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Nelin. Dinam., 2012, Volume 8, Number 4, Pages 735–762 (Mi nd357)  

This article is cited in 10 scientific papers (total in 10 papers)

Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback

Sergey P. Kuznetsov, Alexey Yu. Jalnine, Igor R. Sataev, Julia V. Sedova

Saratov Branch of Kotelnikov's Institute of Radio-Engineering and Electronics of RAS, Zelenaya 38, Saratov, 410019, Russia

Abstract: We perform a numerical study of the motion of the rattleback, a rigid body with a convex surface on a rough horizontal plane in dependence on the parameters, applying the methods used previously for the treatment of dissipative dynamical systems, and adapted for the nonholonomic model. Charts of dynamical regimes are presented on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body. Presence of characteristic structures in the parameter space, previously observed only for dissipative systems, is demonstrated. A method of calculating for the full spectrum of Lyapunov exponents is developed and implemented. It is shown that analysis of the Lyapunov exponents of chaotic regimes of the nonholonomic model reveals two classes, one of which is typical for relatively high energies, and the second for the relatively small energies. For the model reduced to a three-dimensional map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of the quasi-conservative type, with close in magnitude positive and negative Lyapunov exponents, and the rest one about zero. The transition to chaos through a sequence of period-doubling bifurcations is illustrated, and the observed scaling corresponds to that intrinsic to the dissipative systems. A study of strange attractors is provided, in particularly, phase portraits are presented as well as the Lyapunov exponents, the Fourier spectra, the results of calculating the fractal dimensions.

Keywords: rattleback, rigid body dynamics, nonholonomic mechanics, strange attractor, Lyapunov exponents, bifurcation, fractal dimension

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UDC: 517.925 + 517.93
MSC: 37J60, 37N15, 37G35
Received: 09.09.2012
Revised: 16.10.2012

Citation: Sergey P. Kuznetsov, Alexey Yu. Jalnine, Igor R. Sataev, Julia V. Sedova, “Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback”, Nelin. Dinam., 8:4 (2012), 735–762

Citation in format AMSBIB
\by Sergey~P.~Kuznetsov, Alexey~Yu.~Jalnine, Igor~R.~Sataev, Julia~V.~Sedova
\paper Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback
\jour Nelin. Dinam.
\yr 2012
\vol 8
\issue 4
\pages 735--762

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    This publication is cited in the following articles:
    1. A. S. Gonchenko, S. V. Gonchenko, “O suschestvovanii attraktorov lorentsevskogo tipa v negolonomnoi modeli «keltskogo kamnya»”, Nelineinaya dinam., 9:1 (2013), 77–89  mathnet
    2. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Ierarkhiya dinamiki pri kachenii tverdogo tela bez proskalzyvaniya i vercheniya po ploskosti i sfere”, Nelineinaya dinam., 9:2 (2013), 141–202  mathnet
    3. A. O. Kazakov, “Fenomeny khaoticheskoi dinamiki v zadache o kachenii rok-n-rollera bez vercheniya”, Nelineinaya dinam., 9:2 (2013), 309–325  mathnet
    4. A. S. Gonchenko, “Ob attraktorakh lorentsevskogo tipa v modeli keltskogo kamnya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 2, 3–11  mathnet
    5. E. V. Vetchanin, A. O. Kazakov, “Bifurkatsii i khaos v zadache o dvizhenii dvukh tochechnykh vikhrei v akusticheskoi volne”, Nelineinaya dinam., 10:3 (2014), 329–343  mathnet
    6. A. V. Borisov, A. O. Kazakov, I. R. Sataev, “Regulyarnye i khaoticheskie attraktory v negolonomnoi modeli volchka Chaplygina”, Nelineinaya dinam., 10:3 (2014), 361–380  mathnet
    7. D. I. Popov, R. M. Utemesov, “Malomodovoe priblizhenie v zadache Benara-Releya dlya dvukhfaznoi smesi”, Izvestiya Altaiskogo gosudarstvennogo universiteta, 2014, no. 1-1 (81), 231–235  elib
    8. S. P. Kuznetsov, “Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories”, Regular and Chaotic Dynamics, 20:6 (2015), 649–666  mathnet  crossref
    9. A. P. Kuznetsov, S. P. Kuznetsov, Yu. V. Sedova, “Mayatnikovaya sistema s beskonechnym chislom sostoyanii ravnovesiya i kvaziperiodicheskoi dinamikoi”, Nelineinaya dinam., 12:2 (2016), 223–234  mathnet  elib
    10. S. V. Gonchenko, A. S. Gonchenko, A. O. Kazakov, “Three Types of Attractors and Mixed Dynamics of Nonholonomic Models of Rigid Body Motion”, Proc. Steklov Inst. Math., 308 (2020), 125–140  mathnet  crossref  crossref
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