View source for Chaos theory

{{other uses|Chaos theory (disambiguation)}}
[[File:Lorenz attractor yb.svg|thumb|right|A plot of the [[Lorenz attractor]] for values {{nowrap|''r'' {{=}} 28}}, {{nowrap|σ {{=}} 10}}, {{nowrap|''b'' {{=}} 8/3}}]]
[[File:Double-compound-pendulum.gif|thumb|A [[double pendulum|double rod pendulum]] animation showing chaotic behavior. Starting the pendulum from a slightly different [[initial condition]] would result in a completely different [[trajectory]].  The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions.]]

'''Chaos theory''' is a branch of [[mathematics]] focused on the behavior of [[dynamical system]]s that are highly sensitive to [[initial conditions]]. 'Chaos' is an interdisciplinary theory stating that within the apparent randomness of [[chaotic complex system]]s, there are underlying patterns, constant [[feedback loops]], repetition, [[self-similarity]], [[fractals]], [[self-organization]], and reliance on programming at the initial point known as ''sensitive dependence on initial conditions''. The [[butterfly effect]] describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, e.g. a butterfly flapping its wings in Brazil can cause a hurricane in Texas.<ref name="bo1">{{cite web |url = http://geoffboeing.com/2015/03/chaos-theory-logistic-map/ |author = Boeing |title = Chaos Theory and the Logistic Map |date = 2015 |accessdate = 2015-07-16 }}</ref>

Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems—a response popularly referred to as the butterfly effect—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 |publisher = University of Chicago Press |year = 1993 |isbn = 0-226-42976-8 |page = 32 |ref = harv }}</ref><ref name="Boeing2016Systems">{{cite journal|url=http://geoffboeing.com/publications/nonlinear-chaos-fractals-prediction/|author=Boeing, G.|title=Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction|journal=Systems|date=2016|volume=4|issue=4|pages=37|accessdate=2016-12-02|doi=10.3390/systems4040037}}</ref> This happens even though these systems are [[deterministic system (mathematics)|deterministic]], meaning that their future behavior is fully determined by their initial conditions, with no [[randomness|random]] elements involved.<ref>{{harvnb|Kellert|1993|p=56}}</ref> In other words, the deterministic nature of these systems does not make them predictable.<ref>{{harvnb|Kellert|1993|p=62}}</ref><ref name="WerndlCharlotte">{{cite journal |author = Werndl, Charlotte |title = What are the New Implications of Chaos for Unpredictability? |journal = The British Journal for the Philosophy of Science |volume = 60 |issue = 1 |pages = 195–220 |year = 2009 |url = http://bjps.oxfordjournals.org/cgi/content/abstract/60/1/195 |doi = 10.1093/bjps/axn053 }}</ref> This behavior is known as '''deterministic chaos''', or simply '''chaos'''. The theory was summarized by [[Edward Lorenz]] as:<ref>{{cite web |url = http://mpe2013.org/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 |accessdate = 4 April 2013 }}</ref>

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

Chaotic behavior exists in many natural systems, such as [[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> It also occurs spontaneously in some systems with artificial components, such as [[road traffic]].<ref name="SafonovTomer2002">{{cite journal |last1 = Safonov |first1 = Leonid A. |last2 = Tomer |first2 = Elad |last3 = Strygin |first3 = Vadim V. |last4 = Ashkenazy |first4 = Yosef |last5 = Havlin |first5 = Shlomo |title = Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science |volume = 12 |issue = 4 |year = 2002 |page = 1006 |issn = 1054-1500 |doi = 10.1063/1.1507903 |bibcode = 2002Chaos..12.1006S }}</ref> This behavior can be studied through analysis of a chaotic [[mathematical model]], or through analytical techniques such as [[recurrence plot]]s and [[Poincaré map]]s. Chaos theory has applications in several disciplines, including [[meteorology]], 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.)|first=|publisher=Berghahn Books|year=2005|isbn=|location=Oxford|pages=}}</ref><ref name=":1">{{Cite book|url=https://www.researchgate.net/publication/308792484_Quantum_Anthropology_Man_Cultures_and_Groups_in_a_Quantum_Perspective|title=Quantum anthropology: Man, cultures, and groups in a quantum perspective|last=Trnka R., Lorencova R.|first=|publisher=Charles University Karolinum Press|year=2016|isbn=|location=Prague|pages=}}</ref> [[sociology]], [[physics]],<ref>{{cite journal|last1=Hubler|first1=A|title=Adaptive control of chaotic systems|journal=Swiss Physical Society. Helvetica Physica Acta 62|date=1989|pages=339–342}}</ref> [[environmental science]], [[computer science]], [[engineering]], [[economics]], [[biology]], [[ecology]], and [[philosophy]]. 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 whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then 'appear' to become random.<ref name="Boeing2016Systems" /> The amount of time that 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 solar system, 50 million years. 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 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. Note, however, that the y coordinate is defined modulo one at each step, so the square region is actually depicting a cylinder, and the two points are closer than they look.]]
In common usage, "chaos" means "a state of disorder".<ref>Definition of {{linktext|chaos}} at [[Wiktionary]];</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=0-521-58750-6}}</ref>

