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, 141700, 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

References (36) Cited by (36) Similar articles (20) ↓

A.V. Borisov, I.S. Mamaev “Strange attractors in rattleback dynamics” Phys. Usp. 46 393–403 (2003)
E.N. Rumanov “Critical phenomena far from equilibrium” Phys. Usp. 56 93–102 (2013)
V.I. Klyatskin “Statistical topography and Lyapunov exponents in stochastic dynamical systems” Phys. Usp. 51 395–407 (2008)
G.S. Golitsyn “A N Kolmogorov's 1934 paper is the basis for explaining the statistics of natural phenomena of the macrocosm” Phys. Usp. 67 80–90 (2024)
B.Ya. Shmerlin, M.V. Kalashnik “Rayleigh convective instability in the presence of phase transitions of water vapor. The formation of large-scale eddies and cloud structures” Phys. Usp. 56 473–485 (2013)
A.A. Abrashkin, E.N. Pelinovsky “Gerstner waves and their generalizations in hydrodynamics and geophysics” Phys. Usp. 65 453–467 (2022)
O.V. Rudenko “Nonlinear dynamics of quadratically cubic systems” Phys. Usp. 56 683–690 (2013)
A. Loskutov “Dynamical chaos: systems of classical mechanics” Phys. Usp. 50 939–964 (2007)
V.V. Brazhkin “Why does statistical mechanics 'work' in condensed matter?” Phys. Usp. 64 1049–1057 (2021)
S.N. Gordienko “Irreversibility and the probabilistic treatment of the dynamics of classical particles” Phys. Usp. 42 573–590 (1999)
E.P. Zemskov “Turing patterns and Newell—Whitehead—Segel amplitude equation” Phys. Usp. 57 1035–1037 (2014)
P.N. Svirkunov, M.V. Kalashnik “Phase patterns of dispersive waves from moving localized sources” Phys. Usp. 57 80–91 (2014)
A.N. Pavlov, V.S. Anishchenko “Multifractal analysis of complex signals” Phys. Usp. 50 819–834 (2007)
P.S. Landa, Ya.B. Duboshinskii “Self-oscillatory systems with high-frequency energy sources” Sov. Phys. Usp. 32 723–731 (1989)
A.M. Gaifullin, V.V. Zhvick “Laminar submerged jets of incompressible fluid at large Reynolds numbers” Phys. Usp. 66 1142–1153 (2023)
B.M. Bolotovskii, A.V. Serov “Features of the transition radiation field” Phys. Usp. 52 487–493 (2009)
G.B. Malykin “The relation of Thomas precession to Ishlinskii’s theorem as applied to the rotating image of a relativistically moving body” Phys. Usp. 42 505–509 (1999)
V.V. Kunin, B.E. Nemtsov, Éidman B Ya “Precursor and lateral waves during pulse reflection from the separation boundary of two media” Sov. Phys. Usp. 28 827–841 (1985)
V.M. Rozenbaum, I.V. Shapochkina, L.I. Trakhtenberg “Green's function method in the theory of Brownian motors” Phys. Usp. 62 496–509 (2019)
A.A. Shatskii, I.D. Novikov, N.S. Kardashev “A dynamic model of the wormhole and the Multiverse model” Phys. Usp. 51 457–464 (2008)

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