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Nelin. Dinam., 2012, Volume 8, Number 1, Pages 113–147 (Mi nd307)  

This article is cited in 3 scientific papers (total in 3 papers)

The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem

A. V. Borisov, A. A. Kilin, I. S. Mamaev

Institute of Computer Science, Udmurt State University Universitetskaya 1, Izhevsk, 426034 Russia

Abstract: We consider the problem of the 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 in which the mutual 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

Full text: PDF file (1570 kB)
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Document Type: Article
UDC: 532.5.031
MSC: 76B47
Received: 19.09.2011

Citation: A. V. Borisov, A. A. Kilin, I. S. Mamaev, “The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem”, Nelin. Dinam., 8:1 (2012), 113–147

Citation in format AMSBIB
\Bibitem{BorKilMam12}
\by A.~V.~Borisov, A.~A.~Kilin, I.~S.~Mamaev
\paper The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem
\jour Nelin. Dinam.
\yr 2012
\vol 8
\issue 1
\pages 113--147
\mathnet{http://mi.mathnet.ru/nd307}


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    This publication is cited in the following articles:
    1. E. V. Vetchanin, A. O. Kazakov, “Bifurkatsii i khaos v zadache o dvizhenii dvukh tochechnykh vikhrei v akusticheskoi volne”, Nelineinaya dinam., 10:3 (2014), 329–343  mathnet
    2. M. B. Saikhanov, “Strukturno-dinamicheskaya model fotona i monokhromaticheskogo izlucheniya”, Nelineinyi mir, 12:6 (2014), 48–54  elib
    3. Adecarlos C. Carvalho, Hildeberto E. Cabral, “Lyapunov Orbits in the $n$-Vortex Problem on the Sphere”, Regul. Chaotic Dyn., 20:3 (2015), 234–246  mathnet  crossref  mathscinet  zmath  adsnasa
  • Нелинейная динамика
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