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

This article is cited in 41 scientific papers (total in 41 papers)


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.


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English version:
Physics–Uspekhi, 2007, 50:9, 939–964

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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

Citation in format AMSBIB
\by A.~Yu.~Loskutov
\paper Dynamical chaos: systems of classical mechanics
\jour UFN
\yr 2007
\vol 177
\issue 9
\pages 989--1015
\jour Phys. Usp.
\yr 2007
\vol 50
\issue 9
\pages 939--964

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    1. L. A. Kalyakin, “Asymptotic analysis of autoresonance models”, Russian Math. Surveys, 63:5 (2008), 791–857  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Magnitskii N.A., “On the nature of dynamic chaos in a neighborhood of a separatrix of a conservative system”, Differ. Equ., 45:5 (2009), 662–669  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. Manchein C., Beims M.W., “issipation effects in the ratchetlike Fermi acceleration”, Math. Probl. Eng., 2009, 513023, 9 pp.  crossref  mathscinet  isi  elib  scopus
    4. Loskutov A., Leonel E.D., “Time-Dependent Billiards”, Math. Probl. Eng., 2009, 848619, 4 pp.  crossref  zmath  isi  elib  scopus
    5. Aslanov V.S., “Spatial chaotic vibrations when there is a periodic change in the position of the centre of mass of a body in the atmosphere”, J. Appl. Math. Mech., 73:2 (2009), 179–187  crossref  mathscinet  zmath  isi  scopus
    6. B. E. Petrov, “Statistical characteristics of chaoticity and incomplete chaoticity of irregular oscillations generated by a strongly nonlinear autonomous system with two degrees of freedom”, J Commun Technol Electron, 55:1 (2010), 72  crossref  isi  scopus
    7. A B Ryabov, A Loskutov, “Time-dependent focusing billiards and macroscopic realization of Maxwell's Demon”, J Phys A Math Theor, 43:12 (2010), 125104  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. A. Yu. Loskutov, “Fascination of chaos”, Phys. Usp., 53:12 (2010), 1257–1280  mathnet  crossref  crossref  isi  elib
    9. Loskutov A., Ryabov A., Leonel E.D., “Separation of particles in time-dependent focusing billiards”, Physica A-Statistical Mechanics and Its Applications, 389:23 (2010), 5408–5415  crossref  mathscinet  adsnasa  isi  scopus
    10. A. Yu. Loskutov, A. B. Ryabov, A. K. Krasnova, O. A. Chichigina, “Bilyardy s vozmuschaemymi granitsami i nekotorye ikh svoistva”, Nelineinaya dinam., 6:3 (2010), 573–604  mathnet  elib
    11. Mikoss I., Garcia P., “An Exact Map for a Chaotic Billiard”, International Journal of Modern Physics B, 25:5 (2011), 673–681  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. V. P. Budaev, S. P. Savin, L. M. Zelenyi, “Investigation of intermittency and generalized self-similarity of turbulent boundary layers in laboratory and magnetospheric plasmas: towards a quantitative definition of plasma transport features”, Phys. Usp., 54:9 (2011), 875–918  mathnet  crossref  crossref  adsnasa  isi
    13. Talagaev Yu.V., Tarakanov A.F., “Mnogoparametricheskii analiz na osnove kriteriya melnikova i optimalnoe podavlenie khaosa v periodicheski vozmuschaemykh dinamicheskikh sistemakh”, Izvestiya vysshikh uchebnykh zavedenii. Prikladnaya nelineinaya dinamika, 19:4 (2011), 77–90  zmath  elib
    14. Edson D. Leonel, Carl P. Dettmann, “Recurrence of particles in static and time varying oval billiards”, Physics Letters A, 376:20 (2012), 1669–1674  crossref  mathscinet  zmath  isi  scopus
    15. Alexander Loskutov, Olga Chichigina, Alexandra Krasnova, Igor M. Sokolov, “Superdiffusion in 2D open-horizon billiards with stochastically oscillating boundaries”, EPL, 98:1 (2012), 10006  crossref  isi  scopus
    16. A. K. Krasnova, O. A. Chichigina, “Fermi acceleration as a possible mechanism of rapid diffusion of gold clusters on graphite”, Moscow Univ. Phys, 67:1 (2012), 48  crossref  mathscinet  isi  elib  elib  scopus
