Advertisement

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. Borisov
  • Alexey O. Kazakov
  • Igor R. Sataev
Article

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.

Keywords

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

MSC2010 numbers

37J60 37N15 37G35 70E18 70F25 70H45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Walker, G.T., On a Curious Dynamical Property of Celts, Proc. Cambridge Phil. Soc., 1895, vol. 8,pt. 5, pp. 305–306.Google Scholar
  2. 2.
    Astapov, I. S., On Rotation Stability of Celtic Stone, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1980, no. 2, pp. 97–100 (Russian).Google Scholar
  3. 3.
    Karapetyan, A.V., On Realizing Nonholonomic Constraints by Viscous Friction Forces and Celtic Stones Stability, J. Appl. Math. Mech., 1981, vol. 45, no. 1, pp. 30–36; see also: Prikl. Mat. Mekh., 1981, vol. 45, no. 1, pp. 42–51.CrossRefzbMATHGoogle Scholar
  4. 4.
    Markeev, A.P., The Dynamics of a Rigid Body on an Absolutely Rough Plane, J. Appl. Math. Mech., 1983, vol. 47, no. 4, pp. 473–478; see also: Prikl. Mat. Mekh., 1983, vol. 47, no. 4, pp. 575–582.CrossRefzbMATHGoogle Scholar
  5. 5.
    Kane, T.R. and Levinson, D.A., A Realistic Solution of the Symmetric Top Problem, J. Appl. Mech., 1978, vol. 45, no. 4, pp. 903–909.CrossRefGoogle Scholar
  6. 6.
    Aleshkevich, V. A., Dedenko, L.G., and Karavaev, V. A., Lectures on Solid Mechanics, Moscow: Mosk. Gos. Univ., 1997 (Russian).Google Scholar
  7. 7.
    Shen, J., Schneider, D.A., and Bloch, A.M., Controllability and Motion Planning of a Multibody Chaplygin’s Sphere and Chaplygin’s Top, Int. J. Robust Nonlinear Control, 2008, vol. 18, no. 9, pp. 905–945.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Chaplygin, S. A., On a Ball’s Rolling on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148; see also: Math. Sb., 1903, vol. 24, no. 1, pp. 139–168.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kilin, A.A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291–306.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regul. Chaotic Dyn., 2013, vol. 18, no. 6, pp. 832–859.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    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.CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Routh, E. J., A Treatise on the Dynamics of a System of Rigid Bodies: P. 2. The Advanced Part, 6th ed., New York: Macmillan, 1905; see also: New York: Dover, 1955 (reprint).Google Scholar
  13. 13.
    Lynch, P. and Bustamante, M.D., Precession and Recession of the Rock’n’Roller, J. Phys. A, 2009, vol. 42, no. 42, 425203, 25 pp.CrossRefMathSciNetGoogle Scholar
  14. 14.
    Koiller, J. and Ehlers, K. M., Rubber Rolling over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 127–152.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    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, no. 3, pp. 277–328.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kazakov, A. O., Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 508–520.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kazakov, A. O., On the Chaotic Dynamics of a Rubber Ball with Three Internal Rotors, Nonlinear Dynamics & Mobile Robotics, 2014, vol. 2, no. 1, pp. 73–97.MathSciNetGoogle Scholar
  18. 18.
    Gonchenko, S. V., Turaev, D. V., and Shilnikov, L.P., On Newhouse Domains of Two-Dimensional Diffeomorphisms That Are Close to a Diffeomorphism with a Structurally Unstable Heteroclinic Contour, Proc. Steklov Inst. Math., 1997, vol. 216, pp. 70–118; see also: Tr. Mat. Inst. Steklova, 1997, vol. 216, pp. 76–125.MathSciNetGoogle Scholar
  19. 19.
    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.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    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.CrossRefzbMATHGoogle Scholar
  21. 21.
    Gonchenko, A. S., Gonchenko, S.V., and Kazakov, A.O., On Some New Aspects of Celtic Stone Chaotic Dynamics, Rus. J. Nonlin. Dinam., 2012, vol. 8, no. 3, pp. 507–518 (Russian).Google Scholar
  22. 22.
    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.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Borisov, A.V. and Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393–403; see also: Uspekhi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 408–418.CrossRefMathSciNetGoogle Scholar
  24. 24.
    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.CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Gonchenko, A. S. and Gonchenko, A. S., On Existence of Lorenz-Like Attractors in a Nonholonomic Model of Celtic Stones, Rus. J. Nonlin. Dinam., 2013, vol. 9, no. 1, pp. 77–89 (Russian).MathSciNetGoogle Scholar
  26. 26.
    Gonchenko, A. S., Lorenz-Like Attractors in Nonholonomic Models of Celtic Stone, Nonlinearity, 2015 (to appear).Google Scholar
  27. 27.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).Google Scholar
  28. 28.
    Walker, J., The Amateur Scientist: The Mysterious “Rattleback”: A stone That Spins in One Direction and Then Reverses, Sci. Am., 1979, vol. 241, pp. 172–184.CrossRefGoogle Scholar
  29. 29.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., New Effects in Dynamics of Rattlebacks, Dokl. Phys., 2006, vol. 51, no. 5, pp. 272–275; see also: Dokl. Akad. Nauk, 2006, vol. 408, no. 2, pp. 192–195.CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    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.CrossRefzbMATHGoogle Scholar
  31. 31.
    Hairer, E., Norsett. S.P., and Wanner, G., Solving Ordinary Differential Equations: 1. Nonstiff Problems, Berlin: Springer, 1987.CrossRefGoogle Scholar
  32. 32.
    Kuznetsov, S.P., Dynamical Chaos, 2nd ed., Moscow: Fizmatlit, 2006 (Russian).Google Scholar
  33. 33.
    Afraimovich, V. S. and Shilnikov, L.P., Invariant Two-Dimensional Tori, Their Breakdown and Stochasticity, in Methods of Qualitative Theory of Differential Equations, E.A. Leontovich-Andronova (Ed.), Gorky: Gorky Gos. Univ., 1983, pp. 3–26 (Russian).Google Scholar
  34. 34.
    Gonchenko, S. V., Ovsyannikov, I. I., Simó, C., and Turaev, D., Three-Dimensional Hénon-Like Maps and Wild Lorenz-Like Attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2005, vol. 15, no. 11, pp. 3493–3508.CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Gonchenko, A. S., Gonchenko, S.V., and Shilnikov, L.P., Towards Scenarios of Chaos Appearance in Three-Dimensional Maps, Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 1, pp. 3–28 (Russian).Google Scholar
  36. 36.
    Gonchenko, A. S., Gonchenko, S.V., Kazakov, A.O., and Turaev, D.V., Simple Scenarios of Onset of Chaos in Three-Dimensional Maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2014, vol. 24, no. 8, 1440005, 25 pp.CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    • 2
  • 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

Personalised recommendations