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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.

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Fulltext is also available at DOI: 10.3367/UFNe.0183.201310a.1009
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)
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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

References (56) ↓ Cited by (11) Similar articles (20)

  1. Poincaré H Les Méthodes Nouvelles De La Mécanique Céleste Vol. 3 (Paris: Gauthier-Villars, 1899); Poincaré H New Methods Of Celestial Mechanics Vol. 3 (Woodbury, NY: American Institute of Physics, 1993); Puankare A Izbrannye Trudy Vol. 2 (M.: Nauka, 1972)
  2. Poincaré H Acta Math. 13 A3 (1890)
  3. Nemytskii V V, Stepanov V V Kachestvennaya Teoriya Differentsial’nykh Uravnenii (M. - L.: Gostekhizdat, 1947); Nemytskii V V, Stepanov V V Qualitative Theory Of Differential Equations (New York: Dover Publ., 1989)
  4. Afraimovich V, Ugalde E, Urías J Fractal Dimensions For Poincaré Recurrences (Amsterdam: Elsevier, 2006); Afraimovich V, Ugal’de E, Urias Kh Fraktal’nye Razmernosti dlya Vremen Vozvrashcheniya Puankare (M. - Izhevsk: RKhD, 2011)
  5. Boshernitzan M D Inventiones Math. 113 617 (1993)
  6. Moshchevitin N G Usp. Mat. Nauk 53 (1) 223 (1998); Moshchevitin N G Russ. Math. Surv. 53 219 (1998)
  7. Shkredov I D Matem. Zametki 72 625 (2002); Shkredov I D Math. Notes 72 576 (2002)
  8. Hirata M, Saussol B, Vaienti S Commun. Math. Phys. 206 33 (1999)
  9. Abadi M, Galves A Markov Process. Relat. Fields 7 (1) 97 (2001)
  10. Haydn N, Lacroix Y, Vaienti S Ann. Probab. 33 2043 (2005)
  11. Kac M Bull. Am. Math. Soc. 53 1002 (1947)
  12. Kac M Proc. Of The Summer Seminar, Boulder, Colorado, 1957 Vol. 1 Probability And Related Topics In Physical Sciences (Lectures in Applied Mathematics) Vol. 1 (London: Intersci. Publ., 1959); Kats M Veroyatnost’ i Smezhnye Voprosy v Fizike (M.: Mir, 1965)
  13. Chirikov B V, Shepelyanskii D L "Statistika vozvratov Puankare i struktura stokhasticheskogo sloya nelineinogo rezonansa" Preprint №81-69 (Novosibirsk: In-t yadernoi fiziki SO AN SSSR, 1981)
  14. Chetaev N C.R. Acad. Sci. Paris 187 637 (1928)
  15. Chetaev N G Uchenye Zapiski Kazanskogo Un-ta (2) 198 (1929)
  16. Afraimovich V Chaos 7 (1) 12 (1997)
  17. Afraimovich V, Zaslavsky G M Phys. Rev. E 55 5418 (1997)
  18. Pesin Ya B Dimension Theory In Dynamical Systems: Contemporary Views And Applications (Chicago: Univ. of Chicago Press, 1997)
  19. Penné V, Saussol B, Vaienti S Discrete Contin. Dyn. Syst. A 5 783 (1999)
  20. Sirvent V F Stoch. Dyn. 2 599 (2002)
  21. Marwan N et al. Phys. Rep. 438 237 (2007)
  22. Ngamga E J et al. Phys. Rev. E 85 026217 (2012)
  23. Gao J B Phys. Rev. Lett. 83 3178 (1999)
  24. Baptista M S, Kraut S, Grebogi C Phys. Rev. Lett. 95 094101 (2005)
  25. Altmann E G, Tél T Phys. Rev. Lett. 100 174101 (2008)
  26. Bolotin Yu L i dr. Nelineinaya Dinam. 6 719 (2010)
  27. Adler R L, Konheim A G, McAndrew M H Trans. Amer. Math. Soc. 114 309 (1965)
  28. Afraimovich V S, Lin W-W, Rulkov N F Int. J. Bifurcat. Chaos 10 2323 (2000)
  29. Katok A, Hasselblatt B Introduction To The Modern Theory Of Dynamical Systems (Cambridge: Cambridge Univ. Press, 1995); Katok A B, Khasselblat B Vvedenie v Sovremennuyu Teoriyu Dinamicheskikh Sistem (M.: Faktorial, 1999)
  30. Afraimovich V Taiwanese J. Math. 3 (2) 139 (1999)
  31. Anishchenko V S, Boev Y I Commun. Nonlin. Sci. Numer. Simulat. 18 953 (2013)
  32. Anishchenko V et al. Nonlinear Dynamics Of Chaotic And Stochastic Systems. Tutorial And Modern Development (Berlin: Springer, 2002)
  33. Benzi R, Sutera A, Vulpiani A J. Phys. A Math. Gen. 14 L453 (1981)
  34. Kuznetsov S P Usp. Fiz. Nauk 181 121 (2011); Kuznetsov S P Phys. Usp. 54 119 (2011)
  35. Anishchenko V S, Khairulin M E Pis’ma ZhTF 37 (12) 35 (2011); Anishchenko V S, Khairulin M E Tech. Phys. Lett. 37 561 (2011)
  36. Altmann E G, da Silva E C, Caldas I L Chaos 14 975 (2004)
  37. Anishchenko V S et al. Eur. Phys. J. B 82 219 (2011)
  38. Gammaitoni L et al. Rev. Mod. Phys. 70 223 (1998)
  39. Anishchenko V S i dr. Usp. Fiz. Nauk 169 7 (1999); Anishchenko V S et al. Phys. Usp. 42 7 (1999)
  40. Kramers H A Physica 7 284 (1940)
  41. Anishchenko V S, Neiman A B, Safanova M A J. Stat. Phys. 70 183 (1993)
  42. Freidlin M Physica B 306 10 (2001)
  43. Farmer J D, Ott E, Yorke J A Physica D 7 (1 - 3) 153 (1983)
  44. Anishchenko V S Nelineinaya Dinam. 8 (1) 29 (2012)
  45. Lozi R J. Phys. Colloques 39 C5-9 (1978)
  46. Loskutov A Yu Usp. Fiz. Nauk 180 1305 (2010); Loskutov A Phys. Usp. 53 1257 (2010)
  47. Saussol B, Troubetzkoy S, Vaienti S J. Stat. Phys. 106 623 (2002)
  48. Schuster H G Deterministic Chaos. An Introduction (Weinheim: Physik-Verlag, 1984); Shuster G Determinirovannyi Khaos. Vvedenie (M.: Mir, 1988)
  49. Anishchenko V S i dr. Nelineinaya Dinamika 8 449 (2012)
  50. Baptista M S, Maranhão D M, Sartorelli J C Chaos 19 043115 (2009)
  51. Astakhov S V, Anishchenko V S Phys. Lett. A 376 3620 (2012)
  52. Anishchenko V S Slozhnye Kolebaniya v Prostykh Sistemakh (M.: Nauka, 1990)
  53. Anishchenko V S, Astakhov S Int. J. Bifurcation Chaos 18 2851 (2008)
  54. Anishchenko V S, Astakhov S V, Vadivasova T E Regul. Chaot. Dyn. 15 261 (2010)
  55. Hénon M Commun. Math. Phys. 50 69 (1976)
  56. Pecora L M, Carroll T L Phys. Rev. Lett. 64 821 (1990)
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