An animation of a double-rod pendulum at an intermediate energy showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a vastly different trajectory. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions.
Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:
Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
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 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 inner solar system, 4 to 5 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.
Chaos theory is a method of qualitative and quantitative analysis to investigate the behavior of dynamic systems that cannot be explained and predicted by single data relationships, but must be explained and predicted by whole, continuous data relationships.
The map defined by x -> 4 x (1 - x) and y -> (x + y)mod 1 displays sensitivity to initial x positions. Here, two series of x and y values diverge markedly over time from a tiny initial difference.
In common usage, "chaos" means "a state of disorder". 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:
In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions. In the discrete-time case, this is true for all continuous maps on metric spaces. 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. An alternative and a generally weaker definition of chaos uses only the first two properties in the above list.
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. 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, since in reality, all dynamical systems experience influence from their stochastic environments. Within this picture, the long-range dynamical behavior associated with chaotic dynamics (e.g., the butterfly effect) is a consequence of Goldstone's theorem--in the application to the spontaneous topological supersymmetry breaking.
Sensitivity to initial conditions
Lorenz equations used to generate plots for the y variable. The initial conditions for x and z were kept the same but those for y were changed between 1.001, 1.0001 and 1.00001. The values for , and were 45.92, 16 and 4 respectively. As can be seen from the graph, even the slightest difference in initial values causes significant changes after about 12 seconds of evolution in the three cases. This is an example of sensitive dependence on initial conditions.
Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated by other points that have significantly different future paths or trajectories. Thus, an arbitrarily small change or perturbation of the current trajectory may lead to significantly different future behavior.
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 that prevents the predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system could have been vastly different.
A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system (as is usually the case in practice), then beyond a certain time, the system would no longer be predictable. This is most prevalent in the case of weather, which is generally predictable only about a week ahead. This does not mean that one cannot assert anything about events far in the future--only that some restrictions on the system are present. For example, we do know with weather that the temperature will not naturally reach 100 °C or fall to -130 °C on earth (during the current geologic era), but that does not mean that we can predict exactly which day will have the hottest temperature of the year.
In more mathematical terms, the Lyapunov exponent measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions. More specifically, given two starting trajectories in the phase space that are infinitesimally close, with initial separation , the two trajectories end up diverging at a rate given by
where is the time and is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can exist. The number of Lyapunov exponents is equal to the number of dimensions of the phase space, though it is common to just refer to the largest one. For example, the maximal Lyapunov exponent (MLE) is most often used, because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.
In addition to the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, measure-theoreticalmixing (as discussed in ergodic theory) and properties of a K-system.
A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the sequence, however close, it will not enter the sequence and in fact, will diverge from it. Thus for almost all initial conditions, the variable evolves chaotically with non-periodic behavior.
Six iterations of a set of states passed through the logistic map. The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that mixing occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation . To expand the state-space of the logistic map into two dimensions, a second state, , was created as , if and otherwise.
The map defined by x -> 4 x (1 - x) and y -> (x + y)mod 1 also displays topological mixing. Here, the blue region is transformed by the dynamics first to the purple region, then to the pink and red regions, and eventually to a cloud of vertical lines scattered across the space.
Topological mixing (or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or open set of its phase space eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored 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.
A map is said to be topologically transitive if for any pair of non-empty open sets, there exists such that . Topological transitivity is a weaker version of topological mixing. Intuitively, if a map is topologically transitive then given a point x and a region V, there exists a point y near x whose orbit passes through V. This implies that is impossible to decompose the system into two open sets.
An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies in topological transitivity. The Birkhoff Transitivity Theorem states that if X is a second countable, complete metric space, then topological transitivity implies the existence of a dense set of points in X that have dense orbits.
Density of periodic orbits
For a chaotic system to have denseperiodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. The one-dimensional 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).
Sharkovskii's theorem is the basis of the Li and Yorke (1975) proof that any continuous one-dimensional system that exhibits a regular cycle of period three will also display regular cycles of every other length, as well as completely chaotic orbits.
The Lorenz attractor displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.
Some dynamical systems, like the one-dimensional logistic map defined by 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.
An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the Lorenz weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it is not only one of the first, but it is also one of the most complex, and as such gives rise to a very interesting pattern that, with a little imagination, looks like the wings of a butterfly.
Unlike fixed-point 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.
Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. Universality of one-dimensional maps with parabolic maxima and Feigenbaum constants, is well visible with map proposed as a toy
model for discrete laser dynamics:
where stands for electric field amplitude,  is laser gain as bifurcation parameter. The gradual increase of at interval changes dynamics from regular to chaotic one with qualitatively the same bifurcation diagram as those for logistic map.
