  Using a Ginzburg-Landau model, we study the vortex behavior of a rectangular
thin film superconductor subjected to an applied current fed into a portion of
the sides and an applied magnetic field directed orthogonal to the film.
  Through a center manifold reduction we develop a rigorous bifurcation theory
for the appearance of periodic solutions in certain parameter regimes near the
normal state. The leading order dynamics yield in particular a motion law for
kinematic vortices moving up and down the center line of the sample. We also
present computations that reveal the co-existence and periodic evolution of
kinematic and magnetic vortices.
