Covariant Lyapunov vectors for rigid disk systems

TL;DR

This paper investigates the Lyapunov instability and covariant Lyapunov vectors in a large two-dimensional hard disk system, revealing hyperbolic dynamics and the structure of tangent space perturbations.

Contribution

It provides a detailed numerical analysis of covariant Lyapunov vectors and their properties in a 2D hard disk system, demonstrating hyperbolicity and tangent space structure.

Findings

Covariant vectors are transversal but not orthogonal.

Angles between adjacent vectors rarely vanish, indicating hyperbolicity.

System exhibits Lyapunov modes parallel to the x axis.

Abstract

We carry out extensive computer simulations to study the Lyapunov instability of a two-dimensional hard disk system in a rectangular box with periodic boundary conditions. The system is large enough to allow the formation of Lyapunov modes parallel to the x axis of the box. The Oseledec splitting into covariant subspaces of the tangent space is considered by computing the full set of covariant perturbation vectors co-moving with the flow in tangent-space. These vectors are shown to be transversal, but generally not orthogonal to each other. Only the angle between covariant vectors associated with immediate adjacent Lyapunov exponents in the Lyapunov spectrum may become small, but the probability of this angle to vanish approaches zero. The stable and unstable manifolds are transverse to each other and the system is hyperbolic.