Caustics are curves with the property that a billiard trajectory, oncetangent to it, stays tangent after every reflection at the boundary of thebilliard table. When the billiard table is an ellipse, any nonsingular billiardtrajectory has a caustic, which can be either a confocal ellipse or a confocalhyperbola. Resonant caustics ---the ones whose tangent trajectories are closedpolygons--- are destroyed under generic perturbations of the billiard table. Weprove that none of the resonant elliptical caustics persists under a largeclass of explicit perturbations of the original ellipse. This result followsfrom a standard Melnikov argument and the analysis of the complex singularitiesof certain elliptic functions.
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