Parallels between the dynamics at the noise-perturbed onset of chaos in logistic maps and the dynamics of glass formation

TL;DR

This paper explores the analogy between noise-perturbed chaos in logistic maps and glass formation, analyzing how noise influences dynamics and revealing glass-like behaviors such as aging and subdiffusion.

Contribution

It introduces a detailed characterization of the dynamics at the noise-perturbed edge of chaos in logistic maps, linking it to glassy properties and identifying unique features of the map.

Findings

Demonstrates two-step relaxation and aging in logistic maps

Shows subdiffusion and arrest phenomena under noise perturbation

Establishes parallels between map dynamics and glass formation behaviors

Abstract

We develop the characterization of the dynamics at the noise-perturbed edge of chaos in logistic maps in terms of the quantities normally used to describe glassy properties in structural glass formers. Following the recognition [Phys. Lett. \textbf{A 328}, 467 (2004)] that the dynamics at this critical attractor exhibits analogies with that observed in thermal systems close to vitrification, we determine the modifications that take place with decreasing noise amplitude in ensemble and time averaged correlations and in diffusivity. We corroborate explicitly the occurrence of two-step relaxation, aging with its characteristic scaling property, and subdiffusion and arrest for this system. We also discuss features that appear to be specific of the map.