RSS feeds
ÐóññêèéEnglish
Issue 4, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

2010

 / 

December

←  →

Reviews of topical problems


Fascination of chaos


Lomonosov Moscow State University, Faculty of Physics, Leninskie Gory 1 build. 2, Moscow, 119991, Russian Federation

This review introduces most of the concepts used in the study of chaotic phenomena in nonlinear systems and has as its objective to summarize the current understanding of results from the theory of chaotic dynamical systems and to describe the original ideas underlying the study of deterministic chaos. The presentation relies on informal analysis, with abstract mathematical ideas visualized geometrically or by examples from physics. Hyperbolic dynamics, homoclinic trajectories and tangencies, wild hyperbolic sets, and different types of attractors which appear in dynamical systems are considered. The key aspects of ergodic theory are discussed, and the basic statistical properties of chaotic dynamical systems are described. The fundamental difference between stochastic dynamics and deterministic chaos is explained. The review concludes with an investigation of the possibility of studying complex systems on the basis of the analysis of registered signals, i.e., the generated time series.

Fulltext pdf (404 KB)
Fulltext is also available at DOI: 10.3367/UFNe.0180.201012d.1305
PACS: 02.50.−r, 05.45.−a, 47.52.+j (all)
DOI: 10.3367/UFNe.0180.201012d.1305
URL: https://ufn.ru/en/articles/2010/12/d/
000288495700004
2-s2.0-79955613299
Citation: Loskutov A "Fascination of chaos" Phys. Usp. 53 1257–1280 (2010)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Îðèãèíàë: Ëîñêóòîâ À Þ «Î÷àðîâàíèå õàîñà» ÓÔÍ 180 1305–1329 (2010); DOI: 10.3367/UFNr.0180.201012d.1305

References (182) ↓ Cited by (34) Similar articles (20)

  1. Ginzburg V L Fizika KhKh Veka. Razvitie i Perspektivy (Otv. red. E P Velikhov) (M.: Nauka, 1984) p. 281
  2. Smale S Differential And Combinatorial Topology (Gen. Ed. S S Cairns) (Princeton, NJ: Princeton Univ. Press, 1965) p. 63
  3. Smale S Bull. Am. Soc. 73 747 (1967); Smeil S Usp. Mat. Nauk 25 113 (1970)
  4. Anosov D V Dokl. Akad. Nauk SSSR 145 707 (1962); Anosov D V Sov. Math. Dokl. 3 1068 (1962)
  5. Anosov D V Dokl. Akad. Nauk SSSR 151 1250 (1963); Anosov D V Sov. Math. Dokl. 4 1153 (1963)
  6. Anosov D V Geodezicheskie Potoki na Zamknutykh Rimanovykh Mnogoobraziyakh Otritsatel’noi Krivizny (M.: Nauka, 1967)
  7. Shil’nikov L P Dokl. Akad. Nauk SSSR 160 558 (1965); Shil’nikov L P Sov. Math. Dokl. 6 163 (1965)
  8. Shil’nikov L P Dokl. Akad. Nauk SSSR 172 298 (1967); Shil’nikov L P Sov. Math. Dokl. 8 102 (1967)
  9. Sinai Ya G Dokl. Akad. Nauk SSSR 153 1261 (1963)
  10. Sinai Ya G Usp. Mat. Nauk 25 141 (1970); Sinai Ya G Russ. Math. Surv. 25 137 (1970)
  11. Sinai Ya G, Shil’nikov L P (Red. per. s angl.) Strannye Attraktory. Sb. Statei (M.: Mir, 1981)
