Regular and Chaotic Dynamics

, Volume 18, Issue 5, pp 508–520 | Cite as

Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane

  • Alexey O. KazakovEmail author


We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.


mixed dynamics strange attractor unbalanced ball rubber rolling reversibility twodimensional Poincaré map bifurcation focus saddle invariant manifolds homoclinic tangency Lyapunov’s exponents 

MSC2010 numbers

37J60 37N15 37G35 


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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Institute of computer scienceIzhevskRussia
  2. 2.The Research Institute of Applied Mathematics and CyberneticsNizhny Novgorod State UniversityNizhny NovgorodRussia

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