Statistical descriptions of nonlinear systems at the onset of chaos

TL;DR

This paper explores the statistical behavior of nonlinear systems at the onset of chaos, extending entropy concepts to weakly chaotic regimes and testing various generalized entropies using the logistic map.

Contribution

It introduces a generalized statistical formalism that extends the entropy-sensitivity relationship to weakly chaotic systems, including Tsallis, Abe, and Kaniadakis entropies.

Findings

Entropy growth rate K equals the logarithm of sensitivity or weakly chaotic systems.

Finite-entropy growth constrains the suitable generalized entropies.

The parameter space where generalized descriptions are effective is characterized.

Abstract

Ensemble of initial conditions for nonlinear maps can be described in terms of entropy. This ensemble entropy shows an asymptotic linear growth with rate K. The rate K matches the logarithm of the corresponding asymptotic sensitivity to initial conditions \lambda. The statistical formalism and the equality K=\lambda can be extended to weakly chaotic systems by suitable and corresponding generalizations of the logarithm and of the entropy. Using the logistic map as a test case we consider a wide class of deformed statistical description which includes Tsallis, Abe and Kaniadakis proposals. The physical criterion of finite-entropy growth K strongly restricts the suitable entropies. We study how large is the region in parameter space where the generalized description is useful.