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{{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:3D Lorenz Chaotic Attactor.png|thumb|A plot of the 3D [[Lorenz attractor]]]]
[[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]] 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, despite the deterministic nature of these systems, this 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=978-1-63369-396-8 |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.

==Introduction==
Chaos theory concerns deterministic systems which are predictable for some amount of time and then appear to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: How much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the [[Lyapunov time]]. Some examples of Lyapunov times are: Chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years.<ref>{{Cite journal|last1=Wisdom|first1=Jack|last2=Sussman|first2=Gerald Jay|date=1992-07-03|title=Chaotic Evolution of the Solar System|journal=Science|language=en|volume=257|issue=5066|pages=56–62|doi=10.1126/science.257.5066.56|issn=1095-9203|pmid=17800710|bibcode=1992Sci...257...56S|hdl=1721.1/5961|s2cid=12209977|hdl-access=free}}</ref> In chaotic systems, the uncertainty in a forecast increases [[Exponential growth|exponentially]] with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random.<ref>''Sync: The Emerging Science of Spontaneous Order'', Steven Strogatz, Hyperion, New York, 2003, pages 189–190.</ref>

==Chaotic dynamics==
[[File:Chaos Sensitive Dependence.svg|thumb|The [[Map (mathematics)|map]] defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> and <span style="white-space: nowrap;">''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1</span> displays sensitivity to initial x positions. Here, two series of ''x'' and ''y'' values diverge markedly over time from a tiny initial difference.]]
In common usage, "chaos" means "a state of disorder".<ref>Definition of {{linktext|chaos}} at [[Wiktionary]];</ref><ref>{{Cite web|url=https://www.dictionary.com/browse/chaos|title=Definition of chaos {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-11-24}}</ref> However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by [[Robert L. Devaney]], says that to classify a dynamical system as chaotic, it must have these properties:<ref>{{cite book|title=A First Course in Dynamics: With a Panorama of Recent Developments|last=Hasselblatt|first=Boris|author2=Anatole Katok|year=2003|publisher=Cambridge University Press|isbn=978-0-521-58750-1}}</ref>

# it must be [[sensitive dependence on initial conditions|sensitive to initial conditions]],
# it must be [[Mixing (mathematics)#Topological mixing|topologically transitive]],
# it must have [[dense set|dense]] [[periodic orbit]]s.

In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions.<ref>{{cite book |author=Elaydi, Saber N. |title=Discrete Chaos |publisher=Chapman & Hall/CRC |year=1999 |isbn=978-1-58488-002-8 |page=137 }}</ref><ref>{{cite book |author=Basener, William F. |title=Topology and its applications |publisher=Wiley |year=2006 |isbn=978-0-471-68755-9 |page=42 }}</ref> In the discrete-time case, this is true for all [[Continuous function|continuous]] [[Map (mathematics)|maps]] on [[metric space]]s.<ref>{{cite journal | author1=Banks | author2=Brooks | author3=Cairns | author4=Davis | author5=Stacey | title= On Devaney's definition of chaos | journal=The American Mathematical Monthly | volume=99|issue=4|date=1992| pages=332–334 | doi=10.1080/00029890.1992.11995856 }}</ref> In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition.

If attention is restricted to [[Interval (mathematics)|intervals]], the second property implies the other two.<ref>{{cite journal |author1=Vellekoop, Michel |author2=Berglund, Raoul |title=On Intervals, Transitivity = Chaos |journal=The American Mathematical Monthly |volume=101 |issue=4 |pages=353–5 |date=April 1994 |jstor=2975629 |doi=10.2307/2975629}}</ref> An alternative and a generally weaker definition of chaos uses only the first two properties in the above list.<ref>{{cite book |author1=Medio, Alfredo |author2=Lines, Marji |title=Nonlinear Dynamics: A Primer |url=https://archive.org/details/nonlineardynamic00medi |url-access=limited |publisher=Cambridge University Press |year=2001 |isbn=978-0-521-55874-7 |page=[https://archive.org/details/nonlineardynamic00medi/page/n175 165] }}</ref>

===Sensitivity to initial conditions===
{{Main|Butterfly effect}}
[[File:SensInitCond.gif|thumb|Lorenz equations used to generate plots for the y variable. The initial conditions for ''x'' and ''z'' were kept the same but those for ''y'' were changed between '''1.001''', '''1.0001''' and '''1.00001'''. The values for <math>\rho</math>, <math>\sigma</math> and <math>\beta</math> were '''45.91''', '''16''' and '''4 ''' respectively. As can be seen from the graph, even the slightest difference in initial values causes significant changes after about 12 seconds of evolution in the three cases. This is an example of sensitive dependence on initial conditions.]]

'''Sensitivity to initial conditions''' means that each point in a chaotic system is arbitrarily closely approximated by other points that have significantly different future paths or trajectories. Thus, an arbitrarily small change or perturbation of the current trajectory may lead to significantly different future behavior.<ref name=":1" />

Sensitivity to initial conditions is popularly known as the "[[butterfly effect]]", so-called because of the title of a paper given by [[Edward Lorenz]] in 1972 to the [[American Association for the Advancement of Science]] in Washington, D.C., entitled ''Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?''.<ref>{{Cite web|url=http://news.mit.edu/2008/obit-lorenz-0416|title=Edward Lorenz, father of chaos theory and butterfly effect, dies at 90|website=MIT News|date=16 April 2008 |access-date=2019-11-24}}</ref> The flapping wing represents a small change in the initial condition of the system, which causes a chain of events that prevents the predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system could have been vastly different.

As suggested in Lorenz's book entitled ''The Essence of Chaos'', published in 1993,<ref name="Lorenz"/>{{rp|8}} "sensitive dependence can serve as an acceptable definition of chaos". In the same book, Lorenz defined the butterfly effect as: "The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration."<ref name="Lorenz"/>{{rp|23}} The above definition is consistent with the sensitive dependence of solutions on initial conditions (SDIC). An idealized skiing model was developed to illustrate the sensitivity of time-varying paths to initial positions.<ref name="Lorenz"/>{{rp|189–204}} A predictability horizon can be determined before the onset of SDIC (i.e., prior to significant separations of initial nearby trajectories).<ref>{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |date=2022-05-07 |title=One Saddle Point and Two Types of Sensitivities within the Lorenz 1963 and 1969 Models |journal=Atmosphere |language=en |volume=13 |issue=5 |page=753 |doi=10.3390/atmos13050753 |bibcode=2022Atmos..13..753S |issn=2073-4433|doi-access=free }}</ref>

A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system (as is usually the case in practice), then beyond a certain time, the system would no longer be predictable. This is most prevalent in the case of weather, which is generally predictable only about a week ahead.<ref name="RGW">{{cite book |author=Watts, Robert G. |title=Global Warming and the Future of the Earth |url=https://archive.org/details/globalwarmingfut00watt_399 |url-access=limited |publisher=Morgan & Claypool |year=2007 |page=[https://archive.org/details/globalwarmingfut00watt_399/page/n22 17] }}</ref> This does not mean that one cannot assert anything about events far in the future - only that some restrictions on the system are present. For example, we know that the temperature of the surface of the earth will not naturally reach {{convert|100|C|F}} or fall below {{convert|-130|C|F}} on earth (during the current [[geologic era]]), but we cannot predict exactly which day will have the hottest temperature of the year.

In more mathematical terms, the [[Lyapunov exponent]] measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions.<ref>{{Cite web|url=http://mathworld.wolfram.com/LyapunovCharacteristicExponent.html|title=Lyapunov Characteristic Exponent|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-24}}</ref> More specifically, given two starting [[trajectory|trajectories]] in the [[phase space]] that are infinitesimally close, with initial separation <math>\delta \mathbf{Z}_0</math>, the two trajectories end up diverging at a rate given by

:<math> | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |,</math>

where <math>t</math> is the time and <math>\lambda</math> is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can exist. The number of Lyapunov exponents is equal to the number of dimensions of the phase space, though it is common to just refer to the largest one. For example, the maximal Lyapunov exponent (MLE) is most often used, because it determines the overall predictability of the system. A positive MLE, coupled with the solution's boundedness, is usually taken as an indication that the system is chaotic.<ref name=":2" />

In addition to the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, [[Measure (mathematics)|measure-theoretical]] [[Mixing (mathematics)|mixing]] (as discussed in [[ergodicity|ergodic]] theory) and properties of a [[Kolmogorov automorphism|K-system]].<ref name="WerndlCharlotte" />

===Non-periodicity===
A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the sequence, however close, it will not enter the sequence and in fact, will diverge from it. Thus for [[almost all]] initial conditions, the variable evolves chaotically with non-periodic behavior.

===Topological mixing===
[[File:LogisticTopMixing1-6.gif|thumb|Six iterations of a set of states <math>[x,y]</math> passed through the logistic map. The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that ''mixing'' occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation <math>x_{k+1} = 4  x_k  (1 - x_k )</math>. To expand the state-space of the logistic map into two dimensions, a second state, <math>y</math>, was created as <math>y_{k+1} = x_k + y_k </math>, if <math>x_k + y_k <1</math> and <math>y_{k+1} = x_k + y_k -1</math> otherwise.]]
[[File:Chaos Topological Mixing.png|thumb|The map defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> and <span style="white-space: nowrap;">''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1</span> also displays [[topological mixing]]. Here, the blue region is transformed by the dynamics first to the purple region, then to the pink and red regions, and eventually to a cloud of vertical lines scattered across the space.]]

[[Topological mixing]] (or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or [[open set]] of its [[phase space]] eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored [[dye]]s or fluids is an example of a chaotic system.

Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity.

===Topological transitivity===
A map <math>f:X \to X</math> is said to be topologically transitive if for any pair of non-empty [[open set]]s <math>U, V \subset X</math>, there exists <math>k > 0</math> such that <math>f^{k}(U) \cap V \neq \emptyset</math>. Topological transitivity is a weaker version of [[topological mixing]]. Intuitively, if a map is topologically transitive then given a point ''x'' and a region ''V'', there exists a point ''y'' near ''x'' whose orbit passes through ''V''. This implies that it is impossible to decompose the system into two open sets.<ref name="Devaney">{{harvnb|Devaney|2003}}</ref>

An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies topological transitivity. The Birkhoff Transitivity Theorem states that if ''X'' is a [[Second-countable space|second countable]], [[complete metric space]], then topological transitivity implies the existence of a [[dense set]] of points in ''X'' that have dense orbits.<ref>{{harvnb|Robinson|1995}}</ref>

===Density of periodic orbits===
For a chaotic system to have [[Dense set|dense]] [[periodic orbits]] means that every point in the space is approached arbitrarily closely by periodic orbits.<ref name="Devaney"/> The one-dimensional [[logistic map]] defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> is one of the simplest systems with density of periodic orbits. For example, <math>\tfrac{5-\sqrt{5}}{8}</math>&nbsp;→ <math>\tfrac{5+\sqrt{5}}{8}</math>&nbsp;→ <math>\tfrac{5-\sqrt{5}}{8}</math> (or approximately 0.3454915&nbsp;→ 0.9045085&nbsp;→ 0.3454915) is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by [[Sharkovskii's theorem]]).<ref>{{harvnb|Alligood|Sauer|Yorke|1997}}</ref>

Sharkovskii's theorem is the basis of the Li and Yorke<ref>{{cite journal|last1=Li |first1=T.Y. |last2=Yorke |first2=J.A. |title=Period Three Implies Chaos |journal=[[American Mathematical Monthly]] |volume=82 |pages=985–92 |year=1975 |url=http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |author-link=Tien-Yien Li |doi=10.2307/2318254 |issue=10 |author2-link=James A. Yorke |bibcode=1975AmMM...82..985L |archive-url=https://web.archive.org/web/20091229042210/http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |archive-date=2009-12-29 |jstor=2318254 |citeseerx=10.1.1.329.5038 }}</ref> (1975) proof that any continuous one-dimensional system that exhibits a regular cycle of period three will also display regular cycles of every other length, as well as completely chaotic orbits.

===Strange attractors===
[[File:TwoLorenzOrbits.jpg|thumb|right|The [[Lorenz attractor]] displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.]]

Some dynamical systems, like the one-dimensional [[logistic map]] defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x''),</span> are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an [[attractor]], since then a large set of initial conditions leads to orbits that converge to this chaotic region.<ref>{{cite journal|last1=Strelioff|first1=Christopher|last2=et.|first2=al.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|article-number=044101|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|bibcode = 2006PhRvL..96d4101S }}</ref>

An easy way to visualize a chaotic attractor is to start with a point in the [[basin of attraction]] of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the [[Edward Lorenz|Lorenz]] weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it is not only one of the first, but it is also one of the most complex, and as such gives rise to a very interesting pattern that, with a little imagination, looks like the wings of a butterfly.

Unlike [[Attractor#Fixed point|fixed-point attractors]] and [[limit cycle]]s, the attractors that arise from chaotic systems, known as [[strange attractor]]s, have great detail and complexity. Strange attractors occur in both [[continuous function|continuous]] dynamical systems (such as the Lorenz system) and in some [[discrete mathematics|discrete]] systems (such as the [[Hénon map]]). Other discrete dynamical systems have a repelling structure called a [[Julia set]], which forms at the boundary between basins of attraction of fixed points. Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a [[fractal]] structure, and the [[fractal dimension]] can be calculated for them.

=== Coexisting attractors ===
[[File:Coexisting Attractors.png|thumb|Coexisting chaotic and non-chaotic attractors within the generalized Lorenz model.<ref name=":4" /><ref name=":5" /><ref name=":6" /> There are 128 orbits in different colors, beginning with different initial conditions for dimensionless time between 0.625 and 5 and a heating parameter r = 680. Chaotic orbits recurrently return close to the saddle point at the origin. Nonchaotic orbits eventually approach one of two stable critical points, as shown with large blue dots. Chaotic and nonchaotic orbits occupy different regions of attraction within the phase space.]]

