RSS feeds
РусскийEnglish
Issue 11, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

2011

 / 

February

←  →

Reviews of topical problems


Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics


Kotel'nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Saratov Branch, ul. Zelenaya 38, Saratov, 410019, Russian Federation

Research is reviewed on the identification and construction of physical systems with chaotic dynamics due to uniformly hyperbolic attractors (such as the Plykin attraction or the Smale-Williams solenoid). Basic concepts of the mathematics involved and approaches proposed in the literature for constructing systems with hyperbolic attractors are discussed. Topics covered include periodic pulse-driven models; dynamics models consisting of periodically repeated stages, each described by its own differential equations; the construction of systems of alternately excited coupled oscillators; the use of parametrically excited oscillations; and the introduction of delayed feedback. Some maps, differential equations, and simple mechanical and electronic systems exhibiting chaotic dynamics due to the presence of uniformly hyperbolic attractors are presented as examples.

Fulltext pdf (809 KB)
Fulltext is also available at DOI: 10.3367/UFNe.0181.201102a.0121
PACS: 05.45.−a, 45.50.−j, 84.30.−r (all)
DOI: 10.3367/UFNe.0181.201102a.0121
URL: https://ufn.ru/en/articles/2011/2/a/
000294812700001
2-s2.0-79959921536
2011PhyU...54..119K
Citation: Kuznetsov S P "Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics" Phys. Usp. 54 119–144 (2011)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Кузнецов С П «Динамический хаос и однородно гиперболические аттракторы: от математики к физике» УФН 181 121–149 (2011); DOI: 10.3367/UFNr.0181.201102a.0121

References (111) ↓ Cited by (56) Similar articles (20)

