Chaos

A Dynamical System is chaotic if it

1. Has a Dense collection of points with periodic orbits,

2. Is sensitive to the initial condition of the system (so that initially nearby points can evolve quickly into very different states), and

3. Is Topologically Transitive.

Chaotic systems exhibit irregular, unpredictable behavior (the Butterfly Effect). The boundary between linear and chaotic behavior is characterized by Period Doubling, followed by quadrupling, etc.

An example of a simple physical system which displays chaotic behavior is the motion of a magnetic pendulum over a plane containing two or more attractive magnets. The magnet over which the pendulum ultimately comes to rest (due to frictional damping) is highly dependent on the starting position and velocity of the pendulum (Dickau). Another such system is a double pendulum (a pendulum with another pendulum attached to its end).

See also Accumulation Point, Attractor, Basin of Attraction, Butterfly Effect, Chaos Game, Feigenbaum Constant, Fractal Dimension, Gingerbreadman Map, Hénon-Heiles Equation, Hénon Map, Limit Cycle, Logistic Equation, Lyapunov Characteristic Exponent, Period Three Theorem, Phase Space, Quantum Chaos, Resonance Overlap Method, Sarkovskii's Theorem, Shadowing Theorem, Sink (Map), Strange Attractor

References

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Baker, G. L. and Gollub, J. B. Chaotic Dynamics: An Introduction, 2nd ed. Cambridge: Cambridge University Press, 1996.

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Lorenz, E. N. The Essence of Chaos. Seattle, WA: University of Washington Press, 1996.

Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993.

Ott, E.; Sauer, T.; and Yorke, J. A. Coping with Chaos: Analysis of Chaotic Data and the Exploitation of Chaotic Systems. New York: Wiley, 1994.

Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992.

Poon, L. ``Chaos at Maryland.'' http://www-chaos.umd.edu.

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Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.

Tufillaro, N.; Abbott, T. R.; and Reilly, J. An Experimental Approach to Nonlinear Dynamics and Chaos. Redwood City, CA: Addison-Wesley, 1992.

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Wiggins, S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, 1990.