  12. Mukhin R R Ocherki Po Istorii Dinamicheskogo Khaosa (issledovaniya v SSSR v 1950 - 1980-e Gody) (M.: Vest-Konsalting, 2007)
  13. Loskutov A Yu Usp. Fiz. Nauk 177 989 (2007); Loskutov A Phys. Usp. 50 939 (2007)
  14. Ruelle D, Takens F Commun. Math. Phys. 20 167 (1971)
  15. Ruelle D Lecture Notes Phys. 12 292 (1975)
  16. Rabinovich M I, Sushchik M M Usp. Fiz. Nauk 160 (1) 3 (1990); Rabinovich M I, Sushchik M M Sov. Phys. Usp. 33 1 (1990)
  17. Landa P S, McClintock P V E Phys. Rep. 397 1 (2004)
  18. Hartman P Ordinary Differential Equations (New York: Wiley, 1964); Khartman F Obyknovennye Differentsial’nye Uravneniya (M.: Mir, 1970)
  19. Andronov A A, Pontryagin L S Dokl. Akad. Nauk SSSR 14 247 (1937); Andronov A A, Pontryagin L S C.R. Acad. Sci. USSR 14 247 (1937)
  20. Milnor J Commun. Math. Phys. 99 177 (1985)
  21. Afraimovich V S Nelineinye Volny. Dinamika i Evolyutsiya (Otv. red. A V Gaponov-Grekhov, M I Rabinovich) (M.: Nauka, 1989) p. 16
  22. Guckenheimer J, Holmes P Nonlinear Oscillations: Dynamical Systems, And Bifurcations Of Vector Fields (New York: Springer, 1997); Gukenkheimer Dzh, Kholms F Nelineinye Kolebaniya, Dinamicheskie Sistemy i Bifurkatsii Vektornykh Polei (M. - Izhevsk: Inst. komp. issled., 2002)
  23. Arnol’d V I i dr. Dinamicheskie Sistemy-5 (Itogi nauki i tekhniki. Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya) Vol. 5 (Pod red. V I Arnol’da, R V Gamkrelidze) (M.: VINITI, 1986) p. 5; Arnold V I et al. Dynamical Systems V: Bifurcation Theory And Catastrophe Theory (Encyclopedia of Mathematical Sciences) (Berlin: Springer, 1994) p. 1
  24. Eckmann J-P Rev. Mod. Phys. 53 643 (1981)
  25. Devaney R L An Introduction To Chaotic Dynamical Systems 2nd ed. (New York: Addison-Wesley Publ. Co., 2003)
  26. Banks J et al. Am. Math. Mon. 99 332 (1992)
  27. Assaf D (IV), Gadbois S Am. Math. Mon. 99 865 (1992)
  28. Kolesov A Yu, Rozov N Kh Usp. Mat. Nauk 64 (4) 125 (2009); Kolesov A Yu, Rozov N Kh Russ. Math. Surv. 64 701 (2009)
  29. Sinai Ya G Sovremennye Problemy Ergodicheskoi Teorii (M.: Fizmatlit, 1995)
  30. Katok A B, Hasselblatt B Introduction To The Modern Theory Of Dynamical Systems (Cambridge: Cambridge Univ. Press, 1995); Katok A B, Khasselblat B Vvedenie v Teoriyu Dinamicheskikh Sistem (M.: Izd-vo MTsNMO, 2005)
  31. Katok A B, 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)
  32. Anosov D V, Solodov V V Dinamicheskie Sistemy-9 (Itogi nauki i tekhniki. Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya) Vol. 66 (M.: VINITI, 1991) p. 12; Anosov D V, Solodov V V Dynamical Systems IX (Encyclopedia of Mathematical Sciences) Vol. 66 (Berlin: Springer, 1995) p. 10
  33. Kuznetsov S P Phys. Rev. Lett. 95 144101 (2005)
  34. Kuznetsov S P, Pikovsky A Physica D 232 87 (2007)
  35. Kuznetsov S P Nelineinaya Dinamika 5 403 (2009)
