Regular and Chaotic Dynamics

, Volume 22, Issue 8, pp 955–975 | Cite as

The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration

  • Ivan A. BizyaevEmail author
  • Alexey V. Borisov
  • Ivan S. Mamaev


This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.


nonholonomic mechanics Fermi acceleration Chaplygin sleigh parametric oscillator tensor invariants involution strange attractor Lyapunov exponents reversible systems chaotic dynamics 

MSC2010 numbers

37J60 34A34 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Hess–Appelrot System and Its Nonholonomic Analogs, Proc. Steklov Inst. Math., 2016, vol. 294, pp. 252–275; see also: Tr. Mat. Inst. Steklova, 2016, vol. 294, pp. 268–292.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Kilin, A.A. and Vetchanin, E. V., The Control of the Motion through an Ideal Fluid of a Rigid Body by Means of Two Moving Masses, Nelin. Dinam., 2015, vol. 11, no. 4, pp. 633–645 (Russian).CrossRefzbMATHGoogle Scholar
  4. 4.
    Kozlov, V. V. and Ramodanov, S. M., On the Motion of a Body with a Rigid Hull and Changing Geometry of Masses in an Ideal Fluid, Dokl. Phys., 2002, vol. 47, no. 2, pp. 132–135; see also: Dokl. Akad. Nauk, 2002, vol. 382, no. 4, pp. 478–481.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kozlov, V.V., InvariantMeasures of Smooth Dynamical Systems, Generalized Functions and Summation Methods, Russian Acad. Sci. Izv. Math., 2016, vol. 80, no. 2, pp. 342–358; see also: Izv. Ross. Akad. Nauk. Ser. Mat., 2016, vol. 80, no. 2, pp. 63–80.Google Scholar
  6. 6.
    Kozlov, V.V. and Onishchenko, D. A., The Motion in a Perfect Fluid of a Body Containing a Moving Point Mass, J. Appl. Math. Mech., 2003, vol. 67, no. 4, pp. 553–564; see also: Prikl. Mat. Mekh., 2003, vol. 67, no. 4, pp. 620–633.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kozlov, V.V. and Ramodanov, S. M., The Motion of a Variable Body in an Ideal Fluid, J. Appl. Math. Mech., 2001, vol. 65, no. 4, pp. 579–587; see also: Prikl. Mat. Mekh., 2001, vol. 65, no. 4, pp. 592–601.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kozlov, V.V., Dynamics of Variable Systems and Lie Groups, J. Appl. Math. Mech., 2004, vol. 68, no. 6, pp. 803–808; see also: Prikl. Mat. Mekh., 2004, vol. 68, no. 6, pp. 899–905.MathSciNetCrossRefGoogle Scholar
  9. 9.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fufaev, N.A., On the Possibility of Realizing a Nonholonomic Constraint by Means of Viscous Friction Forces, J. Appl. Math. Mech., 1964, vol. 28, no. 3, pp. 630–632; see also: Prikl. Mat. Mekh., 1964, vol. 28, no. 3, pp. 513–515.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Arnol’d, V. I., Kozlov, V.V., and Ne?ishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.Google Scholar
  12. 12.
    Bizyaev, I.A., The Inertial Motion of a Roller Racer, Regul. Chaotic Dyn., 2017, vol. 22, no. 3, pp. 239–247.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., Dynamics of the Chaplygin Sleigh on a Cylinder, Regul. Chaotic Dyn., 2016, vol. 21, no. 1, pp. 136–146.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bolotin, S. and Treschev, D., Unbounded Growth of Energy in Nonautonomous Hamiltonian Systems, Nonlinearity, 1999, vol. 12, no. 2, pp. 365–388.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Jacobi Integral in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 383–400.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Borisov, A.V. and Mamaev, I. S., An Inhomogeneous Chaplygin Sleigh, Regul. Chaotic Dyn., 2017, vol. 22, no. 4, pp. 435–447.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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
  19. 19.
