disorganized
出典: meddic
 adj.
 混乱した
 関
 confound、confuse、confusion、derange、derangement、disarray、disorient、disrupt、disruption、perturbation、upset
WordNet ［license wordnet］
「lacking order or methodical arrangement or function; "a disorganized enterprise"; "a thousand pages of muddy and disorganized prose"; "she was too disorganized to be an agreeable roommate"」 同
 disorganised
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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/11/09 12:01:47」(JST)
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[Wiki en表示]Chaos theory is a field of study in mathematics, with applications in several disciplines including meteorology, physics, engineering, economics and biology. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering longterm prediction impossible in general.^{[1]} This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.^{[2]} In other words, the deterministic nature of these systems does not make them predictable.^{[3]}^{[4]} This behavior is known as deterministic chaos, or simply chaos. This was summarised by Edward Lorenz as follows:^{[5]}
Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
Chaotic behavior can be observed in many natural systems, such as weather.^{[6]}^{[7]} Explanation of such behavior may be sought through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.
Contents
 1 Chaotic dynamics
 1.1 Sensitivity to initial conditions
 1.2 Topological mixing
 1.3 Density of periodic orbits
 1.4 Strange attractors
 1.5 Minimum complexity of a chaotic system
 2 History
 3 Distinguishing random from chaotic data
 4 Applications
 5 Cultural references
 6 See also
 7 References
 8 Scientific literature
 8.1 Articles
 8.2 Textbooks
 8.3 Semitechnical and popular works
 9 External links
Chaotic dynamics[edit]
In common usage, "chaos" means "a state of disorder".^{[8]} However, in chaos theory, the term is defined more precisely. Although there is no universally accepted mathematical definition of chaos, a commonly used definition says that, for a dynamical system to be classified as chaotic, it must have the following properties:^{[9]}
 it must be sensitive to initial conditions;
 it must be topologically mixing; and
 its periodic orbits must be dense^{[where?]}.
The requirement for sensitive dependence on initial conditions implies that there is a set of initial conditions of positive measure which do not converge to a cycle of any length.
Sensitivity to initial conditions[edit]
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. However, it has been shown that the last two properties in the list above actually imply sensitivity to initial conditions^{[10]}^{[11]} and if attention is restricted to intervals, the second property implies the other two^{[12]} (an alternative, and in general weaker, definition of chaos uses only the first two properties in the above list).^{[13]} It is interesting that the most practically significant condition, that of sensitivity to initial conditions, is actually redundant in the definition, being implied by two (or for intervals, one) purely topological conditions, which are therefore of greater interest to mathematicians.
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 largescale 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 only a finite amount of information about the system (as is usually the case in practice), then beyond a certain time the system will no longer be predictable. This is most familiar in the case of weather, which is generally predictable only about a week ahead.^{[14]}
The Lyapunov exponent characterises the extent of the sensitivity to initial conditions. Quantitatively, two trajectories in phase space with initial separation diverge
where λ is the Lyapunov exponent. The rate of separation can be different for different orientations of the initial separation vector. Thus, there is a whole spectrum of Lyapunov exponents — the number of them is equal to the number of dimensions of the phase space. It is common to just refer to the largest one, i.e. to the Maximal Lyapunov exponent (MLE), because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.
There are also measuretheoretic mathematical conditions (discussed in ergodic theory) such as mixing or being a Ksystem which relate to sensitivity of initial conditions and chaos.^{[4]}
Topological mixing[edit]
Topological mixing (or topological transitivity) 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. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored dyes or fluids is an example of a chaotic system.
Topological mixing is often omitted from popular accounts of chaos, which equate chaos with 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 will eventually become widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behaviour: all points except 0 will tend to positive or negative infinity.
Density of periodic orbits[edit]
Density of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits.^{[15]} The onedimensional logistic map defined by x → 4 x (1 – x) is one of the simplest systems with density of periodic orbits. For example, → → (or approximately 0.3454915 → 0.9045085 → 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).^{[16]}
Sharkovskii's theorem is the basis of the Li and Yorke^{[17]} (1975) proof that any onedimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits.
Strange attractors[edit]
Some dynamical systems, like the onedimensional logistic map defined by x → 4 x (1 – x), are chaotic everywhere, 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, 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 threedimensional model of the Lorenz weather system. The Lorenz attractor is perhaps one of the bestknown 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 which looks like the wings of a butterfly.
Unlike fixedpoint attractors and limit cycles, the attractors which arise from chaotic systems, known as strange attractors, have great detail and complexity. 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, and a fractal dimension can be calculated for them.
Minimum complexity of a chaotic system[edit]
Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. In contrast, for continuous dynamical systems, the Poincaré–Bendixson theorem shows that a strange attractor can only arise in three or more dimensions. Finite dimensional linear systems are never chaotic; for a dynamical system to display chaotic behaviour it has to be either nonlinear, or infinitedimensional.
The Poincaré–Bendixson theorem states that a two dimensional differential equation has very regular behavior. The Lorenz attractor discussed above is generated by a system of three differential equations with a total of seven terms on the right hand side, five of which are linear terms and two of which are quadratic (and therefore nonlinear). Another wellknown chaotic attractor is generated by the Rossler equations with seven terms on the right hand side, only one of which is (quadratic) nonlinear. Sprott^{[18]} found a three dimensional system with just five terms on the right hand side, and with just one quadratic nonlinearity, which exhibits chaos for certain parameter values. Zhang and Heidel^{[19]}^{[20]} 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 means that a continuous dynamical system on the Euclidean plane cannot be chaotic, twodimensional continuous systems with nonEuclidean geometry can exhibit chaotic behaviour.^{[citation needed]} Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinitedimensional.^{[21]} A theory of linear chaos is being developed in a branch of mathematical analysis known as functional analysis.
History[edit]
An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the threebody problem, he found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point.^{[22]}^{[23]} 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.^{[24]} 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.
Much of the earlier 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,^{[25]} A. N. Kolmogorov,^{[26]}^{[27]}^{[28]} M.L. Cartwright and J.E. Littlewood,^{[29]} and Stephen Smale.^{[30]} Except for Smale, these studies were all directly inspired by physics: the threebody 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]} 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 midcentury, 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.
An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961.^{[6]} Lorenz was using a simple digital computer, a Royal McBee LGP30, 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 computer worked with 6digit precision, but the printout rounded variables off to a 3digit number, so a value like 0.506127 was printed as 0.506. 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 longterm outcome.^{[31]} Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modelling cannot in general make longterm weather predictions. Weather is usually predictable only about a week ahead.^{[14]}
In 1963, Benoît Mandelbrot found recurring patterns at every scale in data on cotton prices.^{[32]} Beforehand, he had studied information theory and concluded noise was patterned like a Cantor set: on any scale the proportion of noisecontaining periods to errorfree periods was a constant – thus errors were inevitable and must be planned for by incorporating redundancy.^{[33]} 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).^{[34]}^{[35]} This challenged the idea that changes in price were normally distributed. In 1967, he published "How long is the coast of Britain? Statistical selfsimilarity 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.^{[36]} Arguing that a ball of twine appears to be a point when viewed from far away (0dimensional), a ball when viewed from fairly near (3dimensional), or a curved strand (1dimensional), 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 ("selfsimilarity") is a fractal (for example, 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 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.^{[37]}
Chaos was observed by a number of experimenters before it was recognized; e.g., in 1927 by van der Pol^{[38]} and in 1958 by R.L. Ives.^{[39]}^{[40]} However, as a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on Nov. 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.^{[41]}^{[42]}
In December 1977, the New York Academy of Sciences organized the first symposium on Chaos, attended by David Ruelle, Robert May, James A. 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, California), and the meteorologist Edward Lorenz.
The following year, independently the French Pierre Coullet and Charles Tresser with the article "Iterations d'endomorphismes et groupe de renormalisation" and the American Mitchell Feigenbaum with the article "Quantitative Universality for a Class of Nonlinear Transformations" described logistic maps.^{[43]}^{[44]} They 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, presented his experimental observation of the 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 his brilliant experimental demonstration of the transition to turbulence and chaos in dynamical systems".^{[45]}
Then in 1986, the New York Academy of Sciences coorganized 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.^{[46]} This led to a renewal of physiology in the 1980s through the application of chaos theory, 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^{[47]} describing for the first time selforganized criticality (SOC), considered to be one of the mechanisms by which complexity arises in nature.
Alongside largely labbased approaches such as the Bak–Tang–Wiesenfeld sandpile, many other investigations have focused on largescale natural or social systems that are known (or suspected) to display scaleinvariant 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 scaleinvariant behaviour such as the Gutenberg–Richter law describing the statistical distribution of earthquake sizes, and the Omori law^{[48]} 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). Given the implications of a scalefree 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.
This same year 1987, James Gleick published Chaos: Making a New Science, which became a bestseller and introduced the general principles of chaos theory as well as its history to the broad public, (though his history underemphasized important Soviet contributions)^{[citation needed]}. At first the domain 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 described themselves) claimed that this new theory was an example of such a 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,^{[49]} involving many different disciplines (mathematics, topology, physics, social systems, population biology, biology, meteorology, astrophysics, information theory, computational neuroscience, etc.).
Distinguishing random from chaotic data[edit]
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 roundoff or truncation error. Thus any real time series, even if mostly deterministic, will contain some randomness.^{[50]}^{[51]}
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.^{[50]}^{[52]} 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.
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.^{[53]}
Essentially, all measures of determinism taken from time series rely upon finding the closest states to a given 'test' state (e.g., correlation dimension, Lyapunov exponents, etc.). To define the state of a system one typically relies on phase space embedding methods.^{[54]} 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.
When a nonlinear deterministic system is attended by external fluctuations, its trajectories present serious and permanent distortions. Furthermore, the noise is amplified due to the inherent nonlinearity and reveals totally new dynamical properties. Statistical tests attempting to separate noise from the deterministic skeleton or inversely isolate the deterministic part risk failure. Things become worse when the deterministic component is a nonlinear feedback system.^{[55]} In presence of interactions between nonlinear deterministic components and noise, the resulting nonlinear series can display dynamics that traditional tests for nonlinearity are sometimes not able to capture.^{[56]}
The question of how to distinguish deterministic chaotic systems from stochastic systems has also been discussed in philosophy. It has been shown that they might be observationally equivalent.^{[57]}
Applications[edit]
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Chaos theory is applied in many scientific disciplines, including: geology, mathematics, microbiology, biology, computer science, economics,^{[59]}^{[60]}^{[61]} engineering,^{[62]} finance,^{[63]}^{[64]} algorithmic trading,^{[65]}^{[66]}^{[67]} meteorology, philosophy, physics, politics, population dynamics,^{[68]} psychology, and robotics.