In contrast to single type chaotic solutions, studies using Lorenz models<ref>{{Cite journal |last1=Yorke |first1=James A. |last2=Yorke |first2=Ellen D. |date=1979-09-01 |title=Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model |journal=Journal of Statistical Physics |language=en |volume=21 |issue=3 |pages=263–277 |doi=10.1007/BF01011469 |bibcode=1979JSP....21..263Y |s2cid=12172750 |issn=1572-9613}}</ref><ref>{{Cite book |last1=Shen |first1=Bo-Wen |last2=Pielke Sr. |first2=R. A. |last3=Zeng |first3=X. |last4=Baik |first4=J.-J. |last5=Faghih-Naini |first5=S. |last6=Cui |first6=J. |last7=Atlas |first7=R. |last8=Reyes |first8=T. A. L. |title=13th Chaotic Modeling and Simulation International Conference |chapter=Is Weather Chaotic? Coexisting Chaotic and Non-chaotic Attractors within Lorenz Models |date=2021 |editor-last=Skiadas |editor-first=Christos H. |editor2-last=Dimotikalis |editor2-first=Yiannis |chapter-url=https://link.springer.com/chapter/10.1007/978-3-030-70795-8_57 |series=Springer Proceedings in Complexity |language=en |location=Cham |publisher=Springer International Publishing |pages=805–825 |doi=10.1007/978-3-030-70795-8_57 |isbn=978-3-030-70795-8|s2cid=245197840 }}</ref> have emphasized the importance of considering various types of solutions. For example, coexisting chaotic and non-chaotic may appear within the same model (e.g., the double pendulum system) using the same modeling configurations but different initial conditions. The findings of attractor coexistence, obtained from classical and generalized Lorenz models,<ref name=":4" /><ref name=":5" /><ref name=":6">{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke Sr. |first2=Roger Pielke |last3=Zeng |first3=Xubin |last4=Cui |first4=Jialin |last5=Faghih-Naini |first5=Sara |last6=Paxson |first6=Wei |last7=Kesarkar |first7=Amit |last8=Zeng |first8=Xiping |last9=Atlas |first9=Robert |date=2022-11-12 |title=The Dual Nature of Chaos and Order in the Atmosphere |journal=Atmosphere |language=en |volume=13 |issue=11 |page=1892 |doi=10.3390/atmos13111892 |bibcode=2022Atmos..13.1892S |issn=2073-4433|doi-access=free |hdl=10150/673501 |hdl-access=free }}</ref> suggested a revised view that "the entirety of weather possesses a dual nature of chaos and order with distinct predictability", in contrast to the conventional view of "weather is chaotic".

===Minimum complexity of a chaotic system===
[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|thumb|right|[[Bifurcation diagram]] of the [[logistic map]] <span style="white-space: nowrap;">''x'' → ''r'' ''x'' (1 – ''x'').</span> Each vertical slice shows the attractor for a specific value of ''r''. The diagram displays [[Period-doubling bifurcation|period-doubling]] as ''r'' increases, eventually producing chaos. Darker points are visited more frequently.]]

Discrete chaotic systems, such as the [[logistic map]], can exhibit strange attractors whatever their [[dimension]]ality. In contrast, for [[continuous function (topology)|continuous]] dynamical systems, the [[Poincaré–Bendixson theorem]] shows that a strange attractor can only arise in three or more dimensions. [[Dimension (vector space)|Finite-dimensional]] [[linear system]]s are never chaotic; for a dynamical system to display chaotic behavior, it must be either [[nonlinearity|nonlinear]] or infinite-dimensional.

The [[Poincaré–Bendixson theorem]] states that a two-dimensional differential equation has very regular behavior. The Lorenz attractor discussed below is generated by a system of three [[differential equation]]s such as:

: <math> \begin{align}
\frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma y - \sigma x, \\
\frac{\mathrm{d}y}{\mathrm{d}t} &= \rho x - x z - y, \\
\frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z.
\end{align} </math>

where <math>x</math>, <math>y</math>, and <math>z</math> make up the [[State space representation|system state]], <math>t</math> is time, and <math>\sigma</math>, <math>\rho</math>, <math>\beta</math> are the system [[parameter]]s. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor is generated by the [[Rössler map|Rössler equations]], which have only one nonlinear term out of seven. Sprott<ref>{{cite journal|last=Sprott |first=J.C.|year=1997|title=Simplest dissipative chaotic flow|journal=[[Physics Letters A]]|volume=228|issue=4–5 |pages=271–274|doi=10.1016/S0375-9601(97)00088-1|bibcode = 1997PhLA..228..271S }}</ref> found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel<ref>{{cite journal|last1=Fu |first1=Z. |last2=Heidel |first2=J.|year=1997|title=Non-chaotic behaviour in three-dimensional quadratic systems|journal=[[Nonlinearity (journal)|Nonlinearity]]|volume=10|issue=5 |pages=1289–1303|doi=10.1088/0951-7715/10/5/014
|bibcode = 1997Nonli..10.1289F |s2cid=250757113 }}</ref><ref>{{cite journal|last1=Heidel |first1=J. |last2=Fu |first2=Z.|year=1999|title=Nonchaotic behaviour in three-dimensional quadratic systems II. The conservative case|journal=Nonlinearity|volume=12|issue=3 |pages=617–633|doi=10.1088/0951-7715/12/3/012|bibcode = 1999Nonli..12..617H |s2cid=250853499 }}</ref> showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are [[Asymptotic analysis|asymptotic]] to a two-dimensional surface and therefore solutions are well behaved.

While the Poincaré–Bendixson theorem shows that a continuous dynamical system on the Euclidean [[plane (mathematics)|plane]] cannot be chaotic, two-dimensional continuous systems with [[non-Euclidean geometry]] can still exhibit some chaotic properties.<ref>{{Cite journal |last=Ulcigrai |first=Corinna |date=2021 |title=Slow chaos in surface flows |url=http://link.springer.com/10.1007/s40574-020-00267-0 |journal=Bollettino dell'Unione Matematica Italiana |language=en |volume=14 |issue=1 |pages=231–255 |arxiv=2010.06231 |doi=10.1007/s40574-020-00267-0 |issn=1972-6724}}</ref> 

The above set of three ordinary differential equations has been referred to as the three-dimensional Lorenz model.<ref>{{Cite journal |last=Shen |first=Bo-Wen |date=2014-05-01 |title=Nonlinear Feedback in a Five-Dimensional Lorenz Model |url=https://journals.ametsoc.org/doi/10.1175/JAS-D-13-0223.1 |journal=Journal of the Atmospheric Sciences |language=en |volume=71 |issue=5 |pages=1701–1723 |doi=10.1175/JAS-D-13-0223.1 |bibcode=2014JAtS...71.1701S |s2cid=123683839 |issn=0022-4928|url-access=subscription }}</ref> Since 1963, higher-dimensional Lorenz models have been developed in numerous studies<ref>{{Cite journal |last1=Musielak |first1=Dora E. |last2=Musielak |first2=Zdzislaw E. |last3=Kennamer |first3=Kenny S. |date=2005-03-01 |title=The onset of chaos in nonlinear dynamical systems determined with a new fractal technique |url=https://www.worldscientific.com/doi/abs/10.1142/S0218348X0500274X |journal=Fractals |volume=13 |issue=1 |pages=19–31 |doi=10.1142/S0218348X0500274X |issn=0218-348X|url-access=subscription }}</ref><ref>{{Cite journal |last1=Roy |first1=D. |last2=Musielak |first2=Z. E. |date=2007-05-01 |title=Generalized Lorenz models and their routes to chaos. I. Energy-conserving vertical mode truncations |url=https://www.sciencedirect.com/science/article/pii/S0960077906001937 |journal=Chaos, Solitons & Fractals |language=en |volume=32 |issue=3 |pages=1038–1052 |doi=10.1016/j.chaos.2006.02.013 |bibcode=2007CSF....32.1038R |issn=0960-0779|url-access=subscription }}</ref><ref name=":4">{{Cite journal |last=Shen |first=Bo-Wen |date=2019-03-01 |title=Aggregated Negative Feedback in a Generalized Lorenz Model |journal=International Journal of Bifurcation and Chaos |volume=29 |issue=3 |pages=1950037–1950091 |doi=10.1142/S0218127419500378 |bibcode=2019IJBC...2950037S |s2cid=132494234 |issn=0218-1274|doi-access=free }}</ref><ref name=":5">{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Baik |first4=Jong-Jin |last5=Faghih-Naini |first5=Sara |last6=Cui |first6=Jialin |last7=Atlas |first7=Robert |date=2021-01-01 |title=Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model |journal=Bulletin of the American Meteorological Society |language=EN |volume=102 |issue=1 |pages=E148–E158 |doi=10.1175/BAMS-D-19-0165.1 |bibcode=2021BAMS..102E.148S |s2cid=208369617 |issn=0003-0007|doi-access=free }}</ref> for examining the impact of an increased degree of nonlinearity, as well as its collective effect with heating and dissipations, on solution stability.
===Chaos and linear systems===
Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional.<ref>{{cite journal
 |last1=Bonet |first1=J. |last2=Martínez-Giménez |first2=F. |last3=Peris |first3=A.
 |year=2001
 |title=A Banach space which admits no chaotic operator
 |journal=Bulletin of the London Mathematical Society
 |volume=33
 |issue=2 |pages=196–8
 |doi=10.1112/blms/33.2.196
|s2cid=121429354 }}</ref> A theory of linear chaos is being developed in [[functional analysis]].<ref>{{cite book | title=Linear Chaos | series=Universitext | date=2011 | doi=10.1007/978-1-4471-2170-1 | isbn=978-1-4471-2169-5 | url=https://link.springer.com/book/10.1007/978-1-4471-2170-1 }}</ref>
[[Quantum mechanics]] is also often considered as a prime example of linear non chaotic theory, that dampens out chaotic behaviour in the same manner that [[viscosity]] dampens out [[turbulence]], this is actually not the case for quantum mechanical systems with infinite degrees of freedom, such as strongly correlated systems that do exhibit forms of [[Quantum turbulence|nano scale turbulence]].<ref>{{Cite journal |last=Zaanen |first=Jan |date=2019-05-20 |title=Planckian dissipation, minimal viscosity and the transport in cuprate strange metals |journal=SciPost Physics |language=en |volume=6 |issue=5 |article-number=061 |doi=10.21468/SciPostPhys.6.5.061 |arxiv=1807.10951 |bibcode=2019ScPP....6...61Z |doi-access=free |issn=2542-4653}}</ref><ref>{{cite journal | last1=Guthrie | first1=A. | last2=Kafanov | first2=S. | last3=Noble | first3=M. T. | last4=Pashkin | first4=Yu. A. | last5=Pickett | first5=G. R. | last6=Tsepelin | first6=V. | last7=Dorofeev | first7=A. A. | last8=Krupenin | first8=V. A. | last9=Presnov | first9=D. E. | title=Nanoscale real-time detection of quantum vortices at millikelvin temperatures | journal=Nature Communications | date=2021 | volume=12 | issue=1 | article-number=2645 | doi=10.1038/s41467-021-22909-3 | pmid=33976214 | pmc=8113507 | arxiv=2007.04482 | bibcode=2021NatCo..12.2645G }}</ref>

==Other characteristics of Chaos==
===Infinite dimensional maps===
The straightforward generalization of coupled discrete maps<ref name="Moloney, J V 1986">{{cite journal |title=Solitary waves as fixed points of infinite-dimensional maps for an optical bistable ring cavity: Analysis
|journal=Journal of Mathematical Physics|volume=29 |issue=1 |page=63 |year=1988
|last1= Adachihara |first1=H
|last2= McLaughlin |first2=D W
|last3= Moloney |first3=J V
|last4= Newell |first4=A C
|doi=10.1063/1.528136 |bibcode=1988JMP....29...63A}}</ref> is based upon convolution integral which mediates interaction between spatially distributed maps:
<math>\psi_{n+1}(\vec r,t)  = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}</math>,

where kernel <math>K(\vec r - \vec r^{,},t)</math> is propagator derived as Green function of a relevant physical system,<ref name="Okulov, A Yu 1988">{{cite book |chapter=Spatiotemporal dynamics of a wave packet in nonlinear medium and discrete maps
|title=Proceedings of the Lebedev Physics Institute |language=ru |editor=N.G. Basov |publisher=Nauka |lccn=88174540 |volume=187 |pages=202–222 |year=1988 |last1= Okulov |first1=A Yu |last2=Oraevskiĭ |first2=A N }}</ref>

<math> f [\psi_{n}(\vec r,t) ] </math> might be logistic map alike <math> \psi \rightarrow G \psi [1 - \tanh (\psi)]</math> or [[complex map]]. For examples of complex maps the [[Julia set]] <math> f[\psi] = \psi^2</math> or [[Ikeda map]]
<math> \psi_{n+1} = A + B \psi_n e^{i (|\psi_n|^2 + C)} </math> may serve. When wave propagation problems at distance <math>L=ct</math> with wavelength <math>\lambda=2\pi/k</math> are considered the kernel <math>K</math> may have a form of Green function for [[Schrödinger equation]]:.<ref name="Okulov, A Yu 2000">{{cite journal |title=Spatial soliton laser: geometry and stability
|journal=Optics and Spectroscopy|volume=89 |issue=1 |pages=145–147 |year=2000 |last1= Okulov |first1=A Yu|s2cid=122790937|doi=10.1134/BF03356001  |bibcode=2000OptSp..89..131O}}</ref><ref name="Okulov, A Yu 2020">{{cite journal |title=Structured light entities, chaos and nonlocal maps
|journal=Chaos, Solitons & Fractals|volume=133 |issue=4|article-number=109638 |year=2020|last1= Okulov |first1=A Yu|doi=10.1016/j.chaos.2020.109638|arxiv=1901.09274|bibcode=2020CSF...13309638O|s2cid=118828500}}</ref>

<math> K(\vec r - \vec r^{,},L)  = \frac {ik\exp[ikL]}{2\pi L}\exp[\frac {ik|\vec r-\vec r^{,}|^2}{2 L} ]</math>.

