Regular and Chaotic Dynamics

, Volume 18, Issue 1–2, pp 33–62 | Cite as

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

  • Alexey V. Borisov
  • Alexander A. Kilin
  • Ivan S. Mamaev
Article

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 

MSC2010 numbers

76B47 

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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    • 2
    • 3
  • Alexander A. Kilin
    • 1
    • 2
    • 3
  • Ivan S. Mamaev
    • 1
    • 2
    • 3
  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia
  2. 2.A.A. Blagonravov Mechanical Engineering Research Institute of RASMoscowRussia
  3. 3.Institute of Mathematics and Mechanics of the Ural Branch of RASEkaterinburgRussia

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