Invariant density statistical quantifiers and a temperature for the logistic map

TL;DR

This paper investigates the logistic map's dynamics using invariant density-based statistical quantifiers, introducing a temperature concept and analyzing informational measures to distinguish different dynamical regimes.

Contribution

It introduces a novel temperature definition for the logistic map and uses Fisher information and CR complexity to characterize its dynamical regimes.

Findings

Fisher information peaks in regular regimes

CR complexity varies with dynamical behavior

Proposed temperature correlates with dynamical transitions

Abstract

In this work, we study the dynamics of the logistic map based on a probabilistic characterization in terms of the invariant density. We analyze the relevant regimes of the dynamics (regular, oscillatory, onset chaotic and fully chaotic) in terms of the Fisher information and the Cr\'amer-Rao (CR) complexity. We have found that these informational quantifiers allow to distinguish the dynamical regions of the map, by maximizing the Fisher information in the regular behavior and with the CR complexity exhibiting variations and a maximum near to the Pameau-Maneville scenario. Fisher information as a function of time is examined in the light of Frieden's informational interpretation of the Second Law of Thermodynamics. We apply the Equipartition Theorem to propose a definition of temperature for the logistic map, providing a macroscopic signature of the dynamics.