    Borisov, A. V. and Kuznetsov, S.P., Regular and Chaotic Motions of Chaplygin Sleigh under Periodic Pulsed Torque Impacts, Regul. Chaotic Dyn., 2016, vol. 21, nos. 7–8, pp. 792–803.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How to Control Chaplygin’s Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3–4, pp. 258–272.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How to Control Chaplygin’s Sphere Using Rotors: 2, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 144–158.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Carathéodory, C., Der Schlitten, Z. Angew. Math. Mech., 1933, vol. 13, no. 2, pp. 71–76.CrossRefzbMATHGoogle Scholar
  26. 26.
    Fedorov, Yu.N. and García-Naranjo, L.C., The Hydrodynamic Chaplygin Sleigh, J. Phys. A, 2010, vol. 43, no. 43, 434013, 18 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gonchenko, A. S., Gonchenko, S.V., and Kazakov, A.O., Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 521–538.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gelfreich, V. and Turaev, D., Fermi Acceleration in Non-Autonomous Billiards, J. Phys. A, 2008, vol. 41, no. 21, 212003, 6 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ito, A., Successive Subharmonic Bifurcations and Chaos in a Nonlinear Mathieu Equation, Progr. Theoret. Phys., 1979, vol. 61, no. 3, pp. 815–824.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Izrailev, F. M., Rabinovich, M. I., and Ugodnikov, A. D., Approximate Description of Three-Dimensional Dissipative Systems with Stochastic Behaviour, Phys. Lett. A, 1981, vol. 86, nos. 6–7, pp. 321–325.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Pereira, T. and Turaev, D., Exponential Energy Growth in Adiabatically Changing Hamiltonian Systems, Phys. Rev. E (3), 2015, vol. 91, no. 1, 010910(R), 4 pp.MathSciNetCrossRefGoogle Scholar
  32. 32.
    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
  33. 33.
    Krishnaprasad, P. S. and Tsakiris, D.P., Oscillations, SE(2)-Snakes and Motion Control: A Study of the Roller Racer, Dyn. Syst., 2001, vol. 16, no. 4, pp. 347–397.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kelly, S.D., Fairchild, M. J., Hassing, P.M., and Tallapragada, P., Proportional Heading Control for Planar Navigation: The Chaplygin Beanie and Fishlike Robotic Swimming, in Proc. of the American Control Conf. (Montreal,QC, Canada, June 2012), pp. 4885–4890.Google Scholar
  35. 35.
    Koiller, J., Markarian, R., Oliffson Kamphorst, S., and Pinto de Carvalho, S., Time-Dependent Billiards, Nonlinearity, 1995, vol. 8, no. 6, pp. 983–1003.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kilin, A.A., Pivovarova, E.N., and Ivanova, T.B., Spherical Robot of Combined Type: Dynamics and Control, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 716–728.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Ott, E., Grebogi, C., and Yorke, J.A., Controlling Chaos, Phys. Rev. Lett., 1990, vol. 64, no. 11, pp. 1196–1199.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Leonard, N. E., Periodic Forcing, Dynamics and Control of Underactuated Spacecraft and Underwater Vehicles, in Proc. of the 34th IEEE Conf. on Decision and Control (New Orleans, La., Dec 1995), pp. 3980–3985.Google Scholar
  39. 39.
    Lenz, F., Diakonos, F.K., and Schmelcher, P., Tunable Fermi Acceleration in the Driven Elliptical Billiard, Phys. Rev. Lett., 2008, vol. 100, no. 1, 014103, 4 pp.CrossRefGoogle Scholar
  40. 40.
    Lewis, A. D., Ostrowskiy, J.P., Burdickz, J. W., and Murray, R. M., Nonholonomic Mechanics and Locomotion: The Snakeboard Example, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (San Diego, Calif., May 1994), pp. 2391–2400.Google Scholar
