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



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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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
  2. 2.
    Koiller, J. and Ehlers, K. M., Rubber Rolling over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 127–152.MathSciNetCrossRefMATHGoogle Scholar
  3. 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
  4. 4.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, pp. 227–328.MathSciNetGoogle Scholar
  5. 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
  6. 6.
    Borisov, A.V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490.MathSciNetCrossRefMATHGoogle Scholar
  7. 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. 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
  9. 9.
    Borisov, A. V. and Mamaev, I. S., Rolling of a Rigid Body on a Plane and Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., The Rolling Motion of a Ball on a Surface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201–219.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chavoya-Aceves, O. and Piña, E., Symmetry Lines of the Dynamics of a Heavy Rigid Body with a Fixed Point, Il Nuovo Cimento B, 1989, vol. 103, no. 4, pp. 369–387.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Devaney, R. L., Reversible Diffeomorphisms and Flows, Trans. Amer. Math. Soc., 1976, vol. 218, pp. 89–113.MathSciNetCrossRefMATHGoogle Scholar
  13. 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
  14. 14.
    Borisov, A.V. and Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Uspekhi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 408–418 [Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393–403].MathSciNetGoogle Scholar
  15. 15.
    Borisov, A. V., Kilin, A.A. and Mamaev, I. S., New Effects in Dynamics of Rattlebacks, Doklady Physics.-MAIK Nauka/Interperiodica, 2006, vol. 51, no. 5, pp. 272–275.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Borisov, A.V., Jalnine, A.Yu., Kuznetsov, S.P., Sataev, I.R., and Sedova, J.V., Dynamical Phenomena Occurring Due To Phase Volume Compression in Nonholonomic Model of the Rattleback, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 512–532.MathSciNetCrossRefMATHGoogle Scholar
  17. 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
  18. 18.
    Gonchenko, A. S., Gonchenko, S.V., and Kazakov, A.O., Richness of Chaotic Dynamics in the Nonholonomic Model of Celtic Stone, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 521–538.CrossRefMathSciNetGoogle Scholar
  19. 19.
    Kazakov, A. O., Chaotic Dynamics Phenomena in the Rubber Rock-n-Roller on a Plane Problem, Rus. J. Nonlin. Dyn., 2013, vol. 9, no. 2, pp. 309–325 (Russian).MathSciNetGoogle Scholar
  20. 20.
    Gonchenko, S. V., Simó, C., and Vieiro, A., Richness of Dynamics and Global Bifurcations in Systems with a Homoclinic Figure-Eight, Nonlinearity, 2013, vol. 26, no. 3, pp. 621–678.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lukyanov, V. I. and Shilnikov, L.P., On Some Bifurcations of Dynamical Systems with Homoclinic Structures, Dokl. Akad. Nauk SSSR, 1978, vol. 243, no. 1, pp. 26–29 [Soviet Math. Dokl., 1978, vol. 19, pp. 1314–1318].MathSciNetGoogle Scholar
  22. 22.
    Afraimovich, V. S. and Shilnikov, L.P., On Some Global Bifurcations Connected with the Disappearance of a Fixed Point of Saddle-Node Type, Dokl. Akad. Nauk SSSR, 1974, vol. 219, no. 6, pp. 1281–1285 [Soviet Math. Dokl., 1974, vol. 15, no. 3, pp. 1761–1765].MathSciNetGoogle Scholar
  23. 23.
    Feigenbaum, M. J., Universal Behavior in Nonlinear Systems: Order in Chaos (Los Alamos, N.M., 1982), Phys. D, 1983, vol. 7, nos. 1–3, pp. 16–39.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Borisov, A.V. and Simakov, N.N., Period Doubling Bifurcation in Rigid Body Dynamics, Regul. Chaotic Dyn., 1997, vol. 2, no. 1, pp. 64–74 (Russian).MathSciNetMATHGoogle Scholar
  25. 25.
    Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M., Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them: P. 1: Theory; P. 2: Numerical Application, Meccanica, 1980, vol. 15, pp. 9–30.CrossRefMATHGoogle Scholar
  26. 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
  27. 27.
    Lamb, J. S.W. and Stenkin, O. V., Newhouse Regions for Reversible Systems with Infinitely Many Stable, Unstable and Elliptic Periodic Orbits, Nonlinearity, 2004, vol. 17, no. 4, pp. 1217–1244.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Delshams, A., Gonchenko, S. V., Gonchenko, A. S., Lázaro, J. T., and Sten’kin, O., Abundance of Attracting, Repelling and Elliptic Periodic Orbits in Two-Dimensional Reversible Maps, Nonlinearity, 2013, vol. 26, no. 1, pp. 1–33.CrossRefMATHGoogle Scholar
  29. 29.
    Pikovsky, A. and Topaj, D., Reversibility vs. Synchronization in Oscillator Latties, Phys. D, 2002, vol. 170, pp. 118–130.MathSciNetCrossRefMATHGoogle Scholar

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

Personalised recommendations