Sensitivity function and entropy increase rates for z-logistic map family at the edge of chaos

TL;DR

This paper investigates the behavior of generalized entropies and sensitivity functions at the edge of chaos in the z-logistic map family, revealing consistent entropy production and sensitivity patterns at critical points.

Contribution

It introduces a numerical analysis of generalized nonextensive entropies and their exponential functions at the edge of chaos in the z-logistic map family, extending understanding of entropy and sensitivity behavior.

Findings

Entropy production rates are consistent at the edge of chaos.

Sensitivity functions exhibit specific scaling behaviors.

Results are validated at accumulation points of different cycles.

Abstract

It is well known that, for chaotic systems, the production of relevant entropy (Boltzmann-Gibbs) is always linear and the system has strong (exponential) sensitivity to initial conditions. In recent years, various numerical results indicate that basically the same type of behavior emerges at the edge of chaos if a specific generalization of the entropy and the exponential are used. In this work, we contribute to this scenario by numerically analysing some generalized nonextensive entropies and their related exponential definitions using $z$-logistic map family. We also corroborate our findings by testing them at accumulation points of different cycles.