Localization Properties of Covariant Lyapunov Vectors

TL;DR

This paper investigates the localization and angle properties of covariant Lyapunov vectors in a hard disk system, revealing different behaviors in low-density kinetic regimes and dense states, with implications for understanding chaos.

Contribution

It provides a detailed analysis of covariant Lyapunov vectors' localization and angle distributions across different densities, highlighting new insights into their behavior in chaotic systems.

Findings

Strong localization in low-density regimes

Characteristic angle distribution shapes depending on vector differences

No evidence of exact tangencies in generic chaotic trajectories

Abstract

The Lyapunov exponent spectrum and covariant Lyapunov vectors are studied for a quasi-one-dimensional system of hard disks as a function of density and system size. We characterize the system using the angle distributions between covariant vectors and the localization properties of both Gram-Schmidt and covariant vectors. At low density there is a {\it kinetic regime} that has simple scaling properties for the Lyapunov exponents and the average localization for part of the spectrum. This regime shows strong localization in a proportion of the first Gram-Schmidt and covariant vectors and this can be understood as highly localized configurations dominating the vector. The distribution of angles between neighbouring covariant vectors has characteristic shapes depending upon the difference in vector number, which vary over the continuous region of the spectrum. At dense gas or liquid like densities the behaviour of the covariant vectors are quite different. The possibility of tangencies between different components of the unstable manifold and between the stable and unstable manifolds is explored but it appears that exact tangencies do not occur for a generic chaotic trajectory.