The rate of entropy increase at the edge of chaos

TL;DR

This paper extends the concept of entropy rate to the edge of chaos in the logistic map, showing that non-extensive entropy with a specific q value better describes the system's behavior at this critical point.

Contribution

It introduces a non-extensive entropy framework with a specific q value to characterize the edge of chaos in dissipative systems, generalizing the Kolmogorov-Sinai entropy concept.

Findings

The usual Boltzmann-Gibbs-Shannon entropy is inadequate at the edge of chaos.

A specific q value, q*, characterizes the entropy and sensitivity to initial conditions.

The same q* describes the multifractal spectrum of the attractor.

Abstract

Under certain conditions, the rate of increase of the statistical entropy of a simple, fully chaotic, conservative system is known to be given by a single number, characteristic of this system, the Kolmogorov-Sinai entropy rate. This connection is here generalized to a simple dissipative system, the logistic map, and especially to the chaos threshold of the latter, the edge of chaos. It is found that, in the edge-of-chaos case, the usual Boltzmann-Gibbs-Shannon entropy is not appropriate. Instead, the non-extensive entropy $S_q\equiv \frac{1-\sum_{i=1}^W p_i^q}{q-1}$, must be used. The latter contains a parameter q, the entropic index which must be given a special value $q^*\ne 1$ (for q=1 one recovers the usual entropy) characteristic of the edge-of-chaos under consideration. The same q^* enters also in the description of the sensitivity to initial conditions, as well as in that of the multifractal spectrum of the attractor.