While i was studying about chaos theory, i stumbled upon this, When a ball confined in a square, and at the center is located a circle, is to bounce elastically, the path of the object deviates significantly. thereby causing chaos. I think this is equivalent to a sinai billiard.
I couldn't understand the motion completely, but to start of with a simple case let two balls be located at the y-coordinate at the bottom middle,and the x-postion be with an uncertainty of $\pm \epsilon$ as shown in the above figure. The balls start moving with a velocity purely in the y-direction.
What was claimed was that after successive bounces, THe ratio of the bounces could be determined.That is the ratio of the distance the two balls are from each other, for example After the $5th$ bounces from the square and $10th$ bounces from the square,their distance ratio could be determined. I really have no ideal how to begin with this problem. I know this is chaotic system and the initial errors in the position of the ball affects the later outcome, but i don't know how to proceed. Pleas help? And of course $\epsilon$ can be taken to be very small.