  36. Tucker W C.R. Acad. Sci. Paris I Math. 328 1197 (1999)
  37. Tucker W Found. Comput. Math. 2 53 (2002)
  38. Smale S The Mathematics Of Time. Essays On Dynamical Systems, Economic Processes, And Related Topics (Berlin: Springer, 1980)
  39. Smale S The Chaos Avant-Garde. Memories Of The Early Days Of Chaos Theory (Eds R Abraham, Y Ueda) (Singapore: World Scientific, 2000) p. 7
  40. Holmes P J, Moon F C Trans. ASME E 50 1021 (1983)
  41. Loskutov A Yu, Mikhailov A S Osnovy Teorii Slozhnykh Sistem (M. - Izhevsk: RKhD, Inst. komp. issled., 2007)
  42. Zaslavskii G M Gamil’tonov Khaos i Fraktal’naya Dinamika (M. - Izhevsk: RKhD, Inst. komp. issled., 2010)
  43. Alekseev V M Usp. Mat. Nauk 24 (1) 185 (1969)
  44. Alekseev V M Lektsii Po Nebesnoi Mekhanike (M. - Izhevsk: RKhD, 2001)
  45. Levi M Mem. AMS 214 1 (1981)
  46. Holmes P J. Sound Vib. 84 173 (1982)
  47. Moon F C Chaotic Vibrations (New York: Wiley, 1987); Mun F Khaoticheskie Kolebaniya (M.: Mir, 1990)
  48. Poincaré H Les Méthodes Nouvelles De La Méchanique Céleste (Paris: Gauthier-Villars, 1982); Poincaré H New Methods Of Celestial Mechanics (Washington: NASA, 1960); Puankare A Novye Metody Nebesnoi Mekhaniki (M.: Nauka, 1971)
  49. Shil’nikov L P i dr. Metody Kachestvennoi Teorii v Nelineinoi Dinamike Vol. 1 (M. - Izhevsk: Inst. komp. issled., 2004); Shil’nikov L P i dr. Metody Kachestvennoi Teorii v Nelineinoi Dinamike Vol. 2 (M. - Izhevsk: RKhD, Inst. komp. issled., 2009)
  50. Pesin Ya B Dinamicheskie Sistemy-2 (Itogi nauki i tekhniki. Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya) Vol. 2 (M.: VINITI, 1985) p. 123
  51. Birkhoff G D Collected Mathematical Papers (New York: Am. Math. Soc., 1950)
  52. Birkhoff G D Mém. Pont. Acad. Sci. Novi Lyncaei 1 85 (1935)
  53. Katok A Publ. Math. Inst. Hautes Étud. Sci. 51 137 (1980)
  54. Shil’nikov L P Matem. Sb. 81 92 (1970); Shil’nikov L P Math. USSR Sb. 10 91 (1970)
  55. Arnéodo A, Coullet P, Tresser C Commun. Math. Phys. 79 573 (1981)
  56. Arnéodo A, Coullet P, Tresser C J. Stat. Phys. 27 171 (1982)
  57. Gavrilov N K, Shil’nikov L P Matem. Sb. 88 475 (1972); Gavrilov N K, Shil’nikov L P Math. USSR Sb. 17 467 (1972)
  58. Gavrilov N K, Shil’nikov L P Matem. Sb. 90 139 (1973); Gavrilov N K, Shil’nikov L P Math. USSR Sb. 19 139 (1973)
  59. Gonchenko S V, Shil’nikov L P (Red.) Gomoklinicheskie Kasaniya. Sb. Statei. (M. - Izhevsk: RKhD, Inst. komp. issled., 2007)
  60. Palis J Global Analysis, Proc. Symp. Pure Math. 14 221 (1970)
  61. Hirsch M, Pugh C Global Analysis, Proc. Symp. Pure Math. 14 133 (1970)
  62. Nitecki Z Differentiable Dynamics (Cambridge, Mass.: M.I.T. Press, 1971); Nitetski Z Vvedenie v Differentsial’nuyu Dinamiku (M.: Mir, 1975)
  63. Newhouse S, Palis J Astérisque 31 44 (1976)
  64. Palis J, Takens F Invent. Math. 82 397 (1985)
  65. Palis J, Takens F Ann. Math. 125 337 (1987)
