Point-vortex interaction in an oscillatory deformation field:
Hamiltonian dynamics, harmonic resonance and transition to chaos
Pages: 971 - 995,
Volume 11, Issue 4,
Full Text (10591.7K)
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)
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
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;
Published: April 2009.