Center manifold analysis of a point vortex model of vortex shedding with control

In this paper we use methods of dynamical systems theory to provide a precise mathematical characterization of the behavior of the point vortex Fِppl system with a linear feedback control. The Fِppl system was used in an earlier investigation as a simple model for control design for vortex shedding and numerical studies indicated that the state of the controlled system converges to a closed orbit. In this investigation we prove rigorously that this observed behavior in fact represents periodic oscillations on the center manifold of the closed-loop nonlinear system. This manifold is shown to coincide with the uncontrollable subspace of the linearized system.