Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems

TL;DR

This paper investigates how weak chaos in Hamiltonian systems arises through chaotic stripes, which are initial condition intervals leading to chaos, and explores their growth with system nonlinearity.

Contribution

It introduces the concept of chaotic stripes in Hamiltonian systems and demonstrates their role in the transition from integrable to weakly chaotic behavior.

Findings

Chaotic stripes increase with nonlinear parameter.

Initial conditions within stripes lead to chaotic trajectories.

Arnold diffusion occurs inside these stripes in higher-dimensional systems.

Abstract

The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case the initial conditions are the injection angles. For higher-dimensional systems and small nonlinearities the chaotic stripes are the initial condition inside which Arnold diffusion occurs.