Issues

 / 

2013

 / 

October

  

Reviews of topical problems


Poincaré recurrence theory and its applications to nonlinear physics

,
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)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

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

References (56) Cited by (8) Similar articles (20) ↓

  1. V.S. Anishchenko, A.B. Neiman et alStochastic resonance: noise-enhanced order42 7–36 (1999)
  2. A.A. Koronovskii, O.I. Moskalenko, A.E. Hramov “On the use of chaotic synchronization for secure communication52 1213–1238 (2009)
  3. V.S. Anishchenko, T.E. Vadivasova et alStatistical properties of dynamical chaos48 151–166 (2005)
  4. V.V. Klinshov, V.I. Nekorkin “Synchronization of delay-coupled oscillator networks56 1217–1229 (2013)
  5. A. Loskutov “Fascination of chaos53 1257–1280 (2010)
  6. M.I. Rabinovich, M.K. Muezzinoglu “Nonlinear dynamics of the brain: emotion and cognition53 357–372 (2010)
  7. S.P. Kuznetsov “Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics54 119–144 (2011)
  8. A.L. Virovlyansky, D.V. Makarov, S.V. Prants “Ray and wave chaos in underwater acoustic waveguides55 18–46 (2012)
  9. A.A. Makarov, A.L. Malinovsky, E.A. Ryabov “Intramolecular vibrational redistribution: from high-resolution spectra to real-time dynamics55 977–1007 (2012)
  10. V.I. Klyatskin “Integral characteristics: a key to understanding structure formation in stochastic dynamic systems54 441–464 (2011)
  11. V.M. Fedorov “Earth insolation variation and its incorporation into physical and mathematical climate models62 32–45 (2019)
  12. A.L. Fradkov “Application of cybernetic methods in physics48 103–127 (2005)
  13. V.V. Likhanskii, A.P. Napartovich “Radiation emitted by optically coupled lasers33 (3) 228–252 (1990)
  14. G.N. Borisyuk, R.M. Borisyuk et alModels of neural dynamics in brain information processing — the developments of ’the decade’45 1073–1095 (2002)
  15. V.P. Budaev, S.P. Savin, L.M. Zelenyi “Investigation of intermittency and generalized self-similarity of turbulent boundary layers in laboratory and magnetospheric plasmas: towards a quantitative definition of plasma transport features54 875–918 (2011)
  16. S.V. Dmitriev, E.A. Korznikova et alDiscrete breathers in crystals59 446–461 (2016)
  17. O.V. Maslennikov, V.I. Nekorkin “Adaptive dynamical networks60 694–704 (2017)
  18. M.A. Tsyganov, V.N. Biktashev et alWaves in systems with cross-diffusion as a new class of nonlinear waves50 263–286 (2007)
  19. O.G. Bakunin “Stochastic instability and turbulent transport. Characteristic scales, increments, diffusion coefficients58 252–285 (2015)
  20. K.V. Koshel, S.V. Prants “Chaotic advection in the ocean49 1151–1178 (2006)

The list is formed automatically.

© 1918–2019 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions