Regular and Chaotic Dynamics

, Volume 23, Issue 4, pp 480–502 | Cite as

Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation

  • Alexey V. BorisovEmail author
  • Ivan S. Mamaev
  • Eugeny V. Vetchanin


This paper addresses the problem of self-propulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasi-periodic regimes (attracting tori) and strange attractors. One-parameter bifurcation diagrams are constructed, and Neimark–Sacker bifurcations and period-doubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermi-like acceleration) is possible.


self-propulsion in a fluid motion with speed-up parametric excitation viscous dissipation, circulation period-doubling bifurcation Neimark–Sacker bifurcation Poincaré map chart of dynamical regimes chart of Lyapunov exponents strange attractor KAM curves anisotropic dissipation Fermi-like acceleration 

MSC2010 numbers

70H08 70Exx 76Bxx 76Dxx 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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. 407–418Google Scholar
  2. 2.
    Borisov, A.V., Mamaev, I. S., and Bizyaev, I.A., Dynamical Systems with Non-Integrable Constraints: Vaconomic Mechanics, Sub-Riemannian Geometry, and Non-Holonomic Mechanics, Russian Math. Surveys, 2017, vol. 72, no. 5, pp. 783–840 see also: Uspekhi Mat. Nauk, 2017, vol. 72, no. 5(437), pp. 3–62.MathSciNetCrossRefGoogle Scholar
  3. 3.
    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. 192–195CrossRefzbMATHGoogle Scholar
  4. 4.
    Brendelev, V. N., On the Realization of Constraints in Nonholonomic Mechanics, J. Appl. Math. Mech., 1981, vol. 45, no. 3, pp. 351–355 see also: Prikl. Mat. Mekh., 1981, vol. 45, no. 3, pp. 481–487MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Vetchanin, E. V. and Kilin, A.A., Controlled Motion of a Rigid Body with Internal Mechanisms in an Ideal Incompressible Fluid, Proc. Steklov Inst. Math., 2016, vol. 295, pp. 302–332 see also: Tr. Mat. Inst. Steklova, 2016, vol. 295, pp. 321–351MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Vetchanin, E. V. and Klenov, A. I., Experimental Investigation of the Fall of Helical Bodies in a Fluid, Nelin. Dinam., 2017, vol. 13, no. 4, pp. 585–598(Russian).CrossRefzbMATHGoogle Scholar
  7. 7.
    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. 42–51MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kozlov, V. V., Realization of Nonintegrable Constraints in Classical Mechanics, Sov. Phys. Dokl., 1983, vol. 28, pp. 735–737 see also: Dokl. Akad. Nauk SSSR, 1983, vol. 272, no. 3, pp. 550–554zbMATHGoogle Scholar
  9. 9.
    Kozlov, V. V., On Falling of a Heavy Rigid Body in an Ideal Fluid, Izv. Akad. Nauk SSSR Mekh. Tverd. Tela, 1989, no. 5, pp. 10–17(Russian).Google Scholar
  10. 10.
    Kozlov, V. V., On the Problem of Fall of a Rigid Body in a Resisting Medium, Mosc. Univ. Mech. Bull., 1990, vol. 45, no. 1, pp. 30–36 see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1990, no. 1, pp. 79–86zbMATHGoogle Scholar
  11. 11.
    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. 592–601MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    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
  13. 13.
    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. 345–382MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ramodanov, S. M., Tenenev, V. A., and Treschev, D. V., Self-Propulsion of a Body with Rigid Surface and Variable Coefficient of Lift in a Perfect Fluid, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 547–558MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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. 620–633MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chaplygin, S. A., On the Action of a Plane-Parallel Air Flow upon a Cylindrical Wing Moving within It, in The Selected Works on Wing Theory of Sergei A. Chaplygin, San Francisco: Garbell Research Foundation, 1956, pp. 42–72.Google Scholar