  1. Sinai Ya G, Shil’nikov L P (Red. per.) Strannye Attraktory (M.: Mir, 1981)
  2. Hasselblatt B, Young L S Encyclopedia Of Nonlinear Science (Ed. A Scott) (New York: Routledge, 2005) p. 11
  3. Zaslavskii G M, Sagdeev R Z Vvedenie v Nelineinuyu Fiziku: Ot Mayatnika do Turbulentnosti i Khaosa (M.: Nauka, 1988); Sagdeev R Z, Usikov D A, Zaslavsky G M Nonlinear Physics: From The Pendulum To Turbulence And Chaos (Chur: Harwood Acad. Publ., 1988)
  4. Neimark Yu I, Landa P S Stokhasticheskie i Khaoticheskie Kolebaniya (M.: Nauka, 1988); Neimark Yu I, Landa P S Stochastic And Chaotic Oscillations (Dordrecht: Kluver Acad. Publ., 1992)
  5. Rabinovich M I, Trubetskov D I Vvedenie v Teoriyu Kolebanii i Voln (M. - Izhevsk: RKhD, Inst. komp. issled., 2000); Rabinovich M I, Trubetskov D I Oscillations And Waves In Linear And Nonlinear Systems (Dordrecht: Kluwer Acad. Publ., 1989)
  6. Schuster H G, Just W Deterministic Chaos: An Introduction (Weincheim: Wiley-VCH, 2005); Shuster G Determinirovannyi Khaos (M.: Mir, 1988)
  7. Thompson J M T, Stewart H B Nonlinear Dynamics And Chaos (Chichester: Wiley, 1986)
  8. Dmitriev A S, Kislov V Ya Stokhasticheskie Kolebaniya v Radiofizike i Elektronike (M.: Nauka, 1989)
  9. Loskutov A Yu, Mikhailov A S Osnovy Teorii Slozhnykh Sistem (M. - Izhevsk: RKhD, Inst. komp. issled., 2007)
  10. Anishchenko V S Slozhnye Kolebaniya v Prostykh Sistemakh: Mekhanizmy Vozniknoveniya, Struktura i Svoistva Dinamicheskogo Khaosa v Radiofizicheskikh Sistemakh (M.: Librokom, 2009)
  11. Kuznetsov S P Dinamicheskii Khaos (M.: Fizmatlit, 2006)
  12. Strogatz S H Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Boulder, CO: Westview Press, 2001)
  13. Ott E Chaos In Dynamical Systems (Cambridge: Cambridge Univ. Press, 2002)
  14. Smale S Bull. Amer. Math. Soc. New Ser. 73 747 (1967)
  15. Williams R F Publ. Math. Inst. Hautes Étud. Sci. 43 169 (1974)
  16. Sinai Ya G Nelineinye Volny (Pod red. A V Gaponova-Grekhova) (M.: Nauka, 1979) p. 192
  17. Dinamicheskie Sistemy-9 (Itogi nauki i tekhniki. Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya) Vol. 66 (M.: VINITI, 1991); Dynamical Systems IX (Encyclopedia of Mathematical Sciences) Vol. 66 (Berlin: Springer, 1995)
  18. Eckmann J-P, Ruelle D Rev. Mod. Phys. 57 617 (1985)
  19. Shilnikov L Int. J. Bifurcat. Chaos 7 1953 (1997)
  20. Katok A, Hasselblatt B Introduction To The Modern Theory Of Dynamical Systems (Cambridge: Cambridge Univ. Press, 1996); Katok A B, Khasselblat B Vvedenie v Sovremennuyu Teoriyu Dinamicheskikh Sistem (M.: Faktorial, 1999)
  21. Guckenheimer J, Holmes P Nonlinear Oscillations: Dynamical Systems, And Bifurcations Of Vector Fields (Berlin: Springer, 1990); Gukenkheimer Dzh, Kholms F Nelineinye Kolebaniya, Dinamicheskie Sistemy i Bifurkatsii Vektornykh Polei (M. - Izhevsk: Inst. komp. issled., 2002)
  22. Afraimovich V, Hsu S-B Lectures On Chaotic Dynamical Systems (AMS/IP Studies in Advanced Mathematics) Vol. 28 (Providence, RI: American Mathematical Society, 2003)