# it must be sensitive to initial conditions
# it must be [[topological mixing|topologically mixing]]
# it must have [[dense set|dense]] [[periodic orbit]]s

In some cases, the last two properties in the 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=1-58488-002-3 |page=117 }}</ref><ref>{{cite book |author=Basener, William F. |title=Topology and its applications |publisher=Wiley |year=2006 |isbn=0-471-68755-3 |page=42 }}</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 in general 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 |publisher=Cambridge University Press |year=2001 |isbn=0-521-55874-3 |page=165 }}</ref>

=== Chaos as a spontaneous breakdown of topological supersymmetry ===
{{Main article|Supersymmetric theory of stochastic dynamics}}In continuous time dynamical systems, chaos is the phenomenon of the spontaneous breakdown of topological supersymmetry which is an intrinsic property of evolution operators of all stochastic and deterministic (partial) differential equations.<ref>{{cite journal|date=March 2016|title=Introduction to Supersymmetric Theory of Stochastics|journal=Entropy|volume=18|issue=4|pages=108|doi=10.3390/e18040108|author=Ovchinnikov, I.V.|arxiv = 1511.03393 |bibcode = 2016Entrp..18..108O }}</ref><ref>{{cite journal|year=2016|title=Topological supersymmetry breaking: Definition and stochastic generalization of chaos and the limit of applicability of statistics|journal=Modern Physics Letters B|volume=30|pages=1650086|doi=10.1142/S021798491650086X|author1=Ovchinnikov, I.V.|author2=Schwartz, R. N.|author3=Wang, K. L.|bibcode = 2016MPLB...3050086O |url=http://arxiv.org/pdf/1404.4076}}</ref> This picture of dynamical chaos works not only for deterministic models but also for models with external noise, which is an important generalization from the physical point of view because in reality all dynamical systems experience influence from their stochastic environments. Within this picture, the long-range dynamical behavior associated with chaotic dynamics, e.g., the [[butterfly effect]], is a consequence of the [[Goldstone's theorem]] in the application to the spontaneous topological supersymmetry breaking.

===Sensitivity to initial conditions===
{{Main article|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 '''45.92''','''16''' and '''4 ''' respectively. As can be seen, 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 with significantly different future paths, or trajectories. Thus, an arbitrarily small change, or perturbation, of the current trajectory may lead to significantly different future behavior.

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?''. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.

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 is no longer predictable. This is most familiar 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 |publisher=Morgan & Claypool |year=2007 |page=17 }}</ref> Of course, this does not mean that we cannot say anything about events far in the future; some restrictions on the system are present. With weather, we know that the temperature will not naturally reach 100&nbsp;°C or fall to -130&nbsp;°C on earth (during the current [[geologic era]]), but we can't say exactly what day will have the hottest temperature of the year.

In more mathematical terms, the [[Lyapunov exponent]] measures the sensitivity to initial conditions. 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 t is the time and λ is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents 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 is usually taken as an indication that the system is chaotic.

Also, other properties relate to sensitivity of initial conditions, such as [[Measure (mathematics)|measure-theoretical]] [[Mixing (mathematics)|mixing]] (as discussed in [[ergodicity|ergodic]] theory) and properties of a [[Kolmogorov automorphism|K-system]].<ref name="WerndlCharlotte" />

===Topological mixing===
[[File:LogisticTopMixing1-6.gif|thumb|Six iterations of a set of states <math>[x,y]</math>  passed through the logistic map. (a) the blue plot (legend 1) shows the first iterate (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 '''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.

