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Regular and Chaotic Dynamics, 2013, Volume 18, Issue 1-2, Pages 33–62
DOI: https://doi.org/10.1134/S1560354713010036 (Mi rcd94) 		 
This article is cited in 27 scientific papers (total in 27 papers)

The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem

Alexey V. Borisovabc, Alexander A. Kilinbac, Ivan S. Mamaevcab

a Institute of Computer Science, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia
b A.A. Blagonravov Mechanical Engineering Research Institute of RAS, Bardina str. 4, Moscow, 117334 Russia
c Institute of Mathematics and Mechanics of the Ural Branch of RAS, S. Kovalevskaja str. 16, Ekaterinburg, 620990 Russia
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DOI: https://doi.org/10.1134/S1560354713010036
Abstract: We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.
Keywords: ideal fluid, vortex ring, leapfrogging motion of vortex rings, bifurcation complex, periodic solution, integrability, chaotic dynamics.
Funding agency Grant number
Russian Foundation for Basic Research 11-01-91056-NTsNI_a
Ministry of Education and Science of the Russian Federation 1.2953.2011
14.B37.21.1935
NSh-2519.2012.1
MK-8428.2010.1
This research was supported by the RFBR grant 11-01-91056-NTsNI_a, the Target Programme “Development of Scientific Potential of Higher Schools” (State contract 1.2953.2011, 2012–2014); the Federal Target Programme “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” (State contract 14.B37.21.1935, 2009–2013); grant of leading scientific schools NSh-2519.2012.1. A. A. Kilin’s research was supported by the grant of the President of the Russian Federation for the Support of Young Russian Scientists–Candidates of Science (MK-8428.2010.1).
Received: 19.09.2012
Accepted: 21.12.2012
Bibliographic databases:
Document Type: Article
MSC: 76B47
Language: English
Citation: Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem”, Regul. Chaotic Dyn., 18:1-2 (2013), 33–62
Citation in format AMSBIB
\Bibitem{BorKilMam13}
\by Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev
\paper The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 1-2
\pages 33--62
\mathnet{http://mi.mathnet.ru/rcd94}
\crossref{https://doi.org/10.1134/S1560354713010036}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3040981}
\zmath{https://zbmath.org/?q=an:1273.76067}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000317623400003}
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    • This publication is cited in the following 27 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    References:70
     
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