To find a trapping region for the bouncing ball system we will first find
an upper bound for the next outgoing velocity, 
, by looking
at the previous value, 
. We will then find a lower bound for
. These bounds give us the boundaries for a trapping region
in the bouncing ball's impact map 
, which imply a trapping
region in phase space.
To bound the outgoing velocity, we begin with equation (1.13) in the form
The first term on the right-hand side is easy to bound. To bound the second term, we first look at the average ball velocity between impacts, which is given by
Rearranging this expression gives
Equation (1.26) now becomes
Noting that the average table velocity between impacts is the same as the average ball velocity between impacts (see Prob. 1.14), we find that
If we define
and let 
, then
, or
In this case it is essential that the system be dissipative 
for a trapping region to exist. In the conservative limit no trapping
region exists--it is possible for the ball to reach infinite
heights and velocities when no energy is lost at impact.
To find a lower bound for , we simply realize that the
velocity after impact must always be at least that of the table,
For the bouncing ball system the compact trapping region, D, given by
is simply a strip bounded by 
and 
.
To prove that D is a trapping region,
we also need to show that v cannot approach
asymptotically, and that once inside D, the orbit cannot leave
D (these calculations are left to the reader--see Prob. 1.15).
The previous
calculations show that all orbits of the dissipative bouncing ball system will
eventually enter the region D and be ``trapped'' there.
