Critical Phenomena in Hamiltonian Chaos
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.
KeywordsScale Invariance Critical Phenomenon Chaotic Motion Rotation Number Critical Structure
Unable to display preview. Download preview PDF.
- 4.B. V. Chirikov, Time-Dependent Quantum Systems, Proc. Les Houches Summer School on Chaos and Quantum Physics, Elsevier, 1990.Google Scholar
- 6.G. M. Zaslavsky, Chaos in Dynamic Systems, Harwood, 1985.Google Scholar
- 14.B. V. Chirikov and V. V. Vecheslavov, KAM Integrability, in: Analysis etc., Academic Press, 1990, p. 219.Google Scholar
- 15.V. I. Arnold and A. Avez, Ergodic Problems in Classical Mechanics, Benjamin, 1968.Google Scholar
- 19.F. Vivaldi, private communication.Google Scholar
- 20.I. Kornfeld, S. Fomin and Ya. Sinai, Ergodic Theory, Springer, 1982.Google Scholar
- 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
- 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
- 29.J. Bene, P. Szèpfalusy and A. Fülöp, A generic dynamical phase transition in chaotic Hamiltonian systems, Phys. Rev. Lett. (to appear).Google Scholar
- 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
- 31.B. V. Chirikov, Intrinsic Stochasticity, Proc. Int. Conf. on Plasma Physics, Lausanne, 1984, Vol. II, p. 761.Google Scholar
- 36.aibid (Pisma) 38 (1983) 79;Google Scholar
- 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.P. Lévy, Théorie de l’addition des variables eléatoires, Gauthier-Villiers, Paris, 1937;Google Scholar
- 48.P. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, 1979.Google Scholar