Mathematical billiards are important models of dynamical systems from mathematical physics in which point particles collide elastically with fixed boundaries. Chaotic dynamics emerge when the boundary of the billiard table is dispersing or when it contains focusing arcs at sufficient distance to allow a defocusing effect to occur. This project studies a type of billiard known as an asymmetric lemon billiard, comprised of focusing boundaries which seem to violate the usual defocusing condition. Numerical evidence is obtained showing that chaotic dynamics nevertheless occur for a large range of parameter values, extending beyond the range to which analytic proofs apply. This work was completed at Fairfield University during Summer 2020, and was supported by a grant from the National Science Foundation.
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Hi Camryn and Hailey, is there a mathematical definition of defocusing? Also, did you a specific program for exploring results numerically?
Hi Lisa! Defocusing comes from trajectories hitting a concave boundary. Defocusing proves an orbit unstable. The formal definition states if the distance between 2 boundaries is greater than twice the radius, defocusing will occur.
Also, we used MATLAB! We added code to create the asymmetrical lemon to simulate the iterations.
Thanks for the clarification! I also worked with Professor Demers on a research project when I was an undergraduate at Fairfield 🙂