws]] of [[dynamical system]]s that are highly sensitive to [[initial conditions]]. These were once thought to have completely random states of disorder and irregularities.<ref>{{Cite web|url=https://www.britannica.com/science/chaos-theory|title=chaos theory {{!}} Definition & Facts|website=Encyclopedia Britannica|language=en|access-date=2019-11-24}}</ref> Chaos theory states that within the apparent randomness of [[chaotic complex system]]s, there are underlying patterns, interconnection, constant [[feedback loops]], repetition, [[self-similarity]], [[fractals]] and [[self-organization]].<ref name=":1">{{Cite web|url=https://fractalfoundation.org/resources/what-is-chaos-theory/|title=What is Chaos Theory? – Fractal Foundation|language=en-US|access-date=2019-11-24}}</ref> The [[butterfly effect]], an underlying principle of chaos, describes how a small change in one state of a deterministic [[nonlinear system]] can result in large differences in a later state (meaning there is sensitive dependence on initial conditions).<ref>{{Cite web |url=http://mathworld.wolfram.com/Chaos.html|title=Chaos|last=Weisstein|first=Eric W. |website=mathworld.wolfram.com |language=en|access-date=2019-11-24}}</ref> A metaphor for this behavior is that a butterfly flapping its wings in [[Brazil]] can cause or prevent a [[tornado]] in [[Texas]].<ref>{{Cite web|url=https://geoffboeing.com/2015/03/chaos-theory-logistic-map/|title=Chaos Theory and the Logistic Map|last=Boeing
|first=Geoff|date=26 March 2015|language=en|access-date=2020-05-17}}</ref><ref name="Lorenz">{{Cite book |last=Lorenz |first=Edward |title=The Essence of Chaos |publisher=University of Washington Press |year=1993 |isbn=978-0-295-97514-6 |url=https://books.google.com/books?id=j5Ub6sMCoOsC}}</ref>{{rp|181–184}}<ref name=":7">{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Cui |first4=Jialin |last5=Faghih-Naini |first5=Sara |last6=Paxson |first6=Wei |last7=Atlas |first7=Robert |date=2022-07-04 |title=Three Kinds of Butterfly Effects within Lorenz Models |journal=Encyclopedia |volume=2 |issue=3 |pages=1250–1259 |doi=10.3390/encyclopedia2030084 |issn=2673-8392|doi-access=free }} [[File:CC-BY icon.svg|50px]]  Text was copied from this source, which is available under a [https://creativecommons.org/licenses/by/4.0/ Creative Commons Attribution 4.0 International License].</ref>

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in [[numerical analysis|numerical computation]], can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general.<ref>{{cite book |last = Kellert |first = Stephen H. |title = In the Wake of Chaos: Unpredictable Order in Dynamical Systems |url = https://archive.org/details/inwakeofchaosunp0000kell |url-access = registration |publisher = University of Chicago Press |year = 1993 |isbn = 978-0-226-42976-2 |page = [https://archive.org/details/inwakeofchaosunp0000kell/page/32 32] }}</ref> This can happen even though these systems are [[deterministic system (mathematics)|deterministic]], meaning that their future behavior follows a unique evolution<ref name=":2">{{Citation|last=Bishop|first=Robert|title=Chaos|date=2017|url=https://plato.stanford.edu/archives/spr2017/entries/chaos/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Spring 2017|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-11-24}}</ref> and is fully determined by their initial conditions, with no [[randomness|random]] elements involved.<ref>{{harvnb|Kellert|1993|p=56}}</ref> In other words, the deterministic nature of these systems does not make them predictable.<ref>{{harvnb|Kellert|1993|p=62}}</ref><ref name="WerndlCharlotte">{{cite journal |author = Werndl, Charlotte |title = What are the New Implications of Chaos for Unpredictability? |journal = The British Journal for the Philosophy of Science |volume = 60 |issue = 1 |pages = 195–220 |year = 2009 |doi = 10.1093/bjps/axn053 |arxiv = 1310.1576 |s2cid = 354849 }}</ref> This behavior is known as '''deterministic chaos''', or simply '''chaos'''. The theory was summarized by [[Edward Lorenz]] as:<ref>{{cite web |url = http://mpe.dimacs.rutgers.edu/2013/03/17/chaos-in-an-atmosphere-hanging-on-a-wall/ |title = Chaos in an Atmosphere Hanging on a Wall |last1 = Danforth |first1 = Christopher M. |date = April 2013 |work = Mathematics of Planet Earth 2013 |access-date = 12 June 2018 }}</ref>

{{Blockquote|Chaos: When the present determines the future but the approximate present does not approximately determine the future.}}

Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate.<ref name=Lorenz1961/><ref>{{cite book |last = Ivancevic |first = Vladimir G. |title = Complex nonlinearity: chaos, phase transitions, topology change, and path integrals |year = 2008 |publisher = Springer |isbn = 978-3-540-79356-4 |author2 = Tijana T. Ivancevic }}</ref><ref name=":2" /> It also occurs spontaneously in some systems with artificial components, such as [[road traffic]].<ref name=":1" /> This behavior can be studied through the analysis of a chaotic [[mathematical model]] or through analytical techniques such as [[recurrence plot]]s and [[Poincaré map]]s. Chaos theory has applications in a variety of disciplines, including [[meteorology]],<ref name=":2" /> [[anthropology]],<ref name=":0">{{Cite book|title=On the order of chaos. Social anthropology and the science of chaos|last=Mosko M.S., Damon F.H. (Eds.)|publisher=Berghahn Books|year=2005|location=Oxford}}</ref> [[sociology]], [[environmental science]], [[computer science]], [[engineering]], [[economics]], [[ecology]], and [[pandemic]] [[crisis management]].<ref name="CT-REF-20">{{cite web |url=https://www.researchgate.net/publication/340775886 |title=Covid-19 Pandemic and Chaos Theory: Applications based on a Bibliometric Analysis |last=Piotrowski |first=Chris |website=researchgate.net |access-date=2020-05-13}}</ref><ref name="CT-REF-21">{{cite book |last=Weinberger| first=David |title=Everyday Chaos – Technology, Complexity and How We're Thriving in a New World of Possibility |publisher=Harvard Business Review Press |year=2019 | isbn=9781633693968 |url=https://books.google.com/books?id=R7V2DwAAQBAJ}}</ref> The theory formed the basis for such fields of study as [[dynamical systems|complex dynamical systems]], [[edge of chaos]] theory and [[self-assembly]] processes.