Relaxation and Noise in Chaotic Systems

Abstract

For a class of idealized chaotic systems (hyperbolic systems) correlations decay exponentially in time. This result is asymptotic and rigorous. The decay rate is related to the Ruelle-Pollicott resonances. Nearly all chaotic model systems, that are studied by physicists, are not hyperbolic. For many such systems it is known that exponential decay takes place for a long time. It may not be asymptotic, but it may persist for a very long time, longer than any time of experimental relevance. In this review a heuristic method for calculation of this exponential decay of correlations in time is presented. It can be applied to model systems, where there are no rigorous results concerning this exponential decay. It was tested for several realistic systems (kicked rotor and kicked top) in addition to idealized systems (baker map and perturbed cat map). The method consists of truncation of the evolution operator (Frobenius-Perron operator), and performing all calculations with the resulting finite dimensional matrix. This finite dimensional approximation can be considered as coarse graining, and is equivalent to the effect of noise. The exponential decay rate of the chaotic system is obtained when the dimensionality of the approximate evolution operator is taken to infinity, resulting in infinitely fine resolution, that is equivalent to vanishing noise. The corresponding Ruelle-Pollicott resonances can be calculated for many systems that are beyond the validity of the Ruelle-Pollicott theorem.

Advected fields in maps—III. Passive scalar decay in baker's maps

“…The issue of the link between resonances and the decay of correlations in more general dynamical systems is considered in [37][38][39]. In this sense, we have been investigating when the effect of diffusion on a scalar field is equivalent to the decay of correlations in the scalar field under iteration of T. To summarize our results, we have found that the effect of boundary conditions and the magnitude of the correlation decay rate jj are crucial.…”

“…Also we shall find below that 1 ¼ Oð ffiffi ffi " p Þ (see (47)). This means that in the boundary layer we can neglect 1 b 0 in (28) and b 1 ðÇÞ in (37), and then (28) becomes the integral equation…”

“…Here we have made no assumptions about the size of b 1 or 1 and so have retained terms b 1 ðÇÞ (see (37)) and 1 b 0 (see (28)). …”

Classical nonlinear response of a chaotic system. I. Collective resonances

“…Even in such systems, when the true longtime asymptotic decay of correlations is rather powerlike than exponential, RP resonances may by noticeable, leading to the intermediate asymptotic, exponential decay persisting for a very long time [32].…”

Dissipation time and decay of correlations

“…We notice that the first upper bound does not depend on U at all, but only on the noise. Using the estimate (15), we obtain the following obvious corollary:…”