  66. Sten’kin O V, Shil’nikov L P Matem. Sb. 189 (4) 125 (1998); Sten’kin O V, Shil’nikov L P Sb. Math. 189 603 (1998)
  67. Newhouse S E Global Analysis, Proc. Symp. Pure Math. 14 191 (1970)
  68. Newhouse S E Topology 13 9 (1974)
  69. Newhouse S E Publ. Math. Inst. Hautes 50 101 (1979)
  70. Gonchenko S V, Shil’nikov L P, Turaev D V Chaos 6 15 (1996)
  71. Gonchenko S V, Turaev D V, Shil’nikov L P Trudy MIAN 216 76 (1997); Gonchenko S V, Turaev D V, Shil’nikov L P Proc. Steklov Inst. Math. 216 70 (1997)
  72. Gonchenko S V, Sten’kin S V, Shil’nikov L P Nelineinaya Dinamika 2 3 (2006)
  73. Gonchenko S V, Turaev D V, Shil’nikov L P Dinamicheskie Sistemy-6 (Itogi nauki i tekhniki. Ser. Sovremennaya matematika i ee prilozheniya) Vol. 67 (M.: VINITI, 1999) p. 69; Gonchenko S V, Turaev D V, Shil’nikov L P J. Math. Sci. 105 (1) 1738 (2001)
  74. Gonchenko S V et al. Nonlinearity 10 409 (1997)
  75. Gonchenko S V, Shil’nikov L P, Turaev D V Physica D 62 1 (1993)
  76. Downarowicz T, Newhouse S Invetn. Math. 160 453 (2005)
  77. Kaloshin V Yu Commun. Math. Phys. 211 253 (2000)
  78. Shil’nikov A L, Shil’nikov L P, Turaev D V Int. J. Bifurcat. Chaos 3 1123 (1993)
  79. Newhouse S, Palis J, Takens F Publ. Math. Inst. Hautes Étud. Sci. 57 5 (1983)
  80. Afraimovich V S, Shil’nikov L P Metody Kachestvennoi Teorii Differentsial’nykh Uravnenii (Otv. red. E A Leontovich-Andronova) (Gor’kii: GGU, 1983) p. 3
  81. Ovsyannikov I M, Shil’nikov L P Matem. Sb. 130 552 (1986); Ovsyannikov I M, Shil’nikov L P Math. USSR Sb. 58 557 (1987)
  82. Ovsyannikov I M, Shil’nikov L P Matem. Sb. 182 1043 (1991); Ovsyannikov I M, Shil’nikov L P Math. USSR Sb. 73 415 (1992)
  83. Palis J, Takens F Hyperbolicity And Sensitive Chaotic Dynamics At Homoclinic Bifurcations (Cambridge: Cambridge Univ. Press, 1993)
  84. Shil’nikov L P Matem. Sb. 74 378 (1967); Shil’nikov L P Math. USSR Sb. 3 353 (1967)
  85. Bouen R Metody Simvolicheskoi Dinamiki (M.: Mir, 1979)
  86. Ruelle D Hasard Et Chaos (Paris: O. Jacob, 1991); Ruelle D Chance And Chaos (Princeton, N.J.: Princeton Univ. Press, 1991); Ryuel’ D Sluchainost’ i Khaos (M. - Izhevsk: RKhD, 2001)
  87. Plykin R V, Sataev E A, Shlyachkov S V Dinamicheskie Sistemy-9 (Itogi nauki i tekhniki. Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya) Vol. 66 (M.: VINITI, 1991) p. 100
  88. Gonchenko S V, Turaev D V, Shil’nikov L P Itogi Nauki i Tekhniki (Ser. Sovremennaya matematika i ee prilozheniya) Vol. 7 (M.: VINITI, 2003) p. 92
  89. Mora L, Viana M Acta Math. 71 1 (1993)
  90. Viana M Proc. of XIth Intern. Congress of Mathematical Physics, Paris, France, July 18 - 23, 1994 (Ed. D Iagolnitzer) (Cambridge, MA: Intern. Press, 1995) p. 142
  91. Viana M Bull. Braz. Math. Soc. 24 (1) 13 (1993)
  92. Palis J, Viana M Ann. Math. 140 207 (1994)
  93. Grebogi C et al. Physica D 13 261 (1984)
