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 Regul. Chaotic Dyn., 2015, Volume 20, Issue 5, Pages 605–626 (Mi rcd22)

Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors

Ivan A. Bizyaevab, Alexey V. Borisovb, Alexey O. Kazakovac

a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
b Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
c National Research University Higher School of Economics, ul. Rodionova 136, Nizhny Novgorod, 603093 Russia

Abstract: In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.

Keywords: Suslov problem, nonholonomic constraint, reversal, strange attractor

 Funding Agency Grant Number Russian Science Foundation 14-50-0000515-12-20035 Russian Foundation for Basic Research 15-38-20879 mol_a_ved15-08-09261-a Sections 1, 3 and 7 were prepared by A.V. Borisov under the RSF grant No. 14-50-00005. Sections 2 and 5 were prepared by I.A. Bizyaev within the framework of the RFBR grants No. 15-38-20879 mol_a_ved and No. 15-08-09261-a. The work of A.O. Kazakov (Sections 4 and 6) was supported by RSF grant No. 15-12-20035.

DOI: https://doi.org/10.1134/S1560354715050056

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

Citation: Ivan A. Bizyaev, Alexey V. Borisov, Alexey O. Kazakov, “Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors”, Regul. Chaotic Dyn., 20:5 (2015), 605–626

Citation in format AMSBIB
\Bibitem{BizBorKaz15} \by Ivan A. Bizyaev, Alexey V. Borisov, Alexey O. Kazakov \paper Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors \jour Regul. Chaotic Dyn. \yr 2015 \vol 20 \issue 5 \pages 605--626 \mathnet{http://mi.mathnet.ru/rcd22} \crossref{https://doi.org/10.1134/S1560354715050056} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3412546} \zmath{https://zbmath.org/?q=an:06529975} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000362971400005} \elib{http://elibrary.ru/item.asp?id=24961859} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84944459733} 

• http://mi.mathnet.ru/eng/rcd22
• http://mi.mathnet.ru/eng/rcd/v20/i5/p605

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This publication is cited in the following articles:
1. A. Nanda, P. Singla, M. F. Karami, “Energy harvesting using rattleback: theoretical analysis and simulations of spin resonance”, Journal of Sound and Vibration, 369 (2016), 195–208
2. Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248
3. 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
4. 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
5. 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
6. S. P. Kuznetsov, V. P. Kruglov, “On some simple examples of mechanical systems with hyperbolic chaos”, Proc. Steklov Inst. Math., 297 (2017), 208–234
7. Stefan Rauch-Wojciechowski, Maria Przybylska, “Understanding Reversals of a Rattleback”, Regul. Chaotic Dyn., 22:4 (2017), 368–385
8. 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
9. A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov, D. V. Turaev, “On the phenomenon of mixed dynamics in Pikovsky-Topaj system of coupled rotators”, Physica D, 350 (2017), 45–57
10. Shengda Hu, Manuele Santoprete, “Suslov Problem with the Clebsch–Tisserand Potential”, Regul. Chaotic Dyn., 23:2 (2018), 193–211