Characterizing dynamics with covariant Lyapunov vectors

TL;DR

This paper introduces a general method for determining covariant Lyapunov vectors in various dynamical systems, enabling analysis of hyperbolicity and revealing their localization and spectral properties in spatially extended systems.

Contribution

A novel, universal approach to compute covariant Lyapunov vectors applicable to both discrete and continuous systems, enhancing understanding of system stability and structure.

Findings

Covariant Lyapunov vectors can be used to quantify hyperbolicity.

These vectors exhibit localization and distinct spectral properties in extended systems.

The method applies to a broad class of dynamical systems.

Abstract

A general method to determine covariant Lyapunov vectors in both discrete- and continuous-time dynamical systems is introduced. This allows to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents.