An efficient method for recovering Lyapunov vectors from singular vectors

Abstract

A B S T R A C T Lyapunov vectors are natural generalizations of normal modes for linear disturbances to aperiodic deterministic flows and offer insights into the physical mechanisms of aperiodic flow and the maintenance of chaos. Most standard techniques for computing Lyapunov vectors produce results which are norm-dependent and lack invariance under the linearized flow (except for the leading Lyapunov vector) and these features can make computation and physical interpretation problematic. An efficient, norm-independent method for constructing the n most rapidly growing Lyapunov vectors from n − 1 leading forward and n leading backward asymptotic singular vectors is proposed. The Lyapunov vectors so constructed are invariant under the linearized flow in the sense that, once computed at one time, they are defined, in principle, for all time through the tangent linear propagator. An analogous method allows the construction of the n most rapidly decaying Lyapunov vectors from n decaying forward and n − 1 decaying backward singular vectors. This method is demonstrated using two low-order geophysical models.

Characteristic Lyapunov vectors in chaotic time-delayed systems

“…Adaptation of the method proposed in [14] to this kind of systems -together with the computer capabilities nowadays available-allowed us to reach systems with fairly large delays, which serves to investigate the "thermodinamic limit" of these systems. Our results for the LVs coincide quantitatively with those obtained in extended dissipative systems [23,24].…”

“…(7). In contrast, if u is a constant, like for instance in the case of the MG model, a diffusion term would also eventually appear far from the small gradient limit as a result of the absolute value in ln |∂ x h − u| in (14). Finally, notice that the noise-like term ln(|v|/|u|), which is different from that in (12), will not lead to rare events and its distribution will be exponentially decaying in general.…”

“…Once both backward LVs and forward LVs have been computed the characteristic LVs are easily calculated following the prescriptions in Ref. [14]. One delicate point in delayed systems is that one must make sure that both forward and backward LV sets correspond exactly to the same time interval [t, t + τ ).…”

“…In this paper we calculate the set of characteristic LVs by adapting to DDSs the method recently introduced by Wolfe and Samelson [14]. In order to find the nth characteristic LV one has to compute the first n − 1 forward LVs in addition to the first n backward LVs.…”

“…However, their numerical computation is difficult in highdimensional systems. We adapt to DDSs one of the methods to compute characteristic LVs [14].…”