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Dynamical entropy for systems with stochastic perturbation

Ostruszka1, Pakonski, Slomczynski

  • 1Instytut Fizyki im. Mariana Smoluchowskiego, Uniwersytet Jagiellonski, Reymonta 4, 30-059 Krakow, Poland.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|November 23, 2000
PubMed
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We introduce dynamical entropy for noisy systems, a finite measure quantifying system dynamics. This new metric, derived from total and noise entropy, is non-negative and finite, offering a robust way to study complex systems.

Area of Science:

  • * Dynamical systems theory
  • * Nonlinear dynamics
  • * Statistical mechanics

Background:

  • * Standard characterization of deterministic systems using Kolmogorov-Sinai (KS) entropy faces challenges with additive noise, as KS entropy diverges under certain conditions.
  • * The divergence of KS entropy when partition diameter approaches zero necessitates a new approach for noisy systems.

Purpose of the Study:

  • * To quantitatively characterize the dynamics of deterministic systems perturbed by random additive noise.
  • * To define and analyze a new metric, 'dynamical entropy', for noisy systems.
  • * To explore the relationship between this new metric and the KS entropy in the weak noise limit.

Main Methods:

  • * Analysis of the difference between the total entropy of a noisy system and the entropy of the noise itself.

Related Experiment Videos

  • * Mathematical formulation of dynamical entropy as a finite, non-negative quantity.
  • * Consideration of one-dimensional systems with noise described by a finite-dimensional kernel, enabling finite matrix representation of the Frobenius-Perron operator.

    • Main Results:

    • * A finite and non-negative quantity, termed 'dynamical entropy', is defined for noisy systems.
    • * This dynamical entropy quantifies the complexity of system dynamics under noise perturbation.
    • * For one-dimensional systems with specific noise characteristics, the Frobenius-Perron operator is shown to be representable by a finite matrix.

    Conclusions:

    • * Dynamical entropy provides a well-defined and finite measure for the dynamics of noisy systems, overcoming limitations of traditional KS entropy.
    • * The proposed metric is robust and applicable to systems with additive noise.
    • * A conjecture is presented that in the weak noise limit, dynamical entropy converges to the KS entropy of the corresponding deterministic system.