Regular and Chaotic Dynamics

, Volume 23, Issue 5, pp 595–612 | Cite as

Choreographies in the n-vortex Problem

  • Renato C. CallejaEmail author
  • Eusebius J. Doedel
  • Carlos García-Azpeitia


We consider the equations of motion of n vortices of equal circulation in the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame of reference. We use numerical continuation in a boundary value setting to determine the Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, corresponds to choreographies of n vortices. We include numerical results for all cases, for various values of n, and we provide key details on the computational approach.


n-vortex problem choreographies continuation methods 

MSC2010 numbers

34C25 37G40 47H11 54F45 


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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Renato C. Calleja
    • 1
    Email author
  • Eusebius J. Doedel
    • 2
  • Carlos García-Azpeitia
    • 3
  1. 1.IIMASUniversidad Nacional Autónoma de MéxicoMéxicoMéxico
  2. 2.Concordia UniversityMontreal, QuebecCanada
  3. 3.Facultad de CienciasUniversidad Nacional Autónoma de MéxicoCiudadMéxico

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