From Lyapunov modes to the exponents for hard disk systems

TL;DR

This paper investigates how Lyapunov modes are preserved in the tangent space dynamics of hard disk systems, providing exact results for zero modes and approximate results for others, and introduces a method to determine Lyapunov exponents from these modes.

Contribution

It demonstrates the preservation of Lyapunov modes in hard disk systems and proposes a Gram-Schmidt procedure to compute Lyapunov exponents from the modes.

Findings

Lyapunov modes are exactly preserved for zero modes.

Approximate preservation of modes for transverse and LP modes.

A method to predict Lyapunov exponents from modes.

Abstract

We demonstrate the preservation of the Lyapunov modes by the underlying tangent space dynamics of hard disks. This result is exact for the zero modes and correct to order $\epsilon$ for the transverse and LP modes where $\epsilon$ is linear in the mode number. For sufficiently large mode numbers the dynamics no longer preserves the mode structure. We propose a Gram-Schmidt procedure based on orthogonality with respect to the centre space that determines the values of the Lyapunov exponents for the modes. This assumes a detailed knowledge of the modes, but from that predicts the values of the exponents from the modes. Thus the modes and the exponents contain the same information.