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Uspekhi Fizicheskikh Nauk, 2013, Volume 183, Number 10, Pages 1009–1028
DOI: https://doi.org/10.3367/UFNr.0183.201310a.1009 (Mi ufn4518) 		 
This article is cited in 16 scientific papers (total in 16 papers)

REVIEWS OF TOPICAL PROBLEMS

Poincaré recurrence theory and its applications to nonlinear physics

V. S. Anishchenko, S. V. Astakhov

Physics Faculty, Chernyshevsky Saratov State University
Full-text PDF (1076 kB) Citations (16)
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DOI: https://doi.org/10.3367/UFNr.0183.201310a.1009
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.
Keywords: Poincare recurrences, Afraimovich-Pesin dimension, Kolmogorov entropy, stochastic resonance, synchronization.
Received: October 30, 2012
Revised: March 15, 2013
Accepted: March 19, 2013
English version:
Physics–Uspekhi, 2013, Volume 56, Issue 10, Pages 955–972
DOI: https://doi.org/10.3367/UFNe.0183.201310a.1009
Bibliographic databases:
Document Type: Article
PACS: 05.45.-a
MSC: 37B20, 37C45, 37L30
Language: Russian
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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  • https://www.mathnet.ru/eng/ufn4518
  • https://www.mathnet.ru/eng/ufn/v183/i10/p1009
    • This publication is cited in the following 16 articles:
    1. Stankevich N., Volkov E., “Chaos-Hyperchaos Transition in Three Identical Quorum-Sensing Mean-Field Coupled Ring Oscillators”, Chaos, 31:10 (2021), 103112  crossref  isi
    2. Munoz-Arias M.H., Poggi P.M., Deutsch I.H., “Nonlinear Dynamics and Quantum Chaos of a Family of Kicked P-Spin Models”, Phys. Rev. E, 103:5 (2021), 052212  crossref  isi
    3. 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
    4. Vadim Anishchenko, Nadezhda Semenova, Elena Rybalova, Galina Strelkova, Nonlinear Systems and Complexity, 21, Regularity and Stochasticity of Nonlinear Dynamical Systems, 2018, 19  crossref
    5. Vadim G. Kleparskiy, Alexey I. Razumovskiy, Viktor E. Scheinis, 2018 Eleventh International Conference “Management of large-scale system development” (MLSD, 2018, 1  crossref
    6. N. I. Semenova, T. I. Galaktionova, V. S. Anischenko, “Vozvraty Puankare i razmernost Afraimovicha–Pesina v neavtonomnom konservativnom ostsillyatore”, Izv. Sarat. un-ta. Nov. cer. Ser. Fizika, 16:4 (2016), 195–203  mathnet  crossref
    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. 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
    11. 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
    12. 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
    13. 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
    14. 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
    15. V. M. Anikin, “Vadim Semenovich Anischenko (K 70-letiyu so dnya rozhdeniya)”, Izv. Sarat. un-ta. Nov. cer. Ser. Fizika, 14:1 (2014), 83–86  mathnet
    16. 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
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