===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>{{harvnb|Devaney|2003}}</ref> 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 |format=PDF |author-link=Tien-Yien Li |doi=10.2307/2318254 |issue=10 |author2-link=James A. Yorke |bibcode=1975AmMM...82..985L |deadurl=yes |archiveurl=https://web.archive.org/web/20091229042210/http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |archivedate=2009-12-29 |df= }}</ref> (1975) proof that any 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|doi=10.1103/PhysRevLett.96.044101|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 was 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.

===Minimum complexity of a chaotic system===
[[File:LogisticMap BifurcationDiagram.png|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.]]

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|last=Fu |first=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 }}</ref><ref>{{cite journal|last=Heidel |first=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 }}</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 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 exhibit chaotic behavior.<ref name="Rosario 2006">{{cite book|last=Rosario|first=Pedro|title=Underdetermination of Science: Part I|date=2006|publisher=Lulu.com|isbn=1411693914}}</ref>  Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional.<ref>{{cite journal
 |last=Bonet |first=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
}}</ref>  A theory of linear chaos is being developed in a branch of mathematical analysis known as [[functional analysis]].

=== Jerk systems ===
In [[physics]], [[Jerk (physics)|jerk]] is the third derivative of [[position (vector)|position]], with respect to time.  As such, differential equations of the form
:: <math>J\left(\overset{...}{x},\ddot{x},\dot {x},x\right)=0</math>
are sometimes called ''Jerk equations''. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations, is in a certain sense the minimal setting for solutions showing chaotic behaviour. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are called accordingly hyperjerk systems.<ref>K. E. Chlouverakis and J. C. Sprott, Chaos Solitons & Fractals 28, 739-746 (2005), Chaotic Hyperjerk Systems, http://sprott.physics.wisc.edu/pubs/paper297.htm</ref>

A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits can model solutions. These circuits are known as jerk circuits.

One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. In fact, certain well-known chaotic systems, such as the Lorenz attractor and the [[Rössler map]], are conventionally described as a system of three first-order differential equations that can combine into a single (although rather complicated) jerk equation. Nonlinear jerk systems are in a sense minimally complex systems to show chaotic behaviour; there is no chaotic system involving only two first-order, ordinary differential equations (the system resulting in an equation of second order only).

An example of a jerk equation with nonlinearity in the magnitude of <math>x</math> is:
:<math>\frac{\mathrm{d}^3 x}{\mathrm{d} t^3}+A\frac{\mathrm{d}^2 x}{\mathrm{d} t^2}+\frac{\mathrm{d} x}{\mathrm{d} t}-|x|+1=0.</math>

Here, ''A'' is an adjustable parameter. This equation has a chaotic solution for ''A''=3/5 and can be implemented with the following jerk circuit; the required nonlinearity is brought about by the two diodes:

[[File:JerkCircuit01.png|frameless|upright=1.4|center]]

In the above circuit, all resistors are of equal value, except <math>R_A=R/A=5R/3</math>, and all capacitors are of equal size. The dominant frequency is <math>1/2\pi R C</math>. The output of [[operational amplifier|op amp]] 0 will correspond to the x variable, the output of 1 corresponds to the first derivative of x and the output of 2 corresponds to the second derivative.

==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>

==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).]]

An early proponent of chaos theory was [[Henri Poincaré]]. In the 1880s, while studying the [[three-body problem]], he 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 |pages=1–270 |year=1890 |doi=10.1007/BF02392506 }}</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 chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature, called "[[Hadamard's billiards]]".<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}}</ref> Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one another, with a positive [[Lyapunov exponent]].

Chaos theory began in the field of [[ergodic theory]]. 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 | authorlink=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 }} 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. |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. |issue=1890 |bibcode=1991RSPSA.434...15K}}</ref><ref>{{cite journal| last=Kolmogorov | first=A. N. | year=1954 | title=Preservation of conditionally periodic movements with small change in the Hamiltonian function | journal=Doklady Akademii Nauk SSSR | volume=98 | pages=527–530| bibcode=1979LNP....93...51K| doi=10.1007/BFb0021737| series=Lecture Notes in Physics| isbn=3-540-09120-3}} 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 }}</ref> Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood.{{Citation needed|date=July 2008}} Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits 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]], the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the [[logistic map]]. 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.