  94. Plykin R V Matem. Sb. 94 243 (1974); Plykin R V Math. USSR Sb. 23 233 (1974)
  95. Plykin R V Usp. Mat. Nauk 39 (6) 75 (1984); Plykin R V Russ. Math. Surv. 39 (6) 85 (1984)
  96. Belykh V N Sistemy Fazovoi Sinkhronizatsii (Pod red. V V Shakhgil’dyana, L N Belyustinoi) (M.: Radio i svyaz’, 1982) p. 161
  97. Bunimovich L A Dinamicheskie Sistemy-2 (Itogi nauki i tekhniki. Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya) Vol. 2 (M.: VINITI, 1985) p. 173
  98. Collet P, Levi Y Commun. Math. Phys. 93 461 (1984)
  99. Lozi R J. Physique Colloq. 39 C5-9 (1978)
  100. Sataev E A J. Math. Sci. 95 2564 (1999)
  101. Misiurewicz M Ann. N.Y. Acad. Sci. 357 348 (1980)
  102. Pesin Ya B Ergodic Theory Dyn. Syst. 12 123 (1992)
  103. Lorenz E N J. Atmos. Sci. 20 130 (1963)
  104. Hénon M Commun. Math. Phys. 50 69 (1976)
  105. Benedicks M, Carleson L Ann. Math. 133 73 (1991)
  106. Kuznetsov S P, Seleznev E P Zh. Eksp. Teor. Fiz. 129 400 (2006); Kuznetsov S P, Seleznev E P JETP 102 355 (2006)
  107. Morales C A Ann. Inst. Henri Poincaré 13 589 (1996)
  108. Belykh V, Belykh I, Mosekilde E Int. J. Bifurcat. Chaos 15 3567 (2005)
  109. Kuznetsov S P Chaos 19 013114 (2009)
  110. Sinai Ya G Nelineinye Volny (Otv. red. A V Gaponov-Grekhov) (M.: Nauka, 1979) p. 192
  111. Haken H A Advanced Synergetics (Berlin: Springer-Verlag, 1983); Khaken G Sinergetika (M.: Mir, 1985)
  112. Afraimovich V S, Shilnikov L P Nonlinear Dynamics And Turbulence (Eds G I Barenblatt, G Iooss, D D Joseph) (New York: Pitman, 1983) p. 1
  113. Garrido L, Simó C et al. Lect. Notes Phys. 179 1 (1983)
  114. Anishchenko V S i dr. Usp. Fiz. Nauk 175 163 (2005); Anishchenko V S et al. Phys. Usp. 48 151 (2005)
  115. Afraimovich V S, Bykov V V, Shil’nikov L P Usp. Mat. Nauk 35 (4) 164 (1980)
  116. Bykov V V Metody Kachestvennoi Teorii Differentsial’nykh Uravnenii (Otv. red. E A Leontovich-Andronova) (Gor’kii: GGU, 1980) p. 44
  117. Arnol’d V I Dopolnitel’nye Glavy Teorii Obyknovennykh Differentsial’nykh Uravnenii (M.: Nauka, 1978); Arnold V I Geometrical Methods In The Theory Of Ordinary Differential Equations (New York: Springer-Verlag, 1983)
  118. Markus L, Meyer K R Mem. Am. Math. Soc. 144 1 (1974)
  119. Perry A D, Wiggins S Physica D 71 102 (1994)
  120. Zaslavsky G M Phys. Rep. 371 461 (2002)
  121. Bunimovich L A Nonlinearity 21 (2) T13 (2008)
  122. Chirikov B V, Shepelyansky D L Physica D 13 395 (1984)
  123. Bunimovich L A Chaos 11 802 (2001)
  124. Altmann E G, Motter A E, Kantz H Chaos 15 033105 (2005)
  125. Marchal C The Three-Body Problem (Amsterdam: Elsevier, 1990); Marshal K Zadacha Trekh Tel (M. - Izhevsk: RKhD, 2004)
  126. Bunimovich L A Commun. Math. Phys. 65 295 (1979)
  127. Kornfel’d I P, Sinai Ya G, Fomin S V Ergodicheskaya Teoriya (M.: Nauka, 1980); Cornfeld I P, Fomin S V, Sinai Ya G Ergodic Theory (New York: Springer-Verlag, 1982)
