Characteristics of chaos evolution in one-dimensional disordered nonlinear lattices

TL;DR

This study numerically examines chaos evolution in one-dimensional disordered nonlinear lattices, revealing slow decay of chaos indicators and persistent chaotic behavior without crossover to regular dynamics, across different chaos regimes.

Contribution

It provides new insights into the decay rates of chaos measures and the spatial behavior of chaos seeds in disordered nonlinear lattices, extending previous research.

Findings

Chaos decay follows a power law with exponents around -0.25 and -0.3.

Chaotic dynamics persist without crossover to regular motion.

Chaotic seeds meander within the wave packet, facilitating lattice chaotization.

Abstract

We numerically investigate the characteristics of chaos evolution during wave packet spreading in two typical one-dimensional nonlinear disordered lattices: the Klein-Gordon system and the discrete nonlinear Schr\"{o}dinger equation model. Completing previous investigations \cite{SGF13} we verify that chaotic dynamics is slowing down both for the so-called `weak' and `strong chaos' dynamical regimes encountered in these systems, without showing any signs of a crossover to regular dynamics. The value of the finite-time maximum Lyapunov exponent $\Lambda$ decays in time $t$ as $\Lambda \propto t^{\alpha_{\Lambda}}$, with $\alpha_{\Lambda}$ being different from the $\alpha_{\Lambda}=-1$ value observed in cases of regular motion. In particular, $\alpha_{\Lambda}\approx -0.25$ (weak chaos) and $\alpha_{\Lambda}\approx -0.3$ (strong chaos) for both models, indicating the dynamical differences of the two regimes and the generality of the underlying chaotic mechanisms. The spatiotemporal evolution of the deviation vector associated with $\Lambda$ reveals the meandering of chaotic seeds inside the wave packet, which is needed for obtaining the chaotization of the lattice's excited part.