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UFN, 2007, Volume 177, Number 9, Pages 989–1015 (Mi ufn514)  

This article is cited in 40 scientific papers (total in 40 papers)

METHODOLOGICAL NOTES

Dynamical chaos: systems of classical mechanics

A. Yu. Loskutov

Physics Department, M. V. Lomonosov Moscow State University

Abstract: 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.

DOI: https://doi.org/10.3367/UFNr.0177.200709d.0989

Full text: PDF file (5422 kB)
Full text: http://www.ufn.ru/ru/articles/2007/9/d/
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English version:
Physics–Uspekhi, 2007, 50:9, 939–964

Bibliographic databases:

Document Type: Article
PACS: 05.45.-a, 05.45.Ac
Received: January 31, 2007
Revised: April 25, 2007

Citation: A. Yu. Loskutov, “Dynamical chaos: systems of classical mechanics”, UFN, 177:9 (2007), 989–1015; Phys. Usp., 50:9 (2007), 939–964

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    Citing articles on Google Scholar: Russian citations, English citations
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