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[[Image:Lorenz attractor yb.svg|thumb|right||A plot of the trajectory Lorenz system for values ''r'' = 28, σ = 10, ''b'' = 8/3]]
 
In [[mathematics]] and [[physics]], '''chaos theory''' describes the behavior of certain [[nonlinearity|nonlinear]] [[dynamical system]]s that under certain conditions exhibit a phenomenon known as '''chaos'''. Among the characteristics of chaotic systems, described below, is sensitivity to initial conditions (popularly referred to as the [[butterfly effect]]). As a result of this sensitivity, the behavior of systems that exhibit chaos appears to be random, even though the system is [[deterministic system (philosophy)|deterministic]] in the sense that it is well defined and contains no random parameters. Examples of such systems include the [[Earth's atmosphere|atmosphere]], the [[solar system]], [[plate tectonics]], [[turbulence|turbulent]] fluids, economics, and population growth.
 
   
Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word ''chaos'' is at odds with common parlance, which suggests complete disorder. (See the article on [[chaos (mythology)|mythological chaos]] for a discussion of the origin of the word in mythology, and other uses.) A related field of physics called [[quantum chaos]] theory studies non-deterministic systems that follow the laws of [[quantum mechanics]].
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{{PsyPerspective}}
   
==Chaotic dynamics==
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[[Image:Lorenz attractor yb.svg|thumb|right||A plot of the [[Lorenz attractor]] for values ''r'' = 28, σ = 10, ''b'' = 8/3]]
For a dynamical system to be classified as chaotic, most scientists will agree that it must have the following properties:
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In [[mathematics]] and [[physics]], '''chaos theory''' describes the behavior of certain [[nonlinearity|nonlinear]] [[dynamical system]]s that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the [[butterfly effect]]). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be [[randomness|random]]. This happens even though these systems are [[deterministic system (philosophy)|deterministic]], meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply ''[[chaos]]''.
   
* it must be ''sensitive to initial conditions'',
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== Overview ==
* it must be ''[[topological mixing|topologically mixing]]'', and
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Chaotic behavior has been observed in the laboratory in a variety of systems including [[electrical circuits]], [[lasers]], oscillating [[chemical reactions]], [[fluid dynamics]], and mechanical and magneto-mechanical devices. Observations of chaotic behaviour in nature include the dynamics of satellites in the [[solar system]], the time evolution of the [[magnetic field of celestial bodies]], [[population dynamics|population growth]] in [[ecology]], the dynamics of the [[action potentials]] in neurons, and [[molecular vibration]]s. Everyday examples of chaotic systems include [[meteorology|weather and climate]].<ref>Sneyers, R: "Climate Chaotic Instability: Statistical Determination and Theoretical Background", 8(5):517-532</ref> There is some controversy over the existence of chaotic dynamics in the [[plate tectonics]] and in [[economics]].<ref> Apostolos Serletis and Periklis Gogas,[http://www.informaworld.com/smpp/content~content=a713761243~db=all~order=page ''Purchasing Power Parity Nonlinearity and Chaos''], in: Applied Financial Economics, 10, 615-622, 2000.</ref> <ref> Apostolos Serletis and Periklis Gogas {{PDFlink|[http://mpra.ub.uni-muenchen.de/1576/01/MPRA_paper_1576.pdf ''The North American Gas Markets are Chaotic'']|918&nbsp;[[Kibibyte|KiB]]<!-- application/pdf, 940407 bytes -->}}, in: The Energy Journal, 20, 83-103, 1999.</ref> <ref> Apostolos Serletis and Periklis Gogas, [http://ideas.repec.org/a/eee/reecon/v51y1997i4p359-385.html ''Chaos in East European Black Market Exchange Rates''], in: Research in Economics, 51, 359-385, 1997.</ref>
* its periodic orbits must be ''[[dense set|dense]]''.
 
   
''Sensitivity to initial conditions'' means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.
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Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word ''chaos'' is at odds with common parlance, which suggests complete disorder. A related field of physics called [[quantum chaos]] theory studies systems that follow the laws of [[quantum mechanics]]. Recently, another field, called [[relativistic chaos]],<ref>A. E. Motter, [http://prola.aps.org/abstract/PRL/v91/i23/e231101 ''Relativistic chaos is coordinate invariant''], in: Phys. Rev. Lett. 91, 231101 (2003).</ref> has emerged to describe systems that follow the laws of [[general relativity]].
   
Sensitivity to initial conditions is popularly known as the "[[butterfly effect]]", suggesting that the flapping of a butterfly's wings might create tiny changes in the atmosphere, which could over time cause a [[tornado]] to occur. 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.
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As well as being orderly in the sense of being deterministic, chaotic systems usually have well defined [[statistics]]. For example, the [[Lorenz system]] pictured is chaotic, but has a clearly defined structure. ''Bounded chaos'' is a useful term for describing models of disorder.
   
Sensitivity to initial conditions is often confused with chaos in popular accounts. It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the [[phase space]] of the system. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by the mapping on the real line from ''x'' to ''2x''). This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, it has extremely simple behavior, as all points except 0 tend to infinity. If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial conditions. For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems.
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== History ==
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[[Image:Fractal fern1.png|thumb|right|150px|[[Fractal]] [[fern]] created using [[chaos game]]. Natural forms in nature (ferns, clouds, mountains, etc.) may be recreated through an [[Iterated function system]] (IFS).]]
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The first discoverer of chaos can plausibly be argued to be [[Jacques Hadamard]], who in [[1898]] published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature. In the system studied, [[Hadamard's billiards]], 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]].
   
Even for bounded systems, sensitivity to initial conditions is not identical with chaos. For example, consider the two-dimension torus described by a pair of angles (x,y), each ranging between zero and . Define a mapping that takes any point (x,y) to (2x, y + a), where a is any number
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In the early [[1900]]s [[Henri Poincaré]], while studying the [[three-body problem]], found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Much of the early theory was developed almost entirely by mathematicians, under the name of [[ergodic theory]]. Later studies, also on the topic of nonlinear differential equations, were carried out by [[George David Birkhoff|G.D. Birkhoff]], [[Andrey Nikolaevich Kolmogorov|A.N. Kolmogorov]], [[Mary Lucy Cartwright|M.L. Cartwright]], [[John Edensor Littlewood|J.E. Littlewood]], and [[Stephen Smale]]. 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. 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.
such that a/2π is irrational. Because of the doubling in the first coordinate, the mapping exhibits sensitive dependence on initial conditions.
 
However, because of the irrational rotation in the second coordinate, there are no periodic orbits, and hence the mapping is not chaotic
 
according to the definition above.
 
   
''Topologically mixing'' means that the system will evolve over time so that any given region or [[open set]] of its phase space will eventually overlap with any other given region. Here, "mixing" is really meant to correspond to the standard intuition: the mixing of colored [[dye]]s or fluids is an example of a chaotic system.
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Despite initial insights in the first half of the century, chaos theory became formalized as such only after mid-century, when it first became evident for some scientists that [[linear theory]], the prevailing system theory at that time, simply could not explain the observed behaviour of certain experiments like that of the [[logistic map]]. What had been beforehand excluded as [[measure]] imprecision and simple "[[noise]]" was considered by chaos theories as a full component of the studied systems.
   