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 well-known chaotic attractor is generated by the Rössler equations, which have only one nonlinear term out of seven. Sprott found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two-dimensional surface and therefore solutions are well behaved.
While the Poincaré-Bendixson theorem shows that a continuous dynamical system on the Euclidean plane cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behavior.[self-published source?] Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional. A theory of linear chaos is being developed in a branch of mathematical analysis known as functional analysis.
Infinite dimensional maps
The straightforward generalization of coupled discrete maps is based upon convolution integral which mediates interaction between spatially distributed maps:
where kernel is propagator derived as Green function of a relevant physical system, might be logistic map alike or complex map. For examples of complex maps the Julia set or Ikeda map may serve. When wave propagation problems at distance with wavelength are considered the kernel may have a form of Green function for Schrödinger equation:.
In physics, jerk is the third derivative of position, with respect to time. As such, 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, non-linear differential equations, is in a certain sense the minimal setting for solutions showing chaotic behaviour. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are called accordingly hyperjerk systems.
A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits can model solutions. These circuits are known as jerk circuits.
One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. In fact, certain well-known chaotic systems, such as the Lorenz attractor and the Rössler map, are conventionally described as a system of three first-order differential equations that can combine into a single (although rather complicated) jerk equation. Another 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 is . 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.
Similar circuits only require one diode or no diodes at all.
See also the well-known Chua's circuit, one basis for chaotic true random number generators. The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system.
An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the three-body problem, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point. 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". 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.
Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. What had been attributed to measure imprecision and simple "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.
Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a simple digital computer, a Royal McBeeLGP-30, to run his weather simulation. He wanted to see a sequence of data again, and to save time he started the simulation in the middle of its course. He did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To his surprise, the weather the machine began to predict was completely different from the previous calculation. Lorenz tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome. Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modelling cannot, in general, make precise long-term weather predictions.
In 1963, Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices. Beforehand he had studied information theory and concluded noise was patterned like a Cantor set: on any scale the proportion of noise-containing periods to error-free periods was a constant - thus errors were inevitable and must be planned for by incorporating redundancy. Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards). This challenged the idea that changes in price were normally distributed. In 1967, he published "How long is the coast of Britain? Statistical self-similarity and fractional dimension", showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an infinitesimally small measuring device. Arguing that a ball of twine appears as a point when viewed from far away (0-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand (1-dimensional), he argued that the dimensions of an object are relative to the observer and may be fractional. An object whose irregularity is constant over different scales ("self-similarity") is a fractal (examples include the Menger sponge, the Sierpi?ski 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.
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 Pierre Coullet and Charles Tresser published "Itérations d'endomorphismes et groupe de renormalisation", and Mitchell Feigenbaum's article "Quantitative Universality for a Class of Nonlinear Transformations" finally appeared in a journal, after 3 years of referee rejections. Thus Feigenbaum (1975) and Coullet & Tresser (1978) discovered the universality in chaos, permitting the application of chaos theory to many different phenomena.
Alongside largely lab-based approaches such as the Bak-Tang-Wiesenfeld sandpile, many other investigations have focused on large-scale natural or social systems that are known (or suspected) to display scale-invariant behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including earthquakes, (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the Gutenberg-Richter law describing the statistical distribution of earthquake sizes, and the Omori law describing the frequency of aftershocks), solar flares, fluctuations in economic systems such as financial markets (references to SOC are common in econophysics), landscape formation, forest fires, landslides, epidemics, and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould). Given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of wars. These investigations of SOC have included both attempts at modelling (either developing new models or adapting existing ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or characteristics of natural scaling laws.
In the same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public, though his history under-emphasized important Soviet contributions. 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.
Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trial-and-error type of refinement to interact with their environment, chaos theory has been used to build a predictive model.
Chaotic dynamics have been exhibited by passive walking biped robots.
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. For example, a study on models of Canadian lynx showed there was chaotic behavior in the population growth. 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. 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.
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. Economic and financial systems are fundamentally different from those in the classical natural sciences since the former are inherently stochastic in nature, as they result from the interactions of people, and thus pure deterministic models are unlikely to provide accurate representations of the data. The empirical literature that tests for chaos in economics and finance presents very mixed results, in part due to confusion between specific tests for chaos and more general tests for non-linear relationships.
