Critical Phenomena in Hamiltonian Chaos

  • Boris V. Chirikov
Part of the NATO ASI Series book series (NSSB, volume 264)


An overview of critical phenomena in Hamiltonian dynamics is presented, including the renormalization chaos, based upon a fairly simple resonant theory. First estimates for the critical structure and related statistical anomalies in arbitrary dimensions are given. The results of some numerical experiments with two-dimensional maps are discussed.

The main idea I would like to expose here is the inexhaustible diversity and richness of dynamical chaos whatever description you choose: trajectories, statistics or, recently, renormalization. The importance of this relatively new phenomenon — the dynamical chaos — is in that it presents, even in very simple models to be discussed below, the surprising complexity of the structures and evolution characteristic for a broad range of processes in nature including the highest levels of its organization. Moreover, dynamical chaos is the only stationary source of any new information and, hence, a necessary part of any creative activity, science included. This is a direct implication of the Alekseev-Brudno theorem and Kolmogorov’s development in the information theory (see, e.g., Refs. 1 and 2). Chaos is not always that bad!

Below I restrict myself to classical mechanics only. So-called “quantum chaos” is another story (see, e.g., Ref. 3 and 4). Let me just mention that apart from very exotic examples there is no “true” chaos in quantum mechanics contrary to a common belief. On the other hand, the inavoidable statistical element of quantum mechanics related to measurement is very likely associated with the same classical chaos in the measuring device.

With a bit of imagination and fantasy one may even conjecture that any macroscopic event in this World, which formally is a result of some quantum “measurement,” would be impossible without chaos.

Also, I am not going to consider any dissipative models (very important in practical applications) because they are not as fundamental as Hamiltonian systems. Besides, strictly speaking, the dissipative systems are not purely dynamical as the dissipation is inevitably related to some noise.

In what follows I take a physicist’s approach to the problem, that is my presentation will be based on a simple (sometimes even qualitative) theory combined with the results of extensive numerical (computer) experiments. For a good physical overview of nonlinear dynamics and chaos see books.5,6

The principal concept of such a theory is nonlinear resonance whose quite familiar by now phase space picture is depicted, e.g., in Fig. 4 below. The essential part of this resonance structure is a pair of periodic orbits, the most important being the unstable one as it gives rise to the separatrix and, under almost any perturbation, to a chaotic surrounding layer. This is precisely the place where chaos is dawning.

Again, I have to restrict myself to a simpler case of strong nonlinearity which does not vanish with perturbation. A very interesting weakly nonlinear resonance will be briefly mentioned in Section I.2 below.

The paper is organized as follows. In the next Section I simple models are described which are currently extensively used in the studies of nonlinear phenomena and chaos. They well represent the whole spectrum of complexity classified in Section II. The main Sections III and IV are devoted to a detailed description of the so-called critical phenomena in dynamics which reveal the most complicated behavior presently known.


