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Methodological notes


Dynamical chaos: systems of classical mechanics


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

This article is a methodological manual for those who are interested in chaotic dynamics. An exposition is given on the foundations of the theory of deterministic chaos that originates in classical mechanics systems. Fundamental results obtained in this area are presented, such as elements of the theory of nonlinear resonance and the Kolmogorov-Arnol\’d-Moser theory, the Poincaré-Birkhoff fixed-point theorem, and the Mel\’nikov method. Particular attention is given to the analysis of the phenomena underlying the self-similarity and nature of chaos: splitting of separatrices and homoclinic and heteroclinic tangles. Important properties of chaotic systems — unpredictability, irreversibility, and decay of temporal correlations — are described. Models of classical statistical mechanics with chaotic properties, which have become popular in recent years — billiards with oscillating boundaries — are considered. It is shown that if a billiard has the property of well-developed chaos, then perturbations of its boundaries result in Fermi acceleration. But in nearly-integrable billiard systems, excitations of the boundaries lead to a new phenomenon in the ensemble of particles, separation of particles in accordance their velocities. If the initial velocity of the particles exceeds a certain critical value characteristic of the given billiard geometry, the particles accelerate; otherwise, they decelerate.

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Fulltext is also available at DOI: 10.1070/PU2007v050n09ABEH006341
PACS: 05.45.−a, 05.45.Ac (all)
DOI: 10.1070/PU2007v050n09ABEH006341
URL: https://ufn.ru/en/articles/2007/9/d/
000252639900004
2-s2.0-38349081661
2007PhyU...50..939L
Citation: Loskutov A "Dynamical chaos: systems of classical mechanics" Phys. Usp. 50 939–964 (2007)
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Оригинал: Лоскутов А Ю «Динамический хаос. Системы классической механики» УФН 177 989–1015 (2007); DOI: 10.3367/UFNr.0177.200709d.0989

References (189) ↓ Cited by (48) Similar articles (20)

  1. Poincaré H Calcul Des Probabilités (Paris: Gauthier-Villars, 1912)
  2. Boltzmann L J. Math. 100 201 (1887)
  3. Boltzmann L Vorlesungen über Gastheorie (Leipzig: J.A. Barth, 1896)
  4. Bol’tsman L Stat’i i Rechi (M.: Nauka, 1970)
  5. Ehrenfest P, Ehrenfest T In Enzyklopaedie Der Mathematischen Wissenschaften Vol. IV (Leipzig, 1911), Tl. 32
  6. Erenfest P Otnositel’nost’. Kvanty. Statistika. Sbornik Statei (M.: Nauka, 1972)
  7. Kats M Veroyatnost’ i Smezhnye Voprosy v Fizike (M.: Mir, 1965)
  8. Cohen E G D, Thirring W (Eds) The Bolzmann Equation: Theory And Applications (Acta Physica Austriaca, Suppl. 10) (Wien: Springer-Verlag, 1973)
