Why Instantaneous Values of the "Covariant" Lyapunov Exponents Depend upon the Chosen State-Space Scale

TL;DR

This paper demonstrates through a chaotic harmonic oscillator example that local covariant Lyapunov exponents depend on the chosen state-space scale, revealing scale sensitivity in local stability measures.

Contribution

It provides a clear example showing how local covariant Lyapunov exponents vary with state-space scaling in a chaotic thermostated system.

Findings

Local Lyapunov exponents change with scale transformations.

Covariant exponents are scale-dependent in the model.

The thermostat variable remains unchanged by scaling.

Abstract

We explore a simple example of a chaotic thermostated harmonic-oscillator system which exhibits qualitatively different local Lyapunov exponents for simple scale-model constant-volume transformations of its coordinate q and momentum p : { q,p } --> { (Q/s),(sP) } . The time-dependent thermostat variable zeta(t) is unchanged by such scaling. The original (q,p,zeta) motion and the scale-model (Q,P,zeta) version of the motion are physically identical. But both the local Gram-Schmidt Lyapunov exponents and the related local "covariant" exponents change with the change of scale. Thus this model furnishes a clearcut chaotic time-reversible example showing how and why both the local Lyapunov exponents and covariant exponents vary with the scale factor s .