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Reviews of topical problems


Poincaré recurrence theory and its applications to nonlinear physics

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Physics Department, Saratov State University named after N.G. Chernyshevsky, ul. Astrakhanskaya 83, Saratov, 410012, Russian Federation

Theoretical results on 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. The relationships of the recurrence time sequence dimension with 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.

Text can be downloaded in Russian. English translation is available here.
PACS: 05.45.−a
DOI: 10.3367/UFNe.0183.201310a.1009
URL: https://ufn.ru/en/articles/2013/10/a/
Citation: Anishchenko V S, Astakhov S V "Poincaré recurrence theory and its applications to nonlinear physics" Phys. Usp. 56 955–972 (2013)
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Received: 30th, October 2012, revised: 15th, March 2013, 19th, March 2013

Оригинал: Анищенко В С, Астахов С В «Теория возвратов Пуанкаре и её приложение к задачам нелинейной физики» УФН 183 1009–1028 (2013); DOI: 10.3367/UFNr.0183.201310a.1009

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