Periodic orbits and chaotic-diffusion probability distributions

Itzhack Dana, Vladislav E. Chernov

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1 Scopus citations


Periodic-orbit (PO) formulas for chaotic-diffusion probability distributions (PDs) are examined in the case of the perturbed Arnol'd cat map on the cylinder. This translationally invariant system exhibits a transition from uniform to nonuniform hyperbolicity as the perturbation parameter is increased. Two coarse-grained PDs, describing the "diffusion" between unit cells of the system, are studied: (a) a PD based on PO ensembles; (b) a PD based on generic ensembles. The approximate PO formula for PD (b) gives results which fluctuate around the expected Gaussian distribution for all parameters considered and thus agree qualitatively with results from standard methods. The exact PO formula for PD (a) gives similar results only for sufficiently small parameters. The results for large parameters decrease monotonically relative to the Gaussian distribution. This deviation seems to disappear as the PO period is increased.

Original languageEnglish
Pages (from-to)219-229
Number of pages11
JournalPhysica A: Statistical Mechanics and its Applications
Issue number1-4
StatePublished - 1 Feb 2004

Bibliographical note

Funding Information:
This work was partially supported by the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities. V.E.C. acknowledges the CRDF and Ministry of Education of the Russian Federation for Award #VZ-010-0.


  • Chaotic diffusion
  • Periodic orbits
  • Perturbed Arnol'd cat map
  • Probability distributions
  • Structural stability


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