Аннотация:
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.
Ключевые слова:
rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor.
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.
Поступила в редакцию: 26.08.2014 Принята в печать: 08.09.2014
Образец цитирования:
Alexey V. Borisov, Alexey O. Kazakov, Igor R. Sataev, “The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top”, Regul. Chaotic Dyn., 19:6 (2014), 718–733
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\paper The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 6
\pages 718--733
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