A closer look at coupled logistic maps at the edge of chaos

TL;DR

This paper investigates the probability distributions of coupled logistic maps at the edge of chaos with noise, showing they often fit $q$-Gaussians and relate to nonextensive statistical mechanics, with detailed numerical analysis of parameter effects.

Contribution

It provides a detailed numerical study of coupled logistic maps at the edge of chaos, analyzing their probability distributions and their relation to $q$-Gaussians and nonextensive statistics.

Findings

Distributions exhibit slight asymmetry for some parameters.

$q$-Gaussians fit well over wide parameter ranges.

The $q$ index varies with system parameters.

Abstract

We focus on a linear chain of $N$ first-neighbor-coupled logistic maps at their edge of chaos in the presence of a common noise. This model, characterised by the coupling strength $\epsilon$ and the noise width $\sigma_{max}$, was recently introduced by Pluchino et al [Phys. Rev. E {\bf 87}, 022910 (2013)]. They detected, for the time averaged returns with characteristic return time $\tau$, possible connections with $q$-Gaussians, the distributions which optimise, under appropriate constraints, the nonadditive entropy $S_q$, basis of nonextensive statistics mechanics. We have here a closer look on this model, and numerically obtain probability distributions which exhibit a slight asymmetry for some parameter values, in variance with simple $q$-Gaussians. Nevertheless, along many decades, the fitting with $q$-Gaussians turns out to be numerically very satisfactory for wide regions of the parameter values, and we illustrate how the index $q$ evolves with $(N, \tau, \epsilon, \sigma_{max})$. It is nevertheless instructive on how careful one must be in such numerical analysis. The overall work shows that physical and/or biological systems that are correctly mimicked by the Pluchino et al model are thermostatistically related to nonextensive statistical mechanics when time-averaged relevant quantities are studied.