===Spontaneous order===
Under the right conditions, chaos spontaneously evolves into a lockstep pattern. In the [[Kuramoto model]], four conditions suffice to produce synchronization in a chaotic system.
Examples include the [[coupled oscillation]] of [[Christiaan Huygens]]' pendulums, fireflies, [[neuron]]s, the [[London Millennium Bridge]] resonance, and large arrays of [[Josephson junctions]].<ref>Steven Strogatz, ''Sync: The Emerging Science of Spontaneous Order'', Hyperion, 2003.</ref>

Moreover, from the theoretical physics standpoint, dynamical chaos itself, in its most general manifestation, is a spontaneous order. The essence here is that most orders in nature arise from the [[Spontaneous symmetry breaking|spontaneous breakdown]] of various symmetries. This large family of phenomena includes elasticity, superconductivity, ferromagnetism, and many others. According to the [[supersymmetric theory of stochastic dynamics]], chaos, or more precisely, its stochastic generalization, is also part of this family. The corresponding symmetry being broken is the [[Topological quantum field theory|topological supersymmetry]] which is hidden in all [[Stochastic differential equation|stochastic (partial) differential equations]], and the corresponding [[Phase transition|order parameter]] is a [[Quantum field theory|field-theoretic]] embodiment of the butterfly effect.<ref>{{Cite journal |last=Ovchinnikov |first=I.V. |title=Ubiquitous order known as chaos |date=2024-02-15 |journal=Chaos, Solitons & Fractals |language=en |volume=181 |issue=5 |article-number=114611 |doi=10.1016/j.chaos.2024.114611 |arxiv=2503.17157 |bibcode=2024CSF...18114611O |url=https://www.sciencedirect.com/science/article/abs/pii/S0960077924001620 |issn = 0960-0779}}</ref>
===Combinatorial (or complex) chaos===
There are also definitions of chaos that don't require the sensitivity on initial conditions property, such as combinatorial chaos (I.e. applying recursively a discrete combinatorial action).<ref>{{cite web | title=Science: Mathematician discovers a more complex form of chaos | url=https://www.newscientist.com/article/mg12617232-800-science-mathematician-discovers-a-more-complex-form-of-chaos/ }}</ref> This is also comparable and similar to chaos generated by [[cellular automata]]. This is important because this type of chaos it's also equivalent to a [[turing machine]], you can execute computation with such dynamical systems, and as such the [[halting problem]] is not decidable, therefore some computational algorithms may never end. This is ultimately a very different way for a system to be unpredictable.
<ref>{{cite web | title='Next-Level' Chaos Traces the True Limit of Predictability | date=7 March 2025 | url=https://www.quantamagazine.org/next-level-chaos-traces-the-true-limit-of-predictability-20250307/ }}</ref>

==History==
[[File:Barnsley fern plotted with VisSim.PNG|thumb|upright|[[Barnsley fern]] created using the [[chaos game]]. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an [[iterated function system]] (IFS).]]

[[James Clerk Maxwell]] was the first scientist to emphasize the importance of initial conditions, and he is seen as being one of the earliest to discuss chaos theory, with work in the 1860s and 1870s.<ref>{{Cite journal |last1=Hunt |first1=Brian R. |last2=Yorke |first2=James A. |date=1993 |title=Maxwell on Chaos |url=https://yorke.umd.edu/Yorke_papers_most_cited_and_post2000/1993_04_Hunt_%20Nonlin-Science-Today%20_Maxwell%20on%20Chaos.PDF |journal=Nonlinear Science Today |volume=3 |issue=1}}</ref><ref>{{Cite web |last=Everitt |first=Francis |date=2006-12-01 |title=James Clerk Maxwell: a force for physics |url=https://physicsworld.com/a/james-clerk-maxwell-a-force-for-physics/ |access-date=2023-11-03 |website=Physics World |language=en-GB}}</ref><ref>{{Cite journal |last1=Gardini |first1=Laura |last2=Grebogi |first2=Celso |last3=Lenci |first3=Stefano |date=2020-10-01 |title=Chaos theory and applications: a retrospective on lessons learned and missed or new opportunities |journal=Nonlinear Dynamics |language=en |volume=102 |issue=2 |pages=643–644 |doi=10.1007/s11071-020-05903-0 |s2cid=225246631 |issn=1573-269X|doi-access=free |bibcode=2020NonDy.102..643G |hdl=2164/17003 |hdl-access=free }}</ref> In the 1880s, while studying the [[three-body problem]], [[Henri Poincaré]] found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.<ref>{{cite journal |author=Poincaré, Jules Henri |title=Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt |journal=Acta Mathematica |volume=13 |issue=1–2 |pages=1–270 |year=1890 |doi=10.1007/BF02392506 |doi-access=free }}</ref><ref>{{Cite book|title=The three-body problem and the equations of dynamics: Poincaré's foundational work on dynamical systems theory|last=Poincaré|first=J. Henri|publisher=Springer International Publishing|others=Popp, Bruce D. (Translator)|year=2017|isbn=978-3-319-52898-4|location=Cham, Switzerland|oclc=987302273}}</ref><ref>{{cite book |author1=Diacu, Florin |author2=Holmes, Philip |title=Celestial Encounters: The Origins of Chaos and Stability |publisher=[[Princeton University Press]] |year=1996 }}</ref> In 1898, [[Jacques Hadamard]] published an influential study of the motion of a free particle gliding frictionlessly on a surface of constant negative curvature, called "[[Hadamard's billiards]]". Hadamard was able to show that all trajectories are unstable,<ref>{{cite journal|first = Jacques|last = Hadamard|year = 1898|title = Les surfaces à courbures opposées et leurs lignes géodesiques|journal = Journal de Mathématiques Pures et Appliquées|volume = 4|pages = 27–73|url=https://www.numdam.org/item/JMPA_1898_5_4__27_0.pdf}}</ref> in that all particle trajectories diverge exponentially from one another, with a positive [[Lyapunov exponent]].<ref>{{cite journal|first1=R.|last1=Aurich|first2=M.|last2=Sieber|first3=F.|last3=Steiner|title=Quantum Chaos of the Hadamard–Gutzwiller Model|journal=Physical Review Letters|date=1 August 1988|pages=483–487|volume=61|issue=5|doi=10.1103/PhysRevLett.61.483|pmid=10039347|bibcode=1988PhRvL..61..483A|url=http://bib-pubdb1.desy.de/record/324018/files/PhysRevLett61483.pdf}} Lyapunov exponent was first published in 1897 in french, and later in 1907, is not mentioned by hadamard 1898 which derives a [[hill differential equation]] with negative curvature, and therefore a hyperbolic geometry in phase space, i.e. divergent orbits, but does not actually compute any exponent. In this PRL paper instead the lyapunov exponent is explicit.</ref>

Later studies, also on the topic of nonlinear [[differential equations]], were carried out by [[George David Birkhoff]],<ref>George D. Birkhoff, ''Dynamical Systems,'' vol.&nbsp;9 of the American Mathematical Society Colloquium Publications (Providence, Rhode Island: American Mathematical Society, 1927)</ref> [[Andrey Nikolaevich Kolmogorov]],<ref>{{cite journal| last=Kolmogorov | first=Andrey Nikolaevich | author-link=Andrey Nikolaevich Kolmogorov | year=1941 | title=Local structure of turbulence in an incompressible fluid for very large Reynolds numbers | journal=[[Doklady Akademii Nauk SSSR]] | volume=30 | issue=4 | pages=301–5 |bibcode = 1941DoSSR..30..301K | title-link=turbulence }} Reprinted in: {{cite journal |journal=Proceedings of the Royal Society A |volume=434 |pages=9–13 |year=1991 |doi=10.1098/rspa.1991.0075 |title=The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers |last1=Kolmogorov |first1=A. N. |s2cid=123612939 |issue=1890 |bibcode=1991RSPSA.434....9K}}</ref><ref>{{cite journal| last=Kolmogorov | first=A. N. | year=1941 | title=On degeneration of isotropic turbulence in an incompressible viscous liquid | journal=Doklady Akademii Nauk SSSR | volume=31 | issue=6 | pages=538–540}} Reprinted in: {{cite journal |journal=Proceedings of the Royal Society A |volume=434 |pages=15–17 |year=1991 |doi=10.1098/rspa.1991.0076 |title=Dissipation of Energy in the Locally Isotropic Turbulence |last1=Kolmogorov |first1=A. N. |s2cid=122060992 |issue=1890 |bibcode=1991RSPSA.434...15K}}</ref><ref>{{cite book| last=Kolmogorov | first=A. N. | title=Stochastic Behavior in Classical and Quantum Hamiltonian Systems | chapter=Preservation of conditionally periodic movements with small change in the Hamilton function|pages=51–56| bibcode=1979LNP....93...51K| doi=10.1007/BFb0021737| series=Lecture Notes in Physics| date=1979 | volume=93 | isbn=978-3-540-09120-2}} Translation of ''Doklady Akademii Nauk SSSR'' (1954) 98: 527. See also [[Kolmogorov–Arnold–Moser theorem]]</ref> [[Mary Lucy Cartwright]] and [[John Edensor Littlewood]],<ref>{{cite journal |last1=Cartwright |first1=Mary L. |last2=Littlewood |first2=John E. |title=On non-linear differential equations of the second order, I: The equation ''y''" + ''k''(1−''y''<sup>2</sup>)''y<nowiki>'</nowiki>'' + ''y'' = ''b''λkcos(λ''t'' + ''a''), ''k'' large |journal=Journal of the London Mathematical Society |volume=20 |pages=180–9 |year=1945 |doi=10.1112/jlms/s1-20.3.180 |issue=3 }} See also: [[Van der Pol oscillator]]</ref> and [[Stephen Smale]].<ref>{{cite journal |author=Smale, Stephen |title=Morse inequalities for a dynamical system |journal=Bulletin of the American Mathematical Society |volume=66 |pages=43–49 |date=January 1960 |doi=10.1090/S0002-9904-1960-10386-2 |doi-access=free }}</ref> Experimentalists and mathematicians had encountered turbulence in fluid motion, chaotic behaviour in society and economy, nonperiodic oscillation in radio circuits and fractal patterns in nature without the benefit of a theory to explain what they were seeing.

Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that [[linear theory]] which is [[smoothness|smooth]] and [[Continuous function|continuous]], and which was the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the [[logistic map]] which has [[jump process|jump]] and [[stochastic|erratic]] behaviours. Both of these observations underline the connection of chaos to either [[stochastic]] or [[non-linear]] dynamical systems, but definitely non-[[differentiable]] and non-[[continuous function|continuous]] time evolution.

What had been attributed to measure imprecision and simple "[[Pseudorandom noise|noise]]" was considered by chaos theorists as a full component of the studied systems. In 1959 [[Boris Chirikov|Boris Valerianovich Chirikov]] proposed a criterion for the emergence of classical chaos in Hamiltonian systems ([[Chirikov criterion]]). He applied this criterion to explain some experimental results on [[plasma confinement]] in open mirror traps.<ref>{{Cite journal |last=Chirikov |first=Boris |title=РЕЗОНАНСНЫЕ ПРОЦЕССЫ В МАГНИТНЫХ ЛОВУШКАХ |url=https://www.quantware.ups-tlse.fr/chirikov/refs/chi1959.pdf |journal=Атомная энергия |volume=6}}</ref><ref>{{Cite journal |last=Chirikov |first=B. V. |date=1960-12-01 |title=Resonance processes in magnetic traps |journal=The Soviet Journal of Atomic Energy |language=en |volume=6 |issue=6 |pages=464–470 |doi=10.1007/BF01483352 |s2cid=59483478 |issn=1573-8205}}</ref> This is regarded as the very first physical theory of chaos, which succeeded in explaining a concrete experiment. And Boris Chirikov himself is considered as a pioneer in classical and quantum chaos.<ref>{{Cite journal |last1=Jean |first1=Bellissard |author-link=Jean Bellissard |last2=Dima |first2=Shepelyansky |date=27 February 1998 |title=Boris Chirikov, a pioneer in classical and quantum chaos |url=https://www.quantware.ups-tlse.fr/chirikov/aboutchirikov/poincare1998.pdf |journal=[[Annales Henri Poincaré]] |volume=68 |issue=4 |page=379}}</ref><ref>{{Cite journal |last1=Bellissard |first1=J. |last2=Bohigas |first2=O. |last3=Casati |first3=G. |last4=Shepelyansky |first4=D.L. |date=1 July 1999 |title=A pioneer of chaos |journal=Physica D: Nonlinear Phenomena |volume=131 |issue=1–4 |pages=viii–xv |doi=10.1016/s0167-2789(99)90007-6 |arxiv=cond-mat/9903412 |bibcode=1999PhyD..131D...8B |s2cid=119107150 |issn=0167-2789}}</ref><ref>{{Cite book |first=Dima |last=Shepelyansky |title=Chaos at Fifty Four in 2013 |oclc=859751750}}</ref>

The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated [[iteration]] of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems. As a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on November 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970.<ref>{{harvnb|Abraham|Ueda|2000|loc=See Chapters 3 and 4}}</ref><ref>{{harvnb|Sprott|2003|p=[https://books.google.com/books?id=SEDjdjPZ158C&pg=PA89 89]}}</ref>

[[File:Airplane vortex edit.jpg|thumb|left|[[Turbulence]] in the [[Wingtip vortices|tip vortex]] from an [[airplane]] wing. Studies of the critical point beyond which a system creates turbulence were important for chaos theory, analyzed for example by the [[Soviet physicists|Soviet physicist]] [[Lev Landau]], who developed the [[Landau-Hopf theory of turbulence]]. [[David Ruelle]] and [[Floris Takens]] later predicted, against Landau, that [[fluid turbulence]] could develop through a [[strange attractor]], a main concept of chaos theory.]]

