- Mathematical billiards: Periodic orbits within quadrilaterials
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Abstract:
- Mathematical billiards is a dynamical system that models billiards in an idealised environment. The billiard ball is considered to be a point mass and satisfies the law of reflection when interacting with the boundary; the shape of the mathematical billiard is arbitrary. These simple constraints lead to surprisingly deep and complex dynamics. We focus on billiards with quadrilateral boundaries. The existence of periodic orbits in all polygons is currently one of the most resistant problems in dynamics. This dissertation makes progress on this conjecture and explores the existence of periodic orbits within squares, rectangles and parallelograms. We provide alternative proofs to classical results for square billiards with additional insights and connections to number theory. We also take a dynamical systems approach which enables the use of bifurcation theory and parameter continuation. We introduce a novel continuation formulation which bypasses the extreme degeneracies exhibited by mathematical billiards and use it to compute branches of periodic solutions as a parameter varies the shape of the billiard from a square to a rectangle and parallelogram. The insights gained from the numerical exploration lead us to prove that there exist no period-4 orbits within the parallelogram and to show in a computer-assisted manner the existence of a bifurcation diagram that exhibits a period-adding sequence, where a periodic orbit has its period change under parameter variation in successive jumps of four each time.
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