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 Regul. Chaotic Dyn., 2018, Volume 23, Issue 4, Pages 480–502 (Mi rcd335)

Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation

Alexey V. Borisova, Ivan S. Mamaevba, Eugeny V. Vetchaninab

a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
b Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069 Russia

Abstract: This paper addresses the problem of self-propulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasi-periodic regimes (attracting tori) and strange attractors. One-parameter bifurcation diagrams are constructed, and Neimark – Sacker bifurcations and period-doubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermi-like acceleration) is possible.

Keywords: self-propulsion in a fluid, motion with speed-up, parametric excitation, viscous dissipation, circulation, period-doubling bifurcation, Neimark – Sacker bifurcation, Poincaré map, chart of dynamical regimes, chart of Lyapunov exponents, strange att

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.2404.2017/4.61.2405.2017/4.6 Russian Foundation for Basic Research 15-08-09093-a The work of A.V. Borisov (Introduction and Section 1) was carried out within the framework of the state assignment to the Udmurt State University 1.2404.2017/4.6. The work of E.V. Vetchanin and I. S.Mamaev (Sections 2 and 3) was carried out within the framework of the state assignment to the Izhevsk State Technical University 1.2405.2017/4.6 and was supported by the RFBR grant No 15-08-09093-a.

DOI: https://doi.org/10.1134/S1560354718040081

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Bibliographic databases:

Document Type: Article
MSC: 70H08, 70Exx, 76Bxx, 76Dxx
Language: English

Citation: 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

Citation in format AMSBIB
\Bibitem{BorMamVet18} \by Alexey V. Borisov, Ivan S. Mamaev, Eugeny V. Vetchanin \paper Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation \jour Regul. Chaotic Dyn. \yr 2018 \vol 23 \issue 4 \pages 480--502 \mathnet{http://mi.mathnet.ru/rcd335} \crossref{https://doi.org/10.1134/S1560354718040081} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3836283} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000440806900008} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85051117098} 

• http://mi.mathnet.ru/eng/rcd335
• http://mi.mathnet.ru/eng/rcd/v23/i4/p480

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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. Alexey V. Borisov, Ivan S. Mamaev, Evgeny V. Vetchanin, “Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation”, Regul. Chaotic Dyn., 23:7-8 (2018), 850–874
2. Ivan S. Mamaev, Evgeny V. Vetchanin, “The Self-propulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor”, Regul. Chaotic Dyn., 23:7-8 (2018), 875–886