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Methodological notes


Dynamical chaos: systems of classical mechanics


Lomonosov Moscow State University, Department of Physics, Vorobevy gory, Moscow, 119992, 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.

Text can be downloaded in Russian. English translation is available on IOP Science.
PACS: 05.45.−a, 05.45.Ac (all)
DOI: 10.1070/PU2007v050n09ABEH006341
URL: https://ufn.ru/en/articles/2007/9/d/
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

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  1. A.M. Dykhne, A.A. Snarskii, M.I. Zhenirovskii “Stability and chaos in randomly inhomogeneous two-dimensional media and LC circuits47 821–828 (2004)
  2. A.V. Borisov, I.S. Mamaev “Strange attractors in rattleback dynamics46 393–403 (2003)
  3. S.N. Gordienko “Irreversibility and the probabilistic treatment of the dynamics of classical particles42 573–590 (1999)
  4. V.I. Klyatskin “Statistical topography and Lyapunov exponents in stochastic dynamical systems51 395–407 (2008)
  5. A.I. Musienko, L.I. Manevich “Classical mechanical analogs of relativistic effects47 797–820 (2004)
  6. E.N. Rumanov “Critical phenomena far from equilibrium56 93–102 (2013)
  7. P.S. Landa, Ya.B. Duboshinskii “Self-oscillatory systems with high-frequency energy sources32 723–731 (1989)
  8. A.N. Pavlov, V.S. Anishchenko “Multifractal analysis of complex signals50 819–834 (2007)
  9. O.V. Rudenko “Nonlinear dynamics of quadratically cubic systems56 683–690 (2013)
  10. A.V. Borisov, A.O. Kazakov, S.P. Kuznetsov “Nonlinear dynamics of the rattleback: a nonholonomic model57 453–460 (2014)
  11. E.P. Zemskov “Turing patterns and Newell—Whitehead—Segel amplitude equation57 1035–1037 (2014)
  12. S.V. Vladimirov, Yu.O. Tyshetskiy “On description of a collisionless quantum plasma54 1243–1256 (2011)
  13. G.N. Bochkov, Yu.E. Kuzovlev “Fluctuation-dissipation relations: achievements and misunderstandings56 590–602 (2013)
  14. A.A. Shatskiy, I.D. Novikov, N.S. Kardashev “The Kepler problem and collisions of negative masses54 381–385 (2011)
  15. A.M. Ignatov, A.I. Korotchenko et alOn the interpretation of computer simulation of classical Coulomb plasma38 109–114 (1995)
  16. A.A. Andronov, Yu.A. Ryzhov “An infinity of the classical theory of fluctuations in a nondegenerate electron gas21 873–878 (1978)
  17. P. Paradoksov “How quantum mechanics helps us understand classical mechanics9 618–620 (1967)
  18. L.A. Maksimov, T.V. Khabarova “Properties of acoustic polarization vectors in crystals and the phonon Hall effect53 481–485 (2010)
  19. M.V. Kuzelev, A.A. Rukhadze “Nonrelativistic quantum theory of stimulated Cherenkov radiation and Compton scattering in a plasma54 375–380 (2011)
  20. V.V. Shevchenko “Localization of a stationary electromagnetic field by a planar boundary of a metamaterial54 1131–1142 (2011)

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