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chaos theory
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{{other uses|Chaos theory (disambiguation)|Chaos (disambiguation)}}File:Lorenz attractor yb.svg|thumb|right|A plot of the r {{=}} 28}}, {{nowrap|σ {{=}} 10}}, {{nowrap|b {{=}} 8/3}}File:Double-compound-pendulum.gif|thumb|A double-rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectorytrajectoryChaos 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, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions. The butterfly effect describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, e.g. a butterfly flapping its wings in China can cause a hurricane in Texas.WEB,weblink Boeing, Chaos Theory and the Logistic Map, 2015, 2015-07-16, Small differences in initial conditions, such as those due to rounding errors in numerical computation, yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general.BOOK, Kellert, Stephen H., In the Wake of Chaos: Unpredictable Order in Dynamical Systems, University of Chicago Press, 1993, 978-0-226-42976-2, 32, harv, JOURNAL,weblink Boeing, G., Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction, Systems, 2016, 4, 4, 37, 2016-12-02, 10.3390/systems4040037, 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.{{harvnb|Kellert|1993|p=56}} In other words, the deterministic nature of these systems does not make them predictable.{{harvnb|Kellert|1993|p=62}}JOURNAL, Werndl, Charlotte, What are the New Implications of Chaos for Unpredictability?, The British Journal for the Philosophy of Science, 60, 1, 195–220, 2009,weblink 10.1093/bjps/axn053, 1310.1576, This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:WEB,weblink Chaos in an Atmosphere Hanging on a Wall, Danforth, Christopher M., April 2013, Mathematics of Planet Earth 2013, 12 June 2018, Chaotic behavior exists in many natural systems, such as weather and climate.BOOK, Ivancevic, Vladimir G., Complex nonlinearity: chaos, phase transitions, topology change, and path integrals, 2008, Springer, 978-3-540-79356-4, Tijana T. Ivancevic, It also occurs spontaneously in some systems with artificial components, such as road traffic.JOURNAL, Safonov, Leonid A., Tomer, Elad, Strygin, Vadim V., Ashkenazy, Yosef, Havlin, Shlomo, Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic, Chaos: an Interdisciplinary Journal of Nonlinear Science, 12, 4, 2002, 1006, 1054-1500, 10.1063/1.1507903, 2002Chaos..12.1006S, 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,BOOK, On the order of chaos. Social anthropology and the science of chaos, Mosko M.S., Damon F.H. (Eds.), Berghahn Books, 2005, Oxford, BOOK,weblink Quantum anthropology: Man, cultures, and groups in a quantum perspective, Trnka R., Lorencova R., Charles University Karolinum Press, 2016, Prague, sociology, physics,JOURNAL, Hubler, A, Adaptive control of chaotic systems, Swiss Physical Society. Helvetica Physica Acta 62, 1989, 339–342, 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 self-assembly processes.

Introduction

Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. 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.Sync: The Emerging Science of Spontaneous Order, Steven Strogatz, Hyperion, New York, 2003, pages 189–190.

Chaotic dynamics

File:Chaos Sensitive Dependence.svg|thumb|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. Note, however, that the y coordinate is defined modulo one at each step, so the square region is actually depicting a cylinder, and the two points are closer than they look.]]In common usage, "chaos" means "a state of disorder".Definition of at Wiktionary; 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:BOOK, A First Course in Dynamics: With a Panorama of Recent Developments, Hasselblatt, Boris, Anatole Katok, 2003, Cambridge University Press, 978-0-521-58750-1,
  1. it must be sensitive to initial conditions
  2. it must be topologically mixing
  3. 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.BOOK, Elaydi, Saber N., Discrete Chaos, Chapman & Hall/CRC, 1999, 978-1-58488-002-8, 117, BOOK, Basener, William F., Topology and its applications, Wiley, 2006, 978-0-471-68755-9, 42, 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.JOURNAL, Vellekoop, Michel, Berglund, Raoul, On Intervals, Transitivity = Chaos, The American Mathematical Monthly, 101, 4, 353–5, April 1994, 2975629, 10.2307/2975629, An alternative, and in general weaker, definition of chaos uses only the first two properties in the above list.BOOK, Medio, Alfredo, Lines, Marji, Nonlinear Dynamics: A Primer, Cambridge University Press, 2001, 978-0-521-55874-7, 165,

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.JOURNAL, March 2016, Introduction to Supersymmetric Theory of Stochastics, Entropy, 18, 4, 108, 10.3390/e18040108, Ovchinnikov, I.V., 1511.03393, 2016Entrp..18..108O, JOURNAL, 2016, Topological supersymmetry breaking: Definition and stochastic generalization of chaos and the limit of applicability of statistics, Modern Physics Letters B, 30, 8, 1650086, 10.1142/S021798491650086X, Ovchinnikov, I.V., Schwartz, R. N., Wang, K. L., 2016MPLB...3050086O, 1404.4076, This picture of dynamical chaos works not only for deterministic models but also for models with external noise, which is an important generalization from the physical point of view because in reality all dynamical systems experience influence from their stochastic environments. Within this picture, the long-range dynamical behavior associated with chaotic dynamics, e.g., the butterfly effect, is a consequence of the Goldstone's theorem in the application to the spontaneous topological supersymmetry breaking.

Sensitivity to initial conditions

(File:SensInitCond.gif|thumb|Lorenz equations used to generate plots for the y variable. The initial conditions for x and z were kept the same but those for y were changed between 1.001, 1.0001 and 1.00001. The values for rho, sigma and beta were 45.92, 16 and 4 respectively. As can be seen, even the slightest difference in initial values causes significant changes after about 12 seconds of evolution in the three cases. This is an example of sensitive dependence on initial conditions.)Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated by other points with significantly different future paths, or trajectories. Thus, an arbitrarily small change, or perturbation, of the current trajectory may lead to significantly different future behavior.Sensitivity to initial conditions is popularly known as the "butterfly effect", so-called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C., entitled Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events that prevents the predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system would 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.BOOK, Watts, Robert G., Global Warming and the Future of the Earth, Morgan & Claypool, 2007, 17, 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 Â°C or fall to −130 Â°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
| deltamathbf{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 measure-theoretical mixing (as discussed in ergodic theory) and properties of a K-system.

Topological mixing

File:LogisticTopMixing1-6.gif|thumb|Six iterations of a set of states [x,y] passed through the logistic map. (a) the blue plot (legend 1) shows the first iterate (initial condition), which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that mixing occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation x_{k+1} = 4 x_k (1 - x_k ). To expand the state-space of the logistic map into two dimensions, a second state, y, was created as y_{k+1} = x_k + y_k , if x_k + y_k

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