Chaos could be found in economics by the means of recurrence quantification analysis. In fact, Orlando et al. by the means of the so-called recurrence quantification correlation index were able detect hidden changes in time series. Then, the same technique was employed to detect transitions from laminar (i.e. regular) to turbulent (i.e. chaotic) phases as well as differences between macroeconomic variables and highlight hidden features of economic dynamics. Finally, chaos could help in modeling how economy operate as well as in embedding shocks due to external events such as COVID-19.
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. In celestial mechanics, especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will approach Earth and other planets. 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. 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.
Chaos theory can be applied outside of the natural sciences, but historically nearly all such studies have suffered from lack of reproducibility; poor external validity; and/or inattention to cross-validation, resulting in poor predictive accuracy (if out-of-sample prediction has even been attempted). Glass and Mandell and Selz have found that no EEG study has as yet indicated the presence of strange attractors or other signs of chaotic behavior.
Researchers have continued to apply chaos theory to 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, researchers have found that the group dynamic 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.
Redington and Reidbord (1992) attempted to demonstrate that the human heart could display chaotic traits. They monitored the changes in between-heartbeat intervals for a single psychotherapy patient as she moved through periods of varying emotional intensity during a therapy session. Results were admittedly inconclusive. Not only were there ambiguities in the various plots the authors produced to purportedly show evidence of chaotic dynamics (spectral analysis, phase trajectory, and autocorrelation plots), but also when they attempted to compute a Lyapunov exponent as more definitive confirmation of chaotic behavior, the authors found they could not reliably do so.
In their 1995 paper, Metcalf and Allen maintained that they uncovered in animal behavior a pattern of period doubling leading to chaos. The authors examined a well-known response called schedule-induced polydipsia, by which an animal deprived of food for certain lengths of time will drink unusual amounts of water when the food is at last presented. The control parameter (r) operating here was the length of the interval between feedings, once resumed. The authors were careful to test a large number of animals and to include many replications, and they designed their experiment so as to rule out the likelihood that changes in response patterns were caused by different starting places for r.
Time series and first delay plots provide the best support for the claims made, showing a fairly clear march from periodicity to irregularity as the feeding times were increased. The various phase trajectory plots and spectral analyses, on the other hand, do not match up well enough with the other graphs or with the overall theory to lead inexorably to a chaotic diagnosis. For example, the phase trajectories do not show a definite progression towards greater and greater complexity (and away from periodicity); the process seems quite muddied. Also, where Metcalf and Allen saw periods of two and six in their spectral plots, there is room for alternative interpretations. All of this ambiguity necessitate some serpentine, post-hoc explanation to show that results fit a chaotic model.
By adapting a model of career counseling to include a chaotic interpretation of the relationship between employees and the job market, Amundson and Bright found that better suggestions can be made to people struggling with career decisions. 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.
Some say 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.
Traffic forecasting may benefit from applications of chaos theory. Better predictions of when traffic will occur would allow measures to be taken to disperse it before it would have occurred. Combining chaos theory principles with a few other methods has led to a more accurate short-term prediction model (see the plot of the BML traffic model at right).
Chaos theory has been applied to environmental water cycle data (aka hydrological data), such as rainfall and streamflow. These studies have yielded controversial results, because the methods for detecting a chaotic signature are often relatively subjective. Early studies tended to "succeed" in finding chaos, whereas subsequent studies and meta-analyses called those studies into question and provided explanations for why these datasets are not likely to have low-dimension chaotic dynamics.
^Bonet, J.; Martínez-Giménez, F.; Peris, A. (2001). "A Banach space which admits no chaotic operator". Bulletin of the London Mathematical Society. 33 (2): 196-8. doi:10.1112/blms/33.2.196.
^Adachihara, H; McLaughlin, D W; Moloney, J V; Newell, A C (1988). "Solitary waves as fixed points of infinite-dimensional maps for an optical bistable ring cavity: Analysis". Journal of Mathematical Physics. 29 (1): 63. Bibcode:1988JMP....29...63A. doi:10.1063/1.528136.
^Okulov, A Yu; Oraevski?, A N (1988). "Spatiotemporal dynamics of a wave packet in nonlinear medium and discrete maps". In N.G. Basov (ed.). Proceedings of the Lebedev Physics Institute (in Russian). 187. Nauka. pp. 202-222. LCCN88174540.
^Poincaré, J. Henri (2017). The three-body problem and the equations of dynamics : Poincaré's foundational work on dynamical systems theory. Popp, Bruce D. (Translator). Cham, Switzerland: Springer International Publishing. ISBN9783319528984. OCLC987302273.
