Transition to chaos with conical billiards

We adapt ideas from geometrical optics and classical billiard dynamics to consider particle trajectories with constant velocity on a cone with specular reflections off an elliptical boundary formed by the intersection with a tilted plane, with tilt angle γ. We explore the dynamics as function of γ and the cone deficit angle χ that controls the sharpness of the apex, where a point source of positive Gaussian curvature is concentrated. We find regions of the (γ, χ) -parameter space where, depending on the initial conditions, either (A) the trajectories sample the cone base and avoid the apex region, (B) sample only a portion of the base region while again avoiding the apex or (C) sample the entire cone more uniformly. The special case of an untilted cone displays only type A trajectories which form a caustic at the distance of closest approach to the apex. However, we observe an intricate transition to chaotic dynamics dominated by Type (C) trajectories for sufficiently strong χ and γ. A Poincare map that summarizes trajectories decomposed into the geodesic segments interrupted by specular reflections provides a powerful method for visualizing the transition to chaos. We then analyze the similarities and differences of the path to chaos for conical billiards with other area preserving conservative maps.