==Attractors==
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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. One of the earliest electronic digital computers, [[ENIAC]], was used to run simple weather forecasting models.
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[[Image: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 was important for Chaos theory, analyzed for example by the 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.]]
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An early pioneer of the theory was [[Edward Lorenz]] whose interest in chaos came about accidentally through his work on [[meteorology|weather prediction]] in [[1961]]. Lorenz was using a basic [[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 was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.
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To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The printout rounded variables off to a 3-digit number, but the computer worked with 6-digit numbers. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome. Lorenz's discovery, which gave its name to [[Lorenz attractor]]s, proved that meteorology could not reasonably predict weather beyond a weekly period (at most).
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The year before, [[Benoit Mandelbrot]] found recurring patterns at every scale in data on [[cotton prices]]. 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]]. Mandelbrot described both the [[Noah effect]] (in which sudden discontinuous changes can occur, e.g., in a stock's prices after bad news, thus challenging [[normal distribution]] theory in [[statistics]], aka Bell Curve) and the [[Joseph effect]] (in which persistence of a value can occur for a while, yet suddenly change afterwards). In 1967, he published ''[[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 infinitesimally small measuring device. Arguing that a ball of twine appears to be 1-dimensional (far), 3-dimensional (fairly near), or 1-dimensional (close), 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 (for example, the [[Koch curve]] or "[[snowflake]]", which is infinitely long yet encloses a finite space with dimensions = 1.2618; or the [[Menger sponge]] and the [[Sierpinski gasket]]). In 1975 Mandelbrot published ''[[The Fractal Geometry of Nature]]'', which became a classic of chaos theory. Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model.
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[[Yoshisuke Ueda]] independently identified a chaotic phenomenon as such by using an [[analog computer]] on November 27, 1961. The chaos exhibited by an analog computer is a real phenomenon, in contrast with those that digital computers calculate, which has a different kind of limit on precision. Ueda's supervising professor, Hayashi, did not believe in chaos, and thus he prohibited Ueda from publishing his findings until [[1970]].
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In December 1977 the [[New York Academy of Sciences]] organized the first symposium on Chaos, attended by [[David Ruelle]], [[Robert May]], [[James Yorke]] (coiner of the term "chaos" as used in mathematics), [[Robert Shaw (physicist)|Robert Shaw]] (a physicist, part of the [[Eudaemons]] group with [[J. Doyne Farmer]] and [[Norman Packard]] who tried to find a mathematical method to beat [[roulette]], and then created with them the [[Dynamical Systems Collective]] in [[Santa Cruz]]), and the meteorologist [[Edward Lorenz]].
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The following year, [[Mitchell Feigenbaum]] published the noted article "Quantitative Universality for a Class of Nonlinear Transformations", where he described [[logistic maps]]. Feigenbaum had applied [[fractal geometry]] to the study of natural forms such as coastlines. Feigenbaum notably discovered the universality in chaos, permitting an application of chaos theory to many different phenomena.
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In 1979, [[Albert J. Libchaber]] during a symposium organized in Aspen by [[Pierre Hohenberg]] his experimental observation of the [[bifurcation]] cascade that leads to chaos and turbulence in [[convection|convective]] [[Rayleigh-Benard systems]]. He was awarded the [[Wolf Prize in Physics]] in 1986 along with [[Mitchell J. Feigenbaum]] "for his brilliant experimental demonstration of the transition to turbulence and chaos in dynamical systems".<ref>[http://www.wolffund.org.il/cat.asp?id=25&cat_title=PHYSICS The Wolf Prize in Physics in 1986.]</ref>
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The New York Academy of Sciences then co-organized, in 1986, with the [[National Institute of Mental Health]] and the [[Office of Naval Research]] the first important conference on Chaos in biology and medicine. [[Bernardo Huberman]] thereby presented a mathematical model of the [[eye tracking disorder]] among [[schizophrenics]] <ref> [[Bernardo Huberman]], "A Model for Dysfunctions in Smooth Pursuit Eye Movement" Annals of the New York Academy of Sciences, Vol. 504 Page 260 July 1987, Perspectives in Biological Dynamics and Theoretical Medicine </ref>. Chaos theory thereafter renewed [[physiology]] in the 1980s, for example in the study of pathological [[cardiac cycle]]s.
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In 1987, [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] published a paper in ''[[Physical Review Letters]]'' describing for the first time [[self-organized criticality]] (SOC), considered to be one of the mechanisms by which [[complexity]] arises in nature.
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Alongside largely lab-based approaches such as the [[Bak-Tang-Wiesenfeld sandpile]], many other investigations have centred around large-scale natural or social systems that are known (or suspected) to display [[scale invariance|scale-invariant]] behaviour. 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: [[earthquakes]] (which, long before SOC was discovered, were known as a source of [[scale invariance|scale-invariant]] behaviour such as the [[Gutenberg-Richter law]] describing the statistical distribution of earthquake sizes, and the [[Omori law]] describing the frequency of aftershocks); [[solar flares]]; fluctuations in economic systems such as [[financial markets]] (references to SOC are common in [[econophysics]]); [[landscape formation]]; [[forest fires]]; [[landslides]]; [[epidemics]]; and [[biological evolution]] (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of [[punctuated equilibria|"punctuated equilibria"]] put forward by [[Niles Eldredge]] and [[Stephen Jay Gould]]). Worryingly, given the implications of a [[scale invariance|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 [[wars]]. These "applied" 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.
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The same year, [[James Gleick]] published ''[[Chaos: Making a New Science]]'', which became a best-seller and introduced general principles of chaos theory as well as its history to the broad public. At first the domains of work 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 self-nominated themselves) claimed that this new theory was an example of such as shift, a thesis upheld by J. Gleick.
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The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research, involving many different disciplines (mathematics, [[topology]], physics, population biology, biology, meteorology, astrophysics, [[information theory]], etc.).
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== Chaotic dynamics ==
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For a dynamical system to be classified as chaotic, it must have the following properties:<ref>{{cite book |title=A First Course in Dynamics: With a Panorama of Recent Developments |last=Hasselblatt |first=Boris |coauthors=Anatole Katok |year=2003 |publisher=Cambridge University
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Press |isbn=0521587506 }}</ref>
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* it must be sensitive to initial conditions,
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* it must be [[topological mixing|topologically mixing]], and
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* its [[periodic orbit]]s must be [[dense set|dense]].
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''Sensitivity to initial conditions'' means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behaviour.
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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.
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Sensitivity to initial conditions is often confused with chaos in popular accounts. It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the [[phase space]] of the system. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by the mapping on the real line from ''x'' to ''2x''). This system has sensitive dependence on initial conditions everywhere,
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since any pair of nearby points will eventually become widely separated. However, it has extremely simple behaviour, as all points except 0 tend to infinity. If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial conditions. For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems.
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Even for bounded systems, sensitivity to initial conditions is not identical with chaos. For example, consider the two-dimensional torus described by a pair of angles (''x'',''y''), each ranging between zero and 2π. Define a mapping that takes any point (''x'',''y'') to (''2x'', ''y'' + ''a''), where a is any number
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such that ''a''/2π is irrational. Because of the doubling in the first coordinate, the mapping exhibits sensitive dependence on initial conditions. However, because of the [[irrational rotation]] in the second coordinate, there are no periodic orbits, and hence the mapping is not chaotic according to the definition above.
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''Topologically mixing'' means that the system will evolve over time so that any given region or [[open set]] of its phase space will eventually overlap with any other given region. Here, "mixing" is really meant to correspond to the standard intuition: the mixing of colored [[dye]]s or fluids is an example of a chaotic system.
   
