Poincaré Recurrences from the Perspective of Transient Chaos

Eduardo G. Altmann and Tamás Tél
Phys. Rev. Lett. 100, 174101 – Published 29 April 2008

Abstract

We obtain a description of the Poincaré recurrences of chaotic systems in terms of the ergodic theory of transient chaos. It is based on the equivalence between the recurrence time distribution and an escape time distribution obtained by leaking the system and taking a special initial ensemble. This ensemble is atypical in terms of the natural measure of the leaked system, the conditionally invariant measure. Accordingly, for general initial ensembles, the average recurrence and escape times are different. However, we show that the decay rate of these distributions is always the same. Our results remain valid for Hamiltonian systems with mixed phase space and validate a split of the chaotic saddle in hyperbolic and nonhyperbolic components.

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  • Received 4 December 2007

DOI:https://doi.org/10.1103/PhysRevLett.100.174101

©2008 American Physical Society

Authors & Affiliations

Eduardo G. Altmann1,* and Tamás Tél2

  • 1Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
  • 2Institute for Theoretical Physics, Eötvös University, P.O. Box 32, H-1518 Budapest, Hungary

  • *Current address: Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL, USA. ega@northwestern.edu

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Vol. 100, Iss. 17 — 2 May 2008

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