Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos

Pages: 971 - 995, Volume 11, Issue 4, June 2009

doi:10.3934/dcdsb.2009.11.971       Abstract        Full Text (10591.7K)       Related Articles

Xavier Perrot - LPO, UFR Sciences, UEB/UBO, 6 Avenue Le Gorgeu, 29200 Brest, France (email)
Xavier Carton - LPO, UFR Sciences, UEB/UBO, 6 Avenue Le Gorgeu, 29200 Brest, France (email)

Abstract: We study the Hamiltonian system of two point vortices, embedded in external strain and rotation. This external deformation field mimics the influence of neighboring vortices or currents in complex flows. When the external field is stationary, the equilibria of the two vortices, symmetric with respect to the center of the plane, are determined. The stability analysis indicates that two saddle points lie at the crossing of separatrices, which bound streamfunction lobes having neutral centers.
   When the external field varies periodically with time, resonance becomes possible between the forcing and the oscillation of vortices around the neutral centers. A multiple time-scale expansion provides the slow-time evolution equation for these vortices, which, for weak periodic deformation, oscillate within their original (steady) trajectory. These analytical results accurately compare with numerical integration of the complete equations of motion. As the periodic deformation field increases, this vortex oscillation migrates out of the original trajectories, towards the location of the separatrices. With a periodic external field, these separatrices have given way to heteroclinic trajectories with multiple self-intersections, as shown by the calculation of the Melnikov function.
   Chaos appears in vortex trajectories as they enter the aperiodic domain around the heteroclinic curves. In fact, this chaotic domain progressively fills out the plane, replacing KAM tori and cantori, as the periodic deformation field reaches finite amplitude. The appearance of windows of periodicity is illustrated.

Keywords:  Two-dimensional incompressible flow, vortex interaction, Hamiltonian system, orbits, stability.
Mathematics Subject Classification:  Primary: 76B47, 37J25; Secondary: 76E20.

Received: May 2008;      Revised: October 2008;      Published: April 2009.