Some dynamical systems are chaotic everywhere (see e.g. [[Anosov diffeomorphism]]s) 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 will lead to orbits that converge to this chaotic region.
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=== Attractors ===
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Some dynamical systems are chaotic everywhere (see e.g. [[Anosov diffeomorphism]]s) but in many cases chaotic behaviour is found only in a subset of phase space. The cases of most interest arise when the chaotic behaviour takes place on an [[attractor]], since then a large set of initial conditions will lead to orbits that converge to this chaotic region.
   
 
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.
 
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.
   
[[Image:Damped_driven_chaotic_pendulum_-_double_period_behavior.png|thumb|right|300px|Phase diagram for a damped driven pendulum, with double period motion]]
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[[Image:Damped driven chaotic pendulum - double period behavior.png|thumb|right|300px|Phase diagram for a damped driven pendulum, with double period motion]]
 
For instance, in a system describing a pendulum, the phase space might be two-dimensional, consisting of information about position and velocity. One might plot the ''position'' of a [[pendulum]] against its ''velocity''. A pendulum at rest will be plotted as a point, and one in periodic motion will be plotted as a simple closed curve. When such a plot forms a closed curve, the curve is called an [[orbit (mathematics)|orbit]]. Our pendulum has an infinite number of such orbits, forming a [[pencil (mathematics)|pencil]] of nested ellipses about the origin.
 
For instance, in a system describing a pendulum, the phase space might be two-dimensional, consisting of information about position and velocity. One might plot the ''position'' of a [[pendulum]] against its ''velocity''. A pendulum at rest will be plotted as a point, and one in periodic motion will be plotted as a simple closed curve. When such a plot forms a closed curve, the curve is called an [[orbit (mathematics)|orbit]]. Our pendulum has an infinite number of such orbits, forming a [[pencil (mathematics)|pencil]] of nested ellipses about the origin.
   
==Strange attractors==
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=== Strange attractors ===
 
While most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called ''[[limit cycle]]s'', chaotic motion gives rise to what are known as ''[[strange attractor]]s'', attractors that can have great detail and complexity.
 
While most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called ''[[limit cycle]]s'', chaotic motion gives rise to what are known as ''[[strange attractor]]s'', attractors that can have great detail and complexity.
For instance, a simple three-dimensional model of the [[Edward Lorenz|Lorenz]] weather system gives rise to the famous [[Lorenz attractor]]. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly. Another such attractor is the [[Rössler Map]], which experiences period-two doubling route to chaos, like the logistic map.
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For instance, a simple three-dimensional model of the [[Edward Lorenz|Lorenz]] weather system gives rise to the famous [[Lorenz attractor]]. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly. Another such attractor is the [[Rössler map]], which experiences period-two doubling route to chaos, like the logistic map.
   
 
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.
 
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.
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The initial conditions of three or more bodies interacting through gravitational attraction (see the [[N-body problem|''n''-body problem]]) can be arranged to produce chaotic motion.
 
The initial conditions of three or more bodies interacting through gravitational attraction (see the [[N-body problem|''n''-body problem]]) can be arranged to produce chaotic motion.
[[Image:N-body_problem_(3).gif|thumb|right|300px|Three-body problem]]
 
   
==History==
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=== Minimum complexity of a chaotic system ===
The first discoverer of chaos can plausibly be argued to be [[Jacques Hadamard]], who in 1898 published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature. In the system studied, [[Hadamard's billiards]], Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one-another, with positive [[Lyapunov exponent]].
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[[Image:LogisticMap BifurcationDiagram.png|300px|thumb|right|Bifurcation diagram of a logistic map, displaying chaotic behaviour past a threshold]]
In the early 1900s, [[Henri Poincaré]] while studying the [[three-body problem]], found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Much of the early theory was developed almost entirely by mathematicians, under the name of [[ergodic theory]]. Later studies, also on the topic of nonlinear differential equations, were carried out by [[George David Birkhoff|G.D. Birkhoff]], [[Andrey Nikolaevich Kolmogorov|A.N. Kolmogorov]], [[Mary Lucy Cartwright|M.L. Cartwright]], [[John Edensor Littlewood|J.E. Littlewood]], and [[Stephen Smale]]. 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. 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.
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Simple systems can also produce chaos without relying on [[differential equation]]s. An example is the [[logistic map]], which is a difference equation ([[recurrence relation]]) that describes population growth over time.
   
Chaos theory progressed more rapidly after mid-century, when it first became evident for 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]]. 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. One of the earliest electronic digital computers, [[ENIAC]], was used to run simple weather forecasting models.
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Even the evolution of simple discrete systems, such as [[cellular automata]], can heavily depend on initial conditions. [[Stephen Wolfram]] has investigated a cellular automaton with this property, termed by him ''[[rule 30]]''.
   
An early pioneer of the theory was [[Edward Lorenz]] whose interest in chaos came about accidentally through his work on [[weather]] prediction in [[1961]]. Lorenz was using a basic [[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 was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.
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A minimal model for conservative (reversible) chaotic behavior is provided by [[Arnold's cat map]].
   
To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The printout rounded variables off to a 3-digit number, but the computer worked with 6-digit numbers. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.
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=== Mathematical theory ===
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[[Sarkovskii's theorem]] is the basis of the Li and Yorke (1975) proof that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits.
   
Yoshisuke Ueda independently identified a chaotic phenomenon as such by using an analog computer on November 27, 1961. The chaos exhibited by an analog computer is truly a natural phenomenon, in contrast with those discovered by a digital computer. Ueda's supervising professor, Hayashi, did not believe in chaos throughout his life, and thus he prohibited Ueda from publishing his findings until 1970.
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[[Mathematician]]s have devised many additional ways to make quantitative statements about chaotic systems. These include: [[fractal dimension]] of the attractor, [[Lyapunov exponent]]s, [[recurrence plot]]s, [[Poincaré map]]s, [[bifurcation diagram]]s, and [[transfer operator]].
   
The term '''chaos''' as used in mathematics was coined by the applied mathematician [[James A. Yorke]].
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==Distinguishing random from chaotic data==
   
The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research.
+
It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in practice no time series consists of pure 'signal.' There will always be some form of corrupting noise, even if it is present as round-off or truncation error. Thus any real time series, even if mostly deterministic, will contain some randomness.<ref name=prov>Provenzale A. et al.: "Distinguishing between low-dimensional dynamics and randomness in measured time-series", in: Physica D, 58:31-49, 1992</ref>
   
==Mathematical theory==
+
All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system always evolves in the same way from a given starting point.<ref>Sugihara G. and May R.: "Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series", in: Nature, 344:734-41, 1990</ref><ref name=prov /> Thus, given a time series to test for determinism, one can:
  +
# pick a test state;
  +
# search the time series for a similar or 'nearby' state; and
  +
# compare their respective time evolutions.
   
