Geodesic
The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem
Russian journal of nonlinear dynamics, Tome 8 (2012) no. 1, pp. 113-147

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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, periodic solution, integrability, chaotic dynamics.
Mots-clés : bifurcation complex 
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     title = {The dynamics of vortex rings: {Leapfrogging,} choreographies and the stability problem},
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A. V. Borisov; A. A. Kilin; I. S. Mamaev. The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem. Russian journal of nonlinear dynamics, Tome 8 (2012) no. 1, pp. 113-147. http://geodesic.mathdoc.fr/item/ND_2012_8_1_a7/