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 Trudy Mat. Inst. Steklova, 2020, Volume 308, Pages 135–151 (Mi tm4053)

Three Types of Attractors and Mixed Dynamics of Nonholonomic Models of Rigid Body Motion

S. V. Gonchenkoa, A. S. Gonchenkoa, A. O. Kazakovb

a Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia
b National Research University Higher School of Economics, Bol'shaya Pecherskaya ul. 25/12, Nizhny Novgorod, 603155 Russia

Abstract: We survey recent results on the theory of dynamical chaos from the point of view of topological dynamics. We present the concept of three types of dynamics: conservative, dissipative, and mixed dynamics, and also show several simple examples of attractors and repellers of all three types. Their similarities and differences with other known types of attractors and repellers (maximal and Milnor ones) are discussed. We also present elements of the qualitative theory of mixed dynamics of reversible systems. As examples of such systems we consider three nonholonomic models of rigid body motion: the Suslov top, rubber disk, and Celtic stone. It is shown that they exhibit mixed dynamics of different nature; in particular, the mixed dynamics observed in the model of rubber disk is extremely difficult to distinguish from the conservative one.

 Funding Agency Grant Number Russian Science Foundation 17-11-0104118-71-00127 Russian Foundation for Basic Research 19-01-0060718-31-2005218-29-10081 HSE Basic Research Program Ministry of Science and Higher Education of the Russian Federation 1.3287.2017 This work is supported by the Russian Science Foundation under grants 17-11-01041 (Section 1 and Subsections 4.1, 4.2) and 18-71-00127 (Subsection 4.3) and by the Russian Foundation for Basic Research, project nos. 19-01-00607 (Section 3), 18-31-20052 (Subsection 2.1), and 18-29-10081 (Subsection 2.2). The first two authors also acknowledge the support of the Ministry of Science and Higher Education of the Russian Federation (project no. 1.3287.2017). The third author was also supported in part by the HSE Basic Research Program (2019).

DOI: https://doi.org/10.4213/tm4053

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English version:
Proceedings of the Steklov Institute of Mathematics, 2020, 308, 125–140

Bibliographic databases:

UDC: 517.938
Revised: August 12, 2019
Accepted: October 26, 2019

Citation: S. V. Gonchenko, A. S. Gonchenko, A. O. Kazakov, “Three Types of Attractors and Mixed Dynamics of Nonholonomic Models of Rigid Body Motion”, Differential equations and dynamical systems, Collected papers, Trudy Mat. Inst. Steklova, 308, Steklov Math. Inst. RAS, Moscow, 2020, 135–151; Proc. Steklov Inst. Math., 308 (2020), 125–140

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Bizyaev I.A., Mamaev I.S., “Separatrix Splitting and Nonintegrability in the Nonholonomic Rolling of a Generalized Chaplygin Sphere”, Int. J. Non-Linear Mech., 126 (2020), 103550
2. Stankevich N., Kazakov A., Gonchenko S., “Scenarios of Hyperchaos Occurrence in 4D Rossler System”, Chaos, 30:12 (2020), 123129
3. Kazakov A., “Merger of a Henon-Like Attractor With a Henon-Like Repeller in a Model of Vortex Dynamics”, Chaos, 30:1 (2020), 011105
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