Characterizing Weak Chaos using Time Series of Lyapunov Exponents

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.

arXiv comment8 pages, 6 figures

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• Energy levels repulsion by spin-orbit coupling in two-dimensional Rydberg excitons
• Recurrence-time statistics in non-Hamiltonian volume preserving maps and flows
• Exploring conservative islands using correlated and uncorrelated noise
• Intermittent stickiness synchronization
• Spectral bounds on the entropy flow rate and Lyapunov exponents in differentiable dynamical systems
• Anomalous dynamics and the choice of Poincaré recurrence-set
• Importance sampling with imperfect cloning for the computation of generalized Lyapunov exponents
• Characterizing and quantifying weak chaos in fractional dynamics