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Ford, Joseph

doi:10.1016/0370-1573(81)90185-x

Cited by 3 publications

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“…He showed that the property of mixing and the related local instability of nearly all trajectories of the corresponding dynamical systems underlie this nature. In view of this, M Born [15] (see also [16]) suggested that the behavior of the systems is not predictable in classical mechanics. Later, the dynamics due to such instabilities in the systems came to be known as dynamic stochasticity, or deterministic (dynamic) chaos.…”

Section: Introductionmentioning

confidence: 99%

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Loskutov

2007

PHYS-USP

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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 PoincareÂ ± 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 Ð unpredictability, irreversibility, and decay of temporal correlations Ð are described. Models of classical statistical mechanics with chaotic properties, which have become popular in recent years Ð billiards with oscillating boundaries Ð 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, excita-tions 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. 4.6 Higher-order resonances 5. Elements of the Kolmogorov ± Arnol'd ± Moser theory 947 5.1 The Kolmogorov theorem; 5.2 The Arnol'd diffusion 6. The nature of chaos 949 6.1 Twist map; 6.2 Fixed-point theorem; 6.3 Elliptic and hyperbolic points; 6.4 Splitting of separatrices. Homoclinic tangles 7. The Mel'nikov method 952 7.1 The Mel'nikov function; 7.2 The Duffing oscillator and nonlinear pendulum 8. Principal properties of chaotic systems 954 8.1 Ergodicity and mixing; 8.2 Unpredictability and irreversibility; 8.3 Decay of correlations 9. Billiards 956 9.1 The Lorentz gas; 9.2 Scattering billiards with oscillating boundaries. The Fermi acceleration; 9.3 Focusing billiards with oscillating boundaries. Particle deceleration 10. Conclusion 960 References 961 Physics ± Uspekhi 50 (9) 939 ± 964 (2007) # 2007 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences A Loskutov Physics ± Uspekhi 50 (9) A Loskutov Physics ± Uspekhi 50 (9)We introduce the notation j k 0 a À m 0 nt c and H 0 1 2jH k0m0 1 j. Then Eqns (18) can be rewritten as J ek 0 H 0 1 J sin j Y 21 j k 0 oJ À m 0 n ek 0 dH 0 1 J dJ cos j X Retaining only the resonant term in expansion (17), we express Hamiltonian (16) as

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Section: Introductionmentioning

confidence: 99%