Dynamics of perturbations in disordered chaotic systems

TL;DR

This paper investigates how perturbations evolve in disordered chaotic systems, revealing exponential localization around static pinning centers and a self-organized faceted structure in the perturbation landscape, with three universality classes identified.

Contribution

It introduces a novel analysis of perturbation dynamics in disordered chaotic systems, identifying universality classes based on disorder symmetries and proposing a phenomenological stochastic field theory.

Findings

Perturbations localize exponentially around static pinning centers.

The surface h(x,t) self-organizes into a faceted, scale-invariant structure.

Three universality classes for error propagation are identified.

Abstract

We study the time evolution of perturbations in spatially extended chaotic systems in the presence of quenched disorder. We find that initially random perturbations tend to exponentially localize in space around static pinning centers that are selected by the particular configuration of disorder. The spatial structure of typical perturbations, $\delta u(x,t)$, is analyzed in terms of the Hopf-Cole transform, $h(x,t) \equiv\ln|\delta u(x,t)|$. Our analysis shows that the associated surface $h(x,t)$ self-organizes into a faceted structure with scale-invariant correlations. Scaling analysis of critical roughening exponents reveals that there are three different universality classes for error propagation in disordered chaotic systems that correspond to different symmetries of the underlying disorder. Our conclusions are based on numerical simulations of disordered lattices of coupled chaotic elements and equations for diffusion in random potentials. We propose a phenomenological stochastic field theory that gives some insights on the path for a generalization of these results for a broad class of disordered extended systems exhibiting space-time chaos.