In the high bounce approximation we imagine that the table's displacement amplitude is always small compared to the ball's maximum height. This approximation is depicted in Figure 1.6 where the ball's trajectory is perfectly symmetric about its midpoint, and therefore
The velocity of the ball between the kth and k+1st impacts is given by
At the k+1st impact, the velocity is 
and the time is 
, so
Using equation (1.15) and simplifying, we get
which is the time map in the high bounce approximation.
Figure 1.6: Symmetric orbit in the high bounce approximation.
To find the velocity map in this approximation we begin with the impact relation (eq. (1.7)),
where the last equality follows from the high bounce approximation, equation (1.15). The table's velocity at the k+1st impact can be written as
when the time map, equation (1.18), is used. Equations (1.19) and (1.20) give the velocity map in the high bounce approximation,
The impact equations can be simplified somewhat by changing to the dimensionless quantities
which recasts the time map (eq. (1.18)) and the velocity map (eq. (1.21)) into the explicit mapping form
In the special case where 
, this system of equations is known as the
standard map [4]. The subscripts of 
explicitly show the dependence of the map on the parameters 
and
. The mapping equation (1.25) is easy to solve on a computer.
Given an initial condition 
, the map explicitly
generates the next impact phase and impact velocity as 
, and this in turn generates 
, etc.,
where, in this notation, the superscript n in 
indicates functional
composition (see section 2.2).
Unlike the exact model, both the phase map and the velocity
map are explicit equations in the high bounce approximation.
The high bounce approximation shares many of the same qualitative properties of the exact model for the bouncing ball system, and it will serve as the starting point for several analytic calculations. However, for comparisons with experimental data, it is worthwhile to put the extra effort into numerically solving the exact equations because the high bounce model fails in at least two major ways to model the actual physical system [5]. First, the high bounce model can generate solutions that cannot possibly occur in the real system. These unphysical solutions occur for very small bounces at negative table velocities, where it is possible for the ball to be projected downward beneath the table. That is, the ball can pass through the table in this approximation. Second, this approximation cannot reproduce a large class of real solutions, called ``sticking solutions,'' which are discussed in section 1.4.3. Fundamentally, this is because the map in the high bounce approximation is invertible, whereas the exact model is not invertible. In the exact model there exist some solutions--in particular the sticking solutions--for which two or more orbits are mapped to the same identical point. Thus the map at this point does not have a unique inverse.
