Logistic map trajectory distributions: Renormalization-group, entropy and criticality at the transition to chaos

TL;DR

This paper analyzes the probability density evolution of the logistic map's ensembles near chaos transition, revealing renormalization-group structures and entropy behaviors analogous to phase transitions in statistical mechanics.

Contribution

It introduces a statistical-mechanical perspective on logistic map ensembles, highlighting RG structures and entropy extrema at critical points, which is novel compared to orbit-based studies.

Findings

Densities follow a renormalization-group scaling

Entropies reach extrema at RG fixed points

Entropy behavior resembles a second-order phase transition

Abstract

We study the evolution of the probability density of ensembles of iterates of the logistic map that advance towards and finally remain at attractors of representative dynamical regimes. We consider the mirror families of superstable attractors along the period-doubling cascade, and of chaotic-band attractors along the inverse band-splitting cascade. We examine also their common aperiodic accumulation point. The iteration time progress of the densities of trajectories is determined via the action of the Frobenius-Perron (FP) operator. As a difference with the study of individual orbits, the analysis of ensembles of positions offers a viewpoint from which the nonlinear dynamical features of this iconic model can be better characterized in statistical-mechanical terms. The scaling of the densities along the considered families of attractors conforms to a renormalization-group (RG) structure, while their entropies are seen to attain extrema at the fixed points of the RG flows. Additionally, this entropy as a function of the map control parameter displays the characteristic features of an equation of state of a thermal system undergoing a second-order phase transition. We discuss our results.