  128. Eckmann J-P, Ruelle D Rev. Mod. Phys. 57 617 (1985)
  129. Arnold L, Wihstutz V (Eds) Lyapunov Exponents. Proc Of A Workshop, Bremen, Germany, November 12 - 15, 1984 (Lecture Notes in Mathematics) Vol. 1186 (Berlin: Springer-Verlag, 1986)
  130. Oseledets V I Trudy Mosk. Matem. Ob-va 19 179 (1968); Oseledets V I Trans. Mosc. Math. Soc. 19 197 (1968)
  131. Millionshchikov V M Matem. Sb. 78 (2) 179 (1969); Millionshchikov V M Math. USSR Sb. 7 171 (1969)
  132. Pesin Ya B Usp. Mat. Nauk 32 (4) 55 (1977); Pesin Ya B Russ. Math. Surv. 32 (4) 55 (1977)
  133. Kaplan J L, Yorke J A Lect. Notes Math. 730 228 (1979)
  134. Paladin G, Vulpiani A Phys. Rep. 156 147 (1987)
  135. Martin N F G, England J W Mathematical Theory Of Entropy (Cambridge: Cambridge Univ. Press, 1984); Martin N, Inglend Dzh Matematicheskaya Teoriya Entropii (M.: Mir, 1988)
  136. Ornstein D S Ergodic Theory, Randomness, And Dynamical Systems (New Haven: Yale Univ. Press, 1974); Ornstein D Ergodicheskaya Teoriya, Sluchainost’ i Dinamicheskie Sistemy (M.: Mir, 1978)
  137. Billingsley P Ergodic Theory And Information (New York: Wiley, 1965); Billingslei P Ergodicheskaya Teoriya i Informatsiya (M.: Mir, 1969)
  138. Hentschel H G E, Procaccia I Physica D 8 435 (1983)
  139. Malinetskii G G, Potapov A B Sovremennye Problemy Nelineinoi Dinamiki (M.: URSS, 2000)
  140. Loskutov A Yu, Kozlov A A, Khakhanov Yu M Izv. Vuzov Prikladnaya nelineinaya dinamika 17 (4) 98 (2009)
  141. Grassberger P, Procaccia I Physica D 9 189 (1983)
  142. Grassberger P, Procaccia I Phys. Rev. A 28 2591 (1983)
  143. Sinai Ya G Dinamicheskie Sistemy-2 (Itogi nauki i tekhniki. Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya) Vol. 2 (M.: VINITI, 1985) p. 115
  144. Sinai Ya G Vvedenie v Ergodicheskuyu Teoriyu (M.: FAZIS, 1996)
  145. Sinai Ya G Usp. Mat. Nauk 46 (3) 147 (1991); Sinai Ya G Russ. Math. Surv. 46 (3) 177 (1991)
  146. Halmos P R Lectures On Ergodic Theory (Tokyo: Math. Soc. of Japan, 1956); Khalmosh P R Lektsii Po Ergodicheskoi Teorii (Izhevsk: RKhD, 1999)
  147. Sinai Ya G Usp. Mat. Nauk 27 (4) 21 (1972); Sinai Ya G Russ. Math. Surv. 27 (4) 21 (1972)
  148. Bowen R, Ruelle D Invent. Math. 29 181 (1975)
  149. Ruelle D Am. J. Math. 98 619 (1976)
  150. Bowen R Equilibrium States And The Ergodic Theory Of Anosov Diffeomorphisms (Lecture Notes in Mathematics) Vol. 470 (Berlin: Springer, 1975)
  151. Young L-S J. Stat. Phys. 108 733 (2002)
  152. Benedicks M, Young L-S Invent. Math. 112 541 (1993)
  153. Zermelo E Ann. Phys. 293 485 (1896)
  154. Bol’tsman L Izbrannye Trudy (M.: Nauka, 1984)
  155. Boltzmann L Wiedemann Ann. 60 392 (1896)
  156. Krylov N S Raboty Po Obosnovaniyu Statisticheskoi Fiziki (M.-L.: Izd-vo AN SSSR, 1950); Krylov N S Works On The Foundations Of Statistical Physics (Princeton, N.J.: Princeton Univ. Press, 1979)
