Routes to chaos, universality and glass formation

TL;DR

This paper reviews the universal routes to chaos in dynamical systems, highlighting the role of Tsallis' q-exponentials and drawing parallels between chaos onset and glass formation phenomena.

Contribution

It unifies the understanding of chaos routes using Tsallis' functions and links chaos dynamics with glass formation behaviors through noise perturbations.

Findings

Universal chaos routes described by Tsallis' q-exponentials

Parallel between chaos onset and glass formation behaviors

Noise perturbation reveals glass-like relaxation and aging phenomena

Abstract

We review recent results obtained for the dynamics of incipient chaos. These results suggest a common picture underlying the three universal routes to chaos displayed by the prototypical logistic and circle maps. Namely, the period doubling, intermittency, and quasiperiodicity routes. In these situations the dynamical behavior is exactly describable through infinite families of Tsallis' $q$-exponential functions. Furthermore, the addition of a noise perturbation to the dynamics at the onset of chaos of the logistic map allows to establish parallels with the behavior of supercooled liquids close to glass formation. Specifically, the occurrence of two-step relaxation, aging with its characteristic scaling property, and subdiffusion and arrest is corroborated for such a system.