  17. 17.
    Chernous’ko, F. L., The Optimal Periodic Motions of a Two-Mass System in a Resistant Medium, J. Appl. Math. Mech., 2008, vol. 72, no. 2, pp. 116–125 see also: Prikl. Mat. Mekh., 2008, vol. 72, no. 2, pp. 202–215MathSciNetCrossRefGoogle Scholar
  18. 18.
    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
  19. 19.
    Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration, Regul. Chaotic Dyn., 2017, vol. 22, no. 8, pp. 955–975.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Borisov, A. V., Kozlov, V. V., and Mamaev, I. S., Asymptotic Stability and Associated Problems of Dynamics of Falling Rigid Body, Regul. Chaotic Dyn., 2007, vol. 12, no. 5, pp. 531–565MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Borisov, A. V. and Mamaev, I. S., On the Motion of a Heavy Rigid Body in an Ideal Fluid with Circulation, Chaos, 2006, vol. 16, no. 1, 013118, 7 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Broer, H., Simó, C., and Vitolo, R., Bifurcations and Strange Attractors in the Lorenz-84 Climate Model with Seasonal Forcing, Nonlinearity, 2002, vol. 15, no. 4, pp. 1205–1267MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Eldering, J., Realizing Nonholonomic Dynamics as Limit of Friction Forces, Regul. Chaotic Dyn., 2016, vol. 21, no. 4, pp. 390–409MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hairer, E., Nørsett, S.P., and Wanner, G., Solving Ordinary Differential Equations: 1. Nonstiff Problems, 2nd ed., rev., Springer Series in Computational Mathematics, vol. 8, Berlin: Springer, 1993.Google Scholar
  25. 25.
    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. 716–728MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kirchhoff, G., Vorlesungen über mathematische Physik: Vol. 1. Mechanik, Leipzig: Teubner, 1876.zbMATHGoogle Scholar
  27. 27.
    Klenov, A. I. and Kilin, A.A., Influence of Vortex Structures on the Controlled Motion of an Above-Water Screwless Robot, Regul. Chaotic Dyn., 2016, vol. 21, nos. 7–8, pp. 927–938.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tallapragada, P and Kelly, S. D., Integrability of velocity constraints modeling vortex shedding in ideal fluids, Journal of Computational and Nonlinear Dynamics, 2017, vol. 12, no. 2, 021008, 10 pp.Google Scholar
  29. 29.
    Treschev, D. and Zubelevich, O., Introduction to the Perturbation Theory of Hamiltonian Systems, Springer Monogr. in Math., Berlin: Springer, 2010.Google Scholar
  30. 30.
    Vetchanin, E. V. and Kilin, A. A., Control of Body Motion in an Ideal Fluid Using the Internal Mass and the Rotor in the Presence of Circulation around the Body, J. Dyn. Control Syst., 2017, vol. 23, no. 2, pp. 435–458MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Vetchanin, E. V. and Mamaev, I. S., Dynamics of Two Point Vortices in an External Compressible Shear Flow, Regul. Chaotic Dyn., 2017, vol. 22, no. 8, pp. 893–908MathSciNetCrossRefGoogle Scholar
  32. 32.
    Vetchanin, E. V., Mamaev, I. S., and Tenenev, V. A., The Self-Propulsion of a Body with Moving Internal Masses in a Viscous Fluid, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 100–117.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Bizyaev, I. A., Borisov, A. V. and Kuznetsov, S. P., The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass, arXiv:1807.06340 (2018).Google Scholar
  34. 34.
    Bizyaev, I. A., Borisov, A. V., Kozlov, V. V. and Mamaev, I. S., Fermi-like acceleration and power law energy growth in nonholonomic systems, arXiv:1807.06262 (2018).Google Scholar
  35. 35.
    Fermi, E., On the origin of the cosmic radiation, Physical Review, 1949, vol. 75, no. 8, pp. 1169–1174CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    Email author
  • Ivan S. Mamaev
    • 1
    • 2
  • Eugeny V. Vetchanin
    • 1
    • 2
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Izhevsk State Technical UniversityIzhevskRussia

Personalised recommendations