Issues

 / 

2014

 / 

May

  

Methodological notes


Nonlinear dynamics of the rattleback: a nonholonomic model

 a, b,  c, d,  e
a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russian Federation
b Moscow Institute of Physics and Technology (National Research University), Institutskii per. 9, Dolgoprudny, Moscow Region, 141701, Russian Federation
c Institute of Computer Science, ul. Universitetskaya1, Izhevsk, 426034, Russian Federation
d Lobachevsky State University of Nizhny Novgorod, Faculty of Computational Mathematics and Cybernetics, pr. Gagarina 23, Nizhny Novgorod, 603950, Russian Federation
e Kotel'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.

Fulltext is available at IOP
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/
Citation: Borisov A V, Kazakov A O, Kuznetsov S P "Nonlinear dynamics of the rattleback: a nonholonomic model" Phys. Usp. 57 453–460 (2014)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

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

:   ,   ,    « : » 184 493–500 (2014); DOI: 10.3367/UFNr.0184.201405b.0493

References (36) Cited by (35) Similar articles (14) ↓

  1. A.V. Borisov, I.S. Mamaev “Strange attractors in rattleback dynamics46 393–403 (2003)
  2. E.N. Rumanov “Critical phenomena far from equilibrium56 93–102 (2013)
  3. 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 structures56 473–485 (2013)
  4. V.I. Klyatskin “Statistical topography and Lyapunov exponents in stochastic dynamical systems51 395–407 (2008)
  5. O.V. Rudenko “Nonlinear dynamics of quadratically cubic systems56 683–690 (2013)
  6. V.V. Brazhkin “Why does statistical mechanics 'work' in condensed matter?64 (10) (2021)
  7. A.N. Pavlov, V.S. Anishchenko “Multifractal analysis of complex signals50 819–834 (2007)
  8. S.N. Gordienko “Irreversibility and the probabilistic treatment of the dynamics of classical particles42 573–590 (1999)
  9. E.P. Zemskov “Turing patterns and Newell—Whitehead—Segel amplitude equation57 1035–1037 (2014)
  10. A. Loskutov “Dynamical chaos: systems of classical mechanics50 939–964 (2007)
  11. P.S. Landa, Ya.B. Duboshinskii “Self-oscillatory systems with high-frequency energy sources32 723–731 (1989)
  12. P.N. Svirkunov, M.V. Kalashnik “Phase patterns of dispersive waves from moving localized sources57 80–91 (2014)
  13. R.Z. Muratov “Some useful correspondences in classical magnetostatics, and the multipole representations of the magnetic potential of an ellipsoid55 919–928 (2012)
  14. G.B. Malykin “The relation of Thomas precession to Ishlinskii’s theorem as applied to the rotating image of a relativistically moving body42 505–509 (1999)

The list is formed automatically.

© 1918–2021 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions