A recent appreciation of the singular dynamics at the edge of chaos

TL;DR

This paper explores the complex dynamics at the edge of chaos in the logistic map, revealing a series of $q$-phase transitions linked to Tsallis statistics and drawing parallels with glassy dynamics in supercooled liquids.

Contribution

It identifies and characterizes an infinite family of $q$-phase transitions at the chaos threshold, connecting nonlinear dynamics with nonextensive statistical mechanics.

Findings

Discovery of $q$-phase transitions at the chaos threshold.

Weak sensitivity to initial conditions obeys Tsallis statistics.

Bifurcation gap exhibits glass-like relaxation phenomena.

Abstract

We study the dynamics of iterates at the transition to chaos in the logistic map and find that it is constituted by an infinite family of Mori's $q$-phase transitions. Starting from Feigenbaum's $\sigma $ function for the diameters ratio, we determine the atypical weak sensitivity to initial conditions $\xi _{t}$ associated to each $q$-phase transition and find that it obeys the form suggested by the Tsallis statistics. The specific values of the variable $q$ at which the $q$-phase transitions take place are identified with the specific values for the Tsallis entropic index $q$ in the corresponding $\xi_{t}$. We describe too the bifurcation gap induced by external noise and show that its properties exhibit the characteristic elements of glassy dynamics close to vitrification in supercooled liquids, e.g. two-step relaxation, aging and a relationship between relaxation time and entropy.