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Regul. Chaotic Dyn., 2014, Volume 19, Issue 6, Pages 718–733 (Mi rcd194)  

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

The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplyginís Top

Alexey V. Borisovab, Alexey O. Kazakovc, Igor R. Sataevd

a Moscow Institute of Physics and Technology, Inststitutskii per. 9, Dolgoprudnyi, 141700 Russia
b Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
c Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia
d Saratov Branch of Kotelnikovís Institute of Radio-Engineering and Electronics of RAS ul. Zelenaya 38, Saratov, 410019 Russia

Abstract: In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.

Keywords: rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation
Russian Science Foundation 14-12-00811
The work of A. V. Borisov was supported by the Ministry of Education and Science of the Russian Federation within the framework of the basic part of the state assignment to institutions of higher education. The work of A. O. Kazakov on Section 3 was supported by the grant of the Russian Scientific Foundation No 14-12- 00811, the work of Section 4.1 was partially supported by the grant of the Russian Scientific Foundation 14-41- 00044 and by the grant of the President of the Russian Federation for support of young doctors of science MD-2324.2013.1. The remaining part of the work of A. O. Kazakov was supported by the Ministry of Education and Science (project No 2000). The work of I. R. Sataev was supported by the grant of the President of the Russian Federation for support of leading scientific schools NSh-1726.2014.2.


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MSC: 37J60, 37N15, 37G35, 70E18, 70F25, 70H45
Received: 26.08.2014

Citation: Alexey V. Borisov, Alexey O. Kazakov, Igor R. Sataev, “The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplyginís Top”, Regul. Chaotic Dyn., 19:6 (2014), 718–733

Citation in format AMSBIB
\by Alexey~V.~Borisov, Alexey~O.~Kazakov, Igor~R.~Sataev
\paper The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplyginís Top
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 6
\pages 718--733

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