Regular and Chaotic Dynamics

, Volume 17, Issue 6, pp 512–532 | Cite as

Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback

  • Alexey V. BorisovEmail author
  • Alexey Yu. Jalnine
  • Sergey P. Kuznetsov
  • Igor R. Sataev
  • Julia V. Sedova


We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.


rattleback rigid body dynamics nonholonomic mechanics strange attractor Lyapunov exponents bifurcation fractal dimension 

MSC2010 numbers

74F10 93D20 


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  1. 1.
    Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.zbMATHGoogle Scholar
  2. 2.
    Abraham, R. and Marsden, J.E., Foundations of Mechanics, Providence, R. I.: AMS, Chelsea, 2008.Google Scholar
  3. 3.
    Borisov, A. V. and Mamaev, I. S., The Rolling Motion of a Rigid Body on a Plane and a Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Nonholonomic Dynamical Systems: Integrability, Chaos, Strange Attractors, A. V. Borisov, I. S. Mamaev (Eds.), Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2002 (Russian).zbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Selected Problems of Nonholonomic Mechanics, preprint, Izhevsk: Institute of Computer Science, 2005 (Russian).Google Scholar
  7. 7.
    Walker, J., The Mysterious “Rattleback”: A Stone That Spins in One Direction and Then Reverses, Sci. Amer., 1979, vol. 241, no. 4, pp. 144–149.CrossRefGoogle Scholar
  8. 8.
    Gonchenko, A. S., Gonchenko, S.V., and Shilnikov, L.P., Towards Scenarios of Chaos Appearance in Three-Dimensional Maps, Nelin. Dinam., 2012, vol. 8, no. 1, pp. 3–28 (Russian).Google Scholar
  9. 9.
    Gonchenko, A. S., Gonchenko, S.V., and Kazakov, A.O., On Some Novel Aspects of Chaotic Dynamics of the Rattleback, Nelin. Dinam., 2012, vol. 8, no. 3, pp. 507–518 (Russian).Google Scholar
  10. 10.
    Rabinovich, M. I. and Trubetskov, D. I., Oscillations and Waves: In Linear and Nonlinear Systems, Dordrecht: Kluwer, 1989.CrossRefzbMATHGoogle Scholar
  11. 11.
    Lichtenberg, A. J. and Lieberman, M.A., Regular and Chaotic Dynamics, 2nd ed., Appl. Math. Sci., vol. 38, New York: Springer, 1992.zbMATHGoogle Scholar
  12. 12.
    Schuster, H.G. and Just, W., Deterministic Chaos: An Introduction, 4th ed., Weinheim: Wiley-VCH, 2005.CrossRefzbMATHGoogle Scholar
  13. 13.
    Kuznetsov, S.P., Dynamical Chaos, Moscow: Fizmatlit, 2001 (Russian).Google Scholar
  14. 14.
    Kuznetsov, A.P., Kuznetsov, S.P., Sataev, I.R., and Chua, L. O., Multi-Parameter Criticality in Chua’s Circuit at Period-Doubling Transition to Chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1996, vol. 6, no. 1, pp. 119–148.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kuznetsov, A.P., Sataev, I. R., and Turukina, L.V., Synchronization and Multi-Frequency Oscillations in the Chain of Phase Oscillators, Nelin. Dinam., 2010, vol. 6, no. 4, pp. 693–717 (Russian).Google Scholar
  16. 16.
    Kuznetsov, A.P., Sataev, I. R., and Turukina, L.V., On the Road Towards Multidimensional Tori, Commun. Nonlinear Sci. Numer. Simul., 2011, vol. 16, pp. 2371–2376.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Broer, H., Simó, C., and Vitolo, R., Bifurcations and Strange Attractors in the Lorenz-84 Climate Model with Seasonal Forcing, Nonlinearity, 2002, vol. 15, pp. 1205–1267.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Press, W. H., Flannery, B.P., Teukolsky, S. A., and Vetterling, V. T., Numerical Recipes in C: The Art of Scientific Computing, Cambridge: Cambridge Univ. Press, 1992.Google Scholar
  19. 19.
    Hénon, M., On the Numerical Computation of Poincaré Maps, Phys. D, 1982, vol. 5, nos. 2–3, pp. 412–414.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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
  21. 21.
    Cochran, W.G., Errors of Measurement in Statistics, Technometrics, 1968, vol. 10, no. 4, pp. 637–666.CrossRefzbMATHGoogle Scholar
  22. 22.
    Gallas, J. A. C., Dissecting Shrimps: Results for Some One-Dimensional Physical Systems, Phys. A, 1994, vol. 202, pp. 196–223.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kuznetsov, S.P., Hyperbolic Chaos: A Physicist’s View, Berlin: Springer, 2012.CrossRefzbMATHGoogle Scholar
  24. 24.
    Kaplan, J. L. and Yorke, J. A., A Chaotic Behavior of Multi-Dimensional Differential Equations, in Functional Differential Equations and Approximations of Fixed Points, H. O. Peitgen and H. O. Walther (Eds.), Lecture Notes in Math., vol. 730, Berlin: Springer, 1979, pp. 204–227.CrossRefGoogle Scholar
  25. 25.
    Feigenbaum, M. J., The Universal Metric Properties of Nonlinear Transformations, J. Stat. Phys., 1979, vol. 21, pp. 669–706.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kuznetsov, S. P., Kuznetsov, A.P., and Sataev, I. R., Multiparameter Critical Situations, Universality and Scaling in Two-Dimensional Period-Doubling Maps, J. Stat. Phys., 2005, vol. 121, nos. 5–6, pp. 697–748.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Reick, C., Universal Corrections to Parameter Scaling in Period-Doubling Systems: Multiple Scaling and Crossover, Phys. Rev. A, 1992, vol. 45, pp. 777–792.CrossRefGoogle Scholar
  28. 28.
    Reichl, L.E., The Transition to Chaos: Conservative Classical Systems and Quantum Manifestations, 2nd ed., New York: Springer, 2004.zbMATHGoogle Scholar
  29. 29.
    Jakobson, M.V., Absolutely Continuous Invariant Measures for One-Parameter Families of One-Dimensional Maps, Comm. Math. Phys., 1981, vol. 81, no. 1, pp. 39–88.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Benediks, M. and Carleson, L., Dynamics of the Hénon Map, Ann. Math., 1991, vol. 133, pp. 73–169.CrossRefGoogle Scholar
  31. 31.
    Jenkins, G. M. and Watts, D.G., Spectral Analysis and Its Application, San Francisco: Holden-Day, 1968.Google Scholar
  32. 32.
    Grassberger, P., Generalized Dimensions of Strange Attractors, Phys. Lett. A, 1983, vol. 97, pp. 227–230.MathSciNetCrossRefGoogle Scholar
  33. 33.
    Grassberger, P. and Procaccia, I., Measuring the Strangeness of Strange Attractors, Phys. D, 1983, vol. 9, pp. 189–208.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    Email author
  • Alexey Yu. Jalnine
    • 1
    • 2
  • Sergey P. Kuznetsov
    • 1
    • 2
  • Igor R. Sataev
    • 2
  • Julia V. Sedova
    • 1
    • 2
  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia
  2. 2.Saratov Branch of Kotel’nikov’s Institute of Radio-Engineering and Electronics of RASSaratovRussia

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