[[Sarkovskii's theorem]] is the basis of the Li and Yorke (1975) proof that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits.
+
Define the error as the difference between the time evolution of the 'test' state and the time evolution of the nearby state. A deterministic system will have an error that either remains small (stable, regular solution) or increases exponentially with time (chaos). A stochastic system will have a randomly distributed error.<ref>Casdagli, Martin. "Chaos and Deterministic ''versus'' Stochastic Non-linear Modelling", in: ''Journal Royal Statistics Society'': Series B, 54, nr. 2 (1991), 303-28<!-- Spelled "Modelling" in print, and ''versus'' in italics. --></ref>
   
[[Mathematician]]s have devised many additional ways to make quantitative statements about chaotic systems. These include: [[fractal dimension]] of the attractor, [[Lyapunov exponent]]s, [[recurrence plot]]s, [[Poincaré map]]s, [[bifurcation diagram]]s, [[Transfer operator]]
+
Essentially all measures of determinism taken from time series rely upon finding the closest states to a given 'test' state (i.e., correlation dimension, Lyapunov exponents, etc.). To define the state of a system one typically relies on phase space embedding methods.<ref>Broomhead D. S. and King G. P.: "Extracting Qualitative Dynamics from Experimental Data", in: Physica 20D, 217-36, 1986</ref>
  +
Typically one chooses an embedding dimension, and investigates the propagation of the error between two nearby states. If the error looks random, one increases the dimension. If you can increase the dimension to obtain a deterministic looking error, then you are done. Though it may sound simple it is not really. One complication is that as the dimension increases the search for a nearby state requires a lot more computation time and a lot of data (the amount of data required increases exponentially with embedding dimension) to find a suitably close candidate. If the embedding dimension (number of measures per state) is chosen too small (less than the 'true' value) deterministic data can appear to be random but in theory there is no problem choosing the dimension too large – the method will work.
  +
Practically, anything approaching about 10 dimensions is considered so large that a stochastic description is probably more suitable and convenient anyway.{{Fact|date=December 2007}}
   
== Minimum complexity of a chaotic system ==
+
== Applications ==
[[Image:LogisticMap_BifurcationDiagram.png|300px|thumb|right|Bifurcation diagram of a logistic map, displaying chaotic behavior past a threshold]]
+
Chaos theory is applied in many scientific disciplines: [[mathematics]], [[biology]], [[computer science]], [[economics]], [[engineering]], [[finance]], [[philosophy]], [[physics]], [[politics]], [[population dynamics]], [[psychology]], and [[BEAM robotics|robotics]].<ref> Metaculture.net, [http://metalinks.metaculture.net/science/fractal/applications/default.asp metalinks: Applied Chaos], 2007.</ref>
Simple systems can also produce chaos without relying on [[differential equation]]s. An example is the [[logistic map]], which is a difference equation ([[recurrence relation]]) that describes population growth over time.
 
   
Even the evolvement of simple [[discrete]] systems, such as [[cellular automata]], can heavily depend on initial conditions. [[Stephen Wolfram]] has investigated a cellular automaton with this property, termed by him ''[[rule 30]]''.
+
Chaos theory is also currently being applied to medical studies of [[epilepsy]], specifically to the prediction of seemingly random seizures by observing initial conditions.<ref> Comdig.org, [http://www.comdig.org/index.php?id_issue=1999.06#194 Complexity Digest 199.06]</ref>
   
A minimal model for conservative (reversible) chaotic behavior is provided by [[Arnold's cat map]].
+
== Chaos theory in the media ==
  +
===Movies===
   
==Other examples of chaotic systems==
+
* The 1993 movie ''[[Jurassic Park (film)|Jurassic Park]]''
* [[Double pendulum]]
+
* The 1998 movie ''[[π (film)|π]]''
* [[Logistic map]]
+
* The 2004 movie ''[[The Butterfly Effect]]''
* [[Arnold's cat map]]
+
* The 2006 movie ''[[The Science of Sleep]]''
* [[Henon map|Hénon map]]
+
* The 2006 movie ''[[Chaos (2006 film)|Chaos]]''
* [[Lorenz attractor|Lorenz model]]
 
* [[Horseshoe map|Smale horseshoe]]
 
* [[Dynamical billiards]]
 
* [[Chua's circuit]]
 
* [[Rössler Map]]
 
* [[Economic bubble]]
 
   
==Application==
+
===Books===
  +
* [[Michael Crichton|Michael Crichton's]] ''[[Jurassic Park]]'' and ''[[The Lost World (Michael Crichton)|The Lost World]]''
  +
* [[Ray Bradbury|Ray Bradbury's]] ''[[A Sound of Thunder]]''
   
Chaos theory is applied in many scientific disciplines: [[mathematics]], [[biology]], [[computer science]], [[economics]], [[engineering]], [[philosophy]], [[physics]], [[politics]], [[population dynamics]], [[psychology]], [[BEAM_robotics|robotics]], etc.[http://metalinks.metaculture.net/science/fractal/applications/default.asp]
+
===Theatre===
  +
* [[Tom Stoppard|Tom Stoppard's]] "[[Arcadia (play)|Arcadia]]" (a fictional account of early and contemporary studies; also [[thermodynamics]] and [[determinism]])
   
==See also==
+
== See also ==
  +
{{multicol}}
  +
;Examples of chaotic systems
  +
* [[Arnold's cat map]]
  +
* [[Bouncing Ball Simulation System]]
  +
* [[Chua's circuit]]
  +
* [[Double pendulum]]
  +
* [[Dynamical billiards]]
  +
* [[Economic bubble]]
  +
* [[Henon map|Hénon map]]
  +
* [[Horseshoe map]]
  +
* [[Logistic map]]
  +
* [[Lorenz attractor]]
  +
* [[Rössler Map]]
  +
* [[Standard map]]
  +
* [[Swinging Atwood's Machine]]
  +
{{multicol-break}}
  +
;Other related topics
 
*[[Anosov diffeomorphism]]
 
*[[Anosov diffeomorphism]]
 
*[[Bifurcation theory]]
 
*[[Bifurcation theory]]
*[[Butterfly effect]]
+
*[[Chaos theory in organizational development]]
 
*[[Complexity]]
 
*[[Complexity]]
*[[Dynamical system]]
+
*[[Control of chaos]]
  +
*[[Edge of chaos]]
 
*[[Fractal]]
 
*[[Fractal]]
**[[Benoit Mandelbrot]]
 
 
**[[Mandelbrot set]]
 
**[[Mandelbrot set]]
 
**[[Julia set]]
 
**[[Julia set]]
*[[Edge of chaos]]
+
*[[Mathematical modeling]]
  +
*[[Predictability (measurement)]]
  +
*[[Prediction]]
  +
*[[Probability]]
  +
*[[Stochastic modeling]]
  +
*[[Synchronization of chaos]]
  +
*[[Uncertainty]]
  +
{{multicol-break}}
  +
;People
 
