Some new results on Robnik billiards

Abstract

A Robnik billiard R_lambda is the image in the complex w plane of the unit disc in the z plane under the quadratic conformal map w=z+ lambda z². The parameter range is 0( lambda (1/2. For lambda =1/4 the billiard is strictly convex, its boundary is analytic and it has a zero curvature at one point of the boundary. A rigorous demonstration is given that the billiard R_1/4 has invariant curves in the surface of section. Thus it is not ergodic. The proof is based in KAM theory and makes substantial use of computer algebra. The authors give the first elements of a cascade of bifurcations whose properties show several similarities with numerical results given by Benettin et al. (1980) for conservative dynamical systems and by d'Humieres et al. (1982) in the experimental study of the forced pendulum.