Methodological notes

Dynamical chaos: systems of classical mechanics

Lomonosov Moscow State University, Department of Physics, Leninskie Gory 1 build. 2, Moscow, 119991, 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.

Fulltext is available at IOP
PACS: 05.45.−a, 05.45.Ac (all)
DOI: 10.1070/PU2007v050n09ABEH006341
Citation: Loskutov A "Dynamical chaos: systems of classical mechanics" Phys. Usp. 50 939–964 (2007)
BibTexBibNote ® (generic)BibNote ® (RIS)Medline RefWorks
RT Journal
T1 Dynamical chaos: systems of classical mechanics
A1 Loskutov,A.
PB Physics-Uspekhi
PY 2007
FD 10 Sep, 2007
JF Physics-Uspekhi
JO Phys. Usp.
VO 50
IS 9
SP 939-964
DO 10.1070/PU2007v050n09ABEH006341

Оригинал: Лоскутов А Ю «Динамический хаос. Системы классической механики» УФН 177 989–1015 (2007); DOI: 10.3367/UFNr.0177.200709d.0989

References (189) Cited by (41) Similar articles (20)

© 1918–2021 Uspekhi Fizicheskikh Nauk
Email: Editorial office contacts About the journal Terms and conditions