Abstract
The motion problem for a heavy rigid body on a perfectly rough horizontal plane, which is a classical problem of the nonholonomic system dynamics, is considered. The effect from the loss of stability of a body’s permanent rotation at a certain critical value of its angular velocity is discussed. It is proven that this effect is accompanied by the occurrence of periodic motions of the body with a frequency close to the critical value; that is, the Hopf bifurcation takes place. It is proven by means of the direct calculation of the first Lyapunov coefficient that the corresponding periodic motions are unstable.
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REFERENCES
A. V. Karapetyan, “Hopf bifurcation in the problem of motion of a heavy rigid body on a rough plane,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 19–24 (1985).
A. P. Markeev, “The dynamics of a rigid body on an absolutely rough plane,” J. Appl. Math. Mech. 47, 473–478 (1983).
A. V. Borisov and I. S. Mamaev, “Strange attractors in rattleback dynamics,” Phys. Usp. 46, 393–403 (2003). https://doi.org/10.1070/PU2003v046n04ABEH001306
T. P. Tovstik, “On the dynamics of the Celt rattleback with frictions,” in Proc. 4th Polyakhov Readings, St. Petersburg, Feb. 7–10, 2006 (VVM, St. Petersburg, 2006), pp. 187–196.
T. P. Tovstik, “On the influence of sliding on the Celt rattleback motion,” in Advanced Problems in Mechanics: Proc. 35th Int. Summer School Conf., St. Petersburg, Russia, June 20–28, 2007 (2007), pp. 432–437.
I. G. Malkin, Theory of Stability of Motion (Nauka, Moscow, 1966).
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications (Springer-Verlag, New York, 1976; Mir, Moscow, 1980).
N. N. Bautin, Behavior of Dynamical Systems Near the Boundaries of the Stability Region (Nauka, Moscow, 1984) [in Russian].
G. A. Leonov, N. V. Kuznetsov, and E. V. Kudryashova, “Cycles of two-dimensional systems: Computer calculations, proofs, and experiments,” Vestn. St. Petersburg Univ.: Math. 41, 216 (2008). https://doi.org/10.3103/S1063454108030047
Funding
The work was supported by the Russian Foundation for Basic Research, project no. 20-01-00637.
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In memory of Professor A. V. Karapetyan (1950–2021)
Translated by A. Muravnik
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Kuleshov, A.S., Pikunova, E.N. Analytical Research of the Hopf Bifurcation in the Problem of Motion of the Rattleback. Vestnik St.Petersb. Univ.Math. 55, 203–211 (2022). https://doi.org/10.1134/S1063454122020078
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DOI: https://doi.org/10.1134/S1063454122020078