Nonpersistence of resonant caustics in perturbed elliptic billiards

Abstract

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.

14 pages, 3 figures

References in corpus (2)

• Bifurcations of Liouville Tori in Elliptical Billiards
• The frequency map for billiards inside ellipsoids

Cited by in corpus (8)

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• Coexistence of periodic-2 and periodic-3 caustics for nearly circular analytic billiard maps
• Exponentially small asymptotic formulas for the length spectrum in some billiard tables
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• The first $p$-widths of the unit disk
• On the finite dimensionality of integrable deformations of strictly convex integrable billiard tables
• High-order persistence of resonant caustics in perturbed circular billiards
• On the length and area spectrum of analytic convex domains