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 Regul. Chaotic Dyn., 2014, Volume 19, Issue 6, Pages 718–733 (Mi rcd194)

This article is cited in 30 scientific papers (total in 30 papers)

The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top

Alexey V. Borisovab, Alexey O. Kazakovc, Igor R. Sataevd

a Moscow Institute of Physics and Technology, Inststitutskii per. 9, Dolgoprudnyi, 141700 Russia
b Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
c Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia
d Saratov Branch of Kotelnikov’s Institute of Radio-Engineering and Electronics of RAS ul. Zelenaya 38, Saratov, 410019 Russia

Abstract: In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.

Keywords: rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation MD-2324.2013.12000NSh-1726.2014.2 Russian Science Foundation 14-12-0081114-41-00044 The work of A. V. Borisov was supported by the Ministry of Education and Science of the Russian Federation within the framework of the basic part of the state assignment to institutions of higher education. The work of A. O. Kazakov on Section 3 was supported by the grant of the Russian Scientific Foundation No 14-12- 00811, the work of Section 4.1 was partially supported by the grant of the Russian Scientific Foundation 14-41- 00044 and by the grant of the President of the Russian Federation for support of young doctors of science MD-2324.2013.1. The remaining part of the work of A. O. Kazakov was supported by the Ministry of Education and Science (project No 2000). The work of I. R. Sataev was supported by the grant of the President of the Russian Federation for support of leading scientific schools NSh-1726.2014.2.

DOI: https://doi.org/10.1134/S1560354714060094

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Document Type: Article
MSC: 37J60, 37N15, 37G35, 70E18, 70F25, 70H45
Received: 26.08.2014
Language: English

Citation: Alexey V. Borisov, Alexey O. Kazakov, Igor R. Sataev, “The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top”, Regul. Chaotic Dyn., 19:6 (2014), 718–733

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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5. Alexander P. Ivanov, “On the Control of a Robot Ball Using Two Omniwheels”, Regul. Chaotic Dyn., 20:4 (2015), 441–448
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8. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Dynamics and Control of an Omniwheel Vehicle”, Regul. Chaotic Dyn., 20:2 (2015), 153–172
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11. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204
12. V. A. Tenenev, E. V. Vetchanin, L. F. Ilaletdinov, “Khaoticheskaya dinamika v zadache o padenii tela vintovoi formy v zhidkosti”, Nelineinaya dinam., 12:1 (2016), 99–120
13. I. R. Sataev, A. O. Kazakov, “Stsenarii perekhoda k khaosu v negolonomnoi modeli volchka Chaplygina”, Nelineinaya dinam., 12:2 (2016), 235–250
14. Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248
15. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period”, Regul. Chaotic Dyn., 21:4 (2016), 455–476
16. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hojman Construction and Hamiltonization of Nonholonomic Systems”, SIGMA, 12 (2016), 012, 19 pp.
17. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “The Hess–Appelrot system and its nonholonomic analogs”, Proc. Steklov Inst. Math., 294 (2016), 252–275
18. Alexey V. Borisov, Sergey P. Kuznetsov, “Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts”, Regul. Chaotic Dyn., 21:7-8 (2016), 792–803
19. Alexey V. Borisov, Alexey O. Kazakov, Elena N. Pivovarova, “Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top”, Regul. Chaotic Dyn., 21:7-8 (2016), 885–901
20. Alexey V. Borisov, Alexey O. Kazakov, Igor R. Sataev, “Spiral Chaos in the Nonholonomic Model of a Chaplygin Top”, Regul. Chaotic Dyn., 21:7-8 (2016), 939–954
21. A. S. Gonchenko, S. V. Gonchenko, “Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps”, Physica D, 337 (2016), 43–57
22. V. Kozlov, “The phenomenon of reversal in the Euler–Poincaré–Suslov nonholonomic systems”, J. Dyn. Control Syst., 22:4 (2016), 713–724
23. G. M. Rozenblat, “On the choice of physically realizable parameters when studying the dynamics of spherical and ellipsoidal rigid bodies”, Mech. Sol., 51:4 (2016), 415–423
24. A. Nanda, P. Singla, M. A. Karami, “Energy harvesting using rattleback: theoretical analysis and simulations of spin resonance”, J. Sound Vibr., 369 (2016), 195–208
25. E. V. Vetchanin, A. O. Kazakov, “Bifurcations and chaos in the dynamics of two point vortices in an acoustic wave”, Int. J. Bifurcation Chaos, 26:4 (2016), 1650063
26. T. A. Levanova, A. O. Kazakov, G. V. Osipov, J. Kurths, “Dynamics of ensemble of inhibitory coupled Rulkov maps”, Eur. Phys. J.-Spec. Top., 225:1 (2016), 147–157
27. Stefan Rauch-Wojciechowski, Maria Przybylska, “Understanding Reversals of a Rattleback”, Regul. Chaotic Dyn., 22:4 (2017), 368–385
28. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840
29. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration”, Regul. Chaotic Dyn., 22:8 (2017), 955–975
30. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane”, Regul. Chaotic Dyn., 23:6 (2018), 665–684
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