I'm currently working on an individual project. In which I have created a program to simulate a particle bouncing around a finite bordered region. Where I need help in understanding how I might show chaos in a finite region. For example, one region I am working with is the region left by inscribing a circle in a square. I know this region to be chaotic but I don't know how to show that through some form of data analysis. The data I have is the position and velocity, but due to the elastic nature of the system, the velocity only changes direction.
I started to create phase-plane diagrams for the system but didn't think that truly showed chaos and from there I do not know how to approach this problem.
Go to the section Billiards of this book: http://chaosbook.org/
The idea is to define a Poincare map, with the coordinates $s_n$ (the arc-length position along the boundary for the $n$-th bouncing) and $p_n$ (the momentum component parallel to the boundary). Based on this map you can study the stability of the problem and, therefore, whether it exhibits chaos