^Kolmogorov, A. N. (1941). "On degeneration of isotropic turbulence in an incompressible viscous liquid". Doklady Akademii Nauk SSSR. 31 (6): 538-540. Reprinted in: Kolmogorov, A. N. (1991). "Dissipation of Energy in the Locally Isotropic Turbulence". Proceedings of the Royal Society A. 434 (1890): 15-17. Bibcode:1991RSPSA.434...15K. doi:10.1098/rspa.1991.0076. S2CID122060992.
^Cartwright, Mary L.; Littlewood, John E. (1945). "On non-linear differential equations of the second order, I: The equation y" + k(1-y2)y' + y = b?kcos(?t + a), k large". Journal of the London Mathematical Society. 20 (3): 180-9. doi:10.1112/jlms/s1-20.3.180. See also: Van der Pol oscillator
^Bak, Per; Tang, Chao; Wiesenfeld, Kurt (27 July 1987). "Self-organized 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. PMID10035754. However, the conclusions of this article have been subject to dispute. "?". Archived from the original on 2007-12-14.. See especially: Laurson, Lasse; Alava, Mikko J.; Zapperi, Stefano (15 September 2005). "Letter: Power spectra of self-organized 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.
^Gleick, James (August 26, 2008). Chaos: Making a New Science. Penguin Books. ISBN978-0143113454.
^Kyrtsou, C.; M. Terraza (2003). "Is it possible to study chaotic and ARCH behaviour jointly? Application of a noisy Mackey-Glass equation with heteroskedastic errors to the Paris Stock Exchange returns series". Computational Economics. 21 (3): 257-276. doi:10.1023/A:1023939610962. S2CID154202123.
^Williams, Bill Williams, Justine (2004). Trading chaos : maximize profits with proven technical techniques (2nd ed.). New York: Wiley. ISBN9780471463085.
^Peters, Edgar E. (1994). Fractal market analysis : applying chaos theory to investment and economics (2. print. ed.). New York u.a.: Wiley. ISBN978-0471585244.
^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. ISBN978-0471139386.
^Progonati, E (2018). "Brexit in the Light of Chaos Theory and Some Assumptions About the Future of the European Union". Chaos, complexity and leadership 2018 explorations of chaotic and complexity theory. Springer. ISBN978-3-030-27672-0.
^ abAkhavan, A.; Samsudin, A.; Akhshani, A. (2011-10-01). "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.
^Eduardo, Liz; Ruiz-Herrera, 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:10.1016/0167-9473(95)00056-9.
^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/s0301-2115(96)02628-0. PMID9138960.
^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. arXiv:1005.5384. doi:10.1016/j.procs.2010.12.161.
^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 self-adaptive particle swarm optimization algorithm and chaos theory". Fluid Phase Equilibria. 356: 11-17. doi:10.1016/j.fluid.2013.07.017.
^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.
^Glass, L (1997). "Dynamical disease: The impact of nonlinear dynamics and chaos on cardiology and medicine". In Grebogi, C; Yorke, J. A. (eds.). The impact of chaos on science and society. United Nations University Press.
^Mandell, A. J.; Selz, K. A. (1997). "Is the EEG a strange attractor?". In Grebogi, C; Yorke, J. A. (eds.). The impact of chaos on science and society. United Nations University Press.
^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. ISSN1090-0578.
^Metcalf, B. R.; Allen, J. D. (1995). "In search of chaos in schedule-induced polydipsia". In Abraham, F. D.; Gilgen, A. R. (eds.). Chaos theory in psychology. Greenwood Press.
^Pryor, Robert G. L.; Norman E. Amundson; Jim E. H. Bright (June 2008). "Probabilities and Possibilities: The Strategic Counseling Implications of the Chaos Theory of Careers". The Career Development Quarterly. 56 (4): 309-318. doi:10.1002/j.2161-0045.2008.tb00096.x.
^Thompson, Jamie; Johnstone, James; Banks, Curt (2018). "An examination of initiation rituals in a UK sporting institution and the impact on group development". European Sport Management Quarterly. 18 (5): 544-562. doi:10.1080/16184742.2018.1439984. S2CID149352680.
^Dal Forno, Arianna; Merlone, Ugo (2013). "Chaotic Dynamics in Organization Theory". In Bischi, Gian Italo; Chiarella, Carl; Shusko, Irina (eds.). Global Analysis of Dynamic Models in Economics and Finance. Springer-Verlag. pp. 185-204. ISBN978-3-642-29503-4.
^Wang, Jin; Qixin Shi (February 2013). "Short-term traffic speed forecasting hybrid model based on Chaos-Wavelet Analysis-Support Vector Machine theory". Transportation Research Part C: Emerging Technologies. 27: 219-232. doi:10.1016/j.trc.2012.08.004.