Regular and Chaotic Dynamics

, Volume 19, Issue 6, pp 718–733 | Cite as

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

  • Alexey V. BorisovEmail author
  • Alexey O. Kazakov
  • Igor R. Sataev


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.


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

MSC2010 numbers

37J60 37N15 37G35 70E18 70F25 70H45 


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Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    • 2
    Email author
  • Alexey O. Kazakov
    • 3
  • Igor R. Sataev
    • 4
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  3. 3.Lobachevsky State University of Nizhny NovgorodNizhny NovgorodRussia
  4. 4.Saratov Branch of Kotelnikov’s Institute of Radio-Engineering and Electronics of RASSaratovRussia

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