Chaotic Dynamics and Multistability in the Nonholonomic Model of a Celtic Stone
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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.
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- 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.G.T.Walker, Proc. Cambridge Philos. Soc., 8, 305 (1895).Google Scholar
- 4.I. S. Astapov, Vest. Moscow State Univ., Matem. Mekh., No. 2, 97 (1980).Google Scholar
- 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
- 17.S. Gonchenko, Discont. Nonlin. Complex., 5, No. 4, 365 (2016).Google Scholar
- 19.A.O.Kazakov, arXiv:1801.00150.[math.DS] (2017).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).Google Scholar
- 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
- 27.L.P. Shilnikov, in: Methods of Qualitative Theory of Differential Equations [in Russian], Gorky (1986), p. 150.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.Google Scholar