Modeling a chaotic billiard: The Bunimovich Stadium

The Bunimovich stadium is a chaotic dynamical system in which a single particle, known as a billiard, moves indefinitely within a barrier without loss of momentum. Mathematicians and physicists have been interested in its properties since it was discovered to be chaotic in the 1970’s [5] [3] [4]. The Bunimovich stadium is actively researched [9]. This thesis and its accompanying software, the Bunimovich Stadia Evolution Viewer (BSEV), present a novel visual representation of the the chaotic dynamical system. The goal for the software is to provide insights into the stadium’s properties to aid researchers. This tool allows one to visualize the system’s evolution and see how sets of close billiard trajectories diverge over time. Current tools only provide views of the billiard moving within the boundary. This tool provides a view of the system in terms of the set of all possible collisions, known as the collision space. The collision space of the Bunimovich stadium is equivalent to a cylinder and the new tool presents the visualizations on the cylinder. This view shows the evolution of nearby billiard trajectories in the collision space. Such visualizations may provide insights into an important aspect of the stadium’s chaos, its sensitivity to initial conditions. The BSEV allows users to supply initial conditions for the system and generate images showing the system’s evolution with those inputs. These images specifically aid in visualizing the system’s sensitive dependence. Other viewers do not provide the ability to visualize the system in this way making it difficult for them to provide insights into the chaotic nature of the stadium. The BSEV does not have these limitations and can provide new insights into this chaotic dynamical system. The software for the BSEV can be freely downloaded from the following URL: https://github.com/shoemarw/BunimovichStadiu