Generalized Huberman-Rudnick scaling law and robustness of $q$-Gaussian probability distributions

TL;DR

This paper extends the Huberman-Rudnick scaling law to all periodic windows of the logistic map and demonstrates that $q$-Gaussian distributions are robust near the chaos threshold across these windows.

Contribution

It introduces a universal generalized scaling law for self-similar windows of the logistic map and confirms the robustness of $q$-Gaussian distributions near chaos.

Findings

Convergence to $q$-Gaussian distributions near chaos threshold.

Universal scaling relation for all periodic windows.

Robustness of $q$-Gaussians in probability distributions.

Abstract

We generalize Huberman-Rudnick universal scaling law for all periodic windows of the logistic map and show the robustness of $q$-Gaussian probability distributions in the vicinity of chaos threshold. Our scaling relation is universal for the self-similar windows of the map which exhibit period-doubling subharmonic bifurcations. Using this generalized scaling argument, for all periodic windows, as chaos threshold is approached, a developing convergence to $q$-Gaussian is numerically obtained both in the central regions and tails of the probability distributions of sums of iterates.