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chaos theory
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## 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 inner solar system, 4 to 5 million years.JOURNAL, Wisdom, Jack, Sussman, Gerald Jay, 1992-07-03, Chaotic Evolution of the Solar System, Science, en, 257, 5066, 56â€“62, 10.1126/science.257.5066.56, 1095-9203, 17800710, 1992Sci...257...56S, 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 189â€“190.

## Chaotic dynamics

File:Chaos Sensitive Dependence.svg|thumb|The map defined by x â†’ 4 x (1 â€“ x) and y â†’ (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".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 transitive,
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, 353â€“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

| deltamathbf{Z}(t) | approx e^{lambda t} | delta mathbf{Z}_0 |,
where t is the time and Î» 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 |volume=39 |url=https://books.google.com/books?id=0E667XpBq1UC |year=2000 |publisher=World Scientific |isbn=978-981-238-647-2 |ref=harv|bibcode=2000cagm.book.....A |last1=Abraham |first1=Ralph |display-authors=et al |doi=10.1142/4510 |series=World Scientific Series on Nonlinear Science Series A }}
• BOOK, Michael F. Barnsley, Michael F., Barnsley, Fractals Everywhere,weblink 2000, Morgan Kaufmann, 978-0-12-079069-2,
• BOOK, Richard J., Bird, Chaos and Life: Complexit and Order in Evolution and Thought,weblink 2003, Columbia University Press, 978-0-231-12662-5,
• 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.
• JOURNAL, Cunningham, Lawrence A., From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis, George Washington Law Review, 62, 546, 1994,
• Predrag CvitanoviÄ‡, 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.
• BOOK, 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 Self-Organisation; New Perspectives on Complexity in Nature , National Book Trust, 2003.
• Hans Lauwerier, Fractals, Princeton University Press, 1991.
• Edward Lorenz, The Essence of Chaos, University of Washington Press, 1996.
• Alan Marshall (2002) The Unity of Nature: Wholeness and Disintegration in Ecology and Science, Imperial College Press: London
• David Peak and Michael Frame, Chaos Under Control: The Art and Science of Complexity, Freeman, 1994.
• Heinz-Otto Peitgen and Dietmar Saupe (Eds.), The Science of Fractal Images, Springer 1988, 312 pp.
• Clifford A. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World , St Martins Pr 1991.
• Clifford A. Pickover, Chaos in Wonderland: Visual Adventures in a Fractal World, St Martins Pr 1994.
• Ilya Prigogine and Isabelle Stengers, Order Out of Chaos, Bantam 1984.
• Heinz-Otto 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.
• BOOK, Ian Roulstone, John Norbury, Invisible in the Storm: the role of mathematics in understanding weather,weblink 2013, Princeton University Press, 978-0691152721,
• David Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, 1989.
• Manfred Schroeder, Fractals, Chaos, and Power Laws, Freeman, 1991.
• 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.
• Antonio Sawaya, Financial Time Series Analysis : Chaos and Neurodynamics Approach, Lambert, 2012.

{{Commons category|Chaos theory}}
{{Systems}}{{Chaos theory}}{{Patterns in nature}}

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