Chaotic behavior has been observed in the laboratory in a variety of systems, including electrical circuits,^{[69]} lasers, oscillating chemical reactions, fluid dynamics, and mechanical and magnetomechanical devices, as well as computer models of chaotic processes. Observations of chaotic behavior in nature include changes in weather, the planet orbits in the solar system,^{[70]} 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. There is some controversy over the existence of chaotic dynamics in plate tectonics^{[citation needed]} and in economics.^{[71]}^{[72]}^{[73]}
Chaos theory is currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions.^{[74]}
Quantum chaos theory studies how the correspondence between quantum mechanics and classical mechanics works in the context of chaotic systems.^{[75]} Relativistic chaos describes chaotic systems under general relativity.^{[76]}
The motion of a system of three or more stars interacting gravitationally (the gravitational Nbody problem) is generically chaotic.^{[77]}
In electrical engineering, chaotic systems are used in communications, random number generators, and encryption systems.
In numerical analysis, the NewtonRaphson method of approximating the roots of a function can lead to chaotic iterations if the function has no real roots.^{[78]}
In civil engineering, a traffic model was developed showing that under certain conditions the system dynamics can become chaotic.^{[79]}
Cultural references[edit]
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Chaos theory has been mentioned in movies and works of literature, including Michael Crichton's novel Jurassic Park as well as its film adaptation, the films Chaos and The Butterfly Effect, the Indian movie "Dasavatharam" starring Kamal Hassan, the sitcoms Community and Spaced, Tom Stoppard's play Arcadia and the video games Tom Clancy's Splinter Cell: Chaos Theory and Assassin's Creed (video game). In the computer game The Secret World the Dragon secret society uses chaos theory to achieve political dominance. Ray Bradbury's short story "A Sound of Thunder" explores chaos theory. Chaos theory was the subject of the BBC documentaries High Anxieties — The Mathematics of Chaos directed by David Malone, and The Secret Life of Chaos presented by Jim AlKhalili. Cultural permutations of chaos theory are explored in the book The Unity of Nature by Alan Marshall (Imperial College Press, London, 2002).
See also[edit]




References[edit]
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 ^ See also: Mandelbrot, Benoît B.; Hudson, Richard L. (2004). The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books. p. 201.
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 ^ Buldyrev, S.V.; Goldberger, A.L.; Havlin, S.; Peng, C.K.; Stanley, H.E. (1994). "Fractals in Biology and Medicine: From DNA to the Heartbeat". In Bunde, Armin; Havlin, Shlomo. Fractals in Science. Springer. pp. 49–89. ISBN 3540562206.
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 ^ Abraham & Ueda 2001, See Chapters 3 and 4
 ^ Sprott 2003, p. 89
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 ^ Bak, Per; Tang, Chao; Wiesenfeld, Kurt (27 July 1987). "Selforganized criticality: An explanation of the 1/f noise". Physical Review Letters 59 (4): 381–4. Bibcode:1987PhRvL..59..381B. doi:10.1103/PhysRevLett.59.381. However, the conclusions of this article have been subject to dispute. "?". . See especially: Laurson, Lasse; Alava, Mikko J.; Zapperi, Stefano (15 September 2005). "Letter: Power spectra of selforganized critical sand piles". Journal of Statistical Mechanics: Theory and Experiment 0511. L001.
 ^ Omori, F. (1894). "On the aftershocks of earthquakes". Journal of the College of Science, Imperial University of Tokyo 7: 111–200.
 ^ Motter A. E. and Campbell D. K., Chaos at fifty, Phys. Today 66(5), 2733 (2013).
 ^ ^{a} ^{b} Provenzale, A., et al. (1992). "Distinguishing between lowdimensional dynamics and randomness in measured timeseries". Physica D 58: 31–49. Bibcode:1992PhyD...58...31P. doi:10.1016/01672789(92)901002.
 ^ Brock, W.A. (October 1986). "Distinguishing random and deterministic systems: Abridged version". Journal of Economic Theory 40: 168–195. doi:10.1016/00220531(86)900141.
 ^ Sugihara G., May R. (1990). "Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series" (PDF). Nature 344 (6268): 734–741. Bibcode:1990Natur.344..734S. doi:10.1038/344734a0. PMID 2330029.
 ^ Casdagli, Martin (1991). "Chaos and Deterministic versus Stochastic Nonlinear Modelling". Journal of the Royal Statistical Society, Series B 54 (2): 303–328. JSTOR 2346130.
 ^ Broomhead, D.S.; King, G.P. (June–July 1986). "Extracting qualitative dynamics from experimental data". Physica D 20 (2–3): 217–236. Bibcode:1986PhyD...20..217B. doi:10.1016/01672789(86)90031X.
 ^ Kyrtsou C (2008). "Reexamining the sources of heteroskedasticity: the paradigm of noisy chaotic models". Physica A 387 (27): 6785–9. Bibcode:2008PhyA..387.6785K. doi:10.1016/j.physa.2008.09.008.
 ^ Kyrtsou, C. (2005). "Evidence for neglected linearity in noisy chaotic models". International Journal of Bifurcation and Chaos 15 (10): 3391–4. Bibcode:2005IJBC...15.3391K. doi:10.1142/S0218127405013964.
 ^ Werndl, Charlotte (2009). "Are Deterministic Descriptions and Indeterministic Descriptions Observationally Equivalent?". Studies in History and Philosophy of Modern Physics 40 (3): 232–242. doi:10.1016/j.shpsb.2009.06.004.
 ^ Stephen Coombes (February 2009). "The Geometry and Pigmentation of Seashells". www.maths.nottingham.ac.uk. University of Nottingham. Retrieved 20130410.
 ^ Kyrtsou C., Labys W. (2006). "Evidence for chaotic dependence between US inflation and commodity prices". Journal of Macroeconomics 28 (1): 256–266. doi:10.1016/j.jmacro.2005.10.019.
 ^ Kyrtsou C., Labys W. (2007). "Detecting positive feedback in multivariate time series: the case of metal prices and US inflation". Physica A 377 (1): 227–229. Bibcode:2007PhyA..377..227K. doi:10.1016/j.physa.2006.11.002.
 ^ Kyrtsou, C.; Vorlow, C. (2005). "Complex dynamics in macroeconomics: A novel approach". In Diebolt, C.; Kyrtsou, C. New Trends in Macroeconomics. Springer Verlag.
 ^ Applying Chaos Theory to Embedded Applications
 ^ HristuVarsakelis, D.; Kyrtsou, C. (2008). "Evidence for nonlinear asymmetric causality in US inflation, metal and stock returns". Discrete Dynamics in Nature and Society 2008: 1. doi:10.1155/2008/138547. 138547.
 ^ Kyrtsou, C. and M. Terraza, (2003). "Is it possible to study chaotic and ARCH behaviour jointly? Application of a noisy MackeyGlass equation with heteroskedastic errors to the Paris Stock Exchange returns series". Computational Economics 21 (3): 257–276. doi:10.1023/A:1023939610962.
 ^ Williams, Bill Williams, Justine (2004). Trading chaos : maximize profits with proven technical techniques (2nd ed.). New York: Wiley. ISBN 9780471463085.
 ^ Peters, Edgar E. (1994). Fractal market analysis : applying chaos theory to investment and economics (2. print. ed.). New York u.a.: Wiley. ISBN 9780471585244.
 ^ Peters, / Edgar E. (1996). Chaos and order in the capital markets : a new view of cycles, prices, and market volatility (2nd ed.). New York: John Wiley & Sons. ISBN 9780471139386.
 ^ Dilão, R.; Domingos, T. (2001). "Periodic and QuasiPeriodic Behavior in Resource Dependent Age Structured Population Models". Bulletin of Mathematical Biology 63 (2): 207–230. doi:10.1006/bulm.2000.0213. PMID 11276524.
 ^ Cascais, J.; Dilão, R.; Noronha da Costa, A. (1983). "Chaos and Reverse Bifurcations in a RCL circuit". Physics Letters A 93 (5): 213–6. Bibcode:1983PhLA...93..213C. doi:10.1016/03759601(83)907995.
 ^ Laskar, Jacques (1989). "A numerical experiment on the chaotic behaviour of the Solar System". Nature 338: 237–238. doi:10.1038/338237a0.
 ^ Serletis, Apostolos; Gogas, Periklis (2000). "Purchasing Power Parity Nonlinearity and Chaos". Applied Financial Economics 10 (6): 615–622. doi:10.1080/096031000437962.
 ^ Serletis, Apostolos; Gogas, Periklis (1999). "The North American Gas Markets are Chaotic" (PDF). The Energy Journal 20: 83–103. doi:10.5547/ISSN01956574EJVol20No15.
 ^ Serletis, Apostolos; Gogas, Periklis (1997). "Chaos in East European Black Market Exchange Rates". Research in Economics 51 (4): 359–385. doi:10.1006/reec.1997.0050.
 ^ Victoria White, Office Of Public Information, University Of Florida Health Science Center. "Chaos Theory Helps To Predict Epileptic Seizures, U. Florida".
 ^ Michael Berry, "Quantum Chaology," pp 104–5 of Quantum: a guide for the perplexed by Jim AlKhalili (Weidenfeld and Nicolson 2003)."?".
 ^ Motter, A.E. (2003). "Relativistic chaos is coordinate invariant". Phys. Rev. Lett. 91 (23): 231101. arXiv:grqc/0305020. Bibcode:2003PhRvL..91w1101M. doi:10.1103/PhysRevLett.91.231101.
 ^ Hemsendorf, M.; Merritt, D. (November 2002). "Instability of the Gravitational NBody Problem in the LargeN Limit". The Astrophysical Journal 580 (1): 606–9. arXiv:astroph/0205538. Bibcode:2002ApJ...580..606H. doi:10.1086/343027
 ^ Strang, Gilbert (January 1991). "A chaotic search for i". The College Mathematics Journal 22 (1): 3–12. doi:10.2307/2686733.
 ^ Safonov, Leonid A.; Tomer, Elad; Strygin, Vadim V.; Ashkenazy, Yosef; Havlin, Shlomo (2002). "Multifractal chaotic attractors in a system of delaydifferential equations modeling road traffic". Chaos: An Interdisciplinary Journal of Nonlinear Science 12 (4): 1006. Bibcode:2002Chaos..12.1006S. doi:10.1063/1.1507903. ISSN 10541500.
