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.

PACS: 05.45.−a DOI: 10.3367/UFNe.0183.201310a.1009 URL: https://ufn.ru/en/articles/2013/10/a/ 000329313100001 2013PhyU...56..955A Citation: Anishchenko V S, Astakhov S V "Poincaré recurrence theory and its applications to nonlinear physics" Phys. Usp. 56 955–972 (2013) BibTexBibNote ® (generic)BibNote ® (RIS)Medline RefWorks RT Journal T1 Poincaré recurrence theory and its applications to nonlinear physics A1 Anishchenko,V.S. A1 Astakhov,S.V. PB Physics-Uspekhi PY 2013 FD 10 Oct, 2013 JF Physics-Uspekhi JO Phys. Usp. VO 56 IS 10 SP 955-972 DO 10.3367/UFNe.0183.201310a.1009 LK https://ufn.ru/en/articles/2013/10/a/ Received: 30th, October 2012, revised: 15th, March 2013, accepted: 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