Scale Invariance Critical Phenomenon Chaotic Motion Rotation Number Critical Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    V. M. Alekseev and M. V. Yakobson, Phys. Reports 75 (1981) 287.MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    G. Chaitin, Information, Randomness and Incompleteness, World Scientific, 1990.MATHGoogle Scholar
  3. 3.
    B. V. Chirikov, F. M. Izrailev and D. L. Shepelyansky, Physica D 33 (1988) 77.MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    B. V. Chirikov, Time-Dependent Quantum Systems, Proc. Les Houches Summer School on Chaos and Quantum Physics, Elsevier, 1990.Google Scholar
  5. 5.
    A. Lichtenberg, M. Lieberman, Regular and Stochastic Motion, Springer, 1983.MATHGoogle Scholar
  6. 6.
    G. M. Zaslavsky, Chaos in Dynamic Systems, Harwood, 1985.Google Scholar
  7. 7.
    B. V. Chirikov, Phys. Reports 52 (1979) 263.MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    B. V. Chirikov and V. V. Vecheslavov, Astron. Astroph. 221 (1989) 146.ADSGoogle Scholar
  9. 9.
    G. Casati et al., Phys. Rev. A 36 (1987) 3501.ADSCrossRefGoogle Scholar
  10. 10.
    A. A. Chernikov, R. Z. Sagdeev and G. M. Zaslavsky, Physica D 33 (1988) 65.MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    B. V. Chirikov, Foundations of Physics 16 (1986) 39.MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    M. Eisenman et al., Lecture Notes in Physics 38 (1975) 112;ADSCrossRefGoogle Scholar
  13. 12a.
    J. von Hemmen, ibid., 93 (1979) 232.ADSCrossRefGoogle Scholar
  14. 13.
    B. G. Konopelchenko, Nonlinear Integrable Equations, Lecture Notes in Physics 270 (1987).MATHCrossRefGoogle Scholar
  15. 14.
    B. V. Chirikov and V. V. Vecheslavov, KAM Integrability, in: Analysis etc., Academic Press, 1990, p. 219.Google Scholar
  16. 15.
    V. I. Arnold and A. Avez, Ergodic Problems in Classical Mechanics, Benjamin, 1968.Google Scholar
  17. 16.
    B. V. Chirikov, Proc. Roy. Soc. Lond. A 413 (1987) 145.MathSciNetADSCrossRefGoogle Scholar
  18. 17.
    A. Rechester et al., Phys. Rev. A 23 (1981) 2664.MathSciNetADSCrossRefGoogle Scholar
  19. 18.
    B. V. Chirikov, D. L. Shepelyansky, Radiofizika 29 (1986) 1041.ADSGoogle Scholar
  20. 19.
    F. Vivaldi, private communication.Google Scholar
  21. 20.
    I. Kornfeld, S. Fomin and Ya. Sinai, Ergodic Theory, Springer, 1982.Google Scholar
  22. 21.
    R. MacKay, Physica D 7 (1983) 283.MathSciNetADSMATHCrossRefGoogle Scholar
  23. 22.
    B. V. Chirikov and D. L. Shepelyansky, Physica D 13 (1984) 395.MathSciNetADSMATHCrossRefGoogle Scholar
  24. 23.
    R. Artuso, G. Casati and D. L. Shepelyansky, Breakdown of Universality in Renormalization Dynamics for Critical Invariant Torus, Phys. Rev. Lett. (to appear).Google Scholar
  25. 24.
    B. V. Chirikov, D. L. Shepelyansky, Proc 9th Int. Conf. on Nonlinear Oscillations (Kiev, 1981). Kiev, Naukova Dumka, 1983, Vol. 2, p. 421. English translation available as preprint PPL-TRANS-133, Plasma Physics Lab., Princeton Univ., 1983.Google Scholar
  26. 25.
    S. Channon and J. Lebowitz, Ann. N. Y. Acad., Sci. 357 (1980) 108.ADSCrossRefGoogle Scholar
  27. 26.
    G. Paladin and A. Vulpiani, Phys. Reports 156 (1987) 147.MathSciNetADSCrossRefGoogle Scholar
  28. 27.
    C. Karney, Physica D 8 (1983) 360.MathSciNetADSCrossRefGoogle Scholar
  29. 28.
    P. Grassberger and I. Procaccia, Physica D 13 (1984) 34.MathSciNetADSMATHCrossRefGoogle Scholar
