Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including meteorology, sociology, physics, engineering, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditionsa response 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. The theory was summarized 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 and climate.^{[6]}^{[7]} This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincar maps.
Contents
Introduction[edit]
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. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: How much uncertainty we are willing to tolerate in the forecast; how accurately we are able to measure its current state; and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, ~1 millisecond; weather systems, a couple of days (unproven); the solar system, 50 million years. In chaotic systems the uncertainty in a forecast increases exponentially with elapsed time. Hence doubling the forecast time more than squares the proportional uncertainty in the forecast. This means that 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 to be random.^{[8]}
Chaotic dynamics[edit]
In common usage, "chaos" means "a state of disorder".^{[9]} 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:^{[10]}
 it must be sensitive to initial conditions;
 it must be topologically mixing; and
 it must have dense periodic orbits.
Sensitivity to initial conditions[edit]
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.
It has been shown that in some cases the last two properties in the above actually imply sensitivity to initial conditions,^{[11]}^{[12]} and if attention is restricted to intervals, the second property implies the other two^{[13]} (an alternative, and in general weaker, definition of chaos uses only the first two properties in the above list).^{[14]} It is interesting that the most practically significant property, that of sensitivity to initial conditions, is redundant in the definition, being implied by two (or for intervals, one) purely topological properties, 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 Butterflys Wings in Brazil set off a Tornado in Texas?.^{[15]} 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.^{[16]} Of course this does not mean that we cannot say anything about events far in the future; there are some restrictions on the system. With weather, we know that the temperature will never reach 100 degrees Celsius or fall to 130 degrees Celsius on earth, but we are not able to say exactly what day we 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 end up diverging at a rate given by
where t is the time and is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so there is a whole spectrum of Lyapunov exponents. 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.
There are also other properties that relate to sensitivity of initial conditions, such as measuretheoretical mixing (as discussed in ergodic theory) and properties of a Ksystem.^{[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 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 will eventually become widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 will tend to positive or negative infinity.
Density of periodic orbits[edit]
For a chaotic system to have a dense periodic orbit means that every point in the space is approached arbitrarily closely by periodic orbits.^{[17]} 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).^{[18]}
Sharkovskii's theorem is the basis of the Li and Yorke^{[19]} (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[edit]
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 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, 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 Hnon 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[edit]
Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. In contrast, for continuous dynamical systems, the PoincarBendixson 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 has to be either nonlinear or infinitedimensional.
The PoincarBendixson theorem states that a twodimensional differential equation has very regular behavior. The Lorenz attractor discussed above is generated by a system of three differential equations such as:
where , , and make up the system state, is time, and , , 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 Rossler equations which have only one nonlinear term out of seven. Sprott ^{[20]} found a threedimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel ^{[21]}^{[22]} 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 PoincarBendixson theorem shows that a continuous dynamical system on the Euclidean plane cannot be chaotic, twodimensional continuous systems with nonEuclidean geometry can exhibit chaotic behavior.^{[23]} Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional.^{[24]} A theory of linear chaos is being developed in a branch of mathematical analysis known as functional analysis.
Jerk systems[edit]
In physics, jerk is the third derivative of position, and as such, in mathematics differential equations of the form
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 equation 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.^{[25]}
A jerk system is a system whose behavior is described by a jerk equation, and for certain jerk equations simple electronic circuits may be designed which model the solutions to this equation. 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 Rssler map, are conventionally described as a system of three firstorder differential equations, but which may be combined into a single (although rather complicated) jerk equation. It has been shown, that nonlinear jerk systems are in a sense minimally complex systems to show chaotic behaviour, there is no chaotic system involving only two first order, ordinary differential equations (the system resulting in an equation of second order only).
An example of a jerk equation with nonlinearity in the magnitude of , is:
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 , and all capacitors are of equal size. The dominant frequency will be . The output of op amp 0 will correspond to the x variable, the output of 1 will correspond to the first derivative of x and the output of 2 will correspond to the second derivative.
