  Caustics are curves with the property that a billiard trajectory, once
tangent to it, stays tangent after every reflection at the boundary of the
billiard table. When the billiard table is an ellipse, any nonsingular billiard
trajectory has a caustic, which can be either a confocal ellipse or a confocal
hyperbola. Resonant caustics ---the ones whose tangent trajectories are closed
polygons--- are destroyed under generic perturbations of the billiard table. We
prove that none of the resonant elliptical caustics persists under a large
class of explicit perturbations of the original ellipse. This result follows
from a standard Melnikov argument and the analysis of the complex singularities
of certain elliptic functions.
