Poincare Recurrence and Measure of Hyperbolic and Nonhyperbolic Chaotic Attractors

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Abstract

We study Poincare recurrence of chaotic attractors for regions of finite size. Contrary to the standard case, where the size of the recurrent regions tends to zero, the measure is no longer supported solely by unstable periodic orbits of finite length inside it, but also by other special recurrent trajectories, located outside that region. The presence of the latter leads to a deviation of the distribution of the Poincare first return times from a Poissonian. Consequently, by taking into account the contribution of these special recurrent trajectories, a corrected estimate of the measure is obtained. This has wide experimental implications, as in the laboratory all returns can exclusively be observed for regions of finite size, and only unstable periodic orbits of finite length can be detected.

Original languageEnglish
Pages (from-to)094101-1 - 094101-4
Number of pages4
JournalPhysical Review Letters
Volume95
Issue number9
DOIs
Publication statusPublished - 26 Aug 2005

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trajectories
orbits
deviation
estimates

Keywords

  • unstable periodic-orbits
  • strange sets
  • statistics
  • times

Cite this

Poincare Recurrence and Measure of Hyperbolic and Nonhyperbolic Chaotic Attractors. / Baptista, M S ; Kraut, S ; Grebogi, C .

In: Physical Review Letters, Vol. 95, No. 9, 26.08.2005, p. 094101-1 - 094101-4.

Research output: Contribution to journalArticle

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AB - We study Poincare recurrence of chaotic attractors for regions of finite size. Contrary to the standard case, where the size of the recurrent regions tends to zero, the measure is no longer supported solely by unstable periodic orbits of finite length inside it, but also by other special recurrent trajectories, located outside that region. The presence of the latter leads to a deviation of the distribution of the Poincare first return times from a Poissonian. Consequently, by taking into account the contribution of these special recurrent trajectories, a corrected estimate of the measure is obtained. This has wide experimental implications, as in the laboratory all returns can exclusively be observed for regions of finite size, and only unstable periodic orbits of finite length can be detected.

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