Homoclinic billiard orbitsinside symmetrically perturbed ellipsoids

Abstract

The billiard motion inside an ellipsoid ofℝ3 is completely integrable. Ifthe ellipsoid is not of revolution, thereare many orbits bi-asymptotic to its majoraxis. The set of bi-asymptotic orbits isdescribed from a geometrical, dynamicaland topological point of view. It containseight surfaces, called separatrices.The splitting of the separatrices under symmetric perturbations of theellipsoid is studied using a symplectic discrete version of thePoincaré-Melnikov method, with special emphasis on the followingsituations:close to the flat limit(when the minor axis of the ellipsoid is small enough), close to the oblatelimit(when the ellipsoid is close to an ellipsoid of revolution around its minoraxis)and close to the prolate limit(when the ellipsoid is close to an ellipsoid of revolution around its majoraxis).It is proved that any non-quadratic entire symmetric perturbation breaksthe integrability and splits the separatrices, although (at least) 16symmetric homoclinic orbits persist. Close to the flat limit, these orbitsbecome transverse under very general polynomial perturbations of theellipsoid.Finally, a particular quartic symmetric perturbation is analysed in greatdetail. Close to the flat and to the oblate limits, the 16 symmetrichomoclinic orbits are the unique primary homoclinic orbits. Close to theprolate limit, the number of primary homoclinic orbits undergoesinfinitely many bifurcations. The first bifurcation curves are computednumerically.The planar and high-dimensional cases are also discussed.