[[Edward Lorenz]] was an early pioneer of the theory. His interest in chaos came about accidentally through his work on [[meteorology|weather prediction]] in 1961.<ref>Lorenz, E.N. The statistical prediction of solutions of dynamic equations. In Proceedings of the International Symposium on Numerical Weather Prediction, Tokyo, Japan, 7–13 November 1962; pp. 629–635. {{Citation |title=Numerical solutions to the equations |date=2010-12-02 |work=Numerical Weather and Climate Prediction |pages=17–118 |publisher=Cambridge University Press|doi=10.1017/cbo9780511763243.004 |isbn=978-0-521-51389-0 }}</ref><ref name=Lorenz1961>{{cite journal |author=Lorenz, Edward N. |title=Deterministic non-periodic flow |journal=Journal of the Atmospheric Sciences |volume=20 |pages=130–141 |year=1963 |doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |issue=2 |bibcode=1963JAtS...20..130L|doi-access=free }}</ref> Lorenz and his collaborator [[Ellen Fetter]] and [[Margaret Hamilton (software engineer)|Margaret Hamilton]]<ref>{{Cite web |last=Sokol |first=Joshua |date=May 20, 2019 |title=The Hidden Heroines of Chaos |url=https://www.quantamagazine.org/the-hidden-heroines-of-chaos-20190520/ |access-date=2022-11-09 |website=Quanta Magazine}}</ref> were using a simple digital computer, a [[Royal McBee]] [[LGP-30]], to run weather simulations. They wanted to see a sequence of data again, and to save time they started the simulation in the middle of its course. They did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To their surprise, the weather the machine began to predict was completely different from the previous calculation. They tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome.<ref>{{cite book|title=Chaos: Making a New Science |last=Gleick |first=James |year=1987 |publisher=Cardinal |location=London|page=17|isbn=978-0-434-29554-8|title-link=Chaos: Making a New Science }}</ref> Lorenz's discovery, which gave its name to [[Lorenz attractor]]s, showed that even detailed atmospheric modeling cannot, in general, make precise long-term weather predictions.

In 1963, [[Benoit Mandelbrot]], studying [[information theory]], discovered that noise in many phenomena (including [[stock market|stock prices]] and [[telephone]] circuits) was patterned like a [[Cantor set]], a set of points with infinite roughness and detail.<ref>{{cite journal |author1=Berger J.M. |author2=Mandelbrot B. | year = 1963 | title = A new model for error clustering in telephone circuits | journal = IBM Journal of Research and Development | volume = 7 |issue=3 | pages = 224–236 | doi=10.1147/rd.73.0224}}</ref> Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards).<ref>{{cite book |author=Mandelbrot, B. |title=The Fractal Geometry of Nature |publisher=Freeman |location=New York |year=1977 |page=248 }}</ref><ref>See also: {{cite book |last1=Mandelbrot |first1=Benoît B. |last2=Hudson |first2=Richard L. |title=The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward |url=https://archive.org/details/misbehaviorofmar00beno |url-access=registration |publisher=Basic Books |location=New York |year=2004 |page=[https://archive.org/details/misbehaviorofmar00beno/page/201 201] |isbn=978-0-465-04355-2 }}</ref> In 1967, he published "[[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|How long is the coast of Britain? Statistical self-similarity and fractional dimension]]", showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an [[infinitesimal]]ly small measuring device.<ref>{{cite journal |last=Mandelbrot |first=Benoît |title=How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension |journal=Science |volume=156 |issue=3775 |pages=636–8 |date=5 May 1967 |doi=10.1126/science.156.3775.636 |pmid=17837158 |bibcode=1967Sci...156..636M |s2cid=15662830 |url=http://ena.lp.edu.ua:8080/handle/ntb/52473 |access-date=31 January 2022 |archive-date=19 October 2021 |archive-url=https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473 }}</ref> Arguing that a ball of twine appears as a point when viewed from far away (0-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand (1-dimensional), he argued that the dimensions of an object are relative to the observer and may be fractional. An object whose irregularity is constant over different scales ("self-similarity") is a [[fractal]] (examples include the [[Menger sponge]], the [[Sierpiński gasket]], and the [[Koch curve]] or ''snowflake'', which is infinitely long yet encloses a finite space and has a [[fractal dimension]] of circa 1.2619). In 1982, Mandelbrot published ''[[The Fractal Geometry of Nature]]'', which became a classic of chaos theory.<ref>{{cite book|author=Mandelbrot, B.|title=The Fractal Geometry of Nature|date=1982|place=New York|publisher=Macmillan|isbn=978-0-7167-1186-5|url=https://archive.org/details/fractalgeometryo00beno}}</ref>

In December 1977, the [[New York Academy of Sciences]] organized the first symposium on chaos, attended by David Ruelle, [[Robert May, Baron May of Oxford|Robert May]], [[James A. Yorke]] (coiner of the term "chaos" as used in mathematics), [[Robert Shaw (physicist)|Robert Shaw]], and the meteorologist Edward Lorenz. The following year Pierre Coullet and Charles Tresser published "Itérations d'endomorphismes et groupe de renormalisation", and [[Mitchell Feigenbaum]]'s article "Quantitative Universality for a Class of Nonlinear Transformations" finally appeared in a journal, after 3 years of referee rejections.<ref name="Feigenbaum 25–52">{{cite journal |first=Mitchell |last=Feigenbaum |title=Quantitative universality for a class of nonlinear transformations |journal=Journal of Statistical Physics |volume=19 |issue=1 |pages=25–52 |date=July 1978 |doi=10.1007/BF01020332 |bibcode=1978JSP....19...25F|citeseerx=10.1.1.418.9339 |s2cid=124498882 }}</ref><ref>Coullet, Pierre, and Charles Tresser. "Iterations d'endomorphismes et groupe de renormalisation." Le Journal de Physique Colloques 39.C5 (1978): C5-25</ref> Thus Feigenbaum (1975) and Coullet & Tresser (1978) discovered the [[Universality (dynamical systems)|universality]] in chaos, permitting the application of chaos theory to many different phenomena.

In 1979, [[Albert J. Libchaber]], during a symposium organized in Aspen by [[Pierre Hohenberg]], presented his experimental observation of the [[Bifurcation theory|bifurcation]] cascade that leads to chaos and turbulence in [[Rayleigh–Bénard convection]] systems. He was awarded the [[Wolf Prize in Physics]] in 1986 along with [[Mitchell J. Feigenbaum]] for their inspiring achievements.<ref>{{cite web|url = http://www.wolffund.org.il/cat.asp?id=25&cat_title=PHYSICS|title = The Wolf Prize in Physics in 1986.|access-date = 2008-01-17|archive-date = 2024-05-25|archive-url = https://archive.today/20240525172450/https://www.webcitation.org/65DR3JlSI?url=http://www.wolffund.org.il/cat.asp%3Fid=25}}</ref>

In 1986, the New York Academy of Sciences co-organized with the [[National Institute of Mental Health]] and the [[Office of Naval Research]] the first important conference on chaos in biology and medicine. There, [[Bernardo Huberman]] presented a mathematical model of the [[Eye movement|eye tracking]] dysfunction among people with [[schizophrenia]].<ref>{{cite journal |author-link=Bernardo Huberman |author=Huberman, B.A. |title=A Model for Dysfunctions in Smooth Pursuit Eye Movement |journal=Annals of the New York Academy of Sciences |volume=504 Perspectives in Biological Dynamics and Theoretical Medicine |issue=1 |pages=260–273 |date=July 1987 |doi=10.1111/j.1749-6632.1987.tb48737.x |pmid=3477120 |bibcode = 1987NYASA.504..260H |s2cid=42733652 }}</ref> This led to a renewal of [[physiology]] in the 1980s through the application of chaos theory, for example, in the study of pathological [[cardiac cycle]]s.

In 1987, [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] published a paper in ''[[Physical Review Letters]]''<ref>{{cite journal |author1=Bak, Per |author2=Tang, Chao |author3=Wiesenfeld, Kurt |title=Self-organized criticality: An explanation of the 1/f noise |journal=Physical Review Letters |volume=59 |issue=4 |pages=381–4 |date=27 July 1987 |doi=10.1103/PhysRevLett.59.381 |pmid=10035754 |bibcode=1987PhRvL..59..381B|s2cid=7674321 }} However, the conclusions of this article have been subject to dispute. {{cite web|url=http://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |title=? |archive-url=https://web.archive.org/web/20071214033929/https://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |archive-date=2007-12-14 }}. See especially: {{cite journal |author1=Laurson, Lasse |author2=Alava, Mikko J. |author3=Zapperi, Stefano |title=Letter: Power spectra of self-organized critical sand piles |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=0511 |id=L001 |date=15 September 2005 }}</ref> describing for the first time [[self-organized criticality]] (SOC), considered one of the mechanisms by which [[complexity]] arises in nature.

Alongside largely lab-based approaches such as the [[Bak–Tang–Wiesenfeld sandpile]], many other investigations have focused on large-scale natural or social systems that are known (or suspected) to display [[scale invariance|scale-invariant]] behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including [[earthquake]]s, (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the [[Gutenberg–Richter law]] describing the statistical distribution of earthquake sizes, and the [[Aftershock|Omori law]]<ref>{{cite journal |author=Omori, F. |title=On the aftershocks of earthquakes |journal=Journal of the College of Science, Imperial University of Tokyo |volume=7 |pages=111–200 |year=1894 }}</ref> describing the frequency of aftershocks), [[solar flare]]s, fluctuations in economic systems such as [[financial market]]s (references to SOC are common in [[econophysics]]), landscape formation, [[forest fire]]s, [[landslide]]s, [[epidemic]]s, and [[biological evolution]] (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "[[punctuated equilibrium|punctuated equilibria]]" put forward by [[Niles Eldredge]] and [[Stephen Jay Gould]]). Given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of [[war]]s. These investigations of SOC have included both attempts at modelling (either developing new models or adapting existing ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or characteristics of natural scaling laws.
 
Also in 1987 [[James Gleick]] published ''[[Chaos: Making a New Science]]'', which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public.<ref name=":8">{{cite book|last=Gleick|first=James|title=Chaos: Making a New Science|date=August 26, 2008|publisher=Penguin Books|isbn=978-0-14-311345-4}}</ref> Initially the domain of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of [[nonlinear system]]s analysis. Alluding to [[Thomas Kuhn]]'s concept of a [[paradigm shift]] exposed in ''[[The Structure of Scientific Revolutions]]'' (1962), many "chaologists" (as some described themselves) claimed that this new theory was an example of such a shift, a thesis upheld by Gleick.

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory remains an active area of research,<ref>{{cite journal | last1 = Motter | first1 = A. E. | last2 = Campbell | first2 = D. K. | year = 2013 | title = Chaos at fifty | url = http://www.physicstoday.org/resource/1/phtoad/v66/i5/p27_s1?bypassSSO=1 | journal = Phys. Today | volume = 66 | issue = 5| pages = 27–33 | doi=10.1063/pt.3.1977|arxiv = 1306.5777 |bibcode = 2013PhT....66e..27M | s2cid = 54005470 }}</ref> involving many different disciplines such as [[mathematics]], [[topology]], [[physics]],<ref>{{cite journal|last1=Hubler|first1=A.|last2=Foster|first2=G.|last3=Phelps|first3=K.|title=Managing chaos: Thinking out of the box|journal=Complexity|volume=12|issue=3|page=10|date=2007|doi=10.1002/cplx.20159|bibcode = 2007Cmplx..12c..10H }}</ref> [[social systems]],<ref>{{Cite book|title=Chaos Theory in the Social Sciences: Foundations and Applications|date=1996|publisher=University of Michigan Press|isbn=978-0-472-10638-7|editor-last=Kiel|editor-first=L.|location=Ann Arbor, MI|language=en|doi=10.3998/mpub.14623|editor-last2=Elliott|editor-first2=Euel|hdl = 2027/fulcrum.d504rm03n}}</ref> [[population model]]ing, [[biology]], [[meteorology]], [[astrophysics]], [[information theory]], [[computational neuroscience]], [[pandemic]] [[crisis management]],<ref name="CT-REF-20"/><ref name="CT-REF-21"/> etc.