Scientific literature[edit]
Articles[edit]
 Sharkovskii, A.N. (1964). "Coexistence of cycles of a continuous mapping of the line into itself". Ukrainian Math. J. 16: 61–71.
 Li, T.Y.; Yorke, J.A. (1975). "Period Three Implies Chaos". American Mathematical Monthly 82 (10): 985–92. Bibcode:1975AmMM...82..985L. doi:10.2307/2318254.
 Crutchfield, J.P., Farmer, J.D., Packard, N.H., & Shaw, R.S; Tucker; Morrison (December 1986). "Chaos". Scientific American 255 (6): 38–49 (bibliography p.136). Bibcode:1986SciAm.255...38T 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, but which don't provide article views. The online content is identical to the hardcopy text. Citation variations will be related to country of publication).
 Kolyada, S.F. (2004). "LiYorke sensitivity and other concepts of chaos". Ukrainian Math. J. 56 (8): 1242–57. doi:10.1007/s1125300500554.
 Strelioff, C.; Hübler, A. (2006). "MediumTerm Prediction of Chaos" (PDF). Phys. Rev. Lett. 96 (4): 044101. Bibcode:2006PhRvL..96d4101S. doi:10.1103/PhysRevLett.96.044101. PMID 16486826. 044101.
 Hübler, A.; Foster, G.; Phelps, K. (2007). "Managing Chaos: Thinking out of the Box" (PDF). Complexity 12 (3): 10–13. doi:10.1002/cplx.20159.
Textbooks[edit]
 Alligood, K.T.; Sauer, T.; Yorke, J.A. (1997). Chaos: an introduction to dynamical systems. SpringerVerlag. ISBN 0387946772.
 Baker, G. L. (1996). Chaos, Scattering and Statistical Mechanics. Cambridge University Press. ISBN 0521395119.
 Badii, R.; Politi A. (1997). Complexity: hierarchical structures and scaling in physics. Cambridge University Press. ISBN 0521663857.
 Bunde; Havlin, Shlomo, eds. (1996). Fractals and Disordered Systems. Springer. ISBN 3642848702. and Bunde; Havlin, Shlomo, eds. (1994). Fractals in Science. Springer. ISBN 3540562206.
 Collet, Pierre, and Eckmann, JeanPierre (1980). Iterated Maps on the Interval as Dynamical Systems. Birkhauser. ISBN 0817649263.
 Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems (2nd ed.). Westview Press. ISBN 0813340853.
 Gollub, J. P.; Baker, G. L. (1996). Chaotic dynamics. Cambridge University Press. ISBN 0521476852.
 Guckenheimer, J.; Holmes P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. SpringerVerlag. ISBN 0387908196.
 Gulick, Denny (1992). Encounters with Chaos. McGrawHill. ISBN 0070252033.
 Gutzwiller, Martin (1990). Chaos in Classical and Quantum Mechanics. SpringerVerlag. ISBN 0387971734.
 Hoover, William Graham (1999,2001). Time Reversibility, Computer Simulation, and Chaos. World Scientific. ISBN 9810240732.
 Kautz, Richard (2011). Chaos: The Science of Predictable Random Motion. Oxford University Press. ISBN 9780199594580.
 Kiel, L. Douglas; Elliott, Euel W. (1997). Chaos Theory in the Social Sciences. Perseus Publishing. ISBN 0472084720.
 Moon, Francis (1990). Chaotic and Fractal Dynamics. SpringerVerlag. ISBN 0471545716.
 Ott, Edward (2002). Chaos in Dynamical Systems. Cambridge University Press. ISBN 0521010845.
 Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 0738204536.
 Sprott, Julien Clinton (2003). Chaos and TimeSeries Analysis. Oxford University Press. ISBN 0198508409.
 Tél, Tamás; Gruiz, Márton (2006). Chaotic dynamics: An introduction based on classical mechanics. Cambridge University Press. ISBN 0521839122.
 Thompson J M T, Stewart H B (2001). Nonlinear Dynamics And Chaos. John Wiley and Sons Ltd. ISBN 0471876453.
 Tufillaro; Reilly (1992). An experimental approach to nonlinear dynamics and chaos. AddisonWesley. ISBN 0201554410.
 Zaslavsky, George M. (2005). Hamiltonian Chaos and Fractional Dynamics. Oxford University Press. ISBN 0198526040.
Semitechnical and popular works[edit]
 Christophe Letellier, Chaos in Nature, World Scientific Publishing Company, 2012, ISBN 9789814374422.
 Abraham, Ralph H.; Ueda, Yoshisuke, eds. (2000). The Chaos AvantGarde: Memoirs of the Early Days of Chaos Theory. World Scientific. ISBN 9789812386472.
 Barnsley, Michael F. (2000). Fractals Everywhere. Morgan Kaufmann. ISBN 9780120790692.
 Bird, Richard J. (2003). Chaos and Life: Complexit and Order in Evolution and Thought. Columbia University Press. ISBN 9780231126625.
 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.
 Cunningham, Lawrence A. (1994). "From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis". George Washington Law Review 62: 546.
 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.
 John Gribbin. Deep Simplicity. Penguin Press Science. 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 SelfOrganisation; 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
 HeinzOtto 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.
 HeinzOtto 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.
 Ian Roulstone and John Norbury (2013). Invisible in the Storm: the role of mathematics in understanding weather. Princeton University Press.
 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.
 Sawaya, Antonio (2010). Financial time series analysis : Chaos and neurodynamics approach.
External links[edit]
Wikimedia Commons has media related to Chaos theory. 
 Hazewinkel, Michiel, ed. (2001), "Chaos", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Nonlinear Dynamics Research Group with Animations in Flash
 The Chaos group at the University of Maryland
 The Chaos Hypertextbook. An introductory primer on chaos and fractals
 ChaosBook.org An advanced graduate textbook on chaos (no fractals)
 Society for Chaos Theory in Psychology & Life Sciences
 Nonlinear Dynamics Research Group at CSDC, Florence Italy
 Interactive live chaotic pendulum experiment, allows users to interact and sample data from a real working damped driven chaotic pendulum
 Nonlinear dynamics: how science comprehends chaos, talk presented by Sunny Auyang, 1998.
 Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens
 Gleick's Chaos (excerpt)
 Systems Analysis, Modelling and Prediction Group at the University of Oxford
 A page about the MackeyGlass equation
 High Anxieties — The Mathematics of Chaos (2008) BBC documentary directed by David Malone
 The chaos theory of evolution  article published in Newscientist featuring similarities of evolution and nonlinear systems including fractal nature of life and chaos.
 Jos Leys, Étienne Ghys et Aurélien Alvarez, Chaos, A Mathematical Adventure. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.


Wikipedia preview
出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/09/10 07:04:55」(JST)
wiki en
[Wiki en表示]Chaos theory is a branch of mathematics focused on the behavior of dynamical systems that are highly sensitive to initial conditions. 'Chaos' is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, selfsimilarity, fractals, selforganization, 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 tornado in Texas.^{[1]}
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 longterm prediction of their behavior impossible in general.^{[2]}^{[3]} This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.^{[4]} In other words, the deterministic nature of these systems does not make them predictable.^{[5]}^{[6]} This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:^{[7]}
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.^{[8]}^{[9]} It also occurs spontaneously in some systems with artificial components, such as road traffic.^{[10]} This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in several disciplines, including meteorology, sociology, physics,^{[11]} environmental science, computer science, engineering, economics, biology, ecology, and philosophy. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, selfassembly process.
Contents
 1 Introduction
 2 Chaotic dynamics
 2.1 Chaos as a spontaneous breakdown of topological supersymmetry
 2.2 Sensitivity to initial conditions
 2.3 Topological mixing
 2.4 Density of periodic orbits
 2.5 Strange attractors
 2.6 Minimum complexity of a chaotic system
 2.7 Jerk systems
 3 Spontaneous order
 4 History
 5 Applications
 5.1 Cryptography
 5.2 Robotics
 5.3 Biology
 5.4 Other areas
 6 See also
 7 References
 8 Scientific literature
 8.1 Articles
 8.2 Textbooks
 8.3 Semitechnical and popular works
 9 External links
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.^{[3]} 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 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.^{[12]}
Chaotic dynamics
In common usage, "chaos" means "a state of disorder".^{[13]} 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:^{[14]}
 it must be sensitive to initial conditions
 it must be topologically mixing
 it must have dense periodic orbits
In some cases, the last two properties in the above have been shown to actually imply sensitivity to initial conditions.^{[15]}^{[16]} 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 intervals, the second property implies the other two.^{[17]} An alternative, and in general weaker, definition of chaos uses only the first two properties in the above list.^{[18]}
Chaos as a spontaneous breakdown of topological supersymmetry
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.^{[19]}^{[20]} 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 longrange 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
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", socalled 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 largescale 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.^{[21]} 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 °C or fall to 130 °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 trajectories in the phase space that are infinitesimally close, with initial separation $\delta \mathbf {Z} _{0}$, the two trajectories end up diverging at a rate given by
 $$
 δ Z ( t )  ≈ e λ t  δ Z 0  {\displaystyle \delta \mathbf {Z} (t)\approx e^{\lambda t}\delta \mathbf {Z} _{0}}
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 measuretheoretical mixing (as discussed in ergodic theory) and properties of a Ksystem.^{[6]}
Topological mixing
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 dyes 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 periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits.^{[22]} The onedimensional logistic map defined by x → 4 x (1 – x) is one of the simplest systems with density of periodic orbits. For example, ${\tfrac {5{\sqrt {5}}}{8}}$ → ${\tfrac {5+{\sqrt {5}}}{8}}$ → ${\tfrac {5{\sqrt {5}}}{8}}$ (or approximately 0.3454915 → 0.9045085 → 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).^{[23]}
Sharkovskii's theorem is the basis of the Li and Yorke^{[24]} (1975) proof that any onedimensional 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
Some dynamical systems, like the onedimensional logistic map defined by x → 4 x (1 – x), 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.^{[25]}
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 threedimensional model of the Lorenz weather system. The Lorenz attractor is perhaps one of the bestknown 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 fixedpoint attractors and limit cycles, the attractors that arise from chaotic systems, known as strange attractors, have great detail and complexity. 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, and the fractal dimension can be calculated for them.
Minimum complexity of a chaotic system
Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. In contrast, for continuous dynamical systems, the Poincaré–Bendixson theorem shows that a strange attractor can only arise in three or more dimensions. Finitedimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it must be either nonlinear or infinitedimensional.