  41. 41.
    Liouville, J., Développements sur un chapitre de la Mécanique de Poisson, J. Math. Pures Appl., 1858, vol. 3, pp. 1–25.Google Scholar
  42. 42.
    Lichtenberg, A. J. and Lieberman, M.A., Regular and Chaotic Dynamics, 2nd ed. Appl. Math. Sci., vol. 38, New York: Springer, 1992.Google Scholar
  43. 43.
    Murray, R. M. and Sastry, S. Sh., Nonholonomic Motion Planning: Steering Using Sinusoids, IEEE Trans. Automat. Control, 1993, vol. 38, no. 5, pp. 700–716.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Osborne, J. M. and Zenkov, D. V., Steering the Chaplygin Sleigh by a Moving Mass, in Proc. of the 44th IEEE Conf. on Decision and Control (Seville, Spain, Dec 2005), pp. 1114–1118.CrossRefGoogle Scholar
  45. 45.
    Sprott, J.C., Elegant Chaos: Algebraically Simple Chaotic Flows, Singapore: World Sci., 2010.CrossRefzbMATHGoogle Scholar
  46. 46.
    Tallapragada, P. and Kelly, S.D., Integrability of Velocity Constraints Modeling Vortex Shedding in Ideal Fluids, J. Comput. Nonlinear Dynam., 2017, vol. 12, no. 2, 021008, 7 pp.CrossRefGoogle Scholar
  47. 47.
    Kelly, S.D. and Abrajan-Guerrero, R., Planar Motion Control, Coordination, and Dynamic Entrainment for a Singly Actuated Nonholonomic Robot,–files/start/kellyabrajan-guerrero16cdc.pdf (2016).Google Scholar
  48. 48.
    Vetchanin, E. V. and Kilin, A.A., Free and Controlled Motion of a Body with Moving Internal Mass though a Fluid in the Presence of Circulation around the Body, Dokl. Phys., 2016, vol. 61, no. 1, pp. 32–36; see also: Dokl. Akad. Nauk, 2016, vol. 466, no. 3, pp. 293–297.CrossRefGoogle Scholar
  49. 49.
    Jung, Ch. and Scholz, H.-J., Chaotic Scattering off the Magnetic Dipole, J. Phys. A, 1988, vol. 21, no. 10, pp. 2301–2311.MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Eckhardt, B. and Jung, C., Regular and Irregular Potential Scattering, J. Phys. A, 1986, vol. 19, no. 14, L829–L833.MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Tophøj, L. and Aref, H., Chaotic Scattering of Two Identical Point Vortex Pairs Revisited, Phys. Fluids, 2008, vol. 20, 093605, 10 pp.CrossRefzbMATHGoogle Scholar
  52. 52.
    Bizyaev, I. A., Borisov, A.V., and Kuznetsov, S.P., Chaplygin Sleigh with Periodically Oscillating Internal Mass,Europhys. Lett., 2017, vol. 119, no. 6, 60008, 7 pp.CrossRefGoogle Scholar
  53. 53.
    Feller, W., An Introduction to Probability Theory and its Applications, 3rd ed., vol. 1, New York: Wiley, 1968.Google Scholar
  54. 54.
    Rytov, S.M., Kravtsov, Y.A., Tatarskii, V. I., Principles of Statistical Radiophysics. 1. Elements of Random Process Theory, Berlin: Springer, 1987.zbMATHGoogle Scholar
  55. 55.
    Cox, D. R., Miller, H. D., The Theory of Stochastic Processes, New York: Chapman and Hall/CRC, 2017.zbMATHGoogle Scholar
  56. 56.
    Borisov, A.V., Mamaev, I. S., Bizyaev, I.A., Dynamical systems with non-integrable constraints: vaconomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics, Uspekhi Mat. Nauk, 2017, vol. 72, no. 5(437), pp. 3–62.MathSciNetCrossRefGoogle Scholar
  57. 57.
    Kuznetsov, S.P., Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finite-dimensional Models, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 345–382.MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside, Regul. Chaotic Dyn., 2014, vol. 19, no. 2, pp. 198–213.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • Ivan A. Bizyaev
    • 1
    Email author
  • Alexey V. Borisov
    • 2
  • Ivan S. Mamaev
    • 3
  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  2. 2.Udmurt State UniversityIzhevskRussia
  3. 3.Izhevsk State Technical UniversityIzhevskRussia

Personalised recommendations