There are still a few invariant curves. Now there are lots of elliptic islands. Also, there are some regions with chaotic motion. For k=0, the motion is very simple. Each point just moves along a horizontal line. If you iterate a point enough times, it will seem to fill in the line completely. We call such a line an INVARIANT CURVE or invariant circle. They line up on one straight line. This is a very regular pattern. A regular pattern; they are aligned along straight lines. There are still lots of invariant curves but they are no longer perfectly flat, Instead they are a bit wavy. Also, some circles appear in the top and bottom of the picture. These circles are called ELLIPTIC ISLANDS. Now the motion looks completely chaotic: a point will move all over the phase space. The motion is regular; it is made up of parallel lines. The motion is regular. On the table, we see a pattern formed: an inner circle. In the phase space, the dots all land on one straight line. NOthing yet For most starting point, the motion is chaotic: the orbit will go all over the phase space except for 2 small regions in the middle filled in with elliptic islands. Points that start in an elliptic island will stay there; so there is a small amount of regular motion. 1. The ball bounces irregularly. 2. The ball seems to go all over the table. As it bounces, the ball moves in many different directions. This is different from the square billiard where the path filled up the table but the ball moved in only a few different directions. 3. The dots bounce all around the phase space.