Spontaneous order[edit]
Under the right conditions chaos will spontaneously evolve 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 Millenium Bridge resonance, and large arrays of Josephson junctions.^{[26]}
History[edit]
An early proponent of chaos theory was Henri Poincar. 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.^{[27]}^{[28]} 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".^{[29]} 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 got its start in the field of ergodic theory. Later studies, also on the topic of nonlinear differential equations, were carried out by George David Birkhoff,^{[30]} Andrey Nikolaevich Kolmogorov,^{[31]}^{[32]}^{[33]} Mary Lucy Cartwright and John Edensor Littlewood,^{[34]} and Stephen Smale.^{[35]} 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 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.^{[36]}^{[37]}
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.^{[38]} 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.^{[39]} 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.^{[40]} 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).^{[41]}^{[42]} 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.^{[43]} 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 (examples include the Menger sponge, the Sierpiski 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. Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model.^{[44]}
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.^{[45]}^{[46]} 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 RayleighBnard convection systems. He was awarded the Wolf Prize in Physics in 1986 along with Mitchell J. Feigenbaum for their inspiring achievements.^{[47]}
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.^{[48]} 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^{[49]} 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 BakTangWiesenfeld 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 GutenbergRichter law describing the statistical distribution of earthquake sizes, and the Omori law^{[50]} 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.^{[51]} 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 continues to be a very active area of research,^{[52]} involving many different disciplines (mathematics, topology, physics, social systems, population modeling, 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 a 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 (pseudo)randomness.^{[53]}^{[54]}
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.^{[53]}^{[55]} 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.^{[56]}
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.^{[57]} 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 the dimension can be increased to obtain a deterministically looking error, then analysis is done. Though it may sound simple, 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.^{[58]} 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.^{[59]}
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.^{[60]}
Applications[edit]
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, politics, population dynamics,^{[71]} psychology, and robotics. A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing every day.
Computer science[edit]
Chaos theory is not new to computer science and has been used for many years in cryptography. One type of encryption, secret key or symmetric key, relies on diffusion and confusion, which is modeled well by chaos theory.^{[72]} Another type of computing, DNA computing, when paired with chaos theory, offers a more efficient way to encrypt images and other information.^{[73]} 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.^{[74]} Chaotic dynamics have been exhibited by passive walking biped robots.^{[75]}
Biology[edit]
For over a hundred years, biologists have been keeping track of populations of different species with population models. Most models are deterministic systems, but recently scientists have been able to implement chaotic models in certain populations.^{[76]} For example, a study on models of Canadian lynx showed there was chaotic behavior in the population growth.^{[77]} 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 be learned from looking at the data through the lens of chaos theory.^{[78]} 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.^{[79]}
Other areas[edit]
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.^{[80]} In celestial mechanics, especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will come in range of Earth and other planets.^{[81]} In quantum physics and electrical engineering, the study of large arrays of Josephson junctions benefitted greatly from chaos theory.^{[82]} 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.^{[83]}
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.^{[84]} Modern organizations are increasingly seen as open complex adaptive systems, with fundamental natural nonlinear structures, subject to internal and external forces which may be sources of chaos. The chaos metaphorused in verbal theoriesgrounded 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.^{[85]}
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.^{[86]} Economic and financial systems are fundamentally different from those in the physical and 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.^{[87]}
Traffic forecasting is another area that greatly benefits from applications of chaos theory. Better predictions of when traffic will occur would allow measures to be taken for it to be dispersed before the traffic starts, rather than after. 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).^{[88]}
Chaos theory also finds applications 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.^{[89]}
See also[edit]
References[edit]
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 ^ Kellert 1993, p. 56
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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: 6171.
 Li, T.Y.; Yorke, J.A. (1975). "Period Three Implies Chaos". American Mathematical Monthly 82 (10): 98592. 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): 3849 (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): 124257. doi:10.1007/s1125300500554.
 Strelioff, C.; Hbler, 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.
 Hbler, A.; Foster, G.; Phelps, K. (2007). "Managing Chaos: Thinking out of the Box" (PDF). Complexity 12 (3): 1013. 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, 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.
 Tl, Tams; Gruiz, Mrton (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 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.
 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[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 Cvitanovi, 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. 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[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, tienne Ghys et Aurlien Alvarez, Chaos, A Mathematical Adventure. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.
