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

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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, dissi-pative, 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.

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Funding

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).

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Authors and Affiliations

  1. Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950, Russia

    S. V. Gonchenko & A. S. Gonchenko

  2. National Research University Higher School of Economics, Bol’shaya Pecherskaya ul. 25/12, Nizhny Novgorod, 603155, Russia

    A. O. Kazakov

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  1. S. V. Gonchenko
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  2. A. S. Gonchenko
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Correspondence to S. V. Gonchenko, A. S. Gonchenko or A. O. Kazakov.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2020, Vol. 308, pp. 135–151.

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Gonchenko, S.V., Gonchenko, A.S. & Kazakov, A.O. Three Types of Attractors and Mixed Dynamics of Nonholonomic Models of Rigid Body Motion. Proc. Steklov Inst. Math. 308, 125–140 (2020). https://doi.org/10.1134/S0081543820010101

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