Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics

TL;DR

This paper analyzes the dynamics at the chaos threshold of the logistic map, revealing a structure described by nonextensive statistical mechanics and validating its applicability to critical nonlinear systems.

Contribution

It analytically demonstrates the power-law intertwined trajectories at the chaos threshold and links them to nonextensive statistical mechanics through the $q$-exponential form.

Findings

Sensitivity to initial conditions follows a $q$-exponential form.

Determination of the $q$-index and generalized Lyapunov coefficient.

Validation of nonextensive statistical mechanics at critical points.

Abstract

We uncover the dynamics at the chaos threshold $\mu_{\infty}$ of the logistic map and find it consists of trajectories made of intertwined power laws that reproduce the entire period-doubling cascade that occurs for $\mu <\mu_{\infty}$. We corroborate this structure analytically via the Feigenbaum renormalization group (RG) transformation and find that the sensitivity to initial conditions has precisely the form of a $q$-exponential, of which we determine the $q$-index and the $q$-generalized Lyapunov coefficient $\lambda _{q}$. Our results are an unequivocal validation of the applicability of the non-extensive generalization of Boltzmann-Gibbs (BG) statistical mechanics to critical points of nonlinear maps.