  9. Fermi E, Pasta J, Ulam S "Studies of nonlinear problems" Los Alamos Sci. Laboratory Rep. LA-1940 (Los Alamos: Los Alamos Scientific Laboratory, 1955)
  10. Ford J J. Math. Phys. 2 387 (1961)
  11. Jackson E A J. Math. Phys. 4 551 (1963)
  12. Puankare A Izbrannye Trudy Vol. 1 (M.: Nauka, 1971)
  13. Birkgof D Dinamicheskie Sistemy (Izhevsk: Izd. dom «Udm. un-t» (1999)
  14. Krylov N S Raboty Po Obosnovaniyu Statisticheskoi Fiziki (M.-L.: Izd-vo AN SSSR, 1950)
  15. Born M Usp. Fiz. Nauk 69 173 (1959); Born M Z. Phys. 153 372 (1958)
  16. Ford J "Foreword" Phys. Rep. 75 288 (1981)
  17. Li T-Y, Yorke J A Am. Math. Mon. 82 985 (1975)
  18. Katok A B, Khassel’blat B Vvedenie v Teoriyu Dinamicheskikh Sistem s Obzorom Poslednikh Dostizhenii (M.: Izd-vo MTsNMO, 2005)
  19. Sinai Ya G Imperiya Matematiki (3) 13 (2000)
  20. Anishchenko V S i dr. Usp. Fiz. Nauk 175 163 (2005); Anishchenko V S et al. Phys. Usp. 48 151 (2005)
  21. Kolmogorov A N Dokl. Akad. Nauk SSSR 98 527 (1954)
  22. Arnol’d V I Usp. Mat. Nauk 18 13 (1963)
  23. Arnol’d V I Usp. Mat. Nauk 18 91 (1963)
  24. Mozer Yu v Sb. Matematika. Periodicheskii Sbornik Perevodov Inostrannykh Statei Vol. 6 (M.: Mir, 1962) p. 51
  25. Mozer Yu Lektsii o Gamil’tonovykh Sistemakh (M.: Mir, 1973)
  26. Anosov D V Geodezicheskie Potoki na Zamknutykh Rimanovykh Mnogoobraziyakh Otritsatel’noi Krivizny (M.: Nauka, 1967)
  27. Sinai Ya G Dokl. Akad. Nauk SSSR 153 1261 (1963)
  28. Sinai Ya G Vestn. MGU. Ser. 1. Mat. Mekh. 18 (5) 6 (1963)
  29. Shil’nikov L P Dokl. Akad. Nauk SSSR 160 558 (1965)
  30. Shil’nikov L P Dokl. Akad. Nauk SSSR 172 298 (1967)
  31. Shil’nikov L P i dr. Metody Kachestvennoi Teorii v Nelineinoi Dinamike (M. – Izhevsk: Inst. komp’yut. issled., 2004)
  32. Arnol’d V I v Sb. Itogi Nauki i Tekhniki (Dinamicheskie sistemy, T. 5, Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya, Pod red. V I Arnol’da, R V Gamkrelidze) (M.: VINITI, 1986) p. 5
  33. Kolmogorov A N Dokl. Akad. Nauk SSSR 119 861 (1958)
  34. Kolmogorov A N Dokl. Akad. Nauk SSSR 124 754 (1959)
  35. Sinai Ya G Dokl. Akad. Nauk SSSR 124 768 (1959)
  36. Smale S In Differential And Combinatorial Topology (General Ed. S S Cairns) (Princeton, NJ: Princeton Univ. Press, 1965) p. 63
  37. Smeil S Usp. Mat. Nauk 25 113 (1970)
  38. Anosov D V Dokl. Akad. Nauk SSSR 145 707 (1962); Anosov D V Dokl. Akad. Nauk SSSR 151 1250 (1963)
  39. Bouen R Metody Simvolicheskoi Dinamiki (M.: Mir, 1979)
  40. Katok A, Khassel’blat B Vvedenie v Sovremennuyu Teoriyu Dinamicheskikh Sistem (M.: Faktorial, 1999)
  41. Nitetski Z Vvedenie v Differentsial’nuyu Dinamiku (M.: Mir, 1975)
  42. Lasota A, Mackey V C Chaos, Fractals, And Noise. Stochastic Aspects Of Dynamics 2nd ed. (New York: Springer-Verlag, 1994)
  43. Anosov D V v Sb. Itogi Nauki i Tekhniki (Dinamicheskie sistemy, T. 9, Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya) (M.: VINITI, 1991)
  44. Bunimovich L A, Sinai Ya G Commun. Math. Phys. 78 479 (1981)