== A popular but inaccurate analogy for chaos ==
The sensitive dependence on initial conditions (i.e., butterfly effect) has been illustrated using the following folklore:<ref name=":8" />

<poem style="margin-left: 2em;">
For want of a nail, the shoe was lost.
For want of a shoe, the horse was lost.
For want of a horse, the rider was lost.
For want of a rider, the battle was lost.
For want of a battle, the kingdom was lost.
And all for the want of a horseshoe nail.
</poem>

Based on the above, many people mistakenly believe that the impact of a tiny initial perturbation monotonically increases with time and that any tiny perturbation can eventually produce a large impact on numerical integrations. However, in 2008, Lorenz stated that he did not feel that this verse described true chaos but that it better illustrated the simpler phenomenon of instability and that the verse implicitly suggests that subsequent small events will not reverse the outcome.<ref name=":9">{{Cite web |last=Lorenz |first=E. N. |date=December 2008 |title=The butterfly effect. In Premio Felice Pietro Chisesi E Caterina Tomassoni Award Lecture; University of Rome: Rome, Italy. |url=https://eapsweb.mit.edu/sites/default/files/Tomassoni_2008.pdf |archive-url=https://web.archive.org/web/20230610171122/https://eapsweb.mit.edu/sites/default/files/Tomassoni_2008.pdf |access-date=January 29, 2023 |website= |archive-date=June 10, 2023 }}</ref> Based on the analysis, the verse only indicates divergence, not boundedness.<ref name=":7" /> Boundedness is important for the finite size of a butterfly pattern.<ref name=":7" /> The characteristic of the aforementioned verse was described as "finite-time sensitive dependence".<ref>{{Cite journal |last1=Saiki |first1=Yoshitaka |last2=Yorke |first2=James A. |date=2023-05-02 |title=Can the Flap of a Butterfly's Wings Shift a Tornado into Texas&mdash;Without Chaos? |journal=Atmosphere |language=en |volume=14 |issue=5 |page=821 |doi=10.3390/atmos14050821 |bibcode=2023Atmos..14..821S |issn=2073-4433 |doi-access=free }}</ref>

== Applications ==
[[File:Textile cone.JPG|thumb|left|A [[conus textile]] shell, similar in appearance to [[Rule 30]], a [[cellular automaton]] with chaotic behaviour<ref>{{cite web |url=https://www.maths.nottingham.ac.uk/personal/sc/pdfs/Seashells09.pdf |archive-url=https://web.archive.org/web/20131105134513/https://www.maths.nottingham.ac.uk/personal/sc/pdfs/Seashells09.pdf |archive-date=2013-11-05 |url-status=live |title=The Geometry and Pigmentation of Seashells |author=Stephen Coombes |date=February 2009 |work=www.maths.nottingham.ac.uk |publisher=[[University of Nottingham]] |access-date=2013-04-10}}</ref>]]

Although chaos theory was born from observing weather patterns, it has become applicable to a variety of other situations. Some areas benefiting from chaos theory today are [[geology]], [[mathematics]], [[biology]], [[computer science]], [[economics]],<ref>{{cite journal |author1=Kyrtsou C. |author2=Labys W. | year = 2006 | title = Evidence for chaotic dependence between US inflation and commodity prices | journal = Journal of Macroeconomics | volume = 28 | issue = 1| pages = 256–266 |doi=10.1016/j.jmacro.2005.10.019 }}</ref><ref>{{cite journal | author = Kyrtsou C., Labys W. | year = 2007 | title = Detecting positive feedback in multivariate time series: the case of metal prices and US inflation | doi =10.1016/j.physa.2006.11.002 | journal = Physica A | volume = 377 | issue = 1| pages = 227–229 |bibcode = 2007PhyA..377..227K | last2 = Labys }}</ref><ref>{{cite book |author1=Kyrtsou, C. |author2=Vorlow, C. |chapter=Complex dynamics in macroeconomics: A novel approach |editor1=Diebolt, C. |editor2=Kyrtsou, C. |title=New Trends in Macroeconomics |publisher=Springer Verlag |year=2005 }}</ref> [[engineering]],<ref>{{cite journal |last1=Hernández-Acosta |first1=M. A. |last2=Trejo-Valdez |first2=M. |last3=Castro-Chacón |first3=J. H. |last4=Miguel |first4=C. R. Torres-San |last5=Martínez-Gutiérrez |first5=H. |title=Chaotic signatures of photoconductive Cu 2 ZnSnS 4 nanostructures explored by Lorenz attractors |journal=New Journal of Physics |date=2018 |volume=20 |issue=2 |page=023048 |doi=10.1088/1367-2630/aaad41 |language=en |issn=1367-2630|bibcode=2018NJPh...20b3048H |doi-access=free }}</ref><ref>{{cite web |url = http://www.dspdesignline.com/218101444;jsessionid=Y0BSVTQJJTBACQSNDLOSKH0CJUNN2JVN?pgno=1 |title = Applying Chaos Theory to Embedded Applications |archive-url=https://archive.today/20110809025913/http://www.dspdesignline.com/218101444;jsessionid=Y0BSVTQJJTBACQSNDLOSKH0CJUNN2JVN?pgno=1 |archive-date=9 August 2011 }}</ref> [[finance]],<ref>{{cite journal |author1=Hristu-Varsakelis, D. |author2=Kyrtsou, C. |title=Evidence for nonlinear asymmetric causality in US inflation, metal and stock returns |journal=Discrete Dynamics in Nature and Society |id=138547 |year=2008 |doi=10.1155/2008/138547 |volume=2008 |pages=1–7 |article-number=138547 |doi-access=free }}</ref><ref>{{Cite journal | doi = 10.1023/A:1023939610962 |author1=Kyrtsou, C. |author2=M. Terraza |s2cid=154202123 | year = 2003 | title = Is it possible to study chaotic and ARCH behaviour jointly? Application of a noisy Mackey-Glass equation with heteroskedastic errors to the Paris Stock Exchange returns series | journal = Computational Economics | volume = 21 | issue = 3| pages = 257–276 }}</ref><ref>{{cite book |first1=Justine |last1=Gregory-Williams |first2=Bill |last2=Williams |title=Trading Chaos: Maximize Profits with Proven Technical Techniques |year=2004|publisher=Wiley|location=New York|isbn=978-0-471-46308-5|edition=2nd }}</ref><ref>{{cite book|last=Peters|first=Edgar E.|title=Fractal market analysis: applying chaos theory to investment and economics|year=1994|publisher=Wiley|location=New York u.a.|isbn=978-0-471-58524-4|edition=2. print.}}</ref><ref>{{cite book|last=Peters|first=/ Edgar E.|title=Chaos and order in the capital markets: a new view of cycles, prices, and market volatility|year=1996|publisher=John Wiley & Sons|location=New York|isbn=978-0-471-13938-6|edition=2nd }}</ref> [[meteorology]], [[philosophy]], [[anthropology]],<ref name=":0" /> [[physics]],<ref>{{cite journal|last1=Hubler|first1=A.|last2=Phelps|first2=K.|title=Guiding a self-adjusting system through chaos|journal=Complexity|volume=13|issue=2|page=62|date=2007|doi=10.1002/cplx.20204|bibcode = 2007Cmplx..13b..62W }}</ref><ref>{{cite journal|last1=Gerig|first1=A.|title=Chaos in a one-dimensional compressible flow|journal=Physical Review E |volume=75 |issue=4|article-number=045202|date=2007|doi=10.1103/PhysRevE.75.045202|pmid=17500951|arxiv=nlin/0701050|bibcode = 2007PhRvE..75d5202G |s2cid=45804559}}</ref><ref>{{cite journal|last1=Wotherspoon|first1=T.|last2=Hubler|first2=A.|title=Adaptation to the Edge of Chaos in the Self-Adjusting Logistic Map|journal=The Journal of Physical Chemistry A|volume=113|issue=1|pages=19–22|date=2009|doi=10.1021/jp804420g|pmid=19072712|bibcode = 2009JPCA..113...19W }}</ref> [[politics]],<ref>{{cite journal |last1=Borodkin |first1=Leonid I. |title=Challenges of Instability: The Concepts of Synergetics in Studying the Historical Development of Russia |journal=Ural Historical Journal |date=2019 |volume=63 |issue=2 |pages=127–136 |doi=10.30759/1728-9718-2019-2(63)-127-136|doi-access=free }}</ref><ref>{{cite book |last1=Progonati |first1=E |title=Chaos, complexity and leadership 2018 explorations of chaotic and complexity theory |date=2018 |publisher=Springer |isbn=978-3-030-27672-0 |chapter=Brexit in the Light of Chaos Theory and Some Assumptions About the Future of the European Union}}</ref> [[population dynamics]],<ref>{{cite journal |author1=Dilão, R. |author2=Domingos, T. |s2cid=697164 | year = 2001 | title = Periodic and Quasi-Periodic Behavior in Resource Dependent Age Structured Population Models | journal = Bulletin of Mathematical Biology | volume = 63 |pages = 207–230|doi=10.1006/bulm.2000.0213 | issue = 2 | pmid = 11276524}}</ref> and [[BEAM robotics|robotics]]. A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing.

=== Cryptography ===
{{Main|Chaotic cryptology}}

Chaos theory has been used for many years in [[cryptography]]. In the past few decades, chaos and nonlinear dynamics have been used in the design of hundreds of [[cryptographic primitive]]s. These algorithms include image [[encryption algorithms]], [[hash functions]], [[Cryptographically secure pseudorandom number generator|secure pseudo-random number generators]], [[stream ciphers]], [[Digital watermarking|watermarking]], and [[steganography]].<ref name="Akhavan 1797–1813">{{Cite journal|last1=Akhavan|first1=A.|last2=Samsudin|first2=A.|last3=Akhshani|first3=A.|date=2011-10-01|title=A symmetric image encryption scheme based on combination of nonlinear chaotic maps|journal=Journal of the Franklin Institute|volume=348|issue=8|pages=1797–1813|doi=10.1016/j.jfranklin.2011.05.001}}</ref> The majority of these algorithms are based on uni-modal chaotic maps and a big portion of these algorithms use the control parameters and the initial condition of the chaotic maps as their keys.<ref>{{Cite journal|last1=Behnia|first1=S.|last2=Akhshani|first2=A.|last3=Mahmodi|first3=H.|last4=Akhavan|first4=A.|date=2008-01-01|title=A novel algorithm for image encryption based on mixture of chaotic maps|journal=Chaos, Solitons & Fractals|volume=35|issue=2|pages=408–419|doi=10.1016/j.chaos.2006.05.011|bibcode = 2008CSF....35..408B }}</ref> From a wider perspective, without loss of generality, the similarities between the chaotic maps and the cryptographic systems is the main motivation for the design of chaos based cryptographic algorithms.<ref name="Akhavan 1797–1813"/> One type of encryption, secret key or [[symmetric key]], relies on [[diffusion and confusion]], which is modeled well by chaos theory.<ref>{{cite journal|last=Wang|first=Xingyuan|year=2012|title=An improved key agreement protocol based on chaos|journal=Commun. Nonlinear Sci. Numer. Simul.|volume=15|issue=12|pages=4052–4057|bibcode=2010CNSNS..15.4052W|doi=10.1016/j.cnsns.2010.02.014|author2=Zhao, Jianfeng}}</ref> Another type of computing, [[DNA computing]], when paired with chaos theory, offers a way to encrypt images and other information.<ref>{{cite journal|last=Babaei|first=Majid|s2cid=18407251|year=2013|title=A novel text and image encryption method based on chaos theory and DNA computing|journal=Natural Computing |volume=12|issue=1|pages=101–107|doi=10.1007/s11047-012-9334-9}}</ref> Many of the DNA-Chaos cryptographic algorithms are proven to be either not secure, or the technique applied is suggested to be not efficient.<ref>{{Cite journal|last1=Akhavan|first1=A.|last2=Samsudin|first2=A.|last3=Akhshani|first3=A.|date=2017-10-01|title=Cryptanalysis of an image encryption algorithm based on DNA encoding|journal=Optics & Laser Technology|volume=95|pages=94–99|doi=10.1016/j.optlastec.2017.04.022|bibcode = 2017OptLT..95...94A }}</ref><ref>{{Cite journal|last=Xu|first=Ming|s2cid=125169427|date=2017-06-01|title=Cryptanalysis of an Image Encryption Algorithm Based on DNA Sequence Operation and Hyper-chaotic System|journal=3D Research|language=en|volume=8|issue=2|article-number=15|doi=10.1007/s13319-017-0126-y|issn=2092-6731|bibcode = 2017TDR.....8..126X }}</ref><ref>{{Cite journal |last1=Liu|first1=Yuansheng|last2=Tang|first2=Jie|last3=Xie|first3=Tao|date=2014-08-01|title=Cryptanalyzing a RGB image encryption algorithm based on DNA encoding and chaos map|journal=Optics & Laser Technology|volume=60|pages=111–115|doi=10.1016/j.optlastec.2014.01.015|arxiv=1307.4279|bibcode = 2014OptLT..60..111L |s2cid=18740000}}</ref>

=== Robotics ===
Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trial-and-error type of refinement to interact with their environment, chaos theory has been used to build a [[Predictive modelling|predictive model]].<ref>{{cite journal|last=Nehmzow|first=Ulrich|date=Dec 2005|title=Quantitative description of robot–environment interaction using chaos theory|journal=Robotics and Autonomous Systems|volume=53|issue=3–4|pages=177–193|doi=10.1016/j.robot.2005.09.009|author2=Keith Walker|url=http://cswww.essex.ac.uk/staff/udfn/ftp/ecmrw3.pdf|access-date=2017-10-25|archive-url=https://web.archive.org/web/20170812003513/http://cswww.essex.ac.uk/staff/udfn/ftp/ecmrw3.pdf|archive-date=2017-08-12|citeseerx=10.1.1.105.9178}}</ref>
Chaotic dynamics have been exhibited by [[Passive dynamics|passive walking]] biped robots.<ref>{{cite journal|last=Goswami|first=Ambarish|year=1998|title=A Study of the Passive Gait of a Compass-Like Biped Robot: Symmetry and Chaos|journal=The International Journal of Robotics Research|volume=17|issue=12|pages=1282–1301|doi=10.1177/027836499801701202|author2=Thuilot, Benoit|author3=Espiau, Bernard|citeseerx=10.1.1.17.4861|s2cid=1283494}}</ref>

===Biology===
For over a hundred years, biologists have been keeping track of populations of different species with [[population model]]s. Most models are [[continuous function|continuous]], but recently scientists have been able to implement chaotic models in certain populations.<ref>{{cite journal|last=Eduardo|first=Liz|author2=Ruiz-Herrera, Alfonso|title=Chaos in discrete structured population models|journal=[[SIAM Journal on Applied Dynamical Systems]]|year=2012|volume=11|issue=4|pages=1200–1214|doi=10.1137/120868980}}</ref> For example, a study on models of [[Canada lynx|Canadian lynx]] showed there was chaotic behavior in the population growth.<ref>{{cite journal|last=Lai|first=Dejian|title=Comparison study of AR models on the Canadian lynx data: a close look at BDS statistic|journal=[[Computational Statistics & Data Analysis]]|year=1996|volume=22|issue=4|pages=409–423|doi=10.1016/0167-9473(95)00056-9}}</ref> Chaos can also be found in ecological systems, such as [[hydrology]]. While a chaotic model for hydrology has its shortcomings, there is still much to learn from looking at the data through the lens of chaos theory.<ref>{{cite journal|last=Sivakumar|first=B|title=Chaos theory in hydrology: important issues and interpretations|journal=[[Journal of Hydrology]]|date=31 January 2000|volume=227|issue=1–4|pages=1–20|bibcode=2000JHyd..227....1S|doi=10.1016/S0022-1694(99)00186-9}}</ref> Another biological application is found in [[cardiotocography]]. Fetal surveillance is a delicate balance of obtaining accurate information while being as noninvasive as possible. Better models of warning signs of [[Intrauterine hypoxia|fetal hypoxia]] can be obtained through chaotic modeling.<ref>{{cite journal|last=Bozóki|first=Zsolt|title=Chaos theory and power spectrum analysis in computerized cardiotocography|journal=[[European Journal of Obstetrics & Gynecology and Reproductive Biology]]|date=February 1997|volume=71|issue=2|pages=163–168|doi=10.1016/s0301-2115(96)02628-0|pmid=9138960}}</ref>

