In specifying an individual solution to the bouncing ball system, we need to
know both the initial condition, that is, the initial impact phase and impact
velocity of the ball 
, and the relevant system
parameters, 
, A, and T. Then, to find an
individual trajectory, all we need to do is iterate the mapping for the
appropriate model. However, finding the solution for a single trajectory gives
little insight into the global dynamics of the system.
As stressed in the Introduction, we are not
interested so much in solving an individual orbit,
but rather in understanding the behavior of a large collection of orbits
and, when possible, the system as a whole.
An individual solution can be represented by a curve in phase space. In considering a collection of solutions, we will need to understand the behavior not of a single curve in phase space, but rather of a bundle of adjacent curves, a region in phase space. Similarly, in the impact map we want to consider a collection of initial conditions, a region in the impact map. In general, the future of an orbit is well defined by a flow or mapping. The fate of a region in the phase space or the impact map is defined by the collective futures of each of the individual curves or points, respectively, in the region as is illustrated in Figure 1.7.
Figure 1.7: (a) Evolution of a region in phase space.
(b) Recurrent regions in the phase space of a nonlinear system.
A number of questions can, and will, be asked about the evolution of a region in phase space (or in the impact map). Do regions in the phase space expand or contract as the system evolves? In the bouncing ball system, a bundle of initial conditions will generally contract in area whenever the system is dissipative--a little energy is lost at each impact, and this results in a shrinkage of our initial region, or patch, in phase space (see section 4.4.4 for details). Since this region is shrinking, this raises many questions that will be addressed throughout this book, such as where do all the orbits go, how much of the initial area remains, and what do the orbits do on this remaining set once they get to wherever they're going? This turns out to be a subtle collection of questions. For instance, even the question of what we mean by ``area'' gets tricky because there is more than one useful notion of the area, or measure, of a set. Another related question is, do these regions intersect with themselves as they evolve (see Figure 1.7)? The answer is generally yes, they do intersect, and this observation will lead us to study the rich collection of recurrence structures of a nonlinear system.
A simple question we can answer is: does there exist a closed, simply-connected subset, or region, of the whole phase space (or impact map) such that all the orbits outside this subset eventually enter into it, and, once inside, they never get out again? If such a subset exists, it is called a trapping region . Establishing the existence of a trapping region can simplify our general problem somewhat, because instead of considering all possible initial conditions, we need only consider those initial conditions inside the trapping region, since all other initial conditions will eventually end up there.