Lyapunov exponent

The Lyapunov exponent of a dynamical system (after Alexander Mikhailovich Lyapunov ) describes the rate at which two ( closely spaced ) points in the phase space move apart or closer ( depending on the sign ). Per dimension of the phase space, there is a Lyapunov exponent, which together form the so-called Lyapunov spectrum. Frequently, however, one considers only the largest Lyapunov exponent, as this determines the overall system behavior.

In the one-dimensional Lyapunov exponent of an iterated mapping is defined as follows:

Properties

  • If the largest Lyapunov exponent is positive, the system is usually divergent
  • If it is negative, this corresponds to a phase space contraction, ie the system is dissipative and acts stationary or periodically stable.
  • If the sum of the Lyapunov exponent is zero, it is a conservative system

Importance of the Lyapunov exponent

Kaplan- Yorke conjecture

The Kaplan- Yorke conjecture delivers an estimate for the upper limit of the information dimension with the help of the Lyapunov spectrum. This so-called Kaplan- Yorke dimension is defined as follows:

Wherein the largest natural number, for which the sum is a positive

Lyapunov time

The inverse of the largest Lyapunov exponent, the so-called Lyapunov time or the average prediction time is the time that can make for themselves meaningful predictions about the system behavior.

Swell

  • Kantz, H. and Schreiber, T.: Nonlinear Time Series Analysis. Cambridge University Press, Cambridge 2004, ISBN 0-521-52902-6
  • Theory of dynamical systems
  • Nonlinear Dynamics
Phase velocity Dissipation
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