Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane
- 65 Downloads
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.
Keywordsmixed dynamics strange attractor unbalanced ball rubber rolling reversibility twodimensional Poincaré map bifurcation focus saddle invariant manifolds homoclinic tangency Lyapunov’s exponents
MSC2010 numbers37J60 37N15 37G35
Unable to display preview. Download preview PDF.
- 1.Ehlers, K. and Koiller, J., Rubber Rolling: Geometry and Dynamics of 2–3–5 Distributions, in Proc. IUTAM Symposium 2006 on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow, Russia, 25–30 August 2006), pp. 469–480.Google Scholar
- 3.Hadamard, J., Sur les mouvements de roulement, Mémoires de la Société des sciences physiques et naturelles de Bordeaux, 4 sér., 1895, vol. 5, pp. 397–417.Google Scholar
- 5.Bizyaev, I. A. and Kazakov, A.O., Integrability and Stochastic Behavior in Some Nonholonomic Dynamics Problems, Rus. J. Nonlin. Dyn., 2013, vol. 9, no. 2, pp. 257–265 (Russian).Google Scholar
- 7.Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: The Absence of an Invariant Measure in a System with a Complete Set of Integrals, Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 3, pp. 605–616 [Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571–579].Google Scholar
- 8.Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).Google Scholar
- 13.Gonchenko, S. V., Lamb, J. S. W., Rios, S., Turaev, D., Attractors and Repellers near Generic Reversible Elliptic Points, arXiv:1212.1931v1 [math. DS], 9 Dec. 2012, pp. 1–11.Google Scholar
- 17.Gonchenko, A. S., Gonchenko, S.V., and Kazakov, A.O., On Some New Aspects of Celtic Stone Chaotic Dynamics, Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 3, pp. 507–518 (Russian).Google Scholar
- 26.Felk, E.V. and Savin, A.V. The Effect of Weak Nonlinear Dissipation on the Stochastic Web, in Proc. of the Conf. “Dynamics, Bifurcations and Strange attractors” (Nizhny Novgorod, Russia, 2013), pp. 36–37.Google Scholar