Advertisement

Radiophysics and Quantum Electronics

, Volume 61, Issue 10, pp 773–786 | Cite as

Chaotic Dynamics and Multistability in the Nonholonomic Model of a Celtic Stone

  • A. S. Gonchenko
  • S. V. Gonchenko
  • A. O. KazakovEmail author
  • E. A. Samylina
Article

We study dynamic properties of a Celtic stone moving along a plane. We consider two-parameter families of the corresponding nonholonomic models in which bifurcations leading to changing the types of stable motions of the stone, as well as the chaotic-dynamics onset are analyzed. It shown that the multistability phenomena are observed in such models when stable regimes various types (regular and chaotic) can coexist in the phase space of the system. We also show that chaotic dynamics of the nonholonomic model of a Celtic stone can be rather diverse. In this model, in the corresponding parameter regions, one can observe both spiral strange attractors various types, including the so-called discrete Shilnikov attractors, and mixed dynamics, when an attractor and a repeller intersect and almost coincide. A new scenario of instantaneous transition to the mixed dynamics as a result of the reversible bifurcation of merging of the stable and unstable limit cycles is found.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. S. Gonchenko and A. O. Kazakov, “Secrets of the Celtic-stone dynamics,” in: Nauchn. Obozr., No. 2 (12), 14 (2012).Google Scholar
  2. 2.
    G.T.Walker, Proc. Cambridge Philos. Soc., 8, 305 (1895).Google Scholar
  3. 3.
    J. Walker, Sci. Am., 241, 172 (1979).CrossRefGoogle Scholar
  4. 4.
    I. S. Astapov, Vest. Moscow State Univ., Matem. Mekh., No. 2, 97 (1980).Google Scholar
  5. 5.
    A. V. Karapetyan, Prikl. Mat. Mekh., 45, No. 5, 808 (1981).Google Scholar
  6. 6.
    A.P. Markeev, Prikl. Mat. Mekh., 47, No. 4, 575 (1983).Google Scholar
  7. 7.
    A. V. Borisov and I. S. Mamaev, Physics—Uspekhi, 46, No. 4, 393 (2003).ADSGoogle Scholar
  8. 8.
    A. V. Borisov, A. A. Kilin, and I. S. Mamaev, Doklady Physics, 51, No. 5, 272 (2006).ADSCrossRefGoogle Scholar
  9. 9.
    S.P. Kuznetsov, A.Yu. Zhalnin, I.R. Sataev, and Yu.V. Sedova, Nelin. Din., 8, No. 4, 735 (2012).CrossRefGoogle Scholar
  10. 10.
    A. V. Borisov, A.O.Kazakov, and S.P.Kuznetsov, Physics—Uspekhi, 57, No. 5, 453 (2014).ADSGoogle Scholar
  11. 11.
    A. S. Gonchenko, S.V.Gonchenko, and A.O.Kazakov, Nelin. Din., 8, No. 3, 507 (2012).CrossRefGoogle Scholar
  12. 12.
    A. S. Gonchenko, S.V.Gonchenko, and A.O.Kazakov, Reg. Chaot. Dyn., 18, No. 5, 521 (2013).CrossRefGoogle Scholar
  13. 13.
    S.V. Gonchenko, I. I. Ovsyannikov, C. Simo, and D. Turaev, Int. J. Bifur. Chaos, 15, No. 11, 3493 (2005).CrossRefGoogle Scholar
  14. 14.
    A. S. Gonchenko, S.V.Gonchenko, and L.P. Shilnikov, Nonlin. Din., 8, No. 1, 3 (2012).CrossRefGoogle Scholar
  15. 15.
    A. V. Borisov, A.O.Kazakov, and I.R. Sataev, Reg. Chaot. Dyn., 19, No. 6, 718 (2014).CrossRefGoogle Scholar
  16. 16.
    S.V.Gonchenko and D.V.Turaev, Proc. Steklov. Inst. Math., 297, No. 1, 116 (2017).CrossRefGoogle Scholar
  17. 17.
    S. Gonchenko, Discont. Nonlin. Complex., 5, No. 4, 365 (2016).Google Scholar
  18. 18.
    A. S. Gonchenko, S.V.Gonchenko, A. O. Kazakov, and D.V.Turaev, Physica D, 350, 45 (2017).ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    A.O.Kazakov, arXiv:1801.00150.[math.DS] (2017).Google Scholar
  20. 20.
    A. O. Kazakov, Reg. Chaot. Dyn., 18, No. 5, 508 (2013).CrossRefGoogle Scholar
  21. 21.
    S.P.Kuznetsov, Europhys. Lett., 118, No. 1, 10007 (2017).ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    S.P.Kuznetsov, Reg. Chaot. Dyn., 23, No. 2, 178 (2018).MathSciNetCrossRefGoogle Scholar
  23. 23.
    V. V. Kozlov, Usp. Mekh., 8, No. 3, 85 (1985).Google Scholar
  24. 24.
    N. N. Bautin, Behavior of Dynamical Systems near the Stability Region Boundaries [in Russian], Nauka, Moscow (1984).Google Scholar
  25. 25.
    N.N. Bautin and L.P. Shilnikov, Supplement I to: J.E.Marsden and M.McCracken, The Hopf Bifurcation and its Applications, Springer-Verlag, New York (1976).Google Scholar
  26. 26.
    L. M. Lerman and D. Turaev, Reg. Chaot. Dyn., 17, Nos. 3–4, 318 (2012).CrossRefGoogle Scholar
  27. 27.
    L.P. Shilnikov, in: Methods of Qualitative Theory of Differential Equations [in Russian], Gorky (1986), p. 150.Google Scholar
  28. 28.
    A. S. Gonchenko, S.V.Gonchenko, A. O. Kazakov, and D. Turaev, Int. J. Bifur. Chaos, 24, No. 8, 1440005 (2014).CrossRefGoogle Scholar
  29. 29.
    A. S. Gonchenko and S.V.Gonchenko, Physica D, 337, 43 (2016).ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    A. V. Borisov, A.O. Kazakov and I.R. Sataev, Reg. Chaot. Dyn., 21, Nos. 7–8, 939 (2016).CrossRefGoogle Scholar
  31. 31.
    V. S. Afraimovich and L.P. Shilnikov, “Strange attractors and quasiattractors,” in: G. I. Barenblatt, G. Iooss, and D.D. Joseph, eds., Nonlinear Dynamics and Turbulence, Pitmen, Boston (1983), p. 1.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • A. S. Gonchenko
    • 1
  • S. V. Gonchenko
    • 1
  • A. O. Kazakov
    • 1
    • 2
    Email author
  • E. A. Samylina
    • 1
    • 2
  1. 1.N. I. Lobachevsky State University of Nizhny NovgorodNizhnij NovgorodRussia
  2. 2.Higher School of EconomicsNizhny NovgorodRussia

Personalised recommendations