  23. Hasselblatt B, Katok A A First Course In Dynamics: With A Panorama Of Recent Developments (Cambridge: Cambridge Univ. Press, 2003); Katok A B, Khasselblat B Vvedenie v Teoriyu Dinamicheskikh Sistem: s Obzorom Poslednikh Dostizhenii (M.: Izd-vo MTsNMO, 2005)
  24. Devaney R L An Introduction To Chaotic Dynamical Systems (New York: Westview Press, 2003)
  25. Anishchenko V S i dr. Nelineinye Effekty v Khaoticheskikh i Stokhasticheskikh Sistemakh (M. - Izhevsk: Inst. komp. issled., 2003); Anishchenko V S et al. Nonlinear Dynamics Of Chaotic And Stochastic Systems (Berlin: Springer, 2007)
  26. Barreira L, Pesin Y Smooth Ergodic Theory And Its Applications (Proc. of Symp. in Pure Mathematics) Vol. 69 (Providence, RI: American Mathematical Society, 2001) p. 3
  27. Bonatti C, Díaz L J, Viana M Dynamics Beyond Uniform Hyperbolicity. A Global Geometric And Probobalistic Perspective (Encyclopedia Of Mathematical Sciences, Vol. 102. Mathematical Physics 3) (Berlin: Springer, 2005)
  28. Benedicks M, Carleson L Ann. Math. 133 73 (1991)
  29. Andronov A A, Pontryagin L S Dokl. Akad. Nauk SSSR 14 247 (1937); Andronov A A, Pontryagin L S C.R. (Dokl.) Acad. Sci. USSR 14 247 (1937)
  30. Andronov A A, Vitt A A, Khaikin S E Teoriya Kolebanii (M.: Fizmatgiz, 1959); Andronov A A, Vitt A A, Khaikin S E Theory Of Oscillators (Oxford: Pergamon Press, 1966)
  31. Benettin G et al. Meccanica 15 9 (1980)
  32. Kaplan J L, Yorke J A Functional Differential Equations And Approximation Of Fixed Points (Lecture Notes in Mathematics) Vol. 730 (Eds H-O Peitgen, H-O Walther) (Berlin: Springer-Verlag, 1979) p. 204
  33. Vul E B, Sinai Ya G, Khanin K M Usp. Mat. Nauk 39 (3) 3 (1984); Vul E B, Sinai Y G, Khanin K M Russ. Math. Surv. 39 1 (1984)
  34. Coudene Y Notices Am. Math. Soc. 53 (1) 8 (2006)
  35. Plykin R V Matem. Sb. 94 (2) 243 (1974); Plykin R V Math. USSR Sb. 23 (2) 233 (1974)
  36. Zhirov A Yu Matem. Sb. 185 (6) 3 (1994); Zhirov A Yu Russ. Acad. Sci. Sb. Math. 82 (1) 135 (1995)
  37. Grines V Z, Zhuzhoma E V Regul. Chaotic Dyn. 11 (2) 225 (2006)
  38. Feller W An Introduction To Probability Theory And Its Applications Vol. 1 (New York: Wiley, 1968); Feller V Vvedenie v Teoriyu Veroyatnostei i Ee Prilozheniya Vol. 1 (M.: Mir, 1984)
  39. Adler R L, Konheim A G, McAndrew M H Trans. Am. Math. Soc. 114 309 (1965)
  40. Beck C, Schlögl F Thermodynamics Of Chaotic Systems: An Introduction (Cambridge: Cambridge Univ. Press, 1993)
  41. Kifer Yu I Izv. AN SSSR Ser. matem. 38 1091 (1974); Kifer Ju I Math. USSR Izv. 8 1083 (1974)
  42. Sinai Ya G Usp. Mat. Nauk 27 (4) 21 (1972); Sinai Ya G Russ. Math. Surv. 27 (4) 21 (1972)
  43. Bowen R Equilibrium States And The Ergodic Theory Of Anosov Diffeomorphisms (Lecture Notes in Math.) Vol. 470 (Berlin: Springer-Verlag, 1975)
  44. Ruelle D Am. J. Math. 98 619 (1976)
  45. Kolmogorov A N Dokl. Akad. Nauk SSSR 124 754 (1959)
  46. Sinai Ya G Dokl. Akad. Nauk SSSR 124 768 (1959)
