Skip to main content
SpringerLink
Log in

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

  • Published:
Radiophysics and Quantum Electronics Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. S. Gonchenko and A. O. Kazakov, “Secrets of the Celtic-stone dynamics,” in: Nauchn. Obozr., No. 2 (12), 14 (2012).

  2. G.T.Walker, Proc. Cambridge Philos. Soc., 8, 305 (1895).

    Google Scholar 

  3. J. Walker, Sci. Am., 241, 172 (1979).

    Article  Google Scholar 

  4. I. S. Astapov, Vest. Moscow State Univ., Matem. Mekh., No. 2, 97 (1980).

  5. A. V. Karapetyan, Prikl. Mat. Mekh., 45, No. 5, 808 (1981).

    Google Scholar 

  6. A.P. Markeev, Prikl. Mat. Mekh., 47, No. 4, 575 (1983).

    Google Scholar 

  7. A. V. Borisov and I. S. Mamaev, Physics—Uspekhi, 46, No. 4, 393 (2003).

    ADS  Google Scholar 

  8. A. V. Borisov, A. A. Kilin, and I. S. Mamaev, Doklady Physics, 51, No. 5, 272 (2006).

    Article  ADS  Google Scholar 

  9. S.P. Kuznetsov, A.Yu. Zhalnin, I.R. Sataev, and Yu.V. Sedova, Nelin. Din., 8, No. 4, 735 (2012).

    Article  Google Scholar 

  10. A. V. Borisov, A.O.Kazakov, and S.P.Kuznetsov, Physics—Uspekhi, 57, No. 5, 453 (2014).

    ADS  Google Scholar 

  11. A. S. Gonchenko, S.V.Gonchenko, and A.O.Kazakov, Nelin. Din., 8, No. 3, 507 (2012).

    Article  Google Scholar 

  12. A. S. Gonchenko, S.V.Gonchenko, and A.O.Kazakov, Reg. Chaot. Dyn., 18, No. 5, 521 (2013).

    Article  Google Scholar 

  13. S.V. Gonchenko, I. I. Ovsyannikov, C. Simo, and D. Turaev, Int. J. Bifur. Chaos, 15, No. 11, 3493 (2005).

    Article  Google Scholar 

  14. A. S. Gonchenko, S.V.Gonchenko, and L.P. Shilnikov, Nonlin. Din., 8, No. 1, 3 (2012).

    Article  Google Scholar 

  15. A. V. Borisov, A.O.Kazakov, and I.R. Sataev, Reg. Chaot. Dyn., 19, No. 6, 718 (2014).

    Article  Google Scholar 

  16. S.V.Gonchenko and D.V.Turaev, Proc. Steklov. Inst. Math., 297, No. 1, 116 (2017).

    Article  Google Scholar 

  17. S. Gonchenko, Discont. Nonlin. Complex., 5, No. 4, 365 (2016).

    Google Scholar 

  18. A. S. Gonchenko, S.V.Gonchenko, A. O. Kazakov, and D.V.Turaev, Physica D, 350, 45 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  19. A.O.Kazakov, arXiv:1801.00150.[math.DS] (2017).

  20. A. O. Kazakov, Reg. Chaot. Dyn., 18, No. 5, 508 (2013).

    Article  Google Scholar 

  21. S.P.Kuznetsov, Europhys. Lett., 118, No. 1, 10007 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  22. S.P.Kuznetsov, Reg. Chaot. Dyn., 23, No. 2, 178 (2018).

    Article  MathSciNet  Google Scholar 

  23. V. V. Kozlov, Usp. Mekh., 8, No. 3, 85 (1985).

    Google Scholar 

  24. N. N. Bautin, Behavior of Dynamical Systems near the Stability Region Boundaries [in Russian], Nauka, Moscow (1984).

  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).

  26. L. M. Lerman and D. Turaev, Reg. Chaot. Dyn., 17, Nos. 3–4, 318 (2012).

    Article  Google Scholar 

  27. L.P. Shilnikov, in: Methods of Qualitative Theory of Differential Equations [in Russian], Gorky (1986), p. 150.

  28. A. S. Gonchenko, S.V.Gonchenko, A. O. Kazakov, and D. Turaev, Int. J. Bifur. Chaos, 24, No. 8, 1440005 (2014).

    Article  Google Scholar 

  29. A. S. Gonchenko and S.V.Gonchenko, Physica D, 337, 43 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  30. A. V. Borisov, A.O. Kazakov and I.R. Sataev, Reg. Chaot. Dyn., 21, Nos. 7–8, 939 (2016).

    Article  Google Scholar 

  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.

Download references

Author information

Authors and Affiliations

  1. N. I. Lobachevsky State University of Nizhny Novgorod, Nizhnij Novgorod, Russia, 603022

    A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov & E. A. Samylina

  2. Higher School of Economics, Nizhny Novgorod, Russia

    A. O. Kazakov & E. A. Samylina

Authors
  1. A. S. Gonchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. V. Gonchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. O. Kazakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. A. Samylina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. O. Kazakov.

Additional information

Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 61, No. 10, pp. 867–882, October 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gonchenko, A.S., Gonchenko, S.V., Kazakov, A.O. et al. Chaotic Dynamics and Multistability in the Nonholonomic Model of a Celtic Stone. Radiophys Quantum El 61, 773–786 (2019). https://doi.org/10.1007/s11141-019-09935-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11141-019-09935-4

Search

Navigation