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

Abstract

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 ζ(t) is unchanged by such scaling. The original (qpζ) motion and the scale-model (QP ζ) 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. Key words: Lyapunov Instability, Thermostats, Chaotic Dynamics I. LOCAL AND GLOBAL GRAM-SCHMIDT COVARIANT VECTORS AND EXPONENTSThe popularity of the time-dependent (or "local", or "instantaneous") covariant Lyapunov vectors and their associated exponents as descriptions of chaotic motion seems to us to be linked to a (false) impression extracted from the literature. Some of the literature implies that these descriptors have a special significance independent of such details as the coordinate system used to describe them. A selected literature, some of it quite clear, can be found in References 1-7. If the chosen coordinate system were really insignificant it would be hard to understand a simple, but nonchaotic, counterexample: the one-dimensional harmonic oscillator, which exhibits a strong dependence of its largest local Lyapunov exponent λ 1 (t) on the chosen Cartesian coordinate system [1,8,9].We remind the reader that this local instantaneous Lyapunov exponent λ 1 (t) (the largest of them when time averaged) measures the local rate of divergence of two nearby trajectories. Think of them as a reference trajectory and a satellite trajectory, with the satellite constrained to remain near the reference. It is unnecessary to consider exponents beyond the first to understand why it is that the local Lyapunov exponents, covariant or not, are in fact not scale-independent and do indeed depend upon the chosen coordinate system or set of measurement units. The oftrepeated statement that the local covariant exponents are "norm-independent" should not be misunderstood (as we did) to mean that the exponents are independent of a scale factor, as in a change of units from cgs to MKS.Here we focus on a simple chaotic continuous-flow example [10], the thermostated three-dimensional flow of a harmonic oscillator with coordinate q, momentum p, and friction coefficient ζ(t) in the unscaled (q, p, ζ) phase space:The variation of temperature with coordinate T (q) makes possible dissipation, and phase-volume shrinkage,⊗ < 0, onto a torus, or a strange attractor with fractional dimensionality, or a one-dimensional limit cycle. For the evolution of this model see References 11-14.