TITLE:
Nonpersistence of resonant caustics in perturbed elliptic billiards

AUTHORS:
Sonia Pinto-de-Carvalho (1) and Rafael Ramirez-Ros (2)

(1) Departamento de Matem\'atica, ICEx
    Universidade Federal de Minas Gerais
    CP 702, 30123-970, Belo Horizonte, MG, Brazil

(2) Departament de Matematica Aplicada I
    Universitat Politecnica de Catalunya
    Diagonal 647, 08028 Barcelona, Spain

E-MAIL ADDRESSES:
sonia@mat.ufmg.br, rafael.ramirez@upc.edu

URLs:
www.mat.ufmg.br/~sonia/
www.ma1.upc.edu/~rafael/

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.

KEYWORDS:
Billiards, Caustics, Invariant curves, Melnikov method