Strong and weak chaos in weakly nonintegrable many-body Hamiltonian systems

TL;DR

This paper investigates chaos in weakly coupled nonlinear Hamiltonian lattices, revealing strong chaos from resonances, a weakly chaotic component linked to Arnold diffusion, and subdiffusive wave spreading, with implications for nonlinear dynamics.

Contribution

It provides a detailed numerical analysis of strong and weak chaos regimes, connecting chaos measures to coupling strength and exploring wave packet spreading in disordered lattices.

Findings

Chaos measure proportional to coupling strength and lattice length

Maximal Lyapunov exponent scales with square root of coupling

Subdiffusive spreading of wave packets in disordered lattices

Abstract

We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators, by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.