On the local nature and scaling of chaos in weakly nonlinear disordered chains

TL;DR

This paper investigates how weak nonlinearity induces local chaos in disordered chains, showing that chaos nucleates on rare resonant segments and that its probability can be analytically and numerically estimated with strong agreement.

Contribution

It introduces a local nucleation model for chaos in weakly nonlinear disordered chains and provides an analytical framework validated by numerical simulations.

Findings

Chaos probability is linked to rare resonant segments.

Analytical and numerical estimates of chaos probability agree.

Chaos nucleation occurs locally on specific chain segments.

Abstract

The dynamics of a disordered nonlinear chain can be either regular or chaotic with a certain probability. The chaotic behavior is often associated with the destruction of Anderson localization by the nonlinearity. In the presentwork it is argued that at weak nonlinearity chaos is nucleated locally on rare resonant segments of the chain. Based on this picture, the probability of chaos is evaluated analytically. The same probability is also evaluated by direct numerical sampling of disorder realizations and quantitative agreement between the two results is found.