*[[Mitchell Feigenbaum]]
 
*[[Mitchell Feigenbaum]]
 
*[[Brosl Hasslacher]]
 
*[[Brosl Hasslacher]]
*[[Predictability]]
+
*[[Michel Hénon]]
*[[Chaos Data Analyzer]]
+
*[[Edward Lorenz]]
  +
*[[Aleksandr Lyapunov]]
  +
*[[Benoît Mandelbrot]]
  +
*[[Henri Poincaré]]
  +
*[[Otto Rössler]]
  +
*[[David Ruelle]]
  +
*[[Oleksandr Mikolaiovich Sharkovsky]]
  +
*[[Floris Takens]]
  +
*[[James A. Yorke]]
  +
{{multicol-end}}
   
==References & Bibliography==
+
== References ==
{{wikify}}
+
{{reflist}}
--------
 
   
*Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos." Amer. Math. Monthly 82, 985-992, 1975.
+
== Literature ==
  +
===Articles===
  +
*[[Tien-Yien Li|Li, T. Y.]] and [[James A. Yorke|Yorke, J. A.]] "Period Three Implies Chaos." [[American Mathematical Monthly]] 82, 985-992, 1975.
  +
*Kolyada, S. F. [http://www.springerlink.com/content/q00627510552020g/?p=93e1f3daf93549d1850365a8800afb30&pi=3 "Li-Yorke sensitivity and other concepts of chaos"], Ukrainian Math. J. 56 (2004), 1242--1257.
  +
* [[Claude Shannon|C.E. Shannon]], "[[A Mathematical Theory of Communication]]", ''[[Bell System Technical Journal]]'', vol. 27, pp. 379-423, 623-656, July, October, 1948
   
 
===Textbooks===
 
===Textbooks===
*{{cite book | author=Ott, Edward | title=Chaos in Dynamical Systems | publisher=Cambridge University Press New, York | year=2002 | id=ISBN 0521010845}}
+
*{{cite book | author=Alligood, K. T. | title=Chaos: an introduction to dynamical systems | publisher=Springer-Verlag New York, LLC | year=1997 | id=ISBN 0-387-94677-2}}
*{{cite book | author=Gutzwiller, Martin | title=Chaos in Classical and Quantum Mechanics | publisher=Springer-Verlag New York, LLC | year=1990 | id=ISBN 0387971734}}
+
*{{cite book | author=Baker, G. L. | title=Chaos, Scattering and Statistical Mechanics | publisher=Cambridge University Press | year=1996 | id=ISBN 0-521-39511-9}}
*{{cite book | author=Moon, Francis | title=Chaotic and Fractal Dynamics | publisher=Springer-Verlag New York, LLC | year=1990 | id=ISBN 0471545716}}
+
*{{cite book | author=Badii, R.; Politi A. | title="Complexity: hierarchical structures and scaling in physics" | publisher=Cambridge University Press | year=1997 | id=ISBN 0521663857 | url=http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521663857}}
*{{cite book | author=Tufillaro, Abbott, Reilly | title=An experimental approach to nonlinear dynamics and chaos | publisher=Addison-Wesley New York | year=1992 | id=ISBN 0201554410}}
+
*{{cite book | author=Devaney, Robert L. | title=An Introduction to Chaotic Dynamical Systems, 2nd ed, | publisher=Westview Press | year=2003 | id=ISBN 0-8133-4085-3}}
*{{cite book | author=Gollub, J. P.; Baker, G. L. | title=Chaotic dynamics | publisher=Cambridge University Press | year=1996 | id=ISBN 0521476852}}
+
*{{cite book | author=Gollub, J. P.; Baker, G. L. | title=Chaotic dynamics | publisher=Cambridge University Press | year=1996 | id=ISBN 0-521-47685-2}}
*{{cite book | author=Baker, G. L. | title=Chaos, Scattering and Statistical Mechanics | publisher=Cambridge University Press | year=1996 | id=ISBN 0521395119}}
+
*{{cite book | author=Gutzwiller, Martin | title=Chaos in Classical and Quantum Mechanics | publisher=Springer-Verlag New York, LLC | year=1990 | id=ISBN 0-387-97173-4}}
*{{cite book | author=Alligood, K. T. | title=Chaos: an introduction to dynamical systems | publisher=Springer-Verlag New York, LLC | year=1997 | id=ISBN 0387946772}}
 
*{{cite book | author=Kiel, L. Douglas; Elliott, Euel W. | title=Chaos Theory in the Social Sciences | publisher=Perseus Publishing | year=1997 | id=ISBN 0472084720}}
 
*{{cite book | author=Strogatz, Steven | title=Nonlinear Dynamics and Chaos | publisher=Perseus Publishing | year=2000 | id=ISBN 0738204536}}
 
*{{cite book | author=Sprott, Julien Clinton | title=Chaos and Time-Series Analysis | publisher=Oxford University Press | year=2003 | id=ISBN 0198508409}}
 
 
*{{cite book | author=Hoover, William Graham | title=Time Reversibility, Computer Simulation, and Chaos | publisher=World Scientific | year=1999,2001 | id=ISBN 981-02-4073-2}}
 
*{{cite book | author=Hoover, William Graham | title=Time Reversibility, Computer Simulation, and Chaos | publisher=World Scientific | year=1999,2001 | id=ISBN 981-02-4073-2}}
  +
*{{cite book | author=Kiel, L. Douglas; Elliott, Euel W. | title=Chaos Theory in the Social Sciences | publisher=Perseus Publishing | year=1997 | id=ISBN 0-472-08472-0}}
  +
*{{cite book | author=Moon, Francis | title=Chaotic and Fractal Dynamics | publisher=Springer-Verlag New York, LLC | year=1990 | id=ISBN 0-471-54571-6}}
  +
*{{cite book | author=Ott, Edward | title=Chaos in Dynamical Systems | publisher=Cambridge University Press New, York | year=2002 | id=ISBN 0-521-01084-5}}
  +
*{{cite book | author=Strogatz, Steven | title=Nonlinear Dynamics and Chaos | publisher=Perseus Publishing | year=2000 | id=ISBN 0-7382-0453-6}}
  +
*{{cite book | author=Sprott, Julien Clinton | title=Chaos and Time-Series Analysis | publisher=Oxford University Press | year=2003 | id=ISBN 0-19-850840-9}}
  +
*{{cite book | author=Tél, Tamás; Gruiz, Márton | title=Chaotic dynamics: An introduction based on classical mechanics | publisher=Cambridge University Press | year=2006 | id=ISBN 0-521-83912-2}}
  +
*{{cite book | author=Tufillaro, Abbott, Reilly | title=An experimental approach to nonlinear dynamics and chaos | publisher=Addison-Wesley New York | year=1992 | id=ISBN 0-201-55441-0}}
   