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|2001|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 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}}</ref> Lorenz was using a simple digital computer, a [[Royal McBee]] [[LGP-30]], to run his weather simulation. He wanted to see a sequence of data again, and to save time he started the simulation in the middle of its course. He did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To his surprise, the weather the machine began to predict was completely different from the previous calculation. Lorenz 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=0-434-29554-X}}</ref> Lorenz's discovery, which gave its name to [[Lorenz attractor]]s, showed that even detailed atmospheric modelling cannot, in general, make precise long-term weather predictions.

In 1963, [[Benoit Mandelbrot]] found recurring patterns at every scale in data on cotton prices.<ref>{{cite journal|first = Benoît|last = Mandelbrot|year = 1963|title = The variation of certain speculative prices|journal = Journal of Business|volume = 36|pages = 394–419|doi = 10.1086/294632|issue = 4|jstor=2350970}}</ref> Beforehand he had studied [[information theory]] and concluded noise was patterned like a [[Cantor set]]: on any scale the proportion of noise-containing periods to error-free periods was a constant – thus errors were inevitable and must be planned for by incorporating redundancy.<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 | 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 |publisher=Basic Books |location=New York |year=2004 |page=201 }}</ref> This challenged the idea that changes in price were [[normal distribution|normally distributed]]. 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 }}</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=0716711869}}</ref> Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model.<ref>{{cite book |last1=Buldyrev |first1=S.V. |last2=Goldberger |first2=A.L. | last3=Havlin |first3=S. | authorlink3=Shlomo Havlin|last4=Peng |first4=C.K. | last5=Stanley | first5=H.E. |authorlink5=H. Eugene Stanley|editor1-first=Armin |editor1-last=Bunde  |editor2-first=Shlomo | editor2-last=Havlin | editor2-link=Shlomo Havlin |title=Fractals in Science |publisher=Springer |year=1994 |pages=49–89 |chapter=Fractals in Biology and Medicine: From DNA to the Heartbeat |isbn=3-540-56220-6}}</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, independently [[Pierre Coullet]] and [[Charles Tresser]] with the article "Iterations d'endomorphismes et groupe de renormalisation" and [[Mitchell Feigenbaum]] with the article "Quantitative Universality for a Class of Nonlinear Transformations" described logistic maps.<ref>{{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}}</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>  They notably 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.}}</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 tracking disorder]] among [[schizophrenics]].<ref>{{cite journal |authorlink=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 |pages=260–273 |date=July 1987 |doi=10.1111/j.1749-6632.1987.tb48737.x |bibcode = 1987NYASA.504..260H }}</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 |author=Bak, Per; Tang, Chao; 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 |bibcode=1987PhRvL..59..381B|last2=Tang |last3=Wiesenfeld }} 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=? |work= |publisher= |accessdate= |deadurl=yes |archiveurl=https://web.archive.org/web/20071214033929/https://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |archivedate=2007-12-14 |df= }}. 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.

In the same year, [[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, though his history under-emphasized important Soviet contributions.{{citation needed|reason=No reason given for soviet under-emphasis|date=May 2016}}<ref>{{cite book|last=Gleick|first=James|title=Chaos: Making a New Science|date=August 26, 2008|publisher=Penguin Books|isbn=0143113453}}</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 }}</ref> involving many different disciplines ([[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|date=2007|doi=10.1002/cplx.20159|bibcode = 2007Cmplx..12c..10H }}</ref> [[social systems]], [[population model]]ing, [[biology]], [[meteorology]], [[astrophysics]], [[information theory]], [[computational neuroscience]], etc.).

==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 |title=The Geometry and Pigmentation of Seashells |author=Stephen Coombes |date=February 2009 |work=www.maths.nottingham.ac.uk |publisher=[[University of Nottingham]] |accessdate=2013-04-10}}</ref>]]

