Covariant Lyapunov vectors

TL;DR

This paper reviews the theory and computation of covariant Lyapunov vectors (CLVs), demonstrating their use in analyzing hyperbolicity deviations in chaotic systems through algorithms and numerical experiments.

Contribution

It provides a detailed description of a dynamical algorithm for computing CLVs, analyzes its convergence and performance, and illustrates their application in chaotic system analysis.

Findings

The algorithm converges exponentially in time.

CLVs effectively quantify deviations from hyperbolicity.

Numerical comparisons show the algorithm's efficiency.

Abstract

The recent years have witnessed a growing interest for covariant Lyapunov vectors (CLVs) which span local intrinsic directions in the phase space of chaotic systems. Here we review the basic results of ergodic theory, with a specific reference to the implications of Oseledets' theorem for the properties of the CLVs. We then present a detailed description of a "dynamical" algorithm to compute the CLVs and show that it generically converges exponentially in time. We also discuss its numerical performance and compare it with other algorithms presented in literature. We finally illustrate how CLVs can be used to quantify deviations from hyperbolicity with reference to a dissipative system (a chain of H\'enon maps) and a Hamiltonian model (a Fermi-Pasta-Ulam chain).