    17. Goldman V.M., Novoselov V.I., “Struktura i soderzhanie osnovnykh ponyatii distsipliny źstatisticheskaya termodinamika╗ s pozitsii dostizhenii sovremennoi fiziki i fizicheskogo obrazovaniya”, Fizicheskoe obrazovanie v vuzakh, 18:1 (2012), 12–21  elib
    18. Genri E. Norman, Vladimir V. Stegailov, “Stochastic theory of the classical molecular dynamics method”, Math. Models Comput. Simul., 5:4 (2013), 305–333  mathnet  crossref  mathscinet
    19. Sergey V. Kapranov, Guennadi A. Kouzaev, “Stochastic dynamics of an electric dipole in external electric fields: A perturbed nonlinear pendulum approach”, Physica D: Nonlinear Phenomena, 2013  crossref  mathscinet  isi  scopus
    20. N. A. Magnitskii, “Nonclassical approach to the analysis of Hamiltonian and conservative systems”, Comput Math Model, 2013  crossref  mathscinet  elib  scopus
    21. S.V. Kryuchkov, E.I. Kukhar, D.V. Zav'yalov, “Dynamic chaotization of the electronic subsystem in graphene superlattice”, Physica E: Low-dimensional Systems and Nanostructures, 2013  crossref  isi  scopus
    22. S.V. Kryuchkov, E.I. Kukhar’, D.V. Zav’yalov, “Chaotic behavior of the electrons in graphene superlattice”, Superlattices and Microstructures, 2013  crossref  isi  scopus
    23. V. D. Vinokurova, N. N. Rosanov, “The Fermi-Ulam problem and sticking mode”, Tech. Phys. Lett, 40:11 (2014), 946  crossref  mathscinet  isi  scopus
    24. N. N. Rozanov, N. V. Vysotina, “Soliton in stationary and dynamical traps”, JETP Letters, 100:8 (2014), 508–511  mathnet  crossref  crossref  isi  elib  elib
    25. Rosanov N.N., Sochilin G.B., Vinokurova V.D., Vysotina N.V., “Spatial and Temporal Structures in Cavities With Oscillating Boundaries”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 372:2027, SI (2014), 20140012  crossref  isi  scopus
    26. Kryuchkov S.V. Kukhar E.I., “Propagation of Electromagnetic Solitons in Quantum Superlattices Subjected To the High-Frequency Radiation”, J. Nanoelectron. Optoelectron., 9:4 (2014), 564–569  crossref  isi  scopus
    27. Mokeev A., Korobko E., Bubulis A., “Simulation of Concentration Distribution of Dispersed Particles of Magnetorheological Fluid in the Gap Workpiece-Tool of Finishing Polishing Device”, Mechanika, 2014, no. 2, 221–225  crossref  isi  scopus
    28. N.N.. Rosanov, N.V.. Vysotina, “Fermi-Ulam problem for solitons”, Phys. Rev. A, 91:1 (2015)  crossref  mathscinet  isi  scopus
    29. S. V. Kryuchkov, E. I. Kukhar’, “Possibility of propagation of dissipative solitons in ac-driven superlattice”, Phys. Wave Phen, 23:1 (2015), 21  crossref  isi  scopus
    30. N.N.. Rosanov, N.V.. Vysotina, “Recurrence for motion of solitons of the Bose–Einstein condensate in a dynamic trap”, J. Opt. Soc. Am. B, 32:5 (2015), B20  crossref  isi
    31. S. V. Kryuchkov, E. I. Kukhar', “Alternating current-driven graphene superlattices: Kinks, dissipative solitons, dynamic chaotization”, Chaos, 25:7 (2015), 073116  crossref  mathscinet  zmath  isi  scopus
    32. Yu. E. Kuzovlev, “Why nature needs 1/f noise”, Phys. Usp., 58:7 (2015), 719–729  mathnet  crossref  crossref  adsnasa  isi  elib  elib
    33. N. N. Rosanov, “Absolute stability of dynamic cavities”, Opt. Spectrosc, 119:1 (2015), 124  crossref  isi  scopus
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    35. Rosanov N.N., “on Three-Dimensional Dynamics of Oscillons of Bose–Einstein Condensate”, Opt. Spectrosc., 119:6 (2015), 1000–1003  crossref  isi  scopus
    36. Vinokurova V.D., Rozanov N.N., Fedorov E.G., “On the particle dynamics in a dynamic billiard”, Tech. Phys., 61:7 (2016), 965–970  crossref  isi  elib  scopus
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    39. Aslanov V.S., “Reentry Attitude Dynamics”: Aslanov, VS, Rigid Body Dynamics For Space Applications, Butterworth-Heinemann, 2017, 25–125  crossref  mathscinet  isi
    40. L. F. Petrov, “Search for periodic solutions of highly nonlinear dynamical systems”, Comput. Math. Math. Phys., 58:3 (2018), 384–393  mathnet  crossref  crossref  isi  elib
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