Chaos theory was born from observing weather patterns, but it has become applicable to a variety of other situations. Some areas benefiting from chaos theory today are [[geology]], [[mathematics]], [[microbiology]], [[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>[http://www.dspdesignline.com/218101444;jsessionid=Y0BSVTQJJTBACQSNDLOSKH0CJUNN2JVN?pgno=1 Applying Chaos Theory to Embedded Applications]</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 }}</ref><ref>{{Cite journal | doi = 10.1023/A:1023939610962 |author1=Kyrtsou, C.  |author2=M. Terraza, | 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> [[algorithmic trading]],<ref>{{cite book|last=Williams|first=Bill Williams, Justine|title=Trading chaos : maximize profits with proven technical techniques|year=2004|publisher=Wiley|location=New York|isbn=9780471463085|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=978-0471585244|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=978-0471139386|edition=2nd }}</ref> [[meteorology]], [[philosophy]], anthropology<ref name=":0" /><ref name=":1" />, [[physics]],<ref>{{cite journal|last1=Hubler|first1=A.|last2=Phelps|first2=K.|title=Guiding a self-adjusting system through chaos|journal=Complexity|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|date=2007|doi=10.1103/PhysRevE.75.045202|arxiv=nlin/0701050|bibcode = 2007PhRvE..75d5202G }}</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|date=2009|doi=10.1021/jp804420g|bibcode = 2009JPCA..113...19W }}</ref> [[politics]], [[population dynamics]],<ref>{{cite journal |author1=Dilão, R. |author2=Domingos, T. | 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> [[psychology]],<ref name="SafonovTomer2002"/> 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 ===
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, secure pseudo-random number generators, stream ciphers, watermarking and steganography.<ref name="Akhavan 1797–1813">{{Cite journal|last=Akhavan|first=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|url=http://www.sciencedirect.com/science/article/pii/S0016003211001104|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|last=Behnia|first=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|url=http://www.sciencedirect.com/science/article/pii/S0960077906004681|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|year=2013|title=A novel text and image encryption method based on chaos theory and DNA computing|journal=Natural Computing. an International Journal|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|last=Akhavan|first=A.|last2=Samsudin|first2=A.|last3=Akhshani|first3=A.|date=2017-10-01|title=Cryptanalysis of an image encryption algorithm based on DNA encoding|url=http://www.sciencedirect.com/science/article/pii/S0030399216308635|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|date=2017-06-01|title=Cryptanalysis of an Image Encryption Algorithm Based on DNA Sequence Operation and Hyper-chaotic System|url=https://link.springer.com/article/10.1007/s13319-017-0126-y|journal=3D Research|language=en|volume=8|issue=2|pages=15|doi=10.1007/s13319-017-0126-y|issn=2092-6731|bibcode = 2017TDR.....8..126X }}</ref><ref>{{Cite journal|last=Liu|first=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|url=http://www.sciencedirect.com/science/article/pii/S0030399214000255|journal=Optics & Laser Technology|volume=60|pages=111–115|doi=10.1016/j.optlastec.2014.01.015|arxiv=1307.4279|bibcode = 2014OptLT..60..111L }}</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}}</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}}</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}}</ref>

===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}}</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}}</ref>

Chaos theory can be applied outside of the natural sciences. By adapting a model of career counseling to include a chaotic interpretation of the relationship between employees and the job market, better suggestions can be made to people struggling with career decisions.<ref>{{cite journal|last=Pryor|first=Robert G. L.|author2=Norman E. Aniundson|author3=Jim E. H. Bright|title=Probabilities and Possibilities: The Strategic Counseling Implications of the Chaos Theory of Careers|journal=The Career Development Quarterly|date=June 2008|volume=56|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. The chaos metaphor—used in verbal theories—grounded on mathematical models and psychological aspects of human behavior
provides helpful insights to describing the complexity of small work groups, that go beyond the metaphor itself.<ref>{{cite book | last1 = Dal Forno | first1 = Arianna | last2 = Merlone | first2 = Ugo |editor1-first=Gian Italo |editor1-last=Bischi  |editor2-first=Carl | editor2-last=Chiarella |editor3-first=Irina | editor3-last=Shusko |title=Global Analysis of Dynamic Models in Economics and Finance |publisher=Springer-Verlag |year=2013 |pages=185–204 |chapter= Chaotic Dynamics in Organization Theory |isbn= 978-3-642-29503-4}}</ref>

[[File:BML N=200 P=32.png|400px|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.]]