  157. Pustyl’nikov L D Usp. Mat. Nauk 50 (1) 143 (1995); Pustyl’nikov L D Russ. Math. Surv. 50 (1) 145 (1995)
  158. Bezruchko B P, Smirnov D A Matematicheskoe Modelirovanie i Khaoticheskie Vremennye Ryady (Saratov: Kolledzh, 2005)
  159. Loskutov A, Mironyuk O Int. J. Bifurcat. Chaos 17 3709 (2007)
  160. Ardashev A V, Loskutov A Yu Prakticheskie Aspekty Sovremennykh Metodov Analiza Variabel’nosti Serdechnogo Ritma (M.: Medpraktika, 2010)
  161. Loskutov A, Kotlyarov O Valyutnyi Spekulyant (11) 8 (2008)
  162. Lokutov A Yu Ekonomika Mat. Metody 46 58 (2010)
  163. Gouesbet G, Meunier-Guttin-Cluzel S, Ménard O (Eds) Chaos And Its Reconstruction (New York: Nova Sci. Publ., 2003)
  164. Packard N H et al. Phys. Rev. Lett. 45 712 (1980)
  165. Takens F Lect. Notes Math. 898 336 (1981)
  166. Vautard R, Yiou P, Ghil M Physica D 58 95 (1992)
  167. Danilov D L, Zhiglyavskii A A (Red.) Glavnye Komponenty Vremennykh Ryadov: Metod "Gusenitsa". Sb. Statei (SPb.: S.-Peterb. gos. un-t, 1997)
  168. Farmer J D, Sidorowich J J Phys. Rev. Lett. 59 845 (1987)
  169. Istomin I A, Kotlyarov O L, Loskutov A Yu Teor. Mat. Fiz. 142 148 (2005); Istomin I A, Kotlyarov O L, Loskutov A Yu Theor. Math. Phys. 142 128 (2005)
  170. Loskutov A Yu i dr. Pis’ma Astron. Zhurn. 27 867 (2001); Loskutov A Yu et al. Astron. Lett. 27 745 (2001)
  171. Farmer J D, Sidorowich J J Evolution, Learning, And Cognition (Ed. Y C Lee) (Singapore: World Scientific, 1988) p. 277
  172. Loskutov A Yu i dr. Vestn. Mosk. Un-ta Sep. 3. Fiz. Astron. 6 3 (2002)
  173. Abarbanel H D I et al. Rev. Mod. Phys. 65 1331 (1993)
  174. Franses P H Time Series Models For Business And Economic Forecasting (Cambridge: Cambridge Univ. Press, 1998)
  175. Ghil M et al. Rev. Geophys. 40 (1) 1003 (2002)
  176. Shumway R H, Stoffer D S Time Series Analysis And Its Applications (Berlin: Springer, 2000)
  177. Afraimovich V S, Reiman A M Nelineinye Volny. Dinamika i Evolyutsiya (Otv. red. A V Gaponov-Grekhov, M I Rabinovich) (M.: Nauka, 1989) p. 238
  178. Takens F Nonlinear Dynamics And Turbulence (Eds G I Barenblatt, G Iooss, D D Joseph) (New York: Pitman, 1983) p. 314
  179. Broer H W et al. Structures In Dynamics: Finite Dimensional Deterministic Studies (Amsterdam: North-Holland, 1991); Brur Kh V i dr. Struktury v Dinamike. Konechnomernye Determinirovannye Sistemy (M. - Izhevsk: RKhD, Inst. komp. issled., 2003) p. 119
  180. Mandelbrot B B The Fractal Geometry Of Nature (New York: W.H. Freeman, 1983); Mandel’brot B Fraktal’naya Geometriya Prirody (M.: In-t komp. issled., 2002)
  181. Peitgen H-O, Richter P H The Beauty Of Fractals: Images Of Complex Dynamical Systems (Berlin: Springer-Verlag, 1986); Paitgen Kh-O, Rikhter P Kh Krasota Fraktalov. Obrazy Kompleksnykh Dinamicheskikh Sistem (M.: Mir, 1993)
  182. Ruelle D Math. Intell. 2 126 (1980)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions