Regular and Chaotic Dynamics

, Volume 21, Issue 7–8, pp 792–803 | Cite as

Regular and chaotic motions of a Chaplygin sleigh under periodic pulsed torque impacts

Nonlinear Dynamics & Mobile Robotics

Abstract

For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced three-dimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusive-like random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.

Keywords

Chaplygin sleigh nonholonomic mechanics attractor chaos bifurcation 

MSC2010 numbers

37J60 37C10 34D45 37E30 34C60 

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References

  1. 1.
    Schuster, H.G. and Just, W., Deterministic Chaos: An Introduction, Weinheim: Wiley-VCH, 2005.CrossRefMATHGoogle Scholar
  2. 2.
    Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci., vol. 42, New York: Springer, 1983.CrossRefMATHGoogle Scholar
  3. 3.
    Kuznetsov, S.P., Dynamical Chaos, 2nd ed., Moscow: Fizmatlit, 2006 (Russian).Google Scholar
  4. 4.
    Applications of Chaos and Nonlinear Dynamics in Science and Engineering: Vol. 3, S. Banerjee, L. Rondoni (Eds.), Understanding Complex Systems, Berlin: Springer, 2013.Google Scholar
  5. 5.
    Applications of Chaos and Nonlinear Dynamics in Science and Engineering: Vol. 4, S. Banerjee, L. Rondoni (Eds.), Understanding Complex Systems, Berlin: Springer, 2015.Google Scholar
  6. 6.
    Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities, M. Wiercigroch, B. de Kraker (Eds.), World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol. 28, River Edge,N.J.: World Sci., 2000.Google Scholar
  7. 7.
    Gantmacher, F.R., Lectures in Analytical Mechanics, Moscow: Mir, 1975.Google Scholar
  8. 8.
    Goldstein, H., Poole, Ch.P., Safko, J. L., Classical Mechanics, 3rd ed., Boston,Mass.: Addison-Wesley, 2001.MATHGoogle Scholar
  9. 9.
    Neimark, Ju. I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33, Providence,R.I.: AMS, 1972.MATHGoogle Scholar
  10. 10.
    Borisov, A.V., Mamaev, I. S., and Bizyaev, I.A., Historical and Critical Review of the Development of Nonholonomic Mechanics: The Classical Period, Regul. Chaotic Dyn., 2016, vol. 21, no. 4, pp. 455–476.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ruina, A., Nonholonomic Stability Aspects of Piecewise Holonomic Systems, Rep. Math. Phys., 1998, vol. 42, no. 1–2, pp. 91–100.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    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.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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. 407–418.Google Scholar
  15. 15.
    Borisov, A.V., Kazakov, A.O., and Kuznetsov, S.P., Nonlinear Dynamics of the Rattleback: A Nonholonomic Model, Physics–Uspekhi, 2014, vol. 57, no. 5, pp. 453–460; see also: Uspekhi Fiz. Nauk, 2014, vol. 184, no. 5, pp. 493–500.Google 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.
    Borisov, A.V., Kazakov, A.O., and Sataev, I.R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 718–733.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Chaplygin, S.A., On the Theory ofMotion of Nonholonomic Systems. The Reducing-Multiplier Theorem, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 369–376; see also: Mat. Sb., 1912, vol. 28, no. 2, pp. 303–314.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Carathéodory, C., Der Schlitten, Z. Angew. Math. Mech., 1933, vol. 13, no. 2, pp. 71–76.Google Scholar
  20. 20.
    Borisov, A.V. and Mamaev, I. S., The Dynamics of a Chaplygin Sleigh, J. Appl. Math. Mech., 2009, vol. 73, no. 2, pp. 156–161; see also: Prikl. Mat. Mekh., 2009, vol. 73, no. 2, pp. 219–225.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sagdeev, R. Z., Usikov, D.A., and Zaslavsky, G.M., Nonlinear Physics: From the Pendulum to Turbulence and Chaos, Chur: Harwood Acad. Publ., 1990.Google Scholar
  22. 22.
    Argyris, J., Faust, G., Haase, M., and Friedrich, R., An Exploration of Dynamical Systems and Chaos, 2nd ed., Heidelberg: Springer, 2015.CrossRefMATHGoogle Scholar
  23. 23.
    Ott, E., Grebogi, C., and Yorke, J.A., Controlling Chaos, Phys. Rev. Lett., 1990, vol. 64, no. 11, pp. 1196–1199.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Pyragas, K., Continuous Control of Chaos by Self-Controlling Feedback, Phys. Lett. A, 1992, vol. 170, no. 6, pp. 421–428.CrossRefGoogle Scholar
  25. 25.
    Fradkov, A. L., Evans, R. J., and Andrievsky, B.R., Control of Chaos: Methods and Applications in Mechanics, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2006, vol. 364, no. 1846, pp. 2279–2307.MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Jung, P., Marchegiani, G., and Marchesoni, F., Nonholonomic Diffusion of a Stochastic Sled, Phys. Rev. E, 2016, vol. 93, no. 1, 012606, 9 pp.CrossRefGoogle Scholar
  27. 27.
    Ferraro, S., Jiménez, F., and Martín de Diego, D., New Developments on the Geometric Nonholonomic Integrator, Nonlinearity, 2015, vol. 28, no. 4, pp. 871–900.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Coleman, M. J. and Holmes, P., Motions and Stability of a Piecewise Holonomic System: The Discrete Chaplygin Sleigh, Regul. Chaotic Dyn., 1999, vol. 4, no. 2, pp. 55–77.MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Borisov, A. V., Mamaev, I. S., Kilin, A.A., and Bizyaev, I.A., Qualitative Analysis of the Dynamics of a Wheeled Vehicle, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 739–751.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., On the Hadamard–Hamel Problem and the Dynamics of Wheeled Vehicles, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 752–766.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Karavaev, Yu. L. and Kilin, A.A., The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 134–152.MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Feigenbaum, M. J., Quantitative Universality for a Class of Nonlinear Transformations, J. Stat. Phys., 1978, vol. 19, no. 1, pp. 25–52.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia

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