As Perry points out, [[mathematical model|modeling]] of chaotic [[time series]] in [[theoretical ecology|ecology]] is helped by constraint.<ref name=realdata/>{{rp|176,177}} There is always potential difficulty in distinguishing real chaos from chaos that is only in the model.<ref name=realdata/>{{rp|176,177}} Hence both constraint in the model and or duplicate time series data for comparison will be helpful in constraining the model to something close to the reality, for example Perry & Wall 1984.<ref name=realdata>{{cite book|year=2000|edition=1|publisher=[[Springer Science+Business Media Dordrecht]]|first4=David|first3=Ian|first2=Robert|first1=Joe|last4=Morse|last2=Smith|last1=Perry|last3=Woiwod|editor-first1=Joe N|editor-first2=Robert H|editor-first3=Ian P|editor-first4=David R|editor-last1=Perry|editor-last2=Smith|editor-last3=Woiwod|editor-last4=Morse|pages=xii+226|title=Chaos in Real Data: The Analysis of Non-Linear Dynamics from Short Ecological Time Series|series=[[Population and Community Biology Series]]|doi=10.1007/978-94-011-4010-2|isbn=978-94-010-5772-1|s2cid=37855255}}</ref>{{rp|176,177}} [[Gene-for-gene]] co-evolution sometimes shows chaotic dynamics in [[allele frequencies]].<ref name = "GFG" /> Adding variables exaggerates this: Chaos is more common in [[evolutionary model|models]] incorporating additional variables to reflect additional facets of real populations.<ref name = "GFG" /> [[Robert M. May]] himself did some of these foundational crop co-evolution studies, and this in turn helped shape the entire field.<ref name = "GFG" >
 {{ Cite journal |
       language=en|
           year=1992|
      publisher=[[Nature Publishing Group]]|
         volume=360|
        journal=[[Nature (journal)|Nature]]|
           issn=0028-0836|
          eissn=1476-4687|
         first2=Jeremy|
         first1=John|
     department=Review Article|
          pages=121–125|
          last1=Thompson|
          last2=Burdon|
          title=Gene-for-gene coevolution between plants and parasites|
 issue=6400|
            doi=10.1038/360121a0|
 bibcode=1992Natur.360..121T|
 s2cid=4346920}}
</ref> Even for a steady environment, merely combining one [[crop]] and one [[pathogen]] may result in [[quasiperiodicity|quasi-periodic-]] or [[chaotic oscillation|chaotic-]] oscillations in pathogen [[statistical population|population]].<ref name = "Epidemiology" >
 {{ Cite book |
       language=en|
           year=1998|
        edition=1|
      publisher=[[Springer Science+Business Media Dordrecht]]|
          first=Gareth|
           last=Jones|
 editor-first1=D. Gareth|
 editor-last1=Jones|
          pages=xvi + 460 + 26{{NBSP}}b/w{{NBSP}}{{NBSP}}ill. + 33{{NBSP}}color{{NBSP}}ill|
          title=The Epidemiology of Plant Diseases|
            doi=10.1007/978-94-017-3302-1|
          s2cid=1793087|
           isbn=978-94-017-3302-1}}
</ref>{{ RP |page=169}}

===Economics===
It is possible that economic models can also be improved through an application of chaos theory, but predicting the health of an economic system and what factors influence it most is an extremely complex task.<ref>{{cite journal|last=Juárez|first=Fernando|title=Applying the theory of chaos and a complex model of health to establish relations among financial indicators|journal=Procedia Computer Science|year=2011|volume=3|pages=982–986|doi=10.1016/j.procs.2010.12.161|doi-access=free}}</ref> Economic and financial systems are fundamentally different from those in the classical natural sciences since the former are inherently stochastic in nature, as they result from the interactions of people, and thus pure deterministic models are unlikely to provide accurate representations of the data. The empirical literature that tests for chaos in economics and finance presents very mixed results, in part due to confusion between specific tests for chaos and more general tests for non-linear relationships.<ref>{{cite journal |last=Brooks |first=Chris |author-link=Chris Brooks (academic)|title=Chaos in foreign exchange markets: a sceptical view |journal=Computational Economics|year=1998 |volume=11 |issue=3 |pages=265–281 |issn=1572-9974 |doi=10.1023/A:1008650024944|s2cid=118329463 |url=http://centaur.reading.ac.uk/35988/1/35988.pdf |archive-url=https://web.archive.org/web/20170809072731/http://centaur.reading.ac.uk/35988/1/35988.pdf |archive-date=2017-08-09 |url-status=live }}</ref>

Chaos could be found in economics by the means of [[recurrence quantification analysis]]. In fact, Orlando et al.<ref>{{cite journal |last1=Orlando |first1=Giuseppe |last2=Zimatore |first2=Giovanna |title=RQA correlations on real business cycles time series |journal=Indian Academy of Sciences – Conference Series |date=18 December 2017 |volume=1 |issue=1 |pages=35–41 |doi=10.29195/iascs.01.01.0009|doi-access=free }}</ref> by the means of the so-called recurrence quantification correlation index were able to detect hidden changes in time series. Then, the same technique was employed to detect transitions from laminar (regular) to turbulent (chaotic) phases as well as differences between macroeconomic variables and highlight hidden features of economic dynamics.<ref>{{cite journal |last1=Orlando |first1=Giuseppe |last2=Zimatore |first2=Giovanna |title=Recurrence quantification analysis of business cycles |journal=Chaos, Solitons & Fractals |date=1 May 2018 |volume=110 |pages=82–94 |doi=10.1016/j.chaos.2018.02.032 |bibcode=2018CSF...110...82O |s2cid=85526993 |url=https://www.sciencedirect.com/science/article/abs/pii/S0960077918300924 |language=en |issn=0960-0779|url-access=subscription }}</ref> Finally, chaos theory could help in modeling how an economy operates as well as in embedding shocks due to external events such as COVID-19.<ref>{{cite journal |last1=Orlando |first1=Giuseppe |last2=Zimatore |first2=Giovanna |title=Business cycle modeling between financial crises and black swans: Ornstein–Uhlenbeck stochastic process vs Kaldor deterministic chaotic model |url=https://aip.scitation.org/doi/10.1063/5.0015916 |journal=Chaos: An Interdisciplinary Journal of Nonlinear Science |page=083129 |doi=10.1063/5.0015916 |date=1 August 2020|volume=30 |issue=8 |pmid=32872798 |bibcode=2020Chaos..30h3129O |s2cid=235909725 |url-access=subscription }}</ref>

=== Finite predictability in weather and climate ===
Due to the sensitive dependence of solutions on initial conditions (SDIC), also known as the butterfly effect, chaotic systems like the Lorenz 1963 model imply a finite predictability horizon. This means that while accurate predictions are possible over a finite time period, they are not feasible over an infinite time span. Considering the nature of Lorenz's chaotic solutions, the committee led by Charney et al. in 1966<ref>{{Cite book |date=1966-01-01 |title=The Feasibility of a Global Observation and Analysis Experiment |doi=10.17226/21272|isbn=978-0-309-35922-1 }}</ref> extrapolated a doubling time of five days from a general circulation model, suggesting a predictability limit of two weeks. This connection between the five-day doubling time and the two-week predictability limit was also recorded in a 1969 report by the Global Atmospheric Research Program (GARP).<ref>{{Cite journal |last=GARP |title=GARP topics |journal=Bull. Am. Meteorol. Soc. |volume=50 |pages=136–141}}</ref> To acknowledge the combined direct and indirect influences from the Mintz and Arakawa model and Lorenz's models, as well as the leadership of Charney et al., Shen et al.<ref>{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Zeng |first4=Xiping |date=2024-07-16 |title=Exploring the Origin of the Two-Week Predictability Limit: A Revisit of Lorenz's Predictability Studies in the 1960s |journal=Atmosphere |language=en |volume=15 |issue=7 |page=837 |doi=10.3390/atmos15070837 |doi-access=free |bibcode=2024Atmos..15..837S |issn=2073-4433}}</ref> refer to the two-week predictability limit as the "Predictability Limit Hypothesis," drawing an analogy to Moore's Law.

=== AI-extended modeling framework ===
In AI-driven large language models, responses can exhibit sensitivities to factors like alterations in formatting and variations in prompts. These sensitivities are akin to butterfly effects.<ref>{{Cite arXiv|last1=Salinas |first1=Abel |last2=Morstatter |first2=Fred |date=2024-01-01 |title=The Butterfly Effect of Altering Prompts: How Small Changes and Jailbreaks Affect Large Language Model Performance |class=cs.CL |eprint=2401.03729}}</ref> Although classifying AI-powered large language models as classical deterministic chaotic systems poses challenges, chaos-inspired approaches and techniques (such as ensemble modeling) may be employed to extract reliable information from these expansive language models (see also "[[Butterfly effect in popular culture#AI-powered Large Language Models|Butterfly Effect in Popular Culture]]").

===Other areas===
In chemistry, predicting gas solubility is essential to manufacturing [[polymers]], but models using [[particle swarm optimization]] (PSO) tend to converge to the wrong points. An improved version of PSO has been created by introducing chaos, which keeps the simulations from getting stuck.<ref>{{cite journal|last=Li|first=Mengshan|author2=Xingyuan Huanga|author3=Hesheng Liua|author4=Bingxiang Liub|author5=Yan Wub|author6=Aihua Xiongc|author7=Tianwen Dong|title=Prediction of gas solubility in polymers by back propagation artificial neural network based on self-adaptive particle swarm optimization algorithm and chaos theory|journal=Fluid Phase Equilibria|date=25 October 2013|volume=356|pages=11–17|doi=10.1016/j.fluid.2013.07.017|bibcode=2013FlPEq.356...11L }}</ref> In [[celestial mechanics]], especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will approach Earth and other planets.<ref>{{cite journal|last=Morbidelli|first=A.|title=Chaotic diffusion in celestial mechanics|journal=Regular & Chaotic Dynamics |year=2001|volume=6|issue=4|pages=339–353|doi=10.1070/rd2001v006n04abeh000182}}</ref> Four of the five [[moons of Pluto]] rotate chaotically. In [[quantum physics]] and [[electrical engineering]], the study of large arrays of [[Josephson junctions]] benefitted greatly from chaos theory.<ref>Steven Strogatz, ''Sync: The Emerging Science of Spontaneous Order'', Hyperion, 2003</ref> Closer to home, coal mines have always been dangerous places where frequent natural gas leaks cause many deaths. Until recently, there was no reliable way to predict when they would occur. But these gas leaks have chaotic tendencies that, when properly modeled, can be predicted fairly accurately.<ref>{{cite journal|last=Dingqi|first=Li|author2=Yuanping Chenga|author3=Lei Wanga|author4=Haifeng Wanga|author5=Liang Wanga|author6=Hongxing Zhou|title=Prediction method for risks of coal and gas outbursts based on spatial chaos theory using gas desorption index of drill cuttings|journal=Mining Science and Technology|date=May 2011|volume=21|issue=3|pages=439–443|doi=10.1016/j.mstc.2011.05.010 |bibcode=2011MiSTC..21..439L }}</ref>

Chaos theory can be applied outside of the natural sciences, but historically nearly all such studies have suffered from lack of reproducibility; poor external validity; and/or inattention to cross-validation, resulting in poor predictive accuracy (if out-of-sample prediction has even been attempted). Glass<ref>{{cite book | last1 = Glass | first1 = L |editor1-first=C |editor1-last= Grebogi |editor2-first=J. A. | editor2-last=Yorke |title= The impact of chaos on science and society|publisher= United Nations University Press |year=1997 |chapter= Dynamical disease: The impact of nonlinear dynamics and chaos on cardiology and medicine }}</ref> and Mandell and Selz<ref>{{cite book | last1 = Mandell |first1= A. J. | last2 = Selz |first2= K. A. |editor1-first=C |editor1-last= Grebogi |editor2-first=J. A. | editor2-last=Yorke |title= The impact of chaos on science and society|publisher= United Nations University Press |year=1997 |chapter= Is the EEG a strange attractor? }}</ref> have found that no EEG study has as yet indicated the presence of strange attractors or other signs of chaotic behavior.

Redington and Reidbord (1992) attempted to demonstrate that the human heart could display chaotic traits. They monitored the changes in between-heartbeat intervals for a single psychotherapy patient as she moved through periods of varying emotional intensity during a therapy session. Results were admittedly inconclusive. Not only were there ambiguities in the various plots the authors produced to purportedly show evidence of chaotic dynamics (spectral analysis, phase trajectory, and autocorrelation plots), but also when they attempted to compute a Lyapunov exponent as more definitive confirmation of chaotic behavior, the authors found they could not reliably do so.<ref>{{cite journal|last1=Redington|first1=D. J.|last2=Reidbord|first2=S. P.|s2cid=214722|title=Chaotic dynamics in autonomic nervous system activity of a patient during a psychotherapy session|journal=Biological Psychiatry|date=1992|volume=31|issue=10|pages=993–1007|pmid=1511082|doi=10.1016/0006-3223(92)90093-F}}</ref>

In their 1995 paper, Metcalf and Allen<ref>{{cite book | last1 = Metcalf |first1= B. R. | last2 = Allen |first2= J. D. |editor1-first=F. D. |editor1-last= Abraham |editor2-first=A. R. | editor2-last=Gilgen |title= Chaos theory in psychology |publisher= Greenwood Press |year=1995 |chapter= In search of chaos in schedule-induced polydipsia }}</ref> maintained that they uncovered in animal behavior a pattern of period doubling leading to chaos. The authors examined a well-known response called schedule-induced polydipsia, by which an animal deprived of food for certain lengths of time will drink unusual amounts of water when the food is at last presented. The control parameter (r) operating here was the length of the interval between feedings, once resumed. The authors were careful to test a large number of animals and to include many replications, and they designed their experiment so as to rule out the likelihood that changes in response patterns were caused by different starting places for r.