  30. 29.
    J. Bene, P. Szèpfalusy and A. Fülöp, A generic dynamical phase transition in chaotic Hamiltonian systems, Phys. Rev. Lett. (to appear).Google Scholar
  31. 30.
    B. V. Chirikov and D. L. Shepelyansky, Chaos Border and Statistical Anomalies, in: Renormalization Group, D. V. Shirkov, D. L Kazakov and A. A. Vladimirov Eds., World Scientific, Singapore, 1988, p. 221.Google Scholar
  32. 31.
    B. V. Chirikov, Intrinsic Stochasticity, Proc. Int. Conf. on Plasma Physics, Lausanne, 1984, Vol. II, p. 761.Google Scholar
  33. 32.
    G. Schmidt and J. Bialek, Physica D 5 (1982) 397.MathSciNetADSCrossRefGoogle Scholar
  34. 33.
    J. Greene, J. Math. Phys. 9 (1968) 760;MathSciNetADSMATHCrossRefGoogle Scholar
  35. 33a.
    J. Greene, J. Math. Phys. 20 (1979) 1183.ADSCrossRefGoogle Scholar
  36. 34.
    M. Feigenbaum, J. Stat. Phys. 19 (1978) 25;MathSciNetADSMATHCrossRefGoogle Scholar
  37. 34.
    M. Feigenbaum, J. Stat. Phys. 21 (1979) 669.MathSciNetADSMATHCrossRefGoogle Scholar
  38. 35.
    S. Ostlund et al., Physica D 8 (1983) 303.MathSciNetADSMATHCrossRefGoogle Scholar
  39. 36.
    E. M. Lifshits et al., Zh. Eksp. Teor. Fiz. 59 (1970) 322;ADSGoogle Scholar
  40. 36.a
    ibid (Pisma) 38 (1983) 79;Google Scholar
  41. 36b.
    E. M. Lifshits et al., J. Barrow, Phys. Reports 85 (1982) 1.CrossRefGoogle Scholar
  42. 37.
    B. V. Chirikov, The Nature and Properties of the Dynamic Chaos, Proc. 2d Int. Seminar, “Group Theory Methods in Physics” (Zvenigorod, 1982), Harwood, 1985, Vol. 1, p. 553.Google Scholar
  43. 38.
    I. Dana et al., Phys. Rev. Lett. 62 (1989) 233.MathSciNetADSCrossRefGoogle Scholar
  44. 39.
    R. MacKay et al., Physica D 13 (1984) 55.MathSciNetADSMATHCrossRefGoogle Scholar
  45. 40.
    J. Hanson et al., J. Stat. Phys. 39 (1985) 327.MathSciNetADSMATHCrossRefGoogle Scholar
  46. 41.
    B. V. Chirikov, Lecture Notes in Physics 179 (1983) 29.MathSciNetADSCrossRefGoogle Scholar
  47. 42.
    J. Meiss and E. Ott, Phys. Rev. Lett. 55 (1985) 2741;ADSCrossRefGoogle Scholar
  48. 42.
    J. Meiss and E. Ott, Physica D 20 (1986) 387.MathSciNetADSMATHCrossRefGoogle Scholar
  49. 43.
    P. Lévy, Théorie de l’addition des variables eléatoires, Gauthier-Villiers, Paris, 1937;Google Scholar
  50. 43a.
    T. Geisel et al., Phys. Rev. Lett. 54 (1985) 616;ADSCrossRefGoogle Scholar
  51. 43b.
    R. Pasmanter, Fluid Dynamic Research 3 (1988) 320;ADSCrossRefGoogle Scholar
  52. 43c.
    R. Voss, Physica D 38 (1989) 362;MathSciNetADSCrossRefGoogle Scholar
  53. 43d.
    G. M. Zaslavsky et al., Zh. Exper. Teor. Fiz. 96 (1989) 1563.MathSciNetADSGoogle Scholar
  54. 44.
    H. Mori et al., Prog. Theor. Phys. Suppl., 99, 1 (1989).ADSCrossRefGoogle Scholar
  55. 45.
    J. Greene et al., Physica D 21 (1986) 267.MathSciNetADSMATHCrossRefGoogle Scholar
  56. 46.
    C. Karney et al, ibid 4 (1982) 425.MathSciNetADSMATHCrossRefGoogle Scholar
  57. 47.
    Y. Ichikawa et al., ibid 29 (1987) 247.ADSCrossRefGoogle Scholar
  58. 48.
    P. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, 1979.Google Scholar
  59. 49.
    D. Umberger and D. Farmer, Phys. Rev. Lett. 55 (1985) 661;MathSciNetADSCrossRefGoogle Scholar
  60. 49a.
    C. Grebogi et al., Phys. Lett. A 110 (1985) 1.MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Boris V. Chirikov
    • 1
  1. 1.Institute of Nuclear PhysicsNovosibirskUSSR

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