The Poincaré–Bendixson theorem states that a twodimensional differential equation has very regular behavior. The Lorenz attractor discussed below is generated by a system of three differential equations such as:
 $$
d x d t = σ y − σ x , d y d t = ρ x − x z − y , d z d t = x y − β z . {\displaystyle {\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma y\sigma x,\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=\rho xxzy,\\{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy\beta z.\end{aligned}}}
where $x$, $y$, and $z$ make up the system state, $t$ is time, and $\sigma$, $\rho$, $\beta$ are the system parameters. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another wellknown chaotic attractor is generated by the Rössler equations, which have only one nonlinear term out of seven. Sprott^{[26]} found a threedimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel^{[27]}^{[28]} showed that, at least for dissipative and conservative quadratic systems, threedimensional quadratic systems with only three or four terms on the righthand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a twodimensional surface and therefore solutions are well behaved.
While the Poincaré–Bendixson theorem shows that a continuous dynamical system on the Euclidean plane cannot be chaotic, twodimensional continuous systems with nonEuclidean geometry can exhibit chaotic behavior.^{[29]} Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional.^{[30]} A theory of linear chaos is being developed in a branch of mathematical analysis known as functional analysis.
Jerk systems
In physics, jerk is the third derivative of position, with respect to time. As such, differential equations of the form

 $$
J ( x . . . , x ¨ , x ˙ , x ) = 0 {\displaystyle J\left({\overset {...}{x}},{\ddot {x}},{\dot {x}},x\right)=0}
 $$
are sometimes called Jerk equations. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, nonlinear 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.^{[31]}
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 wellknown chaotic systems, such as the Lorenz attractor and the Rössler map, are conventionally described as a system of three firstorder 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 firstorder, 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 $x$ is:
 $$
d 3 x d t 3 + A d 2 x d t 2 + d x d t −  x  + 1 = 0. {\displaystyle {\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.}
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:
In the above circuit, all resistors are of equal value, except $R_{A}=R/A=5R/3$, and all capacitors are of equal size. The dominant frequency is $1/2\pi RC$. The output of 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, neurons, the London Millennium Bridge resonance, and large arrays of Josephson junctions.^{[32]}
History
An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the threebody problem, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.^{[33]}^{[34]} 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".^{[35]} 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,^{[36]} Andrey Nikolaevich Kolmogorov,^{[37]}^{[38]}^{[39]} Mary Lucy Cartwright and John Edensor Littlewood,^{[40]} and Stephen Smale.^{[41]} Except for Smale, these studies were all directly inspired by physics: the threebody 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]} 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 midcentury, 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 "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.^{[42]}^{[43]}
Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in 1961.^{[8]} Lorenz was using a simple digital computer, a Royal McBee LGP30, 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 6digit precision, but the printout rounded variables off to a 3digit 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 longterm outcome.^{[44]} Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modelling cannot, in general, make precise longterm weather predictions.
In 1963, Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices.^{[45]} Beforehand he had studied information theory and concluded noise was patterned like a Cantor set: on any scale the proportion of noisecontaining periods to errorfree periods was a constant – thus errors were inevitable and must be planned for by incorporating redundancy.^{[46]} 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).^{[47]}^{[48]} This challenged the idea that changes in price were normally distributed. In 1967, he published "How long is the coast of Britain? Statistical selfsimilarity 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.^{[49]} Arguing that a ball of twine appears as a point when viewed from far away (0dimensional), a ball when viewed from fairly near (3dimensional), or a curved strand (1dimensional), 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 ("selfsimilarity") 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.^{[50]} Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model.^{[51]}
In December 1977, the New York Academy of Sciences organized the first symposium on chaos, attended by David Ruelle, Robert May, James A. Yorke (coiner of the term "chaos" as used in mathematics), 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.^{[52]}^{[53]} They notably discovered the 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 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.^{[54]}
In 1986, the New York Academy of Sciences coorganized 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.^{[55]} This led to a renewal of physiology in the 1980s through the application of chaos theory, 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^{[56]} describing for the first time selforganized criticality (SOC), considered one of the mechanisms by which complexity arises in nature.
Alongside largely labbased approaches such as the Bak–Tang–Wiesenfeld sandpile, many other investigations have focused on largescale natural or social systems that are known (or suspected) to display scaleinvariant 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 earthquakes, (which, long before SOC was discovered, were known as a source of scaleinvariant behavior such as the Gutenberg–Richter law describing the statistical distribution of earthquake sizes, and the Omori law^{[57]} 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). Given the implications of a scalefree 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 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 bestseller and introduced the general principles of chaos theory as well as its history to the broad public, though his history underemphasized important Soviet contributions.^{[citation needed]}^{[58]} Initially the domain 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 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,^{[59]} involving many different disciplines (mathematics, topology, physics,^{[60]} social systems, population modeling, biology, meteorology, astrophysics, information theory, computational neuroscience, etc.).
Applications
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,^{[62]}^{[63]}^{[64]} engineering,^{[65]} finance,^{[66]}^{[67]} algorithmic trading,^{[68]}^{[69]}^{[70]} meteorology, philosophy, physics,^{[71]}^{[72]}^{[73]} politics, population dynamics,^{[74]} psychology,^{[10]} and 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 primitives. These algorithms include image encryption algorithms, hash functions, secure pseudorandom number generators, stream ciphers, watermarking and steganography.^{[75]} The majority of these algorithms are based on unimodal chaotic maps and a big portion of these algorithms use the control parameters and the initial condition of the chaotic maps as their keys.^{[76]} 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.^{[75]} One type of encryption, secret key or symmetric key, relies on diffusion and confusion, which is modeled well by chaos theory.^{[77]} Another type of computing, DNA computing, when paired with chaos theory, offers a way to encrypt images and other information.^{[78]} Many of the DNAChaos cryptographic algorithms are proven to be either not secure, or the technique applied is suggested to be not efficient.^{[79]}^{[80]}^{[81]}
Robotics
Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trialanderror type of refinement to interact with their environment, chaos theory has been used to build a predictive model.^{[82]} Chaotic dynamics have been exhibited by passive walking biped robots.^{[83]}
Biology
For over a hundred years, biologists have been keeping track of populations of different species with population models. Most models are continuous, but recently scientists have been able to implement chaotic models in certain populations.^{[84]} For example, a study on models of Canadian lynx showed there was chaotic behavior in the population growth.^{[85]} 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.^{[86]} 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 fetal hypoxia can be obtained through chaotic modeling.^{[87]}
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.^{[88]} In celestial mechanics, especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will approach Earth and other planets.^{[89]} 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.^{[90]} 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.^{[91]}
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.^{[92]} Modern organizations are increasingly seen as open complex adaptive systems 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.^{[93]}
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.^{[94]} 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 nonlinear relationships.^{[95]}
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 shortterm prediction model (see the plot of the BML traffic model at right).^{[96]}
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.^{[97]}
Chaos theory has been applied to environmental water cycle data (aka hydrological data), such as rainfall and streamflow.^{[98]} 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 metaanalyses called those studies into question and provided explanations for why these datasets are not likely to have lowdimension chaotic dynamics.^{[99]}
See also
 Systems science portal
 Mathematics portal
 Examples of chaotic systems
 Advected contours
 Arnold's cat map
 Bouncing ball dynamics
 Chua's circuit
 Cliodynamics
 Coupled map lattice
 Double pendulum
 Duffing equation
 Dynamical billiards
 Economic bubble
 GaspardRice system
 Hénon map
 Horseshoe map
 List of chaotic maps
 Logistic map
 Rössler attractor
 Standard map
 Swinging Atwood's machine
 Tilt A Whirl
 Other related topics
 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
 Illconditioning
 Illposedness
 Nonlinear system
 Patterns in nature
 Predictability
 Quantum chaos
 Santa Fe Institute
 Synchronization of chaos
 Unintended consequence
 People
 Ralph Abraham
 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
 Floris Takens
 James A. Yorke
 George M. Zaslavsky
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 ^ Behnia, S.; Akhshani, A.; Mahmodi, H.; Akhavan, A. (20080101). "A novel algorithm for image encryption based on mixture of chaotic maps". Chaos, Solitons & Fractals. 35 (2): 408–419. doi:10.1016/j.chaos.2006.05.011.
 ^ Wang, Xingyuan; Zhao, Jianfeng (2012). "An improved key agreement protocol based on chaos". Commun. Nonlinear Sci. Numer. Simul. 15 (12): 4052–4057. Bibcode:2010CNSNS..15.4052W. doi:10.1016/j.cnsns.2010.02.014.
 ^ Babaei, Majid (2013). "A novel text and image encryption method based on chaos theory and DNA computing". Natural Computing. an International Journal. 12 (1): 101–107. doi:10.1007/s1104701293349.
 ^ Akhavan, A.; Samsudin, A.; Akhshani, A. (20171001). "Cryptanalysis of an image encryption algorithm based on DNA encoding". Optics & Laser Technology. 95: 94–99. doi:10.1016/j.optlastec.2017.04.022.
 ^ Xu, Ming (20170601). "Cryptanalysis of an Image Encryption Algorithm Based on DNA Sequence Operation and Hyperchaotic System". 3D Research. 8 (2): 15. ISSN 20926731. doi:10.1007/s133190170126y.
 ^ Liu, Yuansheng; Tang, Jie; Xie, Tao (20140801). "Cryptanalyzing a RGB image encryption algorithm based on DNA encoding and chaos map". Optics & Laser Technology. 60: 111–115. arXiv:1307.4279 . doi:10.1016/j.optlastec.2014.01.015.
 ^ Nehmzow, Ulrich; Keith Walker (Dec 2005). "Quantitative description of robot–environment interaction using chaos theory". Robotics and Autonomous Systems. 53 (3–4): 177–193. doi:10.1016/j.robot.2005.09.009.
 ^ Goswami, Ambarish; Thuilot, Benoit; Espiau, Bernard (1998). "A Study of the Passive Gait of a CompassLike Biped Robot: Symmetry and Chaos". The International Journal of Robotics Research. 17 (12): 1282–1301. doi:10.1177/027836499801701202.
 ^ Eduardo, Liz; RuizHerrera, Alfonso (2012). "Chaos in discrete structured population models". SIAM Journal on Applied Dynamical Systems. 11 (4): 1200–1214. doi:10.1137/120868980.
 ^ Lai, Dejian (1996). "Comparison study of AR models on the Canadian lynx data: a close look at BDS statistic". Computational Statistics \& Data Analysis. 22 (4): 409–423. doi:10.1016/01679473(95)000569.
 ^ Sivakumar, B (31 January 2000). "Chaos theory in hydrology: important issues and interpretations". Journal of Hydrology. 227 (1–4): 1–20. Bibcode:2000JHyd..227....1S. doi:10.1016/S00221694(99)001869.
 ^ Bozóki, Zsolt (February 1997). "Chaos theory and power spectrum analysis in computerized cardiotocography". European Journal of Obstetrics & Gynecology and Reproductive Biology. 71 (2): 163–168. doi:10.1016/s03012115(96)026280.
 ^ Li, Mengshan; Xingyuan Huanga; Hesheng Liua; Bingxiang Liub; Yan Wub; Aihua Xiongc; Tianwen Dong (25 October 2013). "Prediction of gas solubility in polymers by back propagation artificial neural network based on selfadaptive particle swarm optimization algorithm and chaos theory". Fluid Phase Equilibria. 356: 11–17. doi:10.1016/j.fluid.2013.07.017.
 ^ Morbidelli, A. (2001). "Chaotic diffusion in celestial mechanics". Regular & Chaotic Dynamics. 6 (4): 339–353. doi:10.1070/rd2001v006n04abeh000182.
 ^ Steven Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion, 2003
 ^ Dingqi, Li; Yuanping Chenga; Lei Wanga; Haifeng Wanga; Liang Wanga; Hongxing Zhou (May 2011). "Prediction method for risks of coal and gas outbursts based on spatial chaos theory using gas desorption index of drill cuttings". Mining Science and Technology. 21 (3): 439–443.
 ^ Pryor, Robert G. L.; Norman E. Aniundson; Jim E. H. Bright (June 2008). "Probabilities and Possibilities: The Strategic Counseling Implications of the Chaos Theory of Careers". The Career Development Quarterly. 56: 309–318. doi:10.1002/j.21610045.2008.tb00096.x.
 ^ Dal Forno, Arianna; Merlone, Ugo (2013). "Chaotic Dynamics in Organization Theory". In Bischi, Gian Italo; Chiarella, Carl; Shusko, Irina. Global Analysis of Dynamic Models in Economics and Finance. SpringerVerlag. pp. 185–204. ISBN 9783642295034.
 ^ Juárez, Fernando (2011). "Applying the theory of chaos and a complex model of health to establish relations among financial indicators". Procedia Computer Science. 3: 982–986. doi:10.1016/j.procs.2010.12.161.
 ^ Brooks, Chris (1998). "Chaos in foreign exchange markets: a sceptical view". Computational Economics. 11: 265–281. ISSN 15729974. doi:10.1023/A:1008650024944.
 ^ Wang, Jin; Qixin Shi (February 2013). "Shortterm traffic speed forecasting hybrid model based on Chaos–Wavelet AnalysisSupport Vector Machine theory". Transportation Research Part C: Emerging Technologies. 27: 219–232. doi:10.1016/j.trc.2012.08.004.
 ^ Dal Forno, Arianna; Merlone, Ugo (2013). "Nonlinear dynamics in work groups with Bion's basic assumptions". Nonlinear Dynamics, Psychology, and Life Sciences. 17 (2): 295–315. ISSN 10900578.
 ^ "Dr. Gregory B. Pasternack  Watershed Hydrology, Geomorphology, and Ecohydraulics :: Chaos in Hydrology". pasternack.ucdavis.edu. Retrieved 20170612.
 ^ Pasternack, Gregory B. (19991101). "Does the river run wild? Assessing chaos in hydrological systems". Advances in Water Resources. 23 (3): 253–260. doi:10.1016/s03091708(99)000081.
Scientific literature
Articles
 Sharkovskii, A.N. (1964). "Coexistence of cycles of a continuous mapping of the line into itself". Ukrainian Math. J. 16: 61–71.
 Li, T.Y.; Yorke, J.A. (1975). "Period Three Implies Chaos". American Mathematical Monthly. 82 (10): 985–92. Bibcode:1975AmMM...82..985L. doi:10.2307/2318254.
 Crutchfield; Tucker; Morrison; J.D.; Packard; N.H.; Shaw; R.S (December 1986). "Chaos". Scientific American. 255 (6): 38–49 (bibliography p.136). Bibcode:1986SciAm.255...38T. 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).
 Kolyada, S.F. (2004). "LiYorke sensitivity and other concepts of chaos". Ukrainian Math. J. 56 (8): 1242–57. doi:10.1007/s1125300500554.
 Day, R.H.; Pavlov, O.V. (2004). "Computing Economic Chaos". Computational Economics. 23 (4): 289–301. SSRN 806124 . doi:10.1023/B:CSEM.0000026787.81469.1f.
 Strelioff, C.; Hübler, A. (2006). "MediumTerm Prediction of Chaos" (PDF). Phys. Rev. Lett. 96 (4): 044101. Bibcode:2006PhRvL..96d4101S. PMID 16486826. doi:10.1103/PhysRevLett.96.044101. 044101. Archived from the original (PDF) on 20130426.
 Hübler, A.; Foster, G.; Phelps, K. (2007). "Managing Chaos: Thinking out of the Box" (PDF). Complexity. 12 (3): 10–13. doi:10.1002/cplx.20159.
 Motter, Adilson E.; Campbell, David K. (2013). "Chaos at 50". Physics Today. 66: 27. doi:10.1063/PT.3.1977.
 Boeing, G. (2016). "Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, SelfSimilarity and the Limits of Prediction". Systems. 4 (4): 37. doi:10.3390/systems4040037.
Textbooks
 Alligood, K.T.; Sauer, T.; Yorke, J.A. (1997). Chaos: an introduction to dynamical systems. SpringerVerlag. ISBN 0387946772.
 Baker, G. L. (1996). Chaos, Scattering and Statistical Mechanics. Cambridge University Press. ISBN 0521395119.
 Badii, R.; Politi A. (1997). Complexity: hierarchical structures and scaling in physics. Cambridge University Press. ISBN 0521663857.
 Bunde; Havlin, Shlomo, eds. (1996). Fractals and Disordered Systems. Springer. ISBN 3642848702. and Bunde; Havlin, Shlomo, eds. (1994). Fractals in Science. Springer. ISBN 3540562206.
 Collet, Pierre, and Eckmann, JeanPierre (1980). Iterated Maps on the Interval as Dynamical Systems. Birkhauser. ISBN 0817649263. CS1 maint: Multiple names: authors list (link)
 Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems (2nd ed.). Westview Press. ISBN 0813340853.
 Feldman, D. P. (2012). Chaos and Fractals: An Elementary Introduction. Oxford University Press. ISBN 9780199566440.
 Gollub, J. P.; Baker, G. L. (1996). Chaotic dynamics. Cambridge University Press. ISBN 0521476852.
 Guckenheimer, John; Holmes, Philip (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. SpringerVerlag. ISBN 0387908196.
 Gulick, Denny (1992). Encounters with Chaos. McGrawHill. ISBN 0070252033.
 Gutzwiller, Martin (1990). Chaos in Classical and Quantum Mechanics. SpringerVerlag. ISBN 0387971734.
 Hoover, William Graham (2001) [1999]. Time Reversibility, Computer Simulation, and Chaos. World Scientific. ISBN 9810240732.
 Kautz, Richard (2011). Chaos: The Science of Predictable Random Motion. Oxford University Press. ISBN 9780199594580.
 Kiel, L. Douglas; Elliott, Euel W. (1997). Chaos Theory in the Social Sciences. Perseus Publishing. ISBN 0472084720.
 Moon, Francis (1990). Chaotic and Fractal Dynamics. SpringerVerlag. ISBN 0471545716.
 Ott, Edward (2002). Chaos in Dynamical Systems. Cambridge University Press. ISBN 0521010845.
 Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 0738204536.
 Sprott, Julien Clinton (2003). Chaos and TimeSeries Analysis. Oxford University Press. ISBN 0198508409.
 Tél, Tamás; Gruiz, Márton (2006). Chaotic dynamics: An introduction based on classical mechanics. Cambridge University Press. ISBN 0521839122.
 Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 9780821883280.
 Thompson JM, Stewart HB (2001). Nonlinear Dynamics And Chaos. John Wiley and Sons Ltd. ISBN 0471876453.
 Tufillaro; Reilly (1992). An experimental approach to nonlinear dynamics and chaos. AddisonWesley. ISBN 0201554410.
 Wiggins, Stephen (2003). Introduction to Applied Dynamical Systems and Chaos. Springer. ISBN 0387001778.
 Zaslavsky, George M. (2005). Hamiltonian Chaos and Fractional Dynamics. Oxford University Press. ISBN 0198526040.
Semitechnical and popular works
 Christophe Letellier, Chaos in Nature, World Scientific Publishing Company, 2012, ISBN 9789814374422.
 Abraham, Ralph H.; Ueda, Yoshisuke, eds. (2000). The Chaos AvantGarde: Memoirs of the Early Days of Chaos Theory. World Scientific. ISBN 9789812386472.
 Barnsley, Michael F. (2000). Fractals Everywhere. Morgan Kaufmann. ISBN 9780120790692.
 Bird, Richard J. (2003). Chaos and Life: Complexit and Order in Evolution and Thought. Columbia University Press. ISBN 9780231126625.
 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.
 Cunningham, Lawrence A. (1994). "From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis". George Washington Law Review. 62: 546.
 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.
 John Gribbin. Deep Simplicity. Penguin Press Science. 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 SelfOrganisation; 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
 HeinzOtto 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.
 HeinzOtto 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.
 Ian Roulstone; John Norbury (2013). Invisible in the Storm: the role of mathematics in understanding weather. 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, 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
Wikimedia Commons has media related to Chaos theory. 
 Hazewinkel, Michiel, ed. (2001) [1994], "Chaos", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Nonlinear Dynamics Research Group with Animations in Flash
 The Chaos group at the University of Maryland
 The Chaos Hypertextbook. An introductory primer on chaos and fractals
 ChaosBook.org An advanced graduate textbook on chaos (no fractals)
 Society for Chaos Theory in Psychology & Life Sciences
 Nonlinear Dynamics Research Group at CSDC, Florence Italy
 Interactive live chaotic pendulum experiment, allows users to interact and sample data from a real working damped driven chaotic pendulum
 Nonlinear dynamics: how science comprehends chaos, talk presented by Sunny Auyang, 1998.
 Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens
 Gleick's Chaos (excerpt)
 Systems Analysis, Modelling and Prediction Group at the University of Oxford
 A page about the MackeyGlass equation
 High Anxieties — The Mathematics of Chaos (2008) BBC documentary directed by David Malone
 The chaos theory of evolution  article published in Newscientist featuring similarities of evolution and nonlinear systems including fractal nature of life and chaos.