Time series and first delay plots provide the best support for the claims made, showing a fairly clear march from periodicity to irregularity as the feeding times were increased. The various phase trajectory plots and spectral analyses, on the other hand, do not match up well enough with the other graphs or with the overall theory to lead inexorably to a chaotic diagnosis. For example, the phase trajectories do not show a definite progression towards greater and greater complexity (and away from periodicity); the process seems quite muddied. Also, where Metcalf and Allen saw periods of two and six in their spectral plots, there is room for alternative interpretations. All of this ambiguity necessitate some serpentine, post-hoc explanation to show that results fit a chaotic model.

By adapting a model of career counseling to include a chaotic interpretation of the relationship between employees and the job market, Amundson and Bright found that better suggestions can be made to people struggling with career decisions.<ref>{{cite journal|last=Pryor|first=Robert G. L.|author2=Norman E. Amundson|author3=Jim E. H. Bright|title=Probabilities and Possibilities: The Strategic Counseling Implications of the Chaos Theory of Careers|author3-link=Jim Bright (psychologist)|journal=The Career Development Quarterly|date=June 2008|volume=56|issue=4|pages=309–318|doi=10.1002/j.2161-0045.2008.tb00096.x}}</ref> Modern organizations are increasingly seen as open [[complex adaptive system]]s with fundamental natural nonlinear structures, subject to internal and external forces that may contribute chaos. For instance, [[team building]] and [[group development]] is increasingly being researched as an inherently unpredictable system, as the uncertainty of different individuals meeting for the first time makes the trajectory of the team unknowable.<ref>{{Cite journal|last1=Thompson|first1=Jamie|last2=Johnstone|first2=James|last3=Banks|first3=Curt|date=2018|title=An examination of initiation rituals in a UK sporting institution and the impact on group development|journal=European Sport Management Quarterly|volume=18|issue=5|pages=544–562|doi=10.1080/16184742.2018.1439984|s2cid=149352680}}</ref>

[[File:BML N=200 P=32.png|right|The red cars and blue cars take turns to move; the red ones only move upwards, and the blue ones move rightwards. Every time, all the cars of the same colour try to move one step if there is no car in front of it. Here, the model has self-organized in a somewhat geometric pattern where there are some traffic jams and some areas where cars can move at top speed.|frameless]]

Traffic forecasting may benefit from applications of chaos theory. Better predictions of when a congestion will occur would allow measures to be taken to disperse it before it would have occurred. Combining chaos theory principles with a few other methods has led to a more accurate short-term prediction model (see the plot of the [[BML traffic model]] at right).<ref>{{cite journal|last=Wang|first=Jin|author2=Qixin Shi|title=Short-term traffic speed forecasting hybrid model based on Chaos–Wavelet Analysis-Support Vector Machine theory|journal=Transportation Research Part C: Emerging Technologies|date=February 2013|volume=27|pages=219–232|doi=10.1016/j.trc.2012.08.004|bibcode=2013TRPC...27..219W }}</ref>

Chaos theory has been applied to environmental [[water cycle]] data (also [[hydrological]] data), such as rainfall and streamflow.<ref>{{Cite web|url=http://pasternack.ucdavis.edu/research/projects/chaos-hydrology/|title=Dr. Gregory B. Pasternack – Watershed Hydrology, Geomorphology, and Ecohydraulics :: Chaos in Hydrology|website=pasternack.ucdavis.edu|language=en|access-date=2017-06-12}}</ref> These studies have yielded controversial results, because the methods for detecting a chaotic signature are often relatively subjective. Early studies tended to "succeed" in finding chaos, whereas subsequent studies and meta-analyses called those studies into question and provided explanations for why these datasets are not likely to have low-dimension chaotic dynamics.<ref>{{Cite journal|last=Pasternack|first=Gregory B.|date=1999-11-01|title=Does the river run wild? Assessing chaos in hydrological systems|journal=Advances in Water Resources|volume=23|issue=3|pages=253–260|doi=10.1016/s0309-1708(99)00008-1|bibcode = 1999AdWR...23..253P }}</ref>

==See also==
{{Portal|Mathematics|Systems science}}

'''Examples of chaotic systems'''
{{Div col|colwidth=18em}}
* [[Contour advection|Advected contours]]
* [[Arnold's cat map]]
* [[Bifurcation theory]]
* [[Bouncing ball dynamics]]
* [[Chua's circuit]]
* [[Cliodynamics]]
* [[Coupled map lattice]]
* [[Double pendulum]]
* [[Duffing equation]]
* [[Dynamical billiards]]
* [[Economic bubble]]
* [[Chaotic scattering#Gaspard–Rice system|Gaspard-Rice system]]
* [[Logistic map]]
* [[Hénon map]]
* [[Horseshoe map]]
* [[List of chaotic maps]]
* [[Rössler attractor]]
* [[Standard map]]
* [[Swinging Atwood's machine]]
* [[Tilt A Whirl]]
{{Div col end}}

'''Other related topics'''

{{Div col|colwidth=18em}}
* [[Amplitude death]]
* [[Anosov diffeomorphism]]
* [[Catastrophe theory]]
* [[Causality]]
* [[Supersymmetric theory of stochastic dynamics|Chaos as topological supersymmetry breaking]]
* [[Chaos machine]]
* [[Chaotic mixing]]
* [[Chaotic scattering]]
* [[Control of chaos]]
* [[Determinism]]
* [[Edge of chaos]]
* [[Emergence]]
* [[Mandelbrot set]]
* [[Kolmogorov–Arnold–Moser theorem]]
* [[Ill-conditioning]]
* [[Ill-posedness]]
* [[Nonlinear system]]
* [[Patterns in nature]]
* [[Predictability]]
* [[Quantum chaos]]
* [[Santa Fe Institute]]
* [[Shadowing lemma]]
* [[Synchronization of chaos]]
* [[Unintended consequence]]
{{Div col end}}

'''People'''
{{Div col|colwidth=18em}}
* [[Ralph Abraham (mathematician)|Ralph Abraham]]
* [[Michael Berry (physicist)|Michael Berry]]
* [[Leon O. Chua]]
* [[Ivar Ekeland]]
* [[Doyne Farmer]]
* [[Martin Gutzwiller]]
* [[Brosl Hasslacher]]
* [[Michel Hénon]]
* [[Aleksandr Lyapunov]]
* [[Norman Packard]]
* [[Otto Rössler]]
* [[David Ruelle]]
* [[Oleksandr Mikolaiovich Sharkovsky]]
* [[Greg Sams]]
* [[Robert Shaw (physicist)|Robert Shaw]]
* [[Floris Takens]]
* [[James A. Yorke]]
* [[George M. Zaslavsky]]
{{Div col end}}

==References==
{{Reflist|30em}}

; Attribution
* {{Free-content attribution
| title = Three Kinds of Butterfly Effects within Lorenz Models
| author = Bo-Wen Shen, Roger A. Pielke, Sr., Xubin Zeng, Jialin Cui, Sara Faghih-Naini, Wei Paxson, and Robert Atlas
| publisher = MDPI
| source = Encyclopedia
| documentURL = https://www.mdpi.com/2673-8392/2/3/84/html?
| license statement URL = https://creativecommons.org/licenses/by/4.0/
| license = CC-BY
}}

==Further reading==
{{Library resources box}}

===Articles===
* {{cite journal |first=A.N. |last=Sharkovskii |author-link=Oleksandr Mykolaiovych Sharkovsky |title=Co-existence of cycles of a continuous mapping of the line into itself |journal=Ukrainian Math. J. |volume=16 |pages=61–71 |year=1964 }}
* {{cite journal |author-link1=Tien-Yien Li |last1=Li |first1=T.Y. |author-link2=James A. Yorke |last2=Yorke |first2=J.A. |title=Period Three Implies Chaos |journal=[[American Mathematical Monthly]] |volume=82 |pages=985–92 |year=1975 |bibcode=1975AmMM...82..985L |doi=10.2307/2318254 |issue=10 |url=http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |jstor=2318254 |citeseerx=10.1.1.329.5038 |access-date=2009-08-12 |archive-date=2009-12-29 |archive-url=https://web.archive.org/web/20091229042210/http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf }}
* {{cite journal|last1=Alemansour|first1=Hamed|last2=Miandoab|first2=Ehsan Maani|last3=Pishkenari|first3=Hossein Nejat|title=Effect of size on the chaotic behavior of nano resonators|journal=Communications in Nonlinear Science and Numerical Simulation|date=March 2017|volume=44|pages=495–505|doi=10.1016/j.cnsns.2016.09.010|bibcode=2017CNSNS..44..495A}}
* {{Cite journal |date = December 1986|title=Chaos |journal=[[Scientific American]] |volume=255 |issue=6 |pages=38–49 (bibliography p.136) |bibcode = 1986SciAm.255d..38T |last2 = Tucker |last3 = Morrison |author1 = Crutchfield|author4 = J.D. Farmer|author5 = Packard|author6=N.H. |author7=Shaw |author8=R.S |author-link1=James P. Crutchfield |author-link5=Norman Packard |author-link7=Robert Shaw (physicist) |author4-link=J. Doyne Farmer |doi=10.1038/scientificamerican1286-46 }} [https://web.archive.org/web/20100612040821/http://cse.ucdavis.edu/~chaos/courses/ncaso/Readings/Chaos_SciAm1986/Chaos_SciAm1986.html Online version] (Note: the volume and page citation cited for the online text differ from that cited here. The citation here is from a photocopy, which is consistent with other citations found online that don't provide article views. The online content is identical to the hardcopy text. Citation variations are related to country of publication).
* {{cite journal |author=Kolyada, S.F. |s2cid=207251437 |title=Li-Yorke sensitivity and other concepts of chaos |journal=Ukrainian Math. J. |volume=56 |pages=1242–57 |year=2004 |doi=10.1007/s11253-005-0055-4 |issue=8 }}
* {{cite journal | last1 = Day| first1 = R.H. | last2 = Pavlov| first2 = O.V. | year = 2004| title = Computing Economic Chaos | ssrn = 806124| journal = Computational Economics | volume = 23 | issue = 4 | pages = 289–301 | doi = 10.1023/B:CSEM.0000026787.81469.1f | arxiv = 2211.02441 | s2cid = 119972392 }}
* {{cite journal|first1=C. |last1=Strelioff |first2=A. |last2=Hübler |title=Medium-Term Prediction of Chaos |journal=Phys. Rev. Lett. |volume=96 |issue=4 |id=044101 |year=2006 |doi=10.1103/PhysRevLett.96.044101 |url=http://www.ccsr.illinois.edu/web/Techreports/2005-08/CCSR-05-4.pdf |pmid=16486826 |article-number=044101 |bibcode=2006PhRvL..96d4101S |archive-url=https://web.archive.org/web/20130426201635/http://www.ccsr.illinois.edu/web/Techreports/2005-08/CCSR-05-4.pdf |archive-date=2013-04-26 }}
* {{cite journal |author1=Hübler, A. |author2=Foster, G. |author3=Phelps, K. |title=Managing Chaos: Thinking out of the Box |journal=Complexity |volume=12 |pages=10–13 |year=2007 |url=http://server17.how-why.com/blog/ManagingChaos.pdf |doi=10.1002/cplx.20159 |issue=3 |bibcode=2007Cmplx..12c..10H |access-date=2011-07-17 |archive-url=https://web.archive.org/web/20121030165633/http://server17.how-why.com/blog/ManagingChaos.pdf |archive-date=2012-10-30 }}
* {{cite journal | last1 = Motter | first1 = Adilson E. | last2 = Campbell | first2 = David K. | year = 2013 | title = Chaos at 50 | journal = Physics Today | volume = 66| issue = 5| page = 27| doi = 10.1063/PT.3.1977 |arxiv = 1306.5777 |bibcode = 2013PhT....66e..27M | s2cid = 54005470 }}

