Symmetry properties of orthogonal and covariant Lyapunov vectors and their exponents

TL;DR

This paper investigates how symmetries like time-reversal and symplectic structure influence covariant Lyapunov vectors and exponents in classical dynamical systems, using examples like the spring pendulum and Hénon-Heiles system.

Contribution

It analyzes the effects of symplectic symmetry and time-reversal invariance on Lyapunov vectors and exponents, providing insights into their transformation properties.

Findings

Symplectic symmetry constrains Lyapunov vectors and exponents.

Time-reversal invariance affects the symmetry properties of Lyapunov spectra.

Transformations between different flow parameterizations are characterized.

Abstract

Lyapunov exponents are indicators for the chaotic properties of a classical dynamical system. They are most naturally defined in terms of the time evolution of a set of so-called covariant vectors, co-moving with the linearized flow in tangent space. Taking a simple spring pendulum and the H\'enon-Heiles system as examples, we demonstrate the consequences of symplectic symmetry and of time-reversal invariance for such vectors, and study the transformation between different parameterizations of the flow.