Now that the ball is moving with the table, what happens as we slowly turn up the table's amplitude while keeping the forcing frequency fixed? Initially, the ball will remain stuck to the table until the table's maximum acceleration is greater than the earth's gravitational acceleration, g. The table's acceleration is given by
The maximum acceleration is
thus 
.
When is greater than g, the ball
becomes unstuck and will fly free from the table until its next impact.
The phase at which the ball becomes initially unstuck occurs when
Even in a system in which
the table's maximum acceleration is much greater than g,
the ball can become stuck.
An infinite number of impacts can occur in a finite stopping time, 
.
The sum
of the times between impacts converges in a finite
time much less than the table's period, T.
The ball gets stuck again at the end of this sequence of impacts
and
moves with the table until it reaches the phase
. This type of sticking solution is an eventually periodic orbit. After its first time of
getting stuck, it will exactly repeat this pattern of getting stuck, and then
released, forever.
However, these sticking solutions are a bit exotic in several respects. Sticking solutions are not invertible; that is, an infinite number of initial conditions can eventually arrive at the same identical sticking solution. It is impossible to run a sticking solution backward in time to find the exact initial condition from which the orbit came. This is because of the geometric convergence of sticking solutions in finite time.
Also, there are an infinite number of different sticking solutions.
Three such solutions are
illustrated in Figure 1.8.
To see
how some of these solutions are formed, let's
turn the table amplitude up a little so that the stopping time,
, is lengthened. Now, it happens that the ball does not get stuck
in the first table period, T, but keeps bouncing on into the second or third
period. However, as it enters each new period, the bounces get progressively
lower so that the ball does eventually get stuck after several periods. Once
stuck, it again gets released when the table's acceleration is greater than g,
and this new pattern repeats itself forever.
Figure 1.8: Sticking solutions in the bouncing ball system.