  45. Bunimovich L A Itogi Nauki i Tekhniki (Dinamicheskie Sistemy, T. 2, Ser. Sovremennye Problemy Matematiki. Fundamental’nye Napravleniya) (M.: VINITI, 1985)
  46. Bunimovich L A Chaos 1 187 (1991)
  47. Tabachnikov A Billiards (Paris: France Math. Soc. Press, 1995)
  48. Dubrovin B A Itogi Nauki i Tekhniki (Dinamicheskie sistemy, T. 4, Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya) (M.: VINITI, 1985) p. 179
  49. Ol’shanetskii M A, Perelomov A M, Semenov-Tyan-Shanskii M A v Sb. Itogi Nauki i Tekhniki (Ser. Dinamicheskie sistemy, T. 7) (M.: VINITI, 1987) p. 86
  50. Trofimov V V, Fomenko A T v Sb. Itogi Nauki i Tekhniki (Dinamicheskie sistemy, T. 7, Ser. Sovremennye problemy matematiki. Fundamental’nye napravleniya) (M.: VINITI, 1987) p. 227
  51. Perelomov A M Integriruemye Sistemy Klassicheskoi Mekhaniki i Algebry Li (M.: Nauka, 1990)
  52. Tabor M Khaos i Integriruemost’ v Nelineinoi Dinamike (M.: Editorial URSS, 2001)
  53. Arnol’d V I, Kozlov V V, Neishtadt A I Matematicheskie Aspekty Klassicheskoi i Nebesnoi Mekhaniki (M.: Editorial URSS, 2002)
  54. Likhtenberg A, Liberman M Regulyarnaya i Stokhasticheskaya Dinamika (M.: Mir, 1984)
  55. Kozlov V V Simmetrii, Topologiya i Rezonansy v Gamil’tonovoi Mekhanike (Izhevsk: Izd-vo Udm. gos. un-ta, 1995)
  56. Trueba J L, Baltanás J P, Sanjuán M A F Chaos, Solitons, Fractals 15 911 (2003)
  57. Dzhanoev A R, Loskutov A, Cao H, Sanjuán M A F Discrete Continuous Dyn. Syst. B 7 275 (2007)
  58. Chirikov B V Atom. Energiya 6 630 (1959)
  59. Chirikov B V Phys. Rep. 52 263 (1979)
  60. Zaslavskii G M Fizika Khaosa v Gamil’tonovykh Sistemakh (M. – Izhevsk: Inst. komp’yut. issled., 2004)
  61. Zaslavskii G M, Sagdeev R Z Vvedenie v Nelineinuyu Fiziku: Ot Mayatnika do Turbulentnosti i Khaosa (M.: Nauka, 1988)
  62. Zaslavskii G M i dr. Slabyi Khaos i Kvaziregulyarnye Struktury (M.: Nauka, 1991)
  63. Morozov A D Rezonansy, Tsikly i Khaos v Kvazikonservativnykh Sistemakh (M. – Izhevsk: RKhD, 2005)
  64. Bogolyubov N N, Mitropol’skii Yu A Asimptoticheskie Metody v Teorii Nelineinykh Kolebanii 2-e izd. (M.: Fizmatgiz, 1958)
  65. Chirikov B V "Issledovaniya po teorii nelineinogo rezonansa i stokhastichnosti" Preprint №267 (Novosibirsk: IYaF AN SSSR, Sib. otd., 1969)
  66. Escande D F Phys. Rep. 121 165 (1985)
  67. Walker G H, Ford J Phys. Rev. 188 416 (1969)
  68. Vainberg S Gravitatsiya i Kosmologiya: Printsipy i Prilozheniya Obshchei Teorii Otnositel’nosti (M.: Mir, 1975)
  69. Diaku F, Kholms F Nebesnye Vstrechi. Istoki Khaosa i Nelineinosti (M. – Izhevsk: Inst. komp’yut. issled., 2004)
  70. de la Yave R Vvedenie v KAM-teoriyu (M. – Izhevsk: Inst. komp’yut. issled., 2003)
  71. Treshchev D V Vvedenie v Teoriyu Vozmushchenii Gamil’tonovykh Sistem (M.: Fazis, 1998)
  72. Arnol’d V I, Avets A Ergodicheskie Problemy Klassicheskoi Mekhaniki (M.: RKhD, 1999)
  73. Lazutkin V F KAM Theory And Semiclassical Approximations To Eigenfunctions (Berlin: Springer-Verlag, 1993)
