Homoclinic billiard orbits inside symmetrically perturbed ellipsoids

The billiard motion inside an ellipsoid of R-3 is completely integrable. If the ellipsoid is not of revolution, there are many orbits bi-asymptotic to its major axis. The set of bi-asymptotic orbits is described from a geomet rical, dynamical and topological point of view. It contains eight surfaces, called separatrices. The splitting of the separatrices under symmetric perturbations of the elli psoid is studied using a symplectic discrete version of the Poincare-Melnik ov method, with special emphasis on the following situations: close to the flat limit (when the minor axis of the ellipsoid is small enough), close to the oblate limit (when the ellipsoid is close to an ellipsoid of revolutio n around its minor axis) and close to the prolate limit (when the ellipsoid is close to an ellipsoid of revolution around its major axis). It is proved that any non-quadratic entire symmetric perturbation breaks th e integrability and splits the separatrices, although (at least) 16 symmetr ic homoclinic orbits persist. Close to the flat limit, these orbits become transverse under very general polynomial perturbations of the ellipsoid. Finally, a particular quartic symmetric perturbation is analysed in great d etail. Close to the flat and to the oblate limits, the 16 symmetric homocli nic orbits are the unique primary homoclinic orbits. Close to the prolate l imit, the number of primary homoclinic orbits undergoes infinitely many bif urcations. The first bifurcation curves are computed numerically. The planar and high-dimensional cases are also discussed.