Large deviations of Lyapunov exponents

TL;DR

This paper explores the large deviations of Lyapunov exponents in dynamical systems, linking these deviations to phase-space structures and demonstrating a numerical method to analyze rare stability or chaos phenomena.

Contribution

It introduces a numerical approach using Lyapunov Weighted Dynamics to sample large deviations and compute the topological pressure in extended dynamical systems.

Findings

Numerical sampling of large deviations reveals stable configurations in celestial mechanics.

The method identifies chaotic breathers in nonlinear media.

It enables computation of topological pressure in dynamical systems.

Abstract

Generic dynamical systems have `typical' Lyapunov exponents, measuring the sensitivity to small perturbations of almost all trajectories. A generic system has also trajectories with exceptional values of the exponents, corresponding to unusually stable or chaotic situations. From a more mathematical point of view, large deviations of Lyapunov exponents characterize phase-space topological structures such as separatrices, homoclinic trajectories and degenerate tori. Numerically sampling such large deviations using the Lyapunov Weighted Dynamics allows one to pinpoint, for example, stable configurations in celestial mechanics or collections of interacting chaotic `breathers' in nonlinear media. Furthermore, we show that this numerical method allows one to compute the topological pressure of extended dynamical systems, a central quantity in the Thermodynamic of Trajectories of Ruelle.