Theory and computation of covariant Lyapunov vectors

TL;DR

This paper reviews the theory and numerical methods for covariant Lyapunov vectors, introduces adjoint vectors, and discusses improved algorithms and hyperbolicity testing for chaotic systems.

Contribution

It provides a comprehensive summary of covariant Lyapunov vectors, introduces the concept of adjoint vectors, and presents an improved computational approach and hyperbolicity tests.

Findings

Introduction of adjoint covariant Lyapunov vectors.

Development of an improved computational algorithm.

Method for testing hyperbolicity without explicit vectors.

Abstract

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe, how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.