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
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. Lichtenberg A J, Lieberman M A Regular And Stochastic Motion (New York: Springer-Verlag, 1983); Likhtenberg A, Liberman M Regulyarnaya i Stokhasticheskaya Dinamika (M.: Mir, 1984)
  2. Zaslavskii G M, Sagdeev R Z Vvedenie v Nelineinuyu Fiziku: Ot Mayatnika do Turbulentnosti i Khaosa (M.: Nauka, 1988); Sagdeev R Z, Usikov D A, Zaslavsky G M Nonlinear Physics: From The Pendulum To Turbulence And Chaos (Chur: Harwood Acad. Publ., 1988)
  3. Rabinovich M I, Trubetskov D I Vvedenie v Teoriyu Kolebanii i Voln (M.: Nauka, 1984); Rabinovich M I, Trubetskov D I Oscillations And Waves In Linear And Nonlinear Systems (Dordrecht: Kluwer Acad. Publ., 1989)
  4. Landa P S Nelineinye Kolebaniya i Volny (M.: Librokom, 2010)
  5. Landau L D, Lifshits E M Mekhanika (M.: Nauka, 1973); Landau L D, Lifshitz E M Mechanics (Oxford: Pergamon Press, 1976)
  6. Arnol’d V I Matematicheskie Metody Klassicheskoi Mekhaniki (M.: Nauka, 1989); Arnold V I Mathematical Methods Of Classical Mechanics (New York: Springer, 1997)
  7. Borisov A V, Mamaev I S (Red.) Negolonomnye Dinamicheskie Sistemy. Integriruemost’, Khaos, Strannye Attraktory (M. - Izhevsk: Inst. komp’yut. issled., 2002)
  8. Neimark Yu I, Fufaev N A Dinamika Negolonomnykh Sistem (M.: Nauka, 1967); Neimark Ju I, Fufaev N A Dynamics Of Nonholonomic Systems (Providence, R.I.: American Mathematical Society, 1972)
  9. Borisov A V, Mamaev I S Regular Chaotic Dynamics 7 177 (2002)
  10. Borisov A V, Mamaev I S, Bizyaev I A Nelineinaya Dinamika 9 141 (2013)
  11. Walker G T Proc. Camb. Phil. Soc. 8 305 (1895)
  12. Walker G T Quart. J. Pure Appl. Math. 28 175 (1896)
  13. Walker J Sci. Am. 241 (10) 144 (1979)
  14. Kozlov V V Uspekhi Mekhaniki 8 (3) 85 (1985)
  15. Karapetyan A V Izv. AN SSSR. Mekh. Tverd. Tela (2) 19 (1985)
  16. Borisov A V, Mamaev I S Usp. Fiz. Nauk 173 407 (2003); Borisov A V, Mamaev I S Phys. Usp. 46 393 (2003)
  17. Borisov A V et al. Regular Chaotic Dynamics 17 512 (2012)
  18. Gonchenko A S, Gonchenko S V, Shil’nikov L P Nelineinaya Dinamika 8 (1) 3 (2012)
  19. Tsai J-C et al. Phys. Rev. Lett. 94 214301 (2005)
  20. Gonchenko A S, Gonchenko S V, Kazakov A O Nelineinaya Dinamika 8 507 (2012)
  21. Schuster H G, Just W Deterministic Chaos (Weinheim: Wiley-VCH, 2005); Shuster G Determinirovannyi Khaos (M.: Mir, 1988)
  22. Kuznetsov S P Dinamicheskii Khaos (M.: Fizmatlit, 2006)
  23. Feigenbaum M J J. Stat. Phys. 21 669 (1979)
  24. Vul E B, Sinai Ya G, Khanin K M Usp. Mat. Nauk 39 (3) 3 (1984); Vul E B, Sinai Ya G, Khanin K M Russ. Math. Surv. 39 1 (1984)
  25. Reick C Phys. Rev. A 45 777 (1992)
  26. Reichl L E The Transition To Chaos: Conservative Classical Systems And Quantum Manifestations (New York: Springer, 2004); Raikhl L E Perekhod k Khaosu v Konservativnykh Klassicheskikh i Kvantovykh Sistemakh (M. - Izhevsk: RKhD, 2008)
  27. Kuznetsov S P, Kuznetsov A P, Sataev I R J. Stat. Phys. 121 697 (2005)
  28. Lorenz E N J. Atmos. Sci. 20 130 (1963); Lorents E Strannye Attraktory (Pod red. Ya G Sinaya, L P Shil’nikova) (M.: Mir, 1981) p. 88
  29. Sparrow C The Lorenz Equations: Bifurcations, Chaos, And Strange Attractors (New York: Springer-Verlag, 1982)
  30. Afraimovich V S, Bykov V V, Shil’nikov L P Dokl. Akad. Nauk SSSR 234 336 (1977); Afraimovich V S, Bykov V V, Shil’nikov L P Sov. Phys. Dokl. 22 253 (1977)
  31. Guckenheimer J, Holmes P Nonlinear Oscillations, Dynamical Systems, And Bifurcations Of Vector Fields (Berlin: Springer, 1990); Gukenkheimer Dzh, Kholms F Nelineinye Kolebaniya, Dinamicheskie Sistemy i Bifurkatsii Vektornykh Polei (M. - Izhevsk: Inst. komp’yut. issled., 2002)
  32. Tucker W Found. Comput. Math. 2 53 (2002)
  33. Gonchenko A S, Gonchenko S V Nelineinaya Dinamika 9 (1) 77 (2013)
  34. Garcia A, Hubbard M Proc. R. Soc. Lond. A 418 165 (1988)
  35. Kane T R, Levinson D A American Society of Mechanical Engineers, Winter Annual Meeting, San Francisco, Calif., Dec. 10 - 15, 1978
  36. Aleshkevich V A, Dedenko L G, Karavaev V A Lektsii Po Mekhanike Tverdogo Tela (M.: Izd-vo MGU, 1997)

© 1918–2022 Uspekhi Fizicheskikh Nauk
Email: Editorial office contacts About the journal Terms and conditions