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 Regul. Chaotic Dyn.: Year: Volume: Issue: Page: Find

 Regul. Chaotic Dyn., 2017, Volume 22, Issue 8, Pages 955–975 (Mi rcd302)

The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration

Ivan A. Bizyaeva, Alexey V. Borisovb, Ivan S. Mamaevc

a Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia
b Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
c Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069 Russia

Abstract: This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.

Keywords: nonholonomic mechanics, Fermi acceleration, Chaplygin sleigh, parametric oscillator, tensor invariants, involution, strange attractor, Lyapunov exponents, reversible systems, chaotic dynamics

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.2404.2017/4.61.2405.2017/4.6 The work of A.V.Borisov (Sections 1 and 2) was carried out within the framework of the state assignment of the Ministry of Education and Science of Russia (1.2404.2017/4.6). The work of I.A.Bizyaev (Section 3) was carried out at MIPT under project 5-100 for state support for leading universities of the Russian Federation. The work of I. S.Mamaev (Section 4) was carried out within the framework of the state assignment of the Ministry of Education and Science of Russia (1.2405.2017/4.6).

DOI: https://doi.org/10.1134/S1560354717080056

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Document Type: Article
MSC: 37J60, 34A34
Language: English

Citation: 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

Citation in format AMSBIB
\Bibitem{BizBorMam17} \by Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev \paper The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration \jour Regul. Chaotic Dyn. \yr 2017 \vol 22 \issue 8 \pages 955--975 \mathnet{http://mi.mathnet.ru/rcd302} \crossref{https://doi.org/10.1134/S1560354717080056} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000425981500005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85042399707} 

• http://mi.mathnet.ru/eng/rcd302
• http://mi.mathnet.ru/eng/rcd/v22/i8/p955

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Fedonyuk V., Tallapragada Ph., “Sinusoidal Control and Limit Cycle Analysis of the Dissipative Chaplygin Sleigh”, Nonlinear Dyn., 93:2 (2018), 835–846
2. Alexey V. Borisov, Ivan S. Mamaev, Eugeny V. Vetchanin, “Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation”, Regul. Chaotic Dyn., 23:4 (2018), 480–502
3. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control”, Regul. Chaotic Dyn., 23:7-8 (2018), 983–994
4. I. S. Mamaev, V. A. Tenenev, E. V. Vetchanin, “Dynamics of a Body with a Sharp Edge in a Viscous Fluid”, Nelineinaya dinam., 14:4 (2018), 473–494