 
===Semitechnical and popular works===
 
===Semitechnical and popular works===
*''The Beauty of Fractals'', by H.-O. Peitgen and P.H. Richter
+
* Ralph H. Abraham and Yoshisuke Ueda (Ed.), ''The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory'', World Scientific Publishing Company, 2001, 232 pp.
*''Chance and Chaos'', by [[David Ruelle]]
+
* [[Michael Barnsley]], ''Fractals Everywhere'', Academic Press 1988, 394 pp.
*''Computers, Pattern, Chaos, and Beauty'', by [[Clifford A. Pickover]]
+
* Richard J Bird, ''Chaos and Life: Complexity and Order in Evolution and Thought'', Columbia University Press 2003, 352 pp.
*''Fractals'', by Hans Lauwerier
+
* [[John Briggs]] and David Peat, ''Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness'', Harper Perennial 1990, 224 pp.
*''Fractals Everywhere'', by [[Michael Barnsley]]
+
* [[John Briggs]] and David Peat, ''Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change'', Harper Perennial 2000, 224 pp.
*''Order Out of Chaos'', by [[Ilya Prigogine]] and Isabelle Stengers
+
* Lawrence A. Cunningham, ''From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis,'' George Washington Law Review, Vol. 62, 1994, 546 pp.
*''Chaos and Life'', by Richard J Bird
+
* Leon Glass and Michael C. Mackey, ''From Clocks to Chaos: The Rhythms of Life,'' Princeton University Press 1988, 272 pp.
*''Does God Play Dice?'', by [[Ian Stewart (mathematician)|Ian Stewart]]
+
* [[James Gleick]], ''[[Chaos: Making a New Science]]'', New York: Penguin, 1988. 368 pp.
*''The Science of Fractal Images'', by Heinz-Otto Peitgen and Dietmar Saupe, Eds.
+
* John Gribbin, ''Deep Simplicity'',
*''Explaining Chaos'', by Peter Smith
+
* L Douglas Kiel, Euel W Elliott (ed.), ''Chaos Theory in the Social Sciences: Foundations and Applications'', University of Michigan Press, 1997, 360 pp.
*''Chaos: Making a New Science'', New York: Penguin, by [[James Gleick]]
+
* Arvind Kumar, ''Chaos, Fractals and Self-Organisation ; New Perspectives on Complexity in Nature '', National Book Trust, 2003.
*''Complexity'', by M. Mitchell Waldrop
+
* Hans Lauwerier, ''Fractals'', Princeton University Press, 1991.
*''Chaos, Fractals and Self-organisation'', by Arvind Kumar
+
* [[Edward Lorenz]], ''The Essence of Chaos'', University of Washington Press, 1996.
*''Chaotic Evolution and Strange Attractors'', by [[David Ruelle]]
+
* Heinz-Otto Peitgen and Dietmar Saupe (Eds.), ''The Science of Fractal Images'', Springer 1988, 312 pp.
*''Sync: The emerging science of spontaneous order'', by Steven Strogatz
+
* [[Clifford A. Pickover]], ''Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World '', St Martins Pr 1991.
*''The Essence of Chaos'', by [[Edward Lorenz]]
+
* [[Ilya Prigogine]] and Isabelle Stengers, ''Order Out of Chaos'', Bantam 1984.
*''Deep Simplicity'', by John Gribbin
+
* H.-O. Peitgen and P.H. Richter, ''The Beauty of Fractals : Images of Complex Dynamical Systems'', Springer 1986, 211 pp.
*''The Road To Chaos'', by Yoshisuke Ueda
+
* [[David Ruelle]], ''Chance and Chaos'', Princeton University Press 1993.
*''The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory'', Edited by Ralph H. Abraham and Yoshisuke Ueda
+
* [[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.
   
==External links==
+
== External links ==
*[http://lagrange.physics.drexel.edu Research Group with Animations in Flash]
+
{{Commonscat|Chaos theory}}
*[http://www.drchaos.net/drchaos/intro.html Nick's Nonlinear Dynamics Archive]
+
*[http://lagrange.physics.drexel.edu Nonlinear Dynamics Research Group] with Animations in Flash
*[http://www.libraryreference.org/chaos.html Chaos Theory and Education]
+
*[http://www.chaos.umd.edu The Chaos group at the University of Maryland]
*[http://www.imho.com/grae/chaos/chaos.html Chaos Theory: A Brief Introduction]
 
*[http://faraday.physics.utoronto.ca/PVB/Harrison/Chaos/Chaos.html An introduction to Chaos by David M. Harrison, Dept. of Physics. Univ. of Toronto]
 
 
*[http://hypertextbook.com/chaos/ The Chaos Hypertextbook]. An introductory primer on chaos and fractals.
 
*[http://hypertextbook.com/chaos/ The Chaos Hypertextbook]. An introductory primer on chaos and fractals.
* ''Chaos Theory in the Social Sciences'', edited by L Douglas Kiel, Euel W Elliott.
 
 
*[http://www.societyforchaostheory.org/ Society for Chaos Theory in Psychology & Life Sciences]
 
*[http://www.societyforchaostheory.org/ Society for Chaos Theory in Psychology & Life Sciences]
*[http://www.cut-the-knot.org/blue/chaos.shtml Emergence of Chaos] at [[cut-the-knot]]
 
 
*[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://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.creatingtechnology.org/papers/chaos.htm Nonlinear dynamics: how science comprehends chaos], talk presented by Sunny Auyang, 1998.
*[http://necsi.org/guide/concepts/chaos.html New England Complex Systems Institute - Concepts: Chaos]
+
*[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)]
  +
*[http://www.eng.ox.ac.uk/samp Systems Analysis, Modelling and Prediction Group] at the University of Oxford.
  +
  +
{{Systems}}
   
 
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[[Category:Fundamental physics concepts]]
 
[[Category:Non-linear systems]]
 
[[Category:Non-linear systems]]
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[[Category:Theories]]
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Lorenz attractor yb

A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3

In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.

Overview Edit

Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions, fluid dynamics, and mechanical and magneto-mechanical devices. Observations of chaotic behaviour in nature include the dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology, the dynamics of the action potentials in neurons, and molecular vibrations. Everyday examples of chaotic systems include weather and climate.[1] There is some controversy over the existence of chaotic dynamics in the plate tectonics and in economics.[2] [3] [4]

Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder. A related field of physics called quantum chaos theory studies systems that follow the laws of quantum mechanics. Recently, another field, called relativistic chaos,[5] has emerged to describe systems that follow the laws of general relativity.

As well as being orderly in the sense of being deterministic, chaotic systems usually have well defined statistics. For example, the Lorenz system pictured is chaotic, but has a clearly defined structure. Bounded chaos is a useful term for describing models of disorder.

History Edit

File:Fractal fern1.png

The first discoverer of chaos can plausibly be argued to be Jacques Hadamard, who in 1898 published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature. In the system studied, Hadamard's billiards, 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.

