Characterizing Weak Chaos using Time Series of Lyapunov Exponents

TL;DR

This paper introduces a new method using time series of finite-time Lyapunov exponents to characterize different dynamical regimes in Hamiltonian systems, improving the understanding of chaos and stickiness.

Contribution

The study presents a novel approach to analyze chaos by examining the number of near-zero Lyapunov exponents, enhancing characterization of stickiness and regime transitions in high-dimensional systems.

Findings

Improved numerical characterization of stickiness in high-dimensional Hamiltonian systems.

Transition probabilities between regimes are linked to phase-space volume.

Lyapunov exponents depend on coupling strength in the studied maps.

Abstract

We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finite- time Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered (stickiness), semi-ordered (or semi-chaotic), and strongly chaotic motion. The dynamics is then investigated looking at the consecutive time spent in each regime, the transition between different regimes, and the regions in the phase-space associated to them. Applying our methodology to a chain of coupled standard maps we obtain: (i) that it allows for an improved numerical characterization of stickiness in high-dimensional Hamiltonian systems, when compared to the previous analyses based on the distribution of recurrence times; (ii) that the transition probabilities between different regimes are determined by the phase-space volume associated to the corresponding regions; (iii) the dependence of the Lyapunov exponents with the coupling strength.