  47. Pesin Ya B Usp. Mat. Nauk 32 (4) 55 (1977); Pesin Ya B Russ. Math. Surv. 32 (4) 55 (1977)
  48. Ruelle D, Takens F Commun. Math. Phys. 20 167 (1971); Ryuel’ D, Takens F Strannye Attraktory (Per. pod red. Ya G Sinaya, L P Shil’nikova) (M.: Mir, 1981) p. 117
  49. Newhouse S, Ruelle D, Takens F Commun. Math. Phys. 64 35 (1978)
  50. Lorenz E N J. Atmos. Sci. 20 130 (1963); Lorents E Strannye Attraktory (Per. pod red. Ya G Sinaya, L P Shil’nikova) (M.: Mir, 1981) p. 88
  51. Tucker W Found. Comput. Math. 2 53 (2002)
  52. Morales C A Ann. Inst. Henri Poincaré Non Linéaire 13 589 (1996)
  53. Lopes A O Geometric Dynamics (Lecture Notes in Mathematics) Vol. 1007 (Ed. J Palis) (Berlin: Springer-Verlag, 1983) p. 498
  54. Hunt T J "Low dimensional dynamics: bifurcations of cantori and realisations of uniform hyperbolicity" PhD Thesis (Cambridge: Univ. of Cambridge, 2000)
  55. Akhiezer N I Elementy Teorii Ellipticheskikh Funktsii (M.: Nauka, 1970); Akhiezer N I Elements Of The Theory Of Elliptic Functions (Providence, RI: American Mathematical Society, 1999)
  56. Aidarova Yu S, Kuznetsov S P Izv. Vuzov Prikladnaya nelineinaya dinamika 16 (3) 176 (2008); Aidarova Yu S, Kuznetsov S P arXiv:0901.2727
  57. Hunt T J , MacKay R S Nonlinearity 16 1499 (2003)
  58. Thurston W P, Weeks J R Sci. Am. 251 (1) 94 (1984); Tersten U P, Uiks Dzh R V Mire Nauki (9) 74 (1984)
  59. Belykh V, Belykh I, Mosekilde E Int. J. Bifurcat. Chaos 15 3567 (2005)
  60. González-Miranda J M Int. J. Bifurcat. Chaos 17 3071 (2007)
  61. Turaev D V, Shil’nikov L P Dokl. Ross. Akad. Nauk 342 596 (1995); Turaev D V, Shil’nikov L P Dokl. Math. 51 404 (1995)
  62. Shil’nikov L P, Turaev D V Comput. Math. Appl. 34 (2 - 4) 173 (1997)
  63. Gavrilov N K, Shilnikov A L Am. Math. Soc. Transl. II 200 165 (1999)
  64. Shilnikov A, Cymbalyuk G Phys. Rev. Lett. 94 048101 (2005)
  65. Glyzin S D, Kolesov A Yu, Rozov N Kh Differ. Equat. 44 (2) 161 (2008)
  66. Kuznetsov S P Regul. Chaotic Dyn. 15 (2-3) 348 (2010)
  67. Halbert J T , Yorke J A, http://www.math.umd.edu/~halbert/taffy-paper-1.pdf
  68. Reichl L E The Transition To Chaos: Conservative Classical Systems And Quantum Manifestations (New York: Springer, 2004)
  69. Heagy J F Physica D 57 436 (1992)
  70. Kuznetsov A P, Savin A V, Savin D V Physica A 387 1464 (2008)
  71. Tyuryukina L V Nelineinyi Mir 8 (2) 72 (2010)
  72. Kuznetsov S P, Tyuryukina L V Izv. Vuzov Prikladnaya nelineinaya dinamika 18 (5) 80 (2010)
  73. Kuznetsov S P Commun. Nonlinear Sci. Numerical Simulation 14 3487 (2009)
  74. Kuznetsov S P Nelineinaya Dinamika 5 (3) 403 (2009)
  75. Kuznetsov S P Chaos 19 013114 (2009)
  76. Klyshko D N Fizicheskie Osnovy Kvantovoi Elektroniki (M.: Nauka, 1986)
  77. Rozenberg G V Usp. Fiz. Nauk 56 77 (1955)
  78. Kuznetsov S P Phys. Rev. Lett. 95 144101 (2005)
  79. Isaeva O B, Jalnine A Yu, Kuznetsov S P Phys. Rev. E 74 046207 (2006)
  80. Isaeva O B, Kuznetsov S P, Osbaldestin A Pis’ma ZhTF 33 (17) 69 (2007); Isaeva O B, Kuznetsov S P, Osbaldestin A Tech. Phys. Lett. 33 748 (2007)
