Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps

TL;DR

This paper introduces a new class of one-dimensional unimodal dissipative maps, investigates their universality class through $q$-Gaussian distributions, and explores their entropy production and sensitivity indices, extending previous work on $z$-logistic maps.

Contribution

It defines the ($z_1,z_2$)-logarithmic map class and analyzes their statistical properties, establishing their relation to $q$-Gaussian attractors and entropy production.

Findings

The new maps exhibit $q$-Gaussian attractor distributions similar to $z$-logistic maps.

The $q$-entropy production aligns with a generalized Pesin-like identity.

Numerical results support the universality of $q$-statistics at the edge of chaos.

Abstract

We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the ($z_1,z_2$)-{\it logarithmic map}, corresponds to a generalization of the $z$-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the $z$-logistic map is numerically consistent with a $q$-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy $S_q$. We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to $q$-Gaussian attractor distributions. We also study the generalized $q$-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The $q$-sensitivity indices are obtained as well. Our results are, like those for the $z$-logistic maps, numerically compatible with the $q$-generalization of a Pesin-like identity for ensemble averages.