Upon plotting a single trajectory of the ball after it rebounded one thousand times off the table cushions. We found that for the first two cases the system was NOT chaotic. We found this by plotting what is known as a Poincare plot.
Non-chaotic dynamics is indicated in this plot by nice well defined structures. However, when we deformed the billiard table into a ``stadium `` shape, the dynamics quickly became chaotic and the Poincare plot turned into mush.