In the early 1900s Henri Poincaré, while studying the three-body problem, found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Much of the early theory was developed almost entirely by mathematicians, under the name of ergodic theory. Later studies, also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff, A.N. Kolmogorov, M.L. Cartwright, J.E. Littlewood, and Stephen Smale. 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. 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 century, chaos theory became formalized as such only after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behaviour of certain experiments like that of the logistic map. What had been beforehand excluded as measure imprecision and simple "noise" was considered by chaos theories 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. One of the earliest electronic digital computers, ENIAC, was used to run simple weather forecasting models.

File:Airplane vortex edit.jpg

An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a basic 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 was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.

To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The printout rounded variables off to a 3-digit number, but the computer worked with 6-digit numbers. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome. Lorenz's discovery, which gave its name to Lorenz attractors, proved that meteorology could not reasonably predict weather beyond a weekly period (at most).

The year before, Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices. 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. Mandelbrot described both the Noah effect (in which sudden discontinuous changes can occur, e.g., in a stock's prices after bad news, thus challenging normal distribution theory in statistics, aka Bell Curve) and the Joseph effect (in which persistence of a value can occur for a while, yet suddenly change afterwards). In 1967, he published 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 infinitesimally small measuring device. Arguing that a ball of twine appears to be 1-dimensional (far), 3-dimensional (fairly near), or 1-dimensional (close), 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 (for example, the Koch curve or "snowflake", which is infinitely long yet encloses a finite space with dimensions = 1.2618; or the Menger sponge and the Sierpinski gasket). In 1975 Mandelbrot published The Fractal Geometry of Nature, which became a classic of chaos theory. Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model.

Yoshisuke Ueda independently identified a chaotic phenomenon as such by using an analog computer on November 27, 1961. The chaos exhibited by an analog computer is a real phenomenon, in contrast with those that digital computers calculate, which has a different kind of limit on precision. Ueda's supervising professor, Hayashi, did not believe in chaos, and thus he prohibited Ueda from publishing his findings until 1970.

In December 1977 the New York Academy of Sciences organized the first symposium on Chaos, attended by David Ruelle, Robert May, James Yorke (coiner of the term "chaos" as used in mathematics), Robert Shaw (a physicist, part of the Eudaemons group with J. Doyne Farmer and Norman Packard who tried to find a mathematical method to beat roulette, and then created with them the Dynamical Systems Collective in Santa Cruz), and the meteorologist Edward Lorenz.

The following year, Mitchell Feigenbaum published the noted article "Quantitative Universality for a Class of Nonlinear Transformations", where he described logistic maps. Feigenbaum had applied fractal geometry to the study of natural forms such as coastlines. Feigenbaum notably discovered the universality in chaos, permitting an application of chaos theory to many different phenomena.

In 1979, Albert J. Libchaber during a symposium organized in Aspen by Pierre Hohenberg his experimental observation of the bifurcation cascade that leads to chaos and turbulence in convective Rayleigh-Benard systems. He was awarded the Wolf Prize in Physics in 1986 along with Mitchell J. Feigenbaum "for his brilliant experimental demonstration of the transition to turbulence and chaos in dynamical systems".[6]

The New York Academy of Sciences then co-organized, in 1986, with the National Institute of Mental Health and the Office of Naval Research the first important conference on Chaos in biology and medicine. Bernardo Huberman thereby presented a mathematical model of the eye tracking disorder among schizophrenics [7]. Chaos theory thereafter renewed physiology in the 1980s, for example in the study of pathological cardiac cycles.

In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters describing for the first time self-organized criticality (SOC), considered to be 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 centred around large-scale natural or social systems that are known (or suspected) to display scale-invariant behaviour. 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: earthquakes (which, long before SOC was discovered, were known as a source of scale-invariant behaviour such as the Gutenberg-Richter law describing the statistical distribution of earthquake sizes, and the Omori law describing the frequency of aftershocks); solar flares; fluctuations in economic systems such as financial markets (references to SOC are common in econophysics); landscape formation; forest fires; landslides; epidemics; and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould). Worryingly, 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 wars. These "applied" 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.

The same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced general principles of chaos theory as well as its history to the broad public. At first the domains of work of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of nonlinear systems analysis. Alluding to Thomas Kuhn's concept of a paradigm shift exposed in The Structure of Scientific Revolutions (1962), many "chaologists" (as some self-nominated themselves) claimed that this new theory was an example of such as shift, a thesis upheld by J. Gleick.

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research, involving many different disciplines (mathematics, topology, physics, population biology, biology, meteorology, astrophysics, information theory, etc.).

Chaotic dynamics Edit

For a dynamical system to be classified as chaotic, it must have the following properties:[8]

Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behaviour.

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.

Sensitivity to initial conditions is often confused with chaos in popular accounts. It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the phase space of the system. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by the mapping on the real line from x to 2x). This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, it has extremely simple behaviour, as all points except 0 tend to infinity. If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial conditions. For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems.

Even for bounded systems, sensitivity to initial conditions is not identical with chaos. For example, consider the two-dimensional torus described by a pair of angles (x,y), each ranging between zero and 2π. Define a mapping that takes any point (x,y) to (2x, y + a), where a is any number such that a/2π is irrational. Because of the doubling in the first coordinate, the mapping exhibits sensitive dependence on initial conditions. However, because of the irrational rotation in the second coordinate, there are no periodic orbits, and hence the mapping is not chaotic according to the definition above.

Topologically mixing means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. Here, "mixing" is really meant to correspond to the standard intuition: the mixing of colored dyes or fluids is an example of a chaotic system.

Attractors Edit

Some dynamical systems are chaotic everywhere (see e.g. Anosov diffeomorphisms) but in many cases chaotic behaviour is found only in a subset of phase space. The cases of most interest arise when the chaotic behaviour takes place on an attractor, since then a large set of initial conditions will lead to orbits that converge to this chaotic region.

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.

Damped driven chaotic pendulum - double period behavior

Phase diagram for a damped driven pendulum, with double period motion

For instance, in a system describing a pendulum, the phase space might be two-dimensional, consisting of information about position and velocity. One might plot the position of a pendulum against its velocity. A pendulum at rest will be plotted as a point, and one in periodic motion will be plotted as a simple closed curve. When such a plot forms a closed curve, the curve is called an orbit. Our pendulum has an infinite number of such orbits, forming a pencil of nested ellipses about the origin.

Strange attractors Edit

While most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called limit cycles, chaotic motion gives rise to what are known as strange attractors, attractors that can have great detail and complexity. For instance, a simple three-dimensional model of the Lorenz weather system gives rise to the famous Lorenz attractor. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly. Another such attractor is the Rössler map, which experiences period-two doubling route to chaos, like the logistic map.

Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some 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.

The Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system if it has three or more dimensions. However, no such restriction applies to discrete systems, which can exhibit strange attractors in two or even one dimensional systems.

The initial conditions of three or more bodies interacting through gravitational attraction (see the n-body problem) can be arranged to produce chaotic motion.

Minimum complexity of a chaotic system Edit

LogisticMap BifurcationDiagram

Bifurcation diagram of a logistic map, displaying chaotic behaviour past a threshold

Simple systems can also produce chaos without relying on differential equations. An example is the logistic map, which is a difference equation (recurrence relation) that describes population growth over time.

