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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

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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.

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* 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.