 Jos Leys, Étienne Ghys et Aurélien Alvarez, Chaos, A Mathematical Adventure. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.
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Wikipedia preview
出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2018/04/05 02:13:38」(JST)
wiki en
[Wiki en表示]Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. 'Chaos' is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, selfsimilarity, fractals, selforganization, 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 China can cause a hurricane in Texas.^{[1]}
Small differences in initial conditions such as those due to rounding errors in numerical computation yield widely diverging outcomes for such dynamical systems, rendering longterm prediction of their behavior impossible in general.^{[2]}^{[3]} This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.^{[4]} In other words, the deterministic nature of these systems does not make them predictable.^{[5]}^{[6]} This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:^{[7]}
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.^{[8]}^{[9]} It also occurs spontaneously in some systems with artificial components, such as road traffic.^{[10]} This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in several disciplines, including meteorology, anthropology,^{[11]}^{[12]} sociology, physics,^{[13]} environmental science, computer science, engineering, economics, biology, ecology, and philosophy. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, and selfassembly processes.
Contents
 1 Introduction
 2 Chaotic dynamics
 2.1 Chaos as a spontaneous breakdown of topological supersymmetry
 2.2 Sensitivity to initial conditions
 2.3 Topological mixing
 2.4 Density of periodic orbits
 2.5 Strange attractors
 2.6 Minimum complexity of a chaotic system
 2.7 Jerk systems
 3 Spontaneous order
 4 History
 5 Applications
 5.1 Cryptography
 5.2 Robotics
 5.3 Biology
 5.4 Other areas
 6 See also
 7 References
 8 Scientific literature
 8.1 Articles
 8.2 Textbooks
 8.3 Semitechnical and popular works
 9 External links
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.^{[3]} 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 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.^{[14]}
Chaotic dynamics
In common usage, "chaos" means "a state of disorder".^{[15]} 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:^{[16]}
 it must be sensitive to initial conditions
 it must be topologically mixing
 it must have dense periodic orbits
In some cases, the last two properties in the above have been shown to actually imply sensitivity to initial conditions.^{[17]}^{[18]} 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 intervals, the second property implies the other two.^{[19]} An alternative, and in general weaker, definition of chaos uses only the first two properties in the above list.^{[20]}
Chaos as a spontaneous breakdown of topological supersymmetry
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.^{[21]}^{[22]} 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 longrange 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
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", socalled 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 largescale 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 prevalent in the case of weather, which is generally predictable only about a week ahead.^{[23]} 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 °C or fall to −130 °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 trajectories in the phase space that are infinitesimally close, with initial separation $\delta \mathbf {Z} _{0}$, the two trajectories end up diverging at a rate given by
 $$
 δ Z ( t )  ≈ e λ t  δ Z 0  , {\displaystyle \delta \mathbf {Z} (t)\approx e^{\lambda t}\delta \mathbf {Z} _{0},}
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 measuretheoretical mixing (as discussed in ergodic theory) and properties of a Ksystem.^{[6]}
Topological mixing
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 dyes 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 periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits.^{[24]} The onedimensional logistic map defined by x → 4 x (1 – x) is one of the simplest systems with density of periodic orbits. For example, ${\tfrac {5{\sqrt {5}}}{8}}$ → ${\tfrac {5+{\sqrt {5}}}{8}}$ → ${\tfrac {5{\sqrt {5}}}{8}}$ (or approximately 0.3454915 → 0.9045085 → 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).^{[25]}
Sharkovskii's theorem is the basis of the Li and Yorke^{[26]} (1975) proof that any continuous onedimensional 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
Some dynamical systems, like the onedimensional logistic map defined by x → 4 x (1 – x), 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.^{[27]}
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 threedimensional model of the Lorenz weather system. The Lorenz attractor is perhaps one of the bestknown 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 fixedpoint attractors and limit cycles, the attractors that arise from chaotic systems, known as strange attractors, have great detail and complexity. 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, and the fractal dimension can be calculated for them.
Minimum complexity of a chaotic system
Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. In contrast, for continuous dynamical systems, the Poincaré–Bendixson theorem shows that a strange attractor can only arise in three or more dimensions. Finitedimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it must be either nonlinear or infinitedimensional.
The Poincaré–Bendixson theorem states that a twodimensional differential equation has very regular behavior. The Lorenz attractor discussed below is generated by a system of three differential equations such as:
 $$
d x d t = σ y − σ x , d y d t = ρ x − x z − y , d z d t = x y − β z . {\displaystyle {\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma y\sigma x,\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=\rho xxzy,\\{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy\beta z.\end{aligned}}}
where $x$, $y$, and $z$ make up the system state, $t$ is time, and $\sigma$, $\rho$, $\beta$ are the system parameters. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another wellknown chaotic attractor is generated by the Rössler equations, which have only one nonlinear term out of seven. Sprott^{[28]} found a threedimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel^{[29]}^{[30]} showed that, at least for dissipative and conservative quadratic systems, threedimensional quadratic systems with only three or four terms on the righthand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a twodimensional surface and therefore solutions are well behaved.
While the Poincaré–Bendixson theorem shows that a continuous dynamical system on the Euclidean plane cannot be chaotic, twodimensional continuous systems with nonEuclidean geometry can exhibit chaotic behavior.^{[31]} Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional.^{[32]} A theory of linear chaos is being developed in a branch of mathematical analysis known as functional analysis.
Jerk systems
In physics, jerk is the third derivative of position, with respect to time. As such, differential equations of the form

 $$
J ( x . . . , x ¨ , x ˙ , x ) = 0 {\displaystyle J\left({\overset {...}{x}},{\ddot {x}},{\dot {x}},x\right)=0}
 $$
are sometimes called Jerk equations. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, nonlinear 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.^{[33]}
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 wellknown chaotic systems, such as the Lorenz attractor and the Rössler map, are conventionally described as a system of three firstorder 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 firstorder, 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 $x$ is:
 $$
d 3 x d t 3 + A d 2 x d t 2 + d x d t −  x  + 1 = 0. {\displaystyle {\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.}
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:
In the above circuit, all resistors are of equal value, except $R_{A}=R/A=5R/3$, and all capacitors are of equal size. The dominant frequency is $1/2\pi RC$. The output of 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, neurons, the London Millennium Bridge resonance, and large arrays of Josephson junctions.^{[34]}
History
An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the threebody problem, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.^{[35]}^{[36]}^{[37]} 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".^{[38]} 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,^{[39]} Andrey Nikolaevich Kolmogorov,^{[40]}^{[41]}^{[42]} Mary Lucy Cartwright and John Edensor Littlewood,^{[43]} and Stephen Smale.^{[44]} Except for Smale, these studies were all directly inspired by physics: the threebody 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]} 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 midcentury, 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 "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.^{[45]}^{[46]}
Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in 1961.^{[8]} Lorenz was using a simple digital computer, a Royal McBee LGP30, 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 6digit precision, but the printout rounded variables off to a 3digit 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 longterm outcome.^{[47]} Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modelling cannot, in general, make precise longterm weather predictions.
In 1963, Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices.^{[48]} Beforehand he had studied information theory and concluded noise was patterned like a Cantor set: on any scale the proportion of noisecontaining periods to errorfree periods was a constant – thus errors were inevitable and must be planned for by incorporating redundancy.^{[49]} 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).^{[50]}^{[51]} This challenged the idea that changes in price were normally distributed. In 1967, he published "How long is the coast of Britain? Statistical selfsimilarity 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.^{[52]} Arguing that a ball of twine appears as a point when viewed from far away (0dimensional), a ball when viewed from fairly near (3dimensional), or a curved strand (1dimensional), 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 ("selfsimilarity") 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.^{[53]} Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model.^{[54]}
In December 1977, the New York Academy of Sciences organized the first symposium on chaos, attended by David Ruelle, Robert May, James A. Yorke (coiner of the term "chaos" as used in mathematics), 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.^{[55]}^{[56]} They notably discovered the 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 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.^{[57]}
In 1986, the New York Academy of Sciences coorganized 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.^{[58]} This led to a renewal of physiology in the 1980s through the application of chaos theory, 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^{[59]} describing for the first time selforganized criticality (SOC), considered one of the mechanisms by which complexity arises in nature.
Alongside largely labbased approaches such as the Bak–Tang–Wiesenfeld sandpile, many other investigations have focused on largescale natural or social systems that are known (or suspected) to display scaleinvariant 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 earthquakes, (which, long before SOC was discovered, were known as a source of scaleinvariant behavior such as the Gutenberg–Richter law describing the statistical distribution of earthquake sizes, and the Omori law^{[60]} 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). Given the implications of a scalefree 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 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 bestseller and introduced the general principles of chaos theory as well as its history to the broad public, though his history underemphasized important Soviet contributions.^{[citation needed]}^{[61]} Initially the domain 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 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,^{[62]} involving many different disciplines (mathematics, topology, physics,^{[63]} social systems, population modeling, biology, meteorology, astrophysics, information theory, computational neuroscience, etc.).
