Once the orbits enter the trapping region, where do they go next? To answer this question we first solve for the motion of a ball bouncing on a stationary table. Then we will imagine slowly turning up the table amplitude.
If the table is stationary, then the high bounce approximation is no longer approximate, but exact. Setting A = 0 in the velocity map, equation (1.21), immediately gives
Using the time map, 
, the coefficient
of restitution is easy to measure [6] by recording three consecutive impact
times, since
To find how long it takes the ball to stop bouncing, consider the sum of the differences of consecutive impact times,
Since 
, 
is the summation of a
geometric
series ,
which can be summed for 
.
After an infinite number of bounces, the ball will come to a halt in a
finite time.
For these equilibrium solutions, all the orbits in the trapping region come to rest on the table. When the table's acceleration is small, the picture does not change much. The ball comes to rest on the oscillating table and then moves in unison with the table from then on.
