I started researching billiard systems in the summer of 2003 as a summer REU project funded by Georgia Tech's VIGRE grant.  I continued working in some capacity every semester since.  My advisors were Mason Porter and Leonid Bunimovich.

What is a Billiard System?

Billiards are a type of nonlinear dynamical system that are investigated for insight into chaos theory, nonlinear dynamics, and ergodic theory.  In classical billiard systems, a point particle is confined to a region in configuration space and collides with the boundary of the region such that the angle of incidence equals the angle of refection.  Depending on the geometry of the particular billiard table, there exist integrable and/or chaotic regions in phase space.  I investigated the existence and stability of periodic orbits in various billiard tables and the effect of perturbations on the characteristics of the integrable islands.  In addition, I studied what happens to a system when two interacting point-particles are introduced inside a billiard system.

Software

First, I developed a series of Matlab files to generate billiard tables and simulate the systems for various initial conditions.  I also made a number of tools to help view the results of the simulation in insightful manners.  I packaged the software as a Graphical User Interface (GUI) for Matlab that can simulate arbitrary billiard tables with any number of initial conditions and iterations.  The software is freely available for anyone interested in numerical simulations of billiards in the research community.  A copy of the software and the accompanying documentation describing how to use the software is found below.  I presented the software and some example output plots at the 2003 Dynamics Days conference.  The poster is also found below.

Scientific Study

Once the simulation software was completed, we were ready to start using it to answer important scientific questions.  We investigated the existence and size of integrable islands in a number of various types of Bunimovich mushrooms, which exhibit a divided phase space containing both integrable and chaotic regions.  Discrete and continuous autocorrelations were calculated for a particle in a mushroom.  The Matlab code was expanded to simulate two particles of finite size in a billiard table.  We found some very interesting properties of this system.  Our results are contained in the document below, which has been submitted to Chaos for publication.  A poster of the work was presented at the American Physical Society meeting in 2005, which appears below.  The results may be featured in a cover story of the Notices of the American Mathematical Society (AMS).