  81. Isaeva O B, Kuznetsov S P, Osbaldestin A H Physica D 237 873 (2008)
  82. Zhalnin A Yu, Kuznetsov S P Zh. Tekh. Fiz. 77 (4) 10 (2007); Jalnine A Yu, Kuznetsov S P Tech. Phys. 52 401 (2007)
  83. Kuptsov P V, Kuznetsov S P Nelineinaya Dinamika 2 (3) 307 (2006)
  84. Kuznetsov S P, Sataev I R Izv. Vuzov Prikladnaya nelineinaya dinamika 14 (5) 3 (2006)
  85. Kuznetsov S P, Sataev I R Phys. Lett. A 365 97 (2007)
  86. Wilczak D SIAM J. Appl. Dyn. Sys. 9 1263 (2010)
  87. Kuptsov P V, Kuznetsov S P, Sataev I R, arXiv:0804.3677
  88. Kuznetsov A P, Kuznetsov S P, Pikovskii A S, Tyuryukina L V Izv. Vuzov Prikladnaya nelineinaya dinamika 15 (6) 75 (2007)
  89. Kuznetsov S P, Pikovsky A Physica D 232 87 (2007)
  90. Kuznetsov S P, Pikovsky A, Sataev I R Proc. of the III Intern. Conf. "Frontiers of Nonlinear Physics" (Nizhny Novgorod: IAP RAS, 2007) p. 66
  91. Louisell W H Coupled Mode And Paramagnetic Electronics (New York: Wiley, 1960); Lyuisell U Svyazannye i Parametricheskie Kolebaniya v Elektronike (M.: IL, 1963)
  92. Akhmanov S A, Khokhlov R V Usp. Fiz. Nauk 88 439 (1966); Akhmanov S A, Khokhlov R V Sov. Phys. Usp. 9 210 (1966)
  93. Kuznetsov A P, Kuznetsov S P, Ryskin N M Nelineinye Kolebaniya (M.: Fizmatlit, 2005)
  94. Kuznetsov S P Zh. Eksp. Teor. Fiz. 133 438 (2008); Kuznetsov S P JETP 106 380 (2008)
  95. Kuznetsov A S, Kuznetsov S P, Sataev I R Zh. Tekh. Fiz. 80 (12) 1 (2010); Kuznetsov A S, Kuznetsov S P, Sataev I R Tech. Phys. 55 1707 (2010)
  96. Kuznetsov S P, Ponomarenko V I Pis’ma ZhTF 34 (18) 1 (2008); Kuznetsov S P, Ponomarenko V I Tech. Phys. Lett. 34 771 (2008)
  97. Kuznetsov S P, Pikovsky A Europhys. Lett. 84 10013 (2008)
  98. Baranov S V, Kuznetsov S P, Ponomarenko V I Izv. Vuzov Prikladnaya nelineinaya dinamika 18 (1) 11 (2010)
  99. Kuznetsov C P, Seleznev E P Zh. Eksp. Teor. Fiz. 129 400 (2006); Kuznetsov S P, Seleznev E P JETP 102 355 (2006)
  100. Emel’yanov V V, Kuznetsov S P, Ryskin N M Pis’ma ZhTF 35 (16) 71 (2009); Emel’yanov V V, Kuznetsov S P, Ryskin N M Tech. Phys. Lett. 35 773 (2009)
  101. Wolf A et al. Physica D 16 285 (1985)
  102. Bunimovich L A, Sinai Ya G Nonlinearity 1 491 (1988)
  103. Bunimovich L A, Sinai Ya G Theory And Application Of Coupled Map Lattices (Ed. K Kaneko) (Singapore: John Wiley, 1993) p. 169
  104. Kuptsov P V, Kuznetsov S P Phys. Rev. E 80 016205 (2009)
  105. Dmitriev A S, Panas A I Dinamicheskii Khaos: Novye Nositeli Informatsii dlya Sistem Svyazi (M.: Fizmatlit, 2002)
  106. Yang T Int. J. Comput. Cognition 2 (2) 81 (2004)
  107. Argyris A et al. Nature 438 343 (2005)
  108. Koronovskii A A, Moskalenko O I, Khramov A E Usp. Fiz. Nauk 179 1281 (2009); Koronovskii A A, Moskalenko O I, Hramov A E Phys. Usp. 52 1213 (2009)
  109. Banerjee S, Yorke J A, Grebogi C Phys. Rev. Lett. 80 3049 (1998)
  110. Elhadj Z, Sprott J C Front. Phys. China 3 (2) 195 (2008)
  111. Drutarovský M, Galajda P Radioengineering 16 (3) 120 (2007)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions