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The reversal and chaotic attractor in the nonholonomic model of Chaplygin’s top

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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.

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Authors and Affiliations

  1. Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia

    Alexey V. Borisov

  2. Moscow Institute of Physics and Technology, Inststitutskii per. 9, Dolgoprudnyi, 141700, Russia

    Alexey V. Borisov

  3. Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950, Russia

    Alexey O. Kazakov

  4. Saratov Branch of Kotelnikov’s Institute of Radio-Engineering and Electronics of RAS, ul. Zelenaya 38, Saratov, 410019, Russia

    Igor R. Sataev

Authors
  1. Alexey V. Borisov
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  2. Alexey O. Kazakov
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  3. Igor R. Sataev
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Correspondence to Alexey V. Borisov.

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Borisov, A.V., Kazakov, A.O. & Sataev, I.R. The reversal and chaotic attractor in the nonholonomic model of Chaplygin’s top. Regul. Chaot. Dyn. 19, 718–733 (2014). https://doi.org/10.1134/S1560354714060094

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  • DOI: https://doi.org/10.1134/S1560354714060094

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