Even the evolution of simple discrete systems, such as cellular automata, can heavily depend on initial conditions. Stephen Wolfram has investigated a cellular automaton with this property, termed by him rule 30.

A minimal model for conservative (reversible) chaotic behavior is provided by Arnold's cat map.

Mathematical theory Edit

Sarkovskii's theorem is the basis of the Li and Yorke (1975) proof that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits.

Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. These include: fractal dimension of the attractor, Lyapunov exponents, recurrence plots, Poincaré maps, bifurcation diagrams, and transfer operator.

Distinguishing random from chaotic dataEdit

It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in practice no time series consists of pure 'signal.' There will always be some form of corrupting noise, even if it is present as round-off or truncation error. Thus any real time series, even if mostly deterministic, will contain some randomness.[9]

All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system always evolves in the same way from a given starting point.[10][9] Thus, given a time series to test for determinism, one can:

  1. pick a test state;
  2. search the time series for a similar or 'nearby' state; and
  3. compare their respective time evolutions.

Define the error as the difference between the time evolution of the 'test' state and the time evolution of the nearby state. A deterministic system will have an error that either remains small (stable, regular solution) or increases exponentially with time (chaos). A stochastic system will have a randomly distributed error.[11]

Essentially all measures of determinism taken from time series rely upon finding the closest states to a given 'test' state (i.e., correlation dimension, Lyapunov exponents, etc.). To define the state of a system one typically relies on phase space embedding methods.[12] Typically one chooses an embedding dimension, and investigates the propagation of the error between two nearby states. If the error looks random, one increases the dimension. If you can increase the dimension to obtain a deterministic looking error, then you are done. Though it may sound simple it is not really. One complication is that as the dimension increases the search for a nearby state requires a lot more computation time and a lot of data (the amount of data required increases exponentially with embedding dimension) to find a suitably close candidate. If the embedding dimension (number of measures per state) is chosen too small (less than the 'true' value) deterministic data can appear to be random but in theory there is no problem choosing the dimension too large – the method will work. Practically, anything approaching about 10 dimensions is considered so large that a stochastic description is probably more suitable and convenient anyway.[How to reference and link to summary or text]

Applications Edit

Chaos theory is applied in many scientific disciplines: mathematics, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, and robotics.[13]

Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions.[14]

Chaos theory in the media Edit

MoviesEdit

BooksEdit

TheatreEdit

See also Edit


References Edit

  1. Sneyers, R: "Climate Chaotic Instability: Statistical Determination and Theoretical Background", 8(5):517-532
  2. Apostolos Serletis and Periklis Gogas,Purchasing Power Parity Nonlinearity and Chaos, in: Applied Financial Economics, 10, 615-622, 2000.
  3. Apostolos Serletis and Periklis Gogas The North American Gas Markets are ChaoticPDF (918 KiB), in: The Energy Journal, 20, 83-103, 1999.
  4. Apostolos Serletis and Periklis Gogas, Chaos in East European Black Market Exchange Rates, in: Research in Economics, 51, 359-385, 1997.
  5. A. E. Motter, Relativistic chaos is coordinate invariant, in: Phys. Rev. Lett. 91, 231101 (2003).
  6. The Wolf Prize in Physics in 1986.
  7. Bernardo Huberman, "A Model for Dysfunctions in Smooth Pursuit Eye Movement" Annals of the New York Academy of Sciences, Vol. 504 Page 260 July 1987, Perspectives in Biological Dynamics and Theoretical Medicine
  8. Hasselblatt, Boris; Anatole Katok (2003). A First Course in Dynamics: With a Panorama of Recent Developments, Cambridge University Press.
  9. 9.0 9.1 Provenzale A. et al.: "Distinguishing between low-dimensional dynamics and randomness in measured time-series", in: Physica D, 58:31-49, 1992
  10. Sugihara G. and May R.: "Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series", in: Nature, 344:734-41, 1990
  11. Casdagli, Martin. "Chaos and Deterministic versus Stochastic Non-linear Modelling", in: Journal Royal Statistics Society: Series B, 54, nr. 2 (1991), 303-28
  12. Broomhead D. S. and King G. P.: "Extracting Qualitative Dynamics from Experimental Data", in: Physica 20D, 217-36, 1986
  13. Metaculture.net, metalinks: Applied Chaos, 2007.
  14. Comdig.org, Complexity Digest 199.06

Literature Edit

ArticlesEdit

TextbooksEdit

  • Alligood, K. T. (1997). Chaos: an introduction to dynamical systems, Springer-Verlag New York, LLC. ISBN 0-387-94677-2.
  • Baker, G. L. (1996). Chaos, Scattering and Statistical Mechanics, Cambridge University Press. ISBN 0-521-39511-9.
  • Badii, R.; Politi A. (1997). "Complexity: hierarchical structures and scaling in physics", Cambridge University Press. ISBN 0521663857.
  • Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems, 2nd ed,, Westview Press. ISBN 0-8133-4085-3.
  • Gollub, J. P.; Baker, G. L. (1996). Chaotic dynamics, Cambridge University Press. ISBN 0-521-47685-2.
  • Gutzwiller, Martin (1990). Chaos in Classical and Quantum Mechanics, Springer-Verlag New York, LLC. ISBN 0-387-97173-4.
  • Hoover, William Graham (1999,2001). Time Reversibility, Computer Simulation, and Chaos, World Scientific. ISBN 981-02-4073-2.
  • Kiel, L. Douglas; Elliott, Euel W. (1997). Chaos Theory in the Social Sciences, Perseus Publishing. ISBN 0-472-08472-0.
  • Moon, Francis (1990). Chaotic and Fractal Dynamics, Springer-Verlag New York, LLC. ISBN 0-471-54571-6.
  • Ott, Edward (2002). Chaos in Dynamical Systems, Cambridge University Press New, York. ISBN 0-521-01084-5.
  • Strogatz, Steven (2000). Nonlinear Dynamics and Chaos, Perseus Publishing. ISBN 0-7382-0453-6.
  • Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis, Oxford University Press. ISBN 0-19-850840-9.
  • Tél, Tamás; Gruiz, Márton (2006). Chaotic dynamics: An introduction based on classical mechanics, Cambridge University Press. ISBN 0-521-83912-2.
  • Tufillaro, Abbott, Reilly (1992). An experimental approach to nonlinear dynamics and chaos, Addison-Wesley New York. ISBN 0-201-55441-0.

Semitechnical and popular worksEdit

  • Ralph H. Abraham and Yoshisuke Ueda (Ed.), The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory, World Scientific Publishing Company, 2001, 232 pp.
  • Michael Barnsley, Fractals Everywhere, Academic Press 1988, 394 pp.
  • Richard J Bird, Chaos and Life: Complexity and Order in Evolution and Thought, Columbia University Press 2003, 352 pp.
  • 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.
  • Lawrence A. Cunningham, From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis, George Washington Law Review, Vol. 62, 1994, 546 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.
  • John Gribbin, Deep Simplicity,
  • 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.
  • 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.
  • H.-O. 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.
  • David Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, 1989.
  • Peter Smith, Explaining Chaos, Cambridge University Press, 1998.
  • 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.

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