  74. Broer H W, Huitema G B, Sevryuk M B Quasi-periodic Motions In Families Of Dynamical Systems: Order Amidst Chaos (Lecture Notes in Math., No. 1645) (Berlin: Springer, 1996)
  75. Broer H W, Huitema G B, Takens F Mem. Am. Math. Soc. 83 1 (1990)
  76. Pöschel J In Smooth Ergodic Theory And Its Applications (Proc. of Symp. in Pure Math., Vol. 69, Eds A Katok et al.) (Providence, RI: Am. Math. Soc., 2001) p. 707
  77. Arnol’d V I Dokl. Akad. Nauk SSSR 156 9 (1964)
  78. Zaslavskii G M, Chirikov B V Usp. Fiz. Nauk 105 3 (1971); Zaslavskii G M, Chirikov B V Sov. Phys. Usp. 14 549 (1972)
  79. Holmes P J, Marsden J E J. Math. Phys. 23 669 (1982)
  80. MacKay R S, Meiss J D, Persival I C Transport In Hamiltonian Systems (London: Queen Mary College, 1983); MacKay R S, Meiss J D, Persival I C Physica D 13 55 (1984)
  81. Tennyson J L, Lieberman M A, Lichtenberg A J AIP Conf. Proc. 57 272 (1979)
  82. Nekhoroshev N N Usp. Mat. Nauk 32 (6) 5 (1977)
  83. Puankare A Izbrannye Trudy T. 1 Novye Metody Nebesnoi Mekhaniki (M.: Nauka, 1971)
  84. Poincaré H Rendicont Circolo Mat. Palermo 33 375 (1912)
  85. Zigel’ K, Mozer Yu Lektsii Po Nebesnoi Mekhanike (M. – Izhevsk: RKhD, 2001)
  86. Berry M V AIP Conf. Proc. 46 16 (1978)
  87. Helleman R H G In Fundamental Problems In Statistical Mechanics Vol. 5 (Ed. E G D Cohen) (Amsterdam: North-Holland, 1980) p. 165
  88. Gukenkheimer Dzh, Kholms F Nelineinyi Kolebaniya, Dinamicheskie Sistemy i Bifurkatsii Vektornykh Polei (M. – Izhevsk: Inst. komp’yut. issled., 2002)
  89. Gel’freikh V G, Lazutkin V F Usp. Mat. Nauk 56 79 (2001)
  90. Shilnikov L in Nonlinear Dynamics, Chaotic and Complex Systems, Proc. of an Intern. Conf., Zakopane, Poland, November 7 – 12, 1995 (Eds E Infeld, R Żelazny, A Galkowski) (Cambridge: Cambridge Univ. Press, 1997) p. 39
  91. Mel’nikov V K Trudy Mosk. Matem. Obshchestva 12 3 (1963)
  92. Duffing G Erzwungene Schwingungen Bei Veränderlicher Eigenfrequenz Und Ihre Technische Bedeutung (Braunschweig: F. Vieweg und Sohn, 1918)
  93. Kuznetsov S P Dinamicheskii Khaos (M.: Fizmatlit, 2001)
  94. Mun F Khaoticheskie Kolebaniya (M.: Mir, 1990)
  95. Neimark Yu I, Landa P S Stokhasticheskie i Khaoticheskie Kolebaniya (M.: Nauka, 1987)
  96. Holmes P Philos. Trans. R. Soc. London A 292 419 (1979)
  97. Cuadros F, Chacón R Phys. Rev. E 47 4628 (1993)
  98. Schwalger T, Dzhanoev A, Loskutov A Chaos 16 023109 (2006)
  99. Holmes P J SIAM J. Appl. Math. 38 65 (1980)
  100. Hopf E Math. Ann. 117 590 (1940)
  101. Prigozhin I Ot Sushchestvuyushchego k Voznikayushchemu: Vremya i Slozhnost’ v Fizicheskikh Naukakh (M.: Nauka, 1985)
  102. Prigozhin I, Stengers I Vremya, Khaos, Kvant: K Resheniyu Paradoksa Vremeni (M.: Progress, 1994)
  103. Mikhailov A S, Loskutov A Foundations In Synergetics II: Complex Patterns (Springer Series in Synergetics, Vol. 52) 2nd ed. (Berlin: Springer, 1996)
  104. Hadamard J J. Math. Pures Appl. 4 27 (1898)
  105. Kozlov V V, Treshchev D V Billiardy. Geneticheskoe Vvedenie v Teoriyu Sistem s Udarami (M.: Izd-vo MGU, 1991)
  106. Bunimovich L A Commun. Math. Phys. 65 295 (1979)
  107. Bunimovich L A Chaos 11 802 (2001)
  108. Sinai Ya G Usp. Mat. Nauk 25 141 (1970)
  109. Bunimovich L A Matem. Sb. 94 49 (1974)
  110. Baldwin P R J. Phys. A 24 L941 (1991)
  111. Chernov N J. Stat. Phys. 88 1 (1997)
  112. Garrido P L J. Stat. Phys. 88 807 (1997)
  113. Loskutov A Yu, Ryabov A B, Akinshin L G Zh. Eksp. Teor. Fiz. 116 1781 (1999)
  114. Loskutov A, Ryabov A B, Akinshin L G J. Phys. A 33 7973 (2000)
  115. Fermi E Phys. Rev. 75 1169 (1949)
  116. Blandford R, Eichler D Phys. Rep. 154 1 (1987)
  117. Ulam S M in Proc. of the 4th Berkeley Symp. on Mathematical Statistics and Probability Vol. 3 (California: California Univ. Press, 1961) p. 315
  118. Brahic A Astron. Astrophys. 12 98 (1971)
  119. Zaslavskii G M Statisticheskaya Neobratimost’ v Nelineinykh Sistemakh (M.: Nauka, 1970)
  120. Lichtenberg A J, Lieberman M A, Cohen R H Physica D 1 291 (1980)
  121. Pustyl’nikov L D Dokl. Akad. Nauk SSSR 292 549 (1987)
  122. Pustyl’nikov L D Matem. Sb. 85 (6) 113 (1994)
  123. Krüger T, Pustyl’nikov L D, Troubetzkoy S E Nonlinearity 8 397 (1995)
  124. Pustyl’nikov L D Usp. Mat. Nauk 50 143 (1995)
  125. Karlis A K et al. Phys. Rev. Lett. 97 194102 (2006)
  126. Koiller J et al. Nonlinearity 8 983 (1995)
  127. Koiller J et al. J. Stat. Phys. 83 127 (1996)
  128. Kamphorst S O, de Carvalho S P Nonlinearity 12 1363 (1999)
  129. Tsang K Y, Ngai K L Phys. Rev. E 56 R17 (1997)
  130. Tsang K Y, Lieberman M A Physica D 11 147 (1984)
  131. Tsang K Y, Lieberman M A Phys. Lett. A 103 175 (1984)
  132. Loskutov A, Ryabov A J. Stat. Phys. 108 995 (2002)
  133. Loskutov A Yu, Ryabov A B v Sb. Nelineinye Volny — 2004 (Pod red A V Gaponova-Grekhova, V I Nekorkina) (N. Novgorod: IPF RAN, 2005) p. 510
  134. de Carvalho R E, de Souza F C, Leonel E D J. Phys. A 39 3561 (2006)
  135. de Carvalho R E, Souza F C, Leonel E D Phys. Rev. E 73 066229 (2006)
  136. Ryabov A B, Loskutov A Phys. Rev. Lett., in preparation
  137. Loskutov A Yu, Mikhailov A S Osnovy Teorii Slozhnykh Sistem (M. – Izhevsk: RKhD, Inst. komp’yut. issled., 2007)
  138. Zaslavsky G M Chaos 5 653 (1995)
  139. Kuznetsov L, Zaslavsky G M Phys. Rep. 288 457 (1997)
  140. Kuksin S B Nearly Integrable Infinite-Dimensional Hamiltonian Systems (Berlin: Springer-Verlag, 1993)
  141. Pöschel J Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 119 (1996)
  142. Bambusi D Math. Z. 230 345 (1999)
  143. Bambusi D, Nekhoroshev N N Physica D 122 73 (1998)
  144. Sevryuk M B Physica D 112 132 (1998)
  145. Zaslavskii G M Usp. Fiz. Nauk 156 193 (1988); Zaslavskii G M et al. Sov. Phys. Usp. 31 887 (1988)
  146. Alekseev V M Lektsii Po Nebesnoi Mekhanike (M. – Izhevsk: RKhD, 2001)
  147. Robutel P Cel. Mech. Dyn. Astron. 62 219 (1995)
  148. Xia Z Ann. Math. 135 411 (1992)
