Two stories outside Boltzmann-Gibbs statistics: Mori's q-phase transitions and glassy dynamics at the onset of chaos

TL;DR

This paper explores non-Boltzmann-Gibbs statistical behaviors in dynamical systems, revealing q-phase transitions in chaos and glassy dynamics at the onset of chaos, with implications for understanding complex systems.

Contribution

It demonstrates the presence of Mori's q-phase transitions and glassy dynamics in chaotic systems, linking these phenomena to nonextensive Tsallis statistics and bifurcation structures.

Findings

Identification of Mori singularities with specific Tsallis q-values

Observation of glass-like features such as aging and two-step relaxation

Comparison of bifurcation gap properties to supercooled liquids

Abstract

First, we analyze trajectories inside the Feigenbaum attractor and obtain the atypical weak sensitivity to initial conditions and loss of information associated to their dynamics. We identify the Mori singularities in its Lyapunov spectrum with the appearance of a special value for the entropic index q of the Tsallis statistics. Secondly, the dynamics of iterates at the noise-perturbed transition to chaos is shown to exhibit the characteristic elements of the glass transition, e.g. two-step relaxation, aging, subdiffusion and arrest. The properties of the bifurcation gap induced by the noise are seen to be comparable to those of a supercooled liquid above a glass transition temperature.