Nonlinear dynamics of the rattleback: a nonholonomic model

A.V. Borisov^a, b, A.O. Kazakov^c, d, S.P. Kuznetsov^e
^aUdmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russian Federation
^bMoscow Institute of Physics and Technology (National Research University), Institutskii per. 9, Dolgoprudny, Moscow Region, 141701, Russian Federation
^cInstitute of Computer Science, ul. Universitetskaya 1, Izhevsk, 426034, Russian Federation
^dLobachevsky State University of Nizhny Novgorod, Faculty of Computational Mathematics and Cybernetics, pr. Gagarina 23, Nizhny Novgorod, 603950, Russian Federation
^eKotel'nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Saratov Branch, ul. Zelenaya 38, Saratov, 410019, Russian Federation

For a solid body of convex form moving on a rough horizontal plane that is known as a rattleback, numerical simulations are used to discuss and illustrate dynamical phenomena that are characteristic of the motion due to a nonholonomic nature of the mechanical system; the relevant feature is the nonconservation of the phase volume in the course of the dynamics. In such a system, a local compression of the phase volume can produce behavior features similar to those exhibited by dissipative systems, such as stable equilibrium points corresponding to stationary rotations; limit cycles (rotations with oscillations); and strange attractors. A chart of dynamical regimes is plotted in a plane whose axes are the total mechanical energy and the relative angle between the geometric and dynamic principal axes of the body. The transition to chaos through a sequence of Feigenbaum period doubling bifurcations is demonstrated. A number of strange attractors are considered, for which phase portraits, Lyapunov exponents, and Fourier spectra are presented.

PACS: 05.45.−a, 45.10.−b, 45.40.−f (all)
DOI: 10.3367/UFNe.0184.201405b.0493
URL: https://ufn.ru/en/articles/2014/5/b/
Web of Science

000340732000002
Scopus

2-s2.0-84905968176
ADS NASA

2014PhyU...57..453B
Citation: Borisov A V, Kazakov A O, Kuznetsov S P "Nonlinear dynamics of the rattleback: a nonholonomic model" Phys. Usp. 57 453–460 (2014)

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Received: 29th, August 2013, revised: 1st, October 2013, accepted: 8th, October 2013

Оригинал: Борисов А В, Казаков А О, Кузнецов С П «Нелинейная динамика кельтского камня: неголономная модель» УФН 184 493–500 (2014); DOI: 10.3367/UFNr.0184.201405b.0493

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