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}}</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 |authorlink=Chris Brooks (academic)|title=Chaos in foreign exchange markets: a sceptical view |journal=Computational Economics|year=1998 |volume=11 |pages=265–281 |ISSN=1572-9974 |doi=10.1023/A:1008650024944}}</ref>

Traffic forecasting also benefits from applications of chaos theory. Better predictions of when traffic will occur lets measures 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}}</ref>

Chaos theory can be applied in psychology. For example, in modeling group behavior in which heterogeneous members may behave as if sharing to different degrees what in [[Wilfred Bion]]'s theory is a  basic assumption, the group dynamics is the result of the individual dynamics of the members: each individual reproduces the group dynamics in a different scale, and the chaotic behavior of the group is reflected in each member.<ref name="Dal FornoMerlone2013">{{cite journal | last1 = Dal Forno | first1 = Arianna | last2 = Merlone | first2 = Ugo | title = Nonlinear dynamics in work groups with Bion's basic assumptions | journal = Nonlinear Dynamics, Psychology, and Life Sciences | volume = 17| issue=2 | year = 2013 | pages = 295–315 | issn = 1090-0578  }}</ref>

Chaos theory has been applied to environmental [[water cycle]] data (aka 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|url=http://www.sciencedirect.com/science/article/pii/S0309170899000081|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|Systems science|Mathematics}}
;Examples of chaotic systems
{{colbegin||18em}}
* [[Contour advection|Advected contours]]
* [[Arnold's cat map]]
* [[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]]
* [[Hénon map]]
* [[Horseshoe map]]
* [[List of chaotic maps]]
* [[Logistic map]]
* [[Rössler attractor]]
* [[Standard map]]
* [[Swinging Atwood's machine]]
* [[Tilt A Whirl]]
{{colend}}
;Other related topics
{{colbegin||18em}}
* [[Amplitude death]]
* [[Anosov diffeomorphism]]
* [[Bifurcation theory]]
* [[Butterfly effect]]
* [[Catastrophe theory]]
* [[Chaos theory in organizational development]]
* [[Chaos machine]]
* [[Chaotic mixing]]
* [[Chaotic scattering]]
* [[Complexity]]
* [[Control of chaos]]
* [[Edge of chaos]]
* [[Emergence]]
* [[Fractal]]
** [[Julia set]]
** [[Mandelbrot set]]
* [[Kolmogorov–Arnold–Moser theorem]]
* [[Ill-conditioning]]
* [[Ill-posedness]]
* [[Nonlinear system]] 
* [[Patterns in nature]]
* [[Predictability]]
* [[Quantum chaos]]
* [[Santa Fe Institute]]
* [[Synchronization of chaos]]
* [[Unintended consequence]]
{{colend}}
;People
{{colbegin||18em}}
* [[Ralph Abraham (mathematician)|Ralph Abraham]]
* [[Michael Berry (physicist)|Michael Berry]]
* [[Leon O. Chua]]
* [[Ivar Ekeland]]
* [[Doyne Farmer]]
* [[Mitchell Feigenbaum]]
* [[Martin Gutzwiller]]
* [[Brosl Hasslacher]]
* [[Michel Hénon]]
* [[Andrey Nikolaevich Kolmogorov]]
* [[Edward Lorenz]]
* [[Aleksandr Lyapunov]]
* [[Ian Malcolm (Jurassic Park character)]]
* [[Benoit Mandelbrot]]
* [[Norman Packard]]
* [[Henri Poincaré]]
* [[Otto Rössler]]
* [[David Ruelle]]
* [[Oleksandr Mikolaiovich Sharkovsky]]
* [[Robert Shaw (physicist)|Robert Shaw]]
* [[Floris Takens]]
* [[James A. Yorke]]
* [[George M. Zaslavsky]]
{{colend}}

