The bifurcation of the unstable periodic orbits in bounded and unbounded three-disk billiards

Abstract

The bifurcation of unstable periodic orbits (UPOS) in bounded and unbounded billiards are investigated. The billiard systems studied in this paper consist of three disks and have C3 nu symmetry. It is found numerically that these systems have essentially two different types of bifurcation for changing the structure of the UPOS. The first type of bifurcation is caused by the tangential collision of the trajectory with a convex boundary segment. The second type of bifurcation is caused by the collision of the trajectory with a vertex point, where two smooth boundary segments meet at a finite angle. The vertex points on the boundary play the central geometrical role of organizing the UPOS in these billiard systems.