The non-extensive version of the Kolmogorov-Sinai entropy at work

TL;DR

This paper extends the Kolmogorov-Sinai entropy to a non-extensive form to analyze the chaotic threshold of the logistic map, revealing a linear entropy evolution with a specific non-extensive index.

Contribution

It introduces a non-extensive entropy framework to better understand the dynamics at the chaos threshold of the logistic map, addressing multifractal complexities.

Findings

Entropy evolution is linear over time at the chaos threshold.

Non-extensive index Q < 1 aligns with heuristic predictions.

Method effectively captures multifractal dynamics.

Abstract

We address the problem of applying the Kolmogorov-Sinai method of entropic analysis, expressed in a generalized non-extensive form, to the dynamics of the logistic map at the chaotic threshold, which is known to be characterized by a power law rather than exponential sensitivity to initial conditions. The computer treatment is made difficult, if not impossible, by the multifractal nature of the natural invariant distribution: Thus the statistical average is carried out on the power index. The resulting entropy time evolution becomes a smooth and linear function of time with the non-extensive index Q < 1 prescribed by the heuristic arguments of earlier work, thereby showing how to make the correct entropic prediction in the spirit of the single-trajectory approach of Kolmogorov.