2004
DOI: 10.1063/1.1629191
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Statistics of Poincaré recurrences for maps with integrable and ergodic components

Abstract: Recurrence gives powerful tools to investigate the statistical properties of dynamical systems. We present in this paper some applications of the statistics of first return times to characterize the mixed behavior of dynamical systems in which chaotic and regular motion coexist. Our analysis is local: we take a neighborhood A of a point x and consider the conditional distribution of the points leaving A and for which the first return to A, suitably normalized, is bigger than t. When the measure of A shrinks to… Show more

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Cited by 24 publications
(30 citation statements)
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“…Quantities like Lyapunov exponents [283,70,284,285] and the indicators related to the RTS [286,287,288,289] have been used for a long time for such a task. Nevertheless, in the recent past, the need for computing stability properties with faster algorithms and for systems with many degrees of freedom resulted in a renewed interest in the technique and different dynamical indicators have been introduced.…”
Section: Extremes As Dynamical Indicatorsmentioning
confidence: 99%
“…Quantities like Lyapunov exponents [283,70,284,285] and the indicators related to the RTS [286,287,288,289] have been used for a long time for such a task. Nevertheless, in the recent past, the need for computing stability properties with faster algorithms and for systems with many degrees of freedom resulted in a renewed interest in the technique and different dynamical indicators have been introduced.…”
Section: Extremes As Dynamical Indicatorsmentioning
confidence: 99%
“…In that direction, the authors of Ref. [11] considered the first recurrence time in order to characterize the statistical properties of near-integrable systems. For points at the boundary of the chaotic sea and invariant tori, they posed that the statistic of the first return times is sensitive to capture the different qualitative behaviors and gives information about the relative weights played by both dynamics.…”
Section: Introductionmentioning
confidence: 99%
“…This choice was made because it seems capable to capture some fundamental features of the dynamics of the underlying systems. For instance, it is known that for a wide class of systems having a strongly mixing dynamics [6] the spectrum of return times decays exponentially; on the other hand, recently it has been shown [7] that for a particular integrable system the distribution of Poincaré recurrences follows an algebraic decay law.…”
Section: Introductionmentioning
confidence: 99%