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 Regul. Chaotic Dyn., 2013, Volume 18, Issue 5, Pages 508–520 (Mi rcd136)

Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane

Alexey O. Kazakovab

a The Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University, pr. Gagarina 23, Nizhny Novgorod, 603950, Russia
b Institute of computer science, ul. Universitetskaya 1, Izhevsk, 426034, Russia

Abstract: We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.

Keywords: mixed dynamics, strange attractor, unbalanced ball, rubber rolling, reversibility, two-dimensional Poincaré map, bifurcation, focus, saddle, invariant manifolds, homoclinic tangency, Lyapunov’s exponents

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-0058913-01-97028-povolzhye Ministry of Education and Science of the Russian Federation 14.B37.21.036114.B37.21.0863 This work was supported by the RFBR grants No. 13-01-00589 and 13-01-97028-povolzhye, the Federal Target Program “Personnel” No.14.B37.21.0361, and by the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” (Contract No. 14.B37.21.0863).

DOI: https://doi.org/10.1134/S1560354713050043

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Document Type: Article
MSC: 37J60, 37N15, 37G35
Language: English

Citation: Alexey O. Kazakov, “Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane”, Regul. Chaotic Dyn., 18:5 (2013), 508–520

Citation in format AMSBIB
\Bibitem{Kaz13} \by Alexey O. Kazakov \paper Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane \jour Regul. Chaotic Dyn. \yr 2013 \vol 18 \issue 5 \pages 508--520 \mathnet{http://mi.mathnet.ru/rcd136} \crossref{https://doi.org/10.1134/S1560354713050043} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3117259} \zmath{https://zbmath.org/?q=an:06292756} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000325810200004} 

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• http://mi.mathnet.ru/eng/rcd/v18/i5/p508

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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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