Applications
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,^{[65]}^{[66]}^{[67]} engineering,^{[68]} finance,^{[69]}^{[70]} algorithmic trading,^{[71]}^{[72]}^{[73]} meteorology, philosophy, anthropology,^{[11]}^{[12]} physics,^{[74]}^{[75]}^{[76]} politics, population dynamics,^{[77]} psychology,^{[10]} and 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 primitives. These algorithms include image encryption algorithms, hash functions, secure pseudorandom number generators, stream ciphers, watermarking and steganography.^{[78]} The majority of these algorithms are based on unimodal chaotic maps and a big portion of these algorithms use the control parameters and the initial condition of the chaotic maps as their keys.^{[79]} 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.^{[78]} One type of encryption, secret key or symmetric key, relies on diffusion and confusion, which is modeled well by chaos theory.^{[80]} Another type of computing, DNA computing, when paired with chaos theory, offers a way to encrypt images and other information.^{[81]} Many of the DNAChaos cryptographic algorithms are proven to be either not secure, or the technique applied is suggested to be not efficient.^{[82]}^{[83]}^{[84]}
Robotics
Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trialanderror type of refinement to interact with their environment, chaos theory has been used to build a predictive model.^{[85]} Chaotic dynamics have been exhibited by passive walking biped robots.^{[86]}
Biology
For over a hundred years, biologists have been keeping track of populations of different species with population models. Most models are continuous, but recently scientists have been able to implement chaotic models in certain populations.^{[87]} For example, a study on models of Canadian lynx showed there was chaotic behavior in the population growth.^{[88]} 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.^{[89]} 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 fetal hypoxia can be obtained through chaotic modeling.^{[90]}
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.^{[91]} In celestial mechanics, especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will approach Earth and other planets.^{[92]} 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.^{[93]} 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.^{[94]}
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.^{[95]} Modern organizations are increasingly seen as open complex adaptive systems with fundamental natural nonlinear structures, subject to internal and external forces that may contribute chaos. For instance, team building and group development is increasingly being researched as an inherently unpredictable system, as the uncertainty of different individuals meeting for the first time makes the trajectory of the team unknowable.^{[96]} 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.^{[97]}
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.^{[98]} 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 nonlinear relationships.^{[99]}
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 shortterm prediction model (see the plot of the BML traffic model at right).^{[100]}
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.^{[101]}
Chaos theory has been applied to environmental water cycle data (aka hydrological data), such as rainfall and streamflow.^{[102]} 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 metaanalyses called those studies into question and provided explanations for why these datasets are not likely to have lowdimension chaotic dynamics.^{[103]}
See also
 Systems science portal
 Mathematics portal
 Examples of chaotic systems
 Advected contours
 Arnold's cat map
 Bouncing ball dynamics
 Chua's circuit
 Cliodynamics
 Coupled map lattice
 Double pendulum
 Duffing equation
 Dynamical billiards
 Economic bubble
 GaspardRice system
 Hénon map
 Horseshoe map
 List of chaotic maps
 Logistic map
 Rössler attractor
 Standard map
 Swinging Atwood's machine
 Tilt A Whirl
 Other related topics
 Amplitude death
 Anosov diffeomorphism
 Bifurcation theory
 Butterfly effect
 Catastrophe theory
 Causality
 Chaos theory in organizational development
 Chaos machine
 Chaotic mixing
 Chaotic scattering
 Complexity
 Control of chaos
 Determinism
 Edge of chaos
 Emergence
 Fractal
 Julia set
 Mandelbrot set
 Kolmogorov–Arnold–Moser theorem
 Illconditioning
 Illposedness
 Nonlinear system
 Patterns in nature
 Predictability
 Quantum chaos
 Santa Fe Institute
 Synchronization of chaos
 Unintended consequence
 People
 Ralph Abraham
 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
 Floris Takens
 James A. Yorke
 George M. Zaslavsky
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 ^ Gerig, A. (2007). "Chaos in a onedimensional compressible flow". Physical Review E. 75 (4): 045202. arXiv:nlin/0701050 . Bibcode:2007PhRvE..75d5202G. doi:10.1103/PhysRevE.75.045202.
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 ^ Dilão, R.; Domingos, T. (2001). "Periodic and QuasiPeriodic Behavior in Resource Dependent Age Structured Population Models". Bulletin of Mathematical Biology. 63 (2): 207–230. doi:10.1006/bulm.2000.0213. PMID 11276524.
 ^ ^{a} ^{b} Akhavan, A.; Samsudin, A.; Akhshani, A. (20111001). "A symmetric image encryption scheme based on combination of nonlinear chaotic maps". Journal of the Franklin Institute. 348 (8): 1797–1813. doi:10.1016/j.jfranklin.2011.05.001.
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Scientific literature
Articles
 Sharkovskii, A.N. (1964). "Coexistence of cycles of a continuous mapping of the line into itself". Ukrainian Math. J. 16: 61–71.
 Li, T.Y.; Yorke, J.A. (1975). "Period Three Implies Chaos" (PDF). American Mathematical Monthly. 82 (10): 985–92. Bibcode:1975AmMM...82..985L. doi:10.2307/2318254.
 Alemansour, Hamed; Miandoab, Ehsan Maani; Pishkenari, Hossein Nejat (March 2017). "Effect of size on the chaotic behavior of nano resonators". Communications in Nonlinear Science and Numerical Simulation. 44: 495–505. doi:10.1016/j.cnsns.2016.09.010.
 Crutchfield; Tucker; Morrison; J.D.; Packard; N.H.; Shaw; R.S (December 1986). "Chaos". Scientific American. 255 (6): 38–49 (bibliography p.136). Bibcode:1986SciAm.255d..38T. 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).
 Kolyada, S.F. (2004). "LiYorke sensitivity and other concepts of chaos". Ukrainian Math. J. 56 (8): 1242–57. doi:10.1007/s1125300500554.
 Day, R.H.; Pavlov, O.V. (2004). "Computing Economic Chaos". Computational Economics. 23 (4): 289–301. doi:10.1023/B:CSEM.0000026787.81469.1f. SSRN 806124 .
 Strelioff, C.; Hübler, A. (2006). "MediumTerm Prediction of Chaos" (PDF). Phys. Rev. Lett. 96 (4): 044101. Bibcode:2006PhRvL..96d4101S. doi:10.1103/PhysRevLett.96.044101. PMID 16486826. 044101. Archived from the original (PDF) on 20130426.
 Hübler, A.; Foster, G.; Phelps, K. (2007). "Managing Chaos: Thinking out of the Box" (PDF). Complexity. 12 (3): 10–13. Bibcode:2007Cmplx..12c..10H. doi:10.1002/cplx.20159.
 Motter, Adilson E.; Campbell, David K. (2013). "Chaos at 50". Physics Today. 66 (5): 27. arXiv:1306.5777 . Bibcode:2013PhT....66e..27M. doi:10.1063/PT.3.1977.
 Boeing, G. (2016). "Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, SelfSimilarity and the Limits of Prediction". Systems. 4 (4): 37. doi:10.3390/systems4040037.
Textbooks
 Alligood, K.T.; Sauer, T.; Yorke, J.A. (1997). Chaos: an introduction to dynamical systems. SpringerVerlag. ISBN 0387946772.
 Baker, G. L. (1996). Chaos, Scattering and Statistical Mechanics. Cambridge University Press. ISBN 0521395119.
 Badii, R.; Politi A. (1997). Complexity: hierarchical structures and scaling in physics. Cambridge University Press. ISBN 0521663857.
 Bunde; Havlin, Shlomo, eds. (1996). Fractals and Disordered Systems. Springer. ISBN 3642848702. and Bunde; Havlin, Shlomo, eds. (1994). Fractals in Science. Springer. ISBN 3540562206.
 Collet, Pierre, and Eckmann, JeanPierre (1980). Iterated Maps on the Interval as Dynamical Systems. Birkhauser. ISBN 0817649263. CS1 maint: Multiple names: authors list (link)
 Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems (2nd ed.). Westview Press. ISBN 0813340853.
 Feldman, D. P. (2012). Chaos and Fractals: An Elementary Introduction. Oxford University Press. ISBN 9780199566440.
 Gollub, J. P.; Baker, G. L. (1996). Chaotic dynamics. Cambridge University Press. ISBN 0521476852.
 Guckenheimer, John; Holmes, Philip (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. SpringerVerlag. ISBN 0387908196.
 Gulick, Denny (1992). Encounters with Chaos. McGrawHill. ISBN 0070252033.
 Gutzwiller, Martin (1990). Chaos in Classical and Quantum Mechanics. SpringerVerlag. ISBN 0387971734.
 Hoover, William Graham (2001) [1999]. Time Reversibility, Computer Simulation, and Chaos. World Scientific. ISBN 9810240732.
 Kautz, Richard (2011). Chaos: The Science of Predictable Random Motion. Oxford University Press. ISBN 9780199594580.
 Kiel, L. Douglas; Elliott, Euel W. (1997). Chaos Theory in the Social Sciences. Perseus Publishing. ISBN 0472084720.
 Moon, Francis (1990). Chaotic and Fractal Dynamics. SpringerVerlag. ISBN 0471545716.
 Ott, Edward (2002). Chaos in Dynamical Systems. Cambridge University Press. ISBN 0521010845.
 Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 0738204536.
 Sprott, Julien Clinton (2003). Chaos and TimeSeries Analysis. Oxford University Press. ISBN 0198508409.
 Tél, Tamás; Gruiz, Márton (2006). Chaotic dynamics: An introduction based on classical mechanics. Cambridge University Press. ISBN 0521839122.
 Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 9780821883280.
 Thompson JM, Stewart HB (2001). Nonlinear Dynamics And Chaos. John Wiley and Sons Ltd. ISBN 0471876453.
 Tufillaro; Reilly (1992). An experimental approach to nonlinear dynamics and chaos. AddisonWesley. ISBN 0201554410.
 Wiggins, Stephen (2003). Introduction to Applied Dynamical Systems and Chaos. Springer. ISBN 0387001778.
 Zaslavsky, George M. (2005). Hamiltonian Chaos and Fractional Dynamics. Oxford University Press. ISBN 0198526040.
Semitechnical and popular works
 Christophe Letellier, Chaos in Nature, World Scientific Publishing Company, 2012, ISBN 9789814374422.
 Abraham, Ralph H.; Ueda, Yoshisuke, eds. (2000). The Chaos AvantGarde: Memoirs of the Early Days of Chaos Theory. World Scientific. ISBN 9789812386472.
 Barnsley, Michael F. (2000). Fractals Everywhere. Morgan Kaufmann. ISBN 9780120790692.
 Bird, Richard J. (2003). Chaos and Life: Complexit and Order in Evolution and Thought. Columbia University Press. ISBN 9780231126625.
 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.
 Cunningham, Lawrence A. (1994). "From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis". George Washington Law Review. 62: 546.
 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.
 John Gribbin. Deep Simplicity. Penguin Press Science. 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 SelfOrganisation; 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
 HeinzOtto 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.
 HeinzOtto 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.
 Ian Roulstone; John Norbury (2013). Invisible in the Storm: the role of mathematics in understanding weather. 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, 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
Wikimedia Commons has media related to Chaos theory. 
 Hazewinkel, Michiel, ed. (2001) [1994], "Chaos", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Nonlinear Dynamics Research Group with Animations in Flash
 The Chaos group at the University of Maryland
 The Chaos Hypertextbook. An introductory primer on chaos and fractals
 ChaosBook.org An advanced graduate textbook on chaos (no fractals)
 Society for Chaos Theory in Psychology & Life Sciences
 Nonlinear Dynamics Research Group at CSDC, Florence Italy
 Interactive live chaotic pendulum experiment, allows users to interact and sample data from a real working damped driven chaotic pendulum
 Nonlinear dynamics: how science comprehends chaos, talk presented by Sunny Auyang, 1998.
 Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens
 Gleick's Chaos (excerpt)
 Systems Analysis, Modelling and Prediction Group at the University of Oxford
 A page about the MackeyGlass equation
 High Anxieties — The Mathematics of Chaos (2008) BBC documentary directed by David Malone
 The chaos theory of evolution – article published in Newscientist featuring similarities of evolution and nonlinear systems including fractal nature of life and chaos.
 Jos Leys, Étienne Ghys et Aurélien Alvarez, Chaos, A Mathematical Adventure. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.
 "Chaos Theory", BBC Radio 4 discussion with Susan Greenfield, David Papineau & Neil Johnson (In Our Time, May 16, 2002)
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英文文献
 Antifungal activity of βcarbolines on Penicillium digitatum and Botrytis cinerea.
 Olmedo GM1, Cerioni L1, González MM2, Cabrerizo FM2, Rapisarda VA1, Volentini SI3.
 Food microbiology.Food Microbiol.2017 Apr;62:914. doi: 10.1016/j.fm.2016.09.011. Epub 2016 Sep 16.
 βcarbolines (βCs) are alkaloids widely distributed in nature that have demonstrated antimicrobial properties. Here, we tested in vitro six βCs against Penicillium digitatum and Botrytis cinerea, causal agents of postharvest diseases on fruit and vegetables. Full aromatic βCs (harmine, harmol,
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 Muscle injury, impaired muscle function and insulin resistance in Chromogranin Aknockout mice.
 Tang K1, Pasqua T1, Biswas A1, Mahata S2, Tang J1, Tang A1, Bandyopadhyay GK1, SinhaHikim AP3,4, Chi NW1,5, Webster NJ1,5, Corti A6, Mahata SK7,5.
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 Chromogranin A (CgA) is widely expressed in endocrine and neuroendocrine tissues as well as in the central nervous system. We observed CgA expression (mRNA and protein) in the gastrocnemius (GAS) muscle and found that performance of CgAdeficient ChgaKO mice in treadmill exercise was impaired. Supp
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 Zinc deficiency reduces fertility in C. elegans hermaphrodites and disrupts oogenesis and meiotic progression.
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 Zinc is necessary for successful gametogenesis in mammals; however the role of zinc in the gonad function of nonmammalian species has not been investigated. The genetic tractability, short generation time, and hermaphroditic reproduction of the nematode C. elegans offer distinct advantages for the
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 Familial aggregation of schizotypy in schizophreniaspectrum disorders and its relation to clinical and neurodevelopmental characteristics.
 Soler J1, Ferentinos P2, Prats C3, Miret S4, Giralt M5, Peralta V6, Fañanás L3, FatjóVilas M7.
 Journal of psychiatric research.J Psychiatr Res.2017 Jan;84:214220. doi: 10.1016/j.jpsychires.2016.09.026. Epub 2016 Sep 29.
 INTRODUCTION: This study explored schizotypy as a familial liability marker for schizophreniaspectrum disorders (SSD) by examining: 1) the aggregation of schizotypy in families with a SSD patient, 2) whether familial resemblance of schizotypy is associated with ridge dissociations (RD), another SSD
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関連リンク
 disorganized 【形】整理されていない、組織の乱れた、まとまりのない、散らかっ...  アルクがお届けする進化するオンライン英和・和英辞書データベース。一般的な単語や連語から、イディオム、専門用語、スラングまで幅広く収録。
 He wanted to sign off on every nonmilitary initiative, but he and his staff were too disorganized to deal with all the paperwork. ... In noman'sland these disorganized Germans ran into a British patrol, and again lost heavily, very few ...
 dis·or·gan·ize (dĭsôr′gənīz′) tr.v. dis·or·gan·ized, dis·or·gan·iz·ing, dis·or·gan·iz·es To destroy the organization, systematic arrangement, or unity of. dis·or′gan·i·za′tion (gənĭzā′shən) n. disorganized (dɪsˈɔːɡəˌnaɪzd) or disorganised
関連画像
■★リンクテーブル★
先読み  「confusion」「disruption」 
リンク元  「upset」「confuse」「perturbation」「confound」「disrupt」 
拡張検索  「disorganized schizophrenia」 
関連記事  「disorganize」 
「confusion」
confusionとdelirium
 Confusion, a mental and behavioral state of reduced comprehension, coherence, and capacity to reason, is one of the most common problems encountered in medicine, accounting for a large number of emergency department visits, hospital admissions, and inpatient consultations. Delirium, a term used to describe an acute confusional state, remains a major cause of morbidity and mortality, contributing billions of dollars yearly to health care costs in the United States alone. Delirium often goes unrecognized despite clear evidence that it is usually the cognitive manifestation of serious underlying medical or neurologic illness. (HIM.158)
WordNet ［license wordnet］
「disorder resulting from a failure to behave predictably; "the army retreated in confusion"」WordNet ［license wordnet］
「a mistake that results from taking one thing to be another; "he changed his name in order to avoid confusion with the notorious outlaw"」WordNet ［license wordnet］
「an act causing a disorderly combination of elements with identities lost and distinctions blended; "the confusion of tongues at the Tower of Babel"」WordNet ［license wordnet］
「a mental state characterized by a lack of clear and orderly thought and behavior; "a confusion of impressions"」 同
 mental confusion, confusedness, muddiness, disarray
PrepTutorEJDIC ［license prepejdic］
「『混乱』,乱雑(disorder) / (…と…との)混同《+『of』+『名』+『with』+『名』》 / 当惑,ろうばい」
「disruption」
 n.
 (国家などの)分裂、崩壊。破壊
 混乱、中断、途絶
PrepTutorEJDIC ［license prepejdic］
「混乱,中断,分裂,崩壊[状態]」
「upset」
 v.
 (過去過去分詞も同形)転覆させる、混乱させる
 n.
 関
 confound、confuse、confusion、derange、derangement、disarray、disorganized、disorient、disrupt、disruption、out of、perturbation、sickness
WordNet ［license wordnet］
「used of an unexpected defeat of a team favored to win; "the Bills'' upset victory over the Houston Oilers"」PrepTutorEJDIC ［license prepejdic］
「『…‘を'ひっくり返す』,転覆させる / 〈計画など〉‘を'『だめにする』,だいなしにする,狂わす / 〈人〉‘の'『心お乱す』,'を'ろうばいさせる / 〈体など〉‘の'調子お狂わせる / 《米》(競技などで)〈相手〉'を'思いがけなく負かす / ひっくり返った / (計画などが)だいなしの,狂った / 《捕語にのみ用いて》(精神的に)混乱した,ろうばいした / (体などの)調子がおかしくなった / 〈U〉〈C〉『転倒』,転落 / 〈C〉(計画.心などの)『混乱』,乱れ;(体の)不調 / 〈C〉《話》けんか / 〈C〉《米》(競技などでの)番狂わせ」WordNet ［license wordnet］
「the act of disturbing the mind or body; "his carelessness could have caused an ecological upset"; "she was unprepared for this sudden overthrow of their normal way of living"」WordNet ［license wordnet］
「the act of upsetting something; "he was badly bruised by the upset of his sled at a high speed"」WordNet ［license wordnet］
「a tool used to thicken or spread metal (the end of a bar or a rivet etc.) by forging or hammering or swaging」WordNet ［license wordnet］
「cause to lose one''s composure」WordNet ［license wordnet］
「defeat suddenly and unexpectedly; "The foreign team upset the local team"」WordNet ［license wordnet］
「disturb the balance or stability of; "The hostile talks upset the peaceful relations between the two countries"」WordNet ［license wordnet］
「mildly physically distressed; "an upset stomach"」
「confuse」
 v.
 混乱させる、錯乱する、混同する
 関
 confound、confusion、derange、derangement、disarray、disorganized、disorient、disrupt、disruption、perturbation、upset
WordNet ［license wordnet］
「be confusing or perplexing to; cause to be unable to think clearly; "These questions confuse even the experts"; "This question completely threw me"; "This question befuddled even the teacher"」PrepTutorEJDIC ［license prepejdic］
「(…と)'を'『混同する』,取り違える《+『名』+『with』+『名』》 / 〈人〉'を'『とまどわせる』,当惑させる / 〈論点・情況・順序など〉'を'混乱させる」WordNet ［license wordnet］
「make unclear, indistinct, or blurred; "Her remarks confused the debate"; "Their words obnubilate their intentions"」WordNet ［license wordnet］
「mistake one thing for another; "you are confusing me with the other candidate"; "I mistook her for the secretary"」WordNet ［license wordnet］
「cause to feel embarrassment; "The constant attention of the young man confused her"」 同
 flurry, disconcert, put off
「perturbation」
 n.
 心の動揺、狼狽、不安、心配。不安/心配の原因。(天・理)摂動
 関
 agitate, confound, confuse, confusion, deflection, derange, derangement, disarray, disorganized, disorient, disrupt, disruption, disturbance, perturb, perturbate, surge, swing, unbalance, upset
WordNet ［license wordnet］
「(physics) a secondary influence on a system that causes it to deviate slightly」WordNet ［license wordnet］
「activity that is a malfunction, intrusion, or interruption; "the term `distress'' connotes some degree of perturbation and emotional upset"; "he looked around for the source of the disturbance"; "there was a disturbance of neural function"」PrepTutorEJDIC ［license prepejdic］
「〈U〉動揺,不安;混乱 / 〈C〉不安(混乱)のもと」
「confound」
 ct.
 関
 confuse、confusion、derange、derangement、disarray、disorganized、disorient、disrupt、disruption、embarrass、embarrassment、perplexity、perturbation、upset
PrepTutorEJDIC ［license prepejdic］
「〈情況など〉'を'『混乱させる』,〈人〉'を'困惑させる,面くらわせる / (…と)…'を'『混同する』《+『名』+『with』+『名』》 / 〈計画・希望・敵など〉'を'破る,くじく(defeat) / 《遠回しに腹立たしさなどを表して》〈神〉が…'を'地獄に落とす」
「disrupt」
 v.
 破壊する、分裂させる、混乱させる、乱す
 関
 abolish、break、confound、confuse、confusion、derange、derangement、destroy、destruction、disarray、dismantle、disorganized、disorient、disruptant、disruption、divide、division、fission、fragmentation、lesion、perturb、perturbation、rupture、split、subversion、subvert、upset
WordNet ［license wordnet］
「throw into disorder; "This event disrupted the orderly process"」PrepTutorEJDIC ［license prepejdic］
「…‘を'混乱させる / (一時的に)…‘を'中断させる」
「disorganized schizophrenia」
「disorganize」