===Textbooks===
* {{cite book |last1=Alligood |first1=K.T. |last2=Sauer |first2=T. |last3=Yorke |first3=J.A. |title=Chaos: an introduction to dynamical systems |publisher=Springer-Verlag |year=1997 |isbn=978-0-387-94677-1 |url=https://books.google.com/books?id=48YHnbHGZAgC}}
* {{cite book| author=Baker, G. L.| title=Chaos, Scattering and Statistical Mechanics| publisher=Cambridge University Press| year=1996| isbn=978-0-521-39511-3}}
* {{cite book |author1=Badii, R. |author2=Politi A. |title=Complexity: hierarchical structures and scaling in physics |publisher=Cambridge University Press |year=1997 |isbn=978-0-521-66385-4 |url=http://www.cambridge.org/gb/academic/subjects/physics/statistical-physics/complexity-hierarchical-structures-and-scaling-physics}}
* {{cite book |author1=Collet, Pierre |author2=[[Jean-Pierre Eckmann|Eckmann, Jean-Pierre]] |title=Iterated Maps on the Interval as Dynamical Systems |publisher=Birkhauser |year=1980 |isbn=978-0-8176-4926-5}}
* {{cite book |last=Devaney |first=Robert L. |author-link=Robert L. Devaney |title=An Introduction to Chaotic Dynamical Systems |edition=2nd |publisher=Westview Press |year=2003 |isbn=978-0-8133-4085-2 }}
* {{cite book |last=Robinson |first=Clark |title=Dynamical systems: Stability, symbolic dynamics, and chaos |publisher=CRC Press |year=1995 |isbn=0-8493-8493-1}}
* {{cite book |author1=Feldman, D. P. |title=Chaos and Fractals: An Elementary Introduction |publisher=Oxford University Press |year=2012 |isbn=978-0-19-956644-0 |url=https://dpfeldman.github.io/Chaos/index.html |access-date=2016-12-29 |archive-date=2019-12-31 |archive-url=https://web.archive.org/web/20191231030717/http://chaos.coa.edu/index.html |url-status=live}}
* {{cite book |author1=Gollub, J. P. |author2=Baker, G. L. |title=Chaotic dynamics |publisher=Cambridge University Press |year=1996 |isbn=978-0-521-47685-0 |url=https://books.google.com/books?id=n1qnekRPKtoC}}
* {{cite book |author=Guckenheimer, John| author-link=John Guckenheimer|author2= Holmes, Philip |author-link2=Philip Holmes|title=Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields |publisher=Springer-Verlag |year=1983 |isbn=978-0-387-90819-9}}
* {{cite book| author=Gulick, Denny| title=Encounters with Chaos| publisher=McGraw-Hill| year=1992| isbn=978-0-07-025203-5}}
* {{cite book |author=Gutzwiller, Martin |title=Chaos in Classical and Quantum Mechanics |publisher=Springer-Verlag |year=1990 |isbn=978-0-387-97173-5 |url=https://books.google.com/books?id=fnO3XYYpU54C}}
* {{cite book |author=Hoover, William Graham|author-link=William G Hoover|title=Time Reversibility, Computer Simulation, and Chaos |publisher=World Scientific |orig-date=1999|year=2001 |isbn=978-981-02-4073-8 |url=https://books.google.com/books?id=24kEKsdl0psC}}
* {{cite book |author=Kautz, Richard |title=Chaos: The Science of Predictable Random Motion |publisher=Oxford University Press |year=2011 |isbn=978-0-19-959458-0 |url=https://books.google.com/books?id=x5YbNZjulN0C}}
* {{cite book |author1=Kiel, L. Douglas |author2=Elliott, Euel W. |title=Chaos Theory in the Social Sciences |publisher=Perseus Publishing |year=1997 |isbn=978-0-472-08472-2 |url=https://books.google.com/books?id=K46kkMXnKfcC}}
* {{cite book |author=Moon, Francis |title=Chaotic and Fractal Dynamics |publisher=Springer-Verlag |year=1990 |isbn=978-0-471-54571-2 |url=https://books.google.com/books?id=Ddz-CI-nSKYC}}
* {{cite book|last1=Orlando|first1=Giuseppe|author-link1=Giuseppe Orlando|last2=Pisarchick|first2=Alexander|last3=Stoop|first3=Ruedi|title=Nonlinearities in Economics |chapter=|series=Dynamic Modeling and Econometrics in Economics and Finance|year=2021|volume=29|url=https://link.springer.com/book/10.1007/978-3-030-70982-2#editorsandaffiliations|language=en-gb|doi=10.1007/978-3-030-70982-2|isbn=978-3-030-70981-5|s2cid=239756912}}
* {{cite book |author=Ott, Edward |title=Chaos in Dynamical Systems |publisher=Cambridge University Press |year=2002 |isbn=978-0-521-01084-9 |url=https://books.google.com/books?id=nOLx--zzHSgC}}
* {{cite book| author=Strogatz, Steven| author-link=Steven Strogatz| title=Nonlinear Dynamics and Chaos| publisher=Perseus Publishing| year=2000| isbn=978-0-7382-0453-6| url=https://archive.org/details/nonlineardynamic00stro}}
* {{cite book |last=Sprott |first=Julien Clinton |title=Chaos and Time-Series Analysis |publisher=Oxford University Press |year=2003 |isbn=978-0-19-850840-3 |url=https://books.google.com/books?id=SEDjdjPZ158C}}
* {{cite book |author1=Tél, Tamás |author2=Gruiz, Márton |title=Chaotic dynamics: An introduction based on classical mechanics |publisher=Cambridge University Press |year=2006 |isbn=978-0-521-83912-9 |url=https://books.google.com/books?id=P2JL7s2IvakC}}
* {{cite book| last = Teschl| given = Gerald|author-link=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
* {{cite book|vauthors=Thompson JM, Stewart HB | title=Nonlinear Dynamics And Chaos| publisher=John Wiley and Sons Ltd| year=2001| isbn=978-0-471-87645-8}}
* {{cite book |author1-link=Nicholas Tufillaro |last1=Tufillaro |last2=Reilly |title=An experimental approach to nonlinear dynamics and chaos |series=American Journal of Physics |volume=61 |issue=10 |page=958 |publisher=Addison-Wesley |year=1992 |isbn=978-0-201-55441-0 |bibcode=1993AmJPh..61..958T |doi=10.1119/1.17380 |url=https://archive.org/details/unset0000unse_q2b7 |url-access=limited }}
* {{cite book | last=Wiggins|first=Stephen | title= Introduction to Applied Dynamical Systems and Chaos | publisher= Springer | year= 2003 | isbn= 978-0-387-00177-7 }}
* {{cite book| author=Zaslavsky, George M.| title=Hamiltonian Chaos and Fractional Dynamics| publisher=Oxford University Press| year=2005| isbn=978-0-19-852604-9}}

===Semitechnical and popular works===
* [[Christophe Letellier]], ''Chaos in Nature'', World Scientific Publishing Company, 2012, {{ISBN|978-981-4374-42-2}}.
* {{cite book |editor1-first=Ralph H. |editor1-last=Abraham |editor2-first=Yoshisuke |editor2-last=Ueda |title=The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory |journal=<!--Citation bot --> |volume=39 |url=https://books.google.com/books?id=0E667XpBq1UC |year=2000 |publisher=World Scientific |isbn=978-981-238-647-2 |bibcode=2000cagm.book.....A |doi=10.1142/4510 |series=World Scientific Series on Nonlinear Science Series A }}
* {{cite book |author-link=Michael F. Barnsley |first=Michael F. |last=Barnsley |title=Fractals Everywhere |url=https://books.google.com/books?id=oh7NoePgmOIC |year=2000 |publisher=Morgan Kaufmann |isbn=978-0-12-079069-2}}
* {{cite book |first=Richard J. |last=Bird |title=Chaos and Life: Complexity and Order in Evolution and Thought |url=https://books.google.com/books?id=fv3sltQBS54C |year=2003 |publisher=Columbia University Press |isbn=978-0-231-12662-5}}
* [[John Briggs (author)|John Briggs]] and David Peat, ''Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness'', Harper Perennial 1990, 224 pp.
* John Briggs and David Peat, ''Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change'', Harper Perennial 2000, 224 pp.
* {{cite journal |author=Cunningham, Lawrence A. |title=From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis |journal=George Washington Law Review |volume=62 |page=546 |year=1994 }}
* [[Predrag Cvitanović]], ''Universality in Chaos'', Adam Hilger 1989, 648 pp.
* [[Leon Glass]] and Michael C. Mackey, ''From Clocks to Chaos: The Rhythms of Life,'' Princeton University Press 1988, 272 pp.
* [[James Gleick]], ''[[Chaos: Making a New Science]]'', New York: Penguin, 1988. 368 pp.
* {{cite book |author=John Gribbin |title=Deep Simplicity |series=Penguin Press Science |publisher=Penguin Books}}
* L Douglas Kiel, Euel W Elliott (ed.), ''Chaos Theory in the Social Sciences: Foundations and Applications'', University of Michigan Press, 1997, 360 pp.
* Arvind Kumar, ''Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature '', National Book Trust, 2003.
* Hans Lauwerier, ''Fractals'', Princeton University Press, 1991.
* [[Edward Lorenz]], ''The Essence of Chaos'', University of Washington Press, 1996.
* {{cite book|doi=10.1142/9781860949548|title=The Unity of Nature - Wholeness and Disintegration in Ecology and Science|year=2002|last1=Marshall|first1=Alan|isbn=978-1-86094-954-8}}
* David Peak and Michael Frame, ''Chaos Under Control: The Art and Science of Complexity'', Freeman, 1994.
* [[Heinz-Otto Peitgen]] and [[Dietmar Saupe]] (Eds.), ''The Science of Fractal Images'', Springer 1988, 312 pp.
* [[Nuria Perpinya]], ''Caos, virus, calma. La Teoría del Caos aplicada al desórden artístico, social y político'', Páginas de Espuma, 2021.
* [[Clifford A. Pickover]], ''Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World '', St Martins Pr 1991.
* [[Clifford A. Pickover]], ''Chaos in Wonderland: Visual Adventures in a Fractal World'', St Martins Pr 1994.
* [[Ilya Prigogine]] and [[Isabelle Stengers]], ''Order Out of Chaos'', Bantam 1984.
* {{cite book|doi=10.1007/978-3-642-61717-1|title=The Beauty of Fractals|url=https://archive.org/details/beautyoffractals0000peit|url-access=registration|year=1986|last1=Peitgen|first1=Heinz-Otto|last2=Richter|first2=Peter H.|isbn=978-3-642-61719-5}}
* [[David Ruelle]], ''Chance and Chaos'', Princeton University Press 1993.
* [[Ivars Peterson]], ''Newton's Clock: Chaos in the Solar System'', Freeman, 1993.
* {{cite book |author1=Ian Roulstone |author2=John Norbury |title=Invisible in the Storm: the role of mathematics in understanding weather |url=https://books.google.com/books?id=qnMrFEHMrWwC|year=2013 |publisher=Princeton University Press|isbn=978-0-691-15272-1 }}
* {{cite book|doi=10.1017/CBO9780511608773|title=Chaotic Evolution and Strange Attractors|url=https://archive.org/details/chaoticevolution0000ruel|url-access=registration|year=1989|last1=Ruelle|first1=D.|isbn=978-0-521-36272-6}}
* Manfred Schroeder, ''Fractals, Chaos, and Power Laws'', Freeman, 1991.
* {{cite book|doi=10.1017/CBO9780511554544|title=Explaining Chaos|year=1998|last1=Smith|first1=Peter|isbn=978-0-511-55454-4}}
* [[Ian Stewart (mathematician)|Ian Stewart]], ''Does God Play Dice?: The Mathematics of Chaos '', Blackwell Publishers, 1990.
* [[Steven Strogatz]], ''Sync: The emerging science of spontaneous order'', Hyperion, 2003.
* Yoshisuke Ueda, ''The Road To Chaos'', Aerial Pr, 1993.
* M. Mitchell Waldrop, ''Complexity : The Emerging Science at the Edge of Order and Chaos'', Simon & Schuster, 1992.
* Antonio Sawaya, ''Financial Time Series Analysis : Chaos and Neurodynamics Approach'', Lambert, 2012.

==External links==
{{Commons category}}
* {{springer|title=Chaos|id=p/c021480}}
* [https://web.archive.org/web/20160310065017/http://lagrange.physics.drexel.edu/ Nonlinear Dynamics Research Group] with Animations in Flash
* [http://www.chaos.umd.edu/ The Chaos group at the University of Maryland]
* [https://hypertextbook.com/chaos/ The Chaos Hypertextbook]. An introductory primer on chaos and fractals
* [https://chaosbook.org/ ChaosBook.org] An advanced graduate textbook on chaos (no fractals)
* [https://www.societyforchaostheory.org/ Society for Chaos Theory in Psychology & Life Sciences]
* [https://web.archive.org/web/20130425220329/http://www.csdc.unifi.it/mdswitch.html?newlang=eng Nonlinear Dynamics Research Group at CSDC], [[Florence]], [[Italy]]
* [http://www.creatingtechnology.org/papers/chaos.htm Nonlinear dynamics: how science comprehends chaos], talk presented by Sunny Auyang, 1998.
* [https://www.egwald.ca/nonlineardynamics/ Nonlinear Dynamics]. Models of bifurcation and chaos by Elmer G. Wiens
* [https://around.com/books/chaos/ Gleick's ''Chaos'' (excerpt)] {{Webarchive|url=https://web.archive.org/web/20070202075958/http://www.around.com/chaos.html |date=2007-02-02 }}
* [https://web.archive.org/web/20070428110552/http://www.eng.ox.ac.uk/samp/ Systems Analysis, Modelling and Prediction Group] at the University of Oxford
* [https://web.archive.org/web/20090307094012/http://www.mgix.com/snippets/?MackeyGlass A page about the Mackey-Glass equation]
* [https://www.youtube.com/watch?v=5pKrKdNclYs High Anxieties — The Mathematics of Chaos] (2008) BBC documentary directed by [[David Malone (independent filmmaker)|David Malone]]
* [https://www.newscientist.com/article/mg20827821.000-the-chaos-theory-of-evolution.html The chaos theory of evolution] – article published in Newscientist featuring similarities of evolution and non-linear systems including fractal nature of life and chaos.
* Jos Leys, [[Étienne Ghys]] et Aurélien Alvarez, [https://www.chaos-math.org/en.html ''Chaos, A Mathematical Adventure'']. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.
* [https://www.bbc.co.uk/programmes/p00548f6 "Chaos Theory"], BBC Radio 4 discussion with Susan Greenfield, David Papineau & Neil Johnson (''In Our Time'', May 16, 2002)
* [https://www.youtube.com/watch?v=fDek6cYijxI Chaos: The Science of the Butterfly Effect] (2019) an explanation presented by [[Derek Muller]]

{{Chaos theory}}
{{Patterns in nature}}
{{Authority control}}

{{DEFAULTSORT:Chaos Theory}}
[[Category:Chaos theory| ]]
[[Category:Complex systems theory]]
[[Category:Computational fields of study]]