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

==Scientific literature==

===Articles===
* {{cite journal |first=A.N. |last=Sharkovskii |authorlink=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 |authorlink1=Tien-Yien Li |last1=Li |first1=T.Y. |authorlink2=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 }}
* {{Cite journal |date = December 1986|title=Chaos |journal=[[Scientific American]] |volume=255 |issue=6 |pages=38–49 (bibliography p.136) |postscript=<!--None-->|bibcode = 1986SciAm.255d..38T |last2 = Tucker |last3 = Morrison |author1 = Crutchfield|author4 = J.D.|author5 = Packard|author6=N.H. |author7=Shaw |author8=R.S |authorlink1=James P. Crutchfield |authorlink5=Norman Packard |authorlink7=Robert Shaw (physicist) }} [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. |title=Li-Yorke sensitivity and other concepts of chaos |journal=Ukrainian Math. J. |volume=56 |pages=1242–57 |year=2004 |url=http://www.springerlink.com/content/q00627510552020g/?p=93e1f3daf93549d1850365a8800afb30&pi=3 |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 }}
* {{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 |format=PDF |pmid=16486826 |page=044101 |bibcode=2006PhRvL..96d4101S |deadurl=yes |archiveurl=https://web.archive.org/web/20130426201635/http://www.ccsr.illinois.edu/web/Techreports/2005-08/CCSR-05-4.pdf |archivedate=2013-04-26 |df= }}
* {{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 |format=PDF |doi=10.1002/cplx.20159 |issue=3|bibcode = 2007Cmplx..12c..10H }}
* {{cite journal | last1 = Motter | first1 = Adilson E. | last2 = Campbell | first2 = David K. | year = 2013 | title = Chaos at 50 | url = http://scitation.aip.org/content/aip/magazine/physicstoday/article/66/5/10.1063/PT.3.1977 | journal = Physics Today | volume =  66| issue = | page =  27| doi = 10.1063/PT.3.1977 |arxiv = 1306.5777 |bibcode = 2013PhT....66e..27M }}
* {{cite journal | last1 = Boeing | first1 = G. | year = 2016 | title = Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction | url = http://dx.doi.org/10.3390/systems4040037 | journal = Systems | volume = 4 | issue = 4 | page =  37 | doi = 10.3390/systems4040037 }}

===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=0-387-94677-2 |ref=harv |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=0-521-39511-9}}
* {{cite book |author1=Badii, R. |author2=Politi A. |title=Complexity: hierarchical structures and scaling in physics |publisher=Cambridge University Press |year=1997 |isbn=0-521-66385-7 |url=http://www.cambridge.org/gb/academic/subjects/physics/statistical-physics/complexity-hierarchical-structures-and-scaling-physics}}
* {{cite book |editor1-last=Bunde  |editor2-first=Shlomo | editor2-last=Havlin | editor2-link=Shlomo Havlin |title=Fractals and Disordered Systems |publisher=Springer |year=1996 |isbn=3642848702}} and {{cite book |editor1-last=Bunde  |editor2-first=Shlomo | editor2-last=Havlin | editor2-link=Shlomo Havlin |title=Fractals in Science |publisher=Springer |year=1994 |isbn=3-540-56220-6}}
* {{cite book| author=Collet, Pierre, and [[Jean-Pierre Eckmann|Eckmann, Jean-Pierre]]| title=Iterated Maps on the Interval as Dynamical Systems | publisher=Birkhauser | year=1980 |isbn=0-8176-4926-3}}
* {{cite book |last=Devaney |first=Robert L.|authorlink= Robert L. Devaney|ref=harv |title=An Introduction to Chaotic Dynamical Systems |edition=2nd |publisher=Westview Press |year=2003 |isbn=0-8133-4085-3 |url=https://books.google.com/books?id=CjAnY99LwTgC}}
* {{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=http://chaos.coa.edu/index.html}}
* {{cite book |author1=Gollub, J. P. |author2=Baker, G. L. |title=Chaotic dynamics |publisher=Cambridge University Press |year=1996 |isbn=0-521-47685-2 |url=https://books.google.com/books?id=n1qnekRPKtoC}}
* {{cite book |author=Guckenheimer, John| authorlink=John Guckenheimer|author2= Holmes, Philip |authorlink2=Philip Holmes|title=Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields |publisher=Springer-Verlag  |year=1983 |isbn=0-387-90819-6}}
* {{cite book| author=Gulick, Denny| title=Encounters with Chaos| publisher=McGraw-Hill| year=1992| isbn=0-07-025203-3}}
* {{cite book |author=Gutzwiller, Martin |title=Chaos in Classical and Quantum Mechanics |publisher=Springer-Verlag |year=1990 |isbn=0-387-97173-4 |url=https://books.google.com/books?id=fnO3XYYpU54C}}
* {{cite book |author=Hoover, William Graham|authorlink=William G Hoover|title=Time Reversibility, Computer Simulation, and Chaos |publisher=World Scientific |origyear=1999|year=2001 |isbn=981-02-4073-2 |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=0-472-08472-0 |url=https://books.google.com/books?id=K46kkMXnKfcC}}
* {{cite book |author=Moon, Francis |title=Chaotic and Fractal Dynamics |publisher=Springer-Verlag |year=1990 |isbn=0-471-54571-6 |url=https://books.google.com/books?id=Ddz-CI-nSKYC}}
* {{cite book |author=Ott, Edward |title=Chaos in Dynamical Systems |publisher=Cambridge University Press  |year=2002 |isbn=0-521-01084-5 |url=https://books.google.com/books?id=nOLx--zzHSgC}}
* {{cite book| author=Strogatz, Steven| authorlink=Steven Strogatz| title=Nonlinear Dynamics and Chaos| publisher=Perseus Publishing| year=2000| isbn=0-7382-0453-6}}
* {{cite book |last=Sprott |first=Julien Clinton |title=Chaos and Time-Series Analysis |publisher=Oxford University Press |year=2003 |isbn=0-19-850840-9 |ref=harv |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=0-521-83912-2 |url=https://books.google.com/books?id=P2JL7s2IvakC}}
* {{cite book| last = Teschl| given = Gerald|authorlink=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 = http://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=0-471-87645-3}}
* {{cite book| author1-link=Nicholas Tufillaro |last1=Tufillaro |last2=Reilly| title=An experimental approach to nonlinear dynamics and chaos| publisher=Addison-Wesley | year=1992| isbn=0-201-55441-0}}
* {{cite book | last=Wiggins|first=Stephen  | title= Introduction to Applied Dynamical Systems and Chaos | publisher= Springer | year= 2003 | isbn= 0-387-00177-8 }}
* {{cite book| author=Zaslavsky, George M.| title=Hamiltonian Chaos and Fractional Dynamics| publisher=Oxford University Press| year=2005| isbn=0-19-852604-0}}

