2002
DOI: 10.1007/3-540-46122-1_7
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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 heuri… Show more

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Cited by 9 publications
(13 citation statements)
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“…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.…”
Section: Discussionmentioning
confidence: 88%
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“…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.…”
Section: Discussionmentioning
confidence: 88%
“…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…”
Section: Scalar Decay With the Zero Boundary Conditionmentioning
confidence: 99%
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“…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].…”
Section: Discussionmentioning
confidence: 99%
“…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:…”
Section: Definitions and General Boundsmentioning
confidence: 97%