Computer-Assisted Methods for Analyzing Periodic Orbits in Vibrating Gravitational Billiards

Using rigorous numerical methods, we prove the existence of 608 isolated periodic orbits in a gravitational billiard in a vibrating unbounded parabolic domain. We then perform pseudo-arclength continuation in the amplitude of the parabolic surface's oscillation to compute large, global branches of periodic orbits. These branches are themselves proven rigorously using computer-assisted methods. Our numerical investigations strongly suggest the existence of multiple pitchfork bifurcations in the billiard model. Based on the numerics, physical intuition and existing results for a simplified model, we conjecture that for any pair (k,p), there is a constant xi for which periodic orbits consisting of k impacts per period p cannot be sustained for amplitudes of oscillation below xi. We compute a verified upper bound for the conjectured critical amplitude for (k,p) = (2, 2) using our rigorous pseudo-arclength continuation.