===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 |url=https://books.google.com/books?id=0E667XpBq1UC |year=2000 |publisher=World Scientific |isbn=978-981-238-647-2 |ref=harv}}
* {{cite book |authorlink=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: Complexit 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.
* Alan Marshall (2002) The Unity of Nature: Wholeness and Disintegration in Ecology and Science, Imperial College Press: London
* [[Heinz-Otto Peitgen]] and [[Dietmar Saupe]] (Eds.), ''The Science of Fractal Images'', Springer 1988, 312 pp.
* [[Clifford A. Pickover]], ''Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World '', St Martins Pr 1991.
* [[Ilya Prigogine]] and [[Isabelle Stengers]], ''Order Out of Chaos'', Bantam 1984.
* Heinz-Otto Peitgen and P. H. Richter, ''The Beauty of Fractals : Images of Complex Dynamical Systems'', Springer 1986, 211 pp.
* [[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/?id=qnMrFEHMrWwC|year=2013 |publisher=Princeton University Press|isbn=0691152721 }}
* David Ruelle, ''Chaotic Evolution and Strange Attractors'', Cambridge University Press, 1989.
* Peter Smith, ''Explaining Chaos'', Cambridge University Press, 1998.
* [[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.
* Sawaya, Antonio (2010). ''Financial time series analysis : Chaos and neurodynamics approach''.

==External links==
{{Commons category|Chaos theory}}
* {{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]
* [http://hypertextbook.com/chaos/ The Chaos Hypertextbook]. An introductory primer on chaos and fractals
* [http://chaosbook.org/ ChaosBook.org] An advanced graduate textbook on chaos (no fractals)
* [http://www.societyforchaostheory.org/ Society for Chaos Theory in Psychology & Life Sciences]
* [http://www.csdc.unifi.it/mdswitch.html?newlang=eng Nonlinear Dynamics Research Group at CSDC], [[Florence]] [[Italy]]
* [http://physics.mercer.edu/pendulum/ Interactive live chaotic pendulum experiment], allows users to interact and sample data from a real working damped driven chaotic pendulum
* [http://www.creatingtechnology.org/papers/chaos.htm Nonlinear dynamics: how science comprehends chaos], talk presented by Sunny Auyang, 1998.
* [http://www.egwald.ca/nonlineardynamics/index.php Nonlinear Dynamics]. Models of bifurcation and chaos by Elmer G. Wiens
* [http://www.around.com/chaos.html Gleick's ''Chaos'' (excerpt)]
* [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/user/thedebtgeneration?feature=mhum 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,  [http://www.chaos-math.org/en ''Chaos, A Mathematical Adventure'']. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.

{{Systems}}
{{Chaos theory}}
{{Patterns in nature}}

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