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UFN, 2013, Volume 183, Number 10, Pages 1009–1028 (Mi ufn4518)  

This article is cited in 10 scientific papers (total in 10 papers)

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Poincaré recurrence theory and its applications to nonlinear physics

V. S. Anishchenko, S. V. Astakhov

Physics Faculty, Chernyshevsky Saratov State University

Abstract: Theoretical results concerning the Poincaré recurrence problem and their application to problems in nonlinear physics are reviewed. The effects of noise, nonhyperbolicity, and the size of the recurrence region on the characteristics of the recurrence time sequence are examined. Relations of the recurrence time sequence dimension to the Lyapunov exponents and the Kolmogorov entropy are demonstrated. Methods for calculating the local and global attractor dimensions and the Afraimovich – Pesin dimension are presented. Methods using the Poincaré recurrence times to diagnose the stochastic resonance and the synchronization of chaos are described.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.3367/UFNr.0183.201310a.1009

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English version:
Physics–Uspekhi, 2013, 56:10, 955–972

Bibliographic databases:

PACS: 05.45.-a
MSC: 37B20, 37C45, 37L30
Received: October 30, 2012
Revised: March 15, 2013
Accepted: March 19, 2013

Citation: V. S. Anishchenko, S. V. Astakhov, “Poincaré recurrence theory and its applications to nonlinear physics”, UFN, 183:10 (2013), 1009–1028; Phys. Usp., 56:10 (2013), 955–972

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Klinshov, V. I. Nekorkin, “Synchronization of delay-coupled oscillator networks”, Phys. Usp., 56:12 (2013), 1217–1229  mathnet  crossref  crossref  adsnasa  isi  elib  elib
    2. Ya.I.. Boev, T.E.. Vadivasova, V.S.. Anishchenko, “Poincaré recurrence statistics as an indicator of chaos synchronization”, Chaos, 24:2 (2014), 023110  crossref  mathscinet  zmath  adsnasa  isi
    3. Ya. I. Boev, N. I. Biryukova, V. S. Anischenko, “Statistika vremen vozvrata Puankare v neavtonomnom odnomernom khaoticheskom otobrazhenii”, Nelineinaya dinam., 10:1 (2014), 3–16  mathnet
    4. V. S. Anishchenko, Ya. I. Boev, “The mean Poincaré return time locking: A criterion of chaos induced synchronization”, Tech. Phys. Lett, 40:4 (2014), 306  crossref  mathscinet  adsnasa  isi  elib  scopus
    5. Yaroslav Boev, Nadezhda Semenova, Galina Strelkova, Vadim Anishchenko, “Poincaré Recurrences in a Nonautonomous Chaotic Map”, Int. J. Bifurcation Chaos, 24:08 (2014), 1440016  crossref  mathscinet  zmath  isi  scopus
    6. N.I.. Semenova, T.E.. Vadivasova, G.I.. Strelkova, V.S.. Anishchenko, “Statistical properties of Poincaré Recurrences and Afraimovich-Pesin dimension for the Circle Map”, Communications in Nonlinear Science and Numerical Simulation, 2014  crossref  mathscinet  isi  scopus
    7. Norbert Marwan, Jürgen Kurths, Saskia Foerster, “Analysing spatially extended high-dimensional dynamics by recurrence plots”, Physics Letters A, 2015  crossref  isi  scopus
    8. Physics Reports, 2015  crossref  isi  scopus
    9. Ya. I. Boev, G. I. Strelkova, V. S. Anischenko, “Otsenka razmernosti khaoticheskikh attraktorov s ispolzovaniem vremen vozvrata Puankare”, Nelineinaya dinam., 11:3 (2015), 475–485  mathnet
    10. G. R. Ivanitskii, “The robot and the human. Where's their similarity limit?”, Phys. Usp., 61:9 (2018), 871–895  mathnet  crossref  crossref  adsnasa  isi  elib
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