  149. Saari D G, Xia Z Notices Am. Math. Soc. 42 538 (1995)
  150. Gerver J L J. Differ. Equations 52 76 (1984)
  151. Simo K i dr. Sovremennye Problemy Khaosa i Nelineinosti (M. – Izhevsk: Inst. komp’yut. issled., 2002)
  152. Beletskii V V, Khentov A A Rezonansnye Vrashcheniya Nebesnykh Tel (N. Novgorod: Nizhegor. gumanitarn. tsentr, 1995)
  153. Milani A, Nobili A M Nature 357 569 (1992)
  154. Laskar J Celestial Mech. 64 115 (1996)
  155. Lecar M et al. Annu. Rev. Astron. Astrophys. 39 581 (2001)
  156. Bakker L F, Diacu F N Romanian Astron. J. 3 139 (1993)
  157. Peterson I Newton’s Clock: Chaos In The Solar System (New York: W.H. Freeman, 1993)
  158. Goldreich P, Rappaport N Icarus 166 320 (2003)
  159. Xia Z J. Dyn. Differ. Equations 5 219 (1993)
  160. Xia Z J. Differ. Equations 110 289 (1994)
  161. Niederman L Nonlinearity 9 1703 (1996)
  162. Chierchia L, Gallavotti G Ann. Inst. Poincaré B Phys. Théor. 60 1 (1994)
  163. Celletti A, Ferraz-Mello S (Eds) Periodic, Quasi-Periodic And Chaotic Motions In Celestial Mechanics: Theory And Applications (Dordrecht: Springer, 2006)
  164. Murray C D, Dermott S F Solar System Dynamics (Cambridge: Camridge Univ. Press, 1999)
  165. Marov M Ya Usp. Fiz. Nauk 175 668 (2005); Marov M Ya Phys. Usp. 48 638 (2005)
  166. Kirkwood D Meteoric Astronomy (Philadelphia: J.B. Lippincott & Co. (1867)
  167. Dermott S F, Murray C D Nature 301 201 (1983)
  168. Neishtadt A I Dokl. Akad. Nauk SSSR 295 47 (1987)
  169. Wisdom J Icarus 56 51 (1983)
  170. Kuiper G H In Astrophysics (Ed. J A Hynek) (New York: McGraw-Hill, 1951) p. 357
  171. Fernández J A Mon. Not. R. Astron. Soc. 192 481 (1980)
  172. Oort J H Bull. Astron. Inst. Neth. 11 91 (1950)
  173. Asphaug E et al. Nature 393 437 (1998)
  174. Morbidelli A, Nesvorný D Icarus 139 295 (1999)
  175. Fridman A M, Gor’kavyi N N Physics Of Planetary Rings: Celestial Mechanics Of Continuous Media (New York: Springer, 1999)
  176. Kozai Y (Ed.) The Stability of the Solar System and of Small Stellar Systems, Symp. No. 62 (Copernicus Symp. I ), Warsaw, Poland, September 5 – 8, 1973 (Dordrecht: D. Reidel, 1974)
  177. Moser J Math. Intelligencer 1 65 (1978)
  178. Laskar J Astron. Astrophys. 287 L9 (1994)
  179. Steane A et al. Phys. Rev. Lett. 74 4972 (1995)
  180. Saif F et al. Phys. Rev. A 58 4779 (1998)
  181. Milovanov A V, Zelenyi L M Phys. Rev. E 64 052101 (2001)
  182. Michalek G, Ostrowski M, Schlickeiser R Solar. Phys. 184 339 (1999)
  183. Malkov M A Phys. Rev. E 58 4911 (1998)
  184. Kobayakawa K, Honda Y S, Samura T Phys. Rev. D 66 083004 (2002)
  185. Veltri A, Carbone V Phys. Rev. Lett. 92 143901 (2004)
  186. Lanzanó G et al. Phys. Rev. Lett. 83 4518 (1999)
  187. Frischat S D, Doron E Phys. Rev. E 57 1421 (1998)
  188. Dembowski C et al. Phys. Rev. Lett. 84 867 (2000)
  189. Hofferbert R et al. Phys. Rev. E 71 046201 (2005)
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