As we increase the table's amplitude we often see that the orbit jumps from a sticking solution to a simple periodic motion. Figure 1.9 shows the convergence of a trajectory of the bouncing ball system to a period one orbit .
Figure 1.9: Convergence to a period one orbit.
The ball's motion converges toward a periodic orbit with a period exactly equal to that of the table, hence the term period one orbit (see Prob. 1.1).
What happens to the period one solution as the forcing amplitude of the table increases further? We discover that the period one orbit bifurcates (literally, splits in two) to the period two orbit illustrated in Figure 1.10.
Figure 1.10: Period two orbit of a bouncing ball.
Now the ball's motion is still periodic, but it bounces high, then low, then high again, requiring twice the table's period to complete a full cycle. If we gradually increase the table's amplitude still further we next discover a period four orbit, and then a period eight orbit, and so on. In this period doubling cascade we only see orbits of period
and not, for instance, period three, five, or six.
The
amplitude ranges for which each of these period 
orbits is observable,
however, gets smaller and smaller.
Eventually it converges to a critical table
amplitude, beyond which the bouncing ball system exhibits
the nonperiodic behavior
illustrated in Figure 1.11.
Figure 1.11: Chaotic orbit of a bouncing ball.
This last type of motion found at the end of the period doubling cascade never settles down to a periodic orbit and is, in fact, our first physical example of a chaotic trajectory known as a strange attractor. This motion is an attractor because it is the asymptotic solution arising from many different initial conditions: different motions of the system are attracted to this particular motion. At this point, the term strange is used to distinguish this motion from other motions such as periodic orbits or equilibrium points. A more precise definition of the term strange is given in section 3.8.
At still higher table amplitudes many other types of strange and periodic motions are possible, a few of which are illustrated in Figure 1.12.
Figure 1.12: A zoo of periodic and chaotic motions seen in the bouncing ball
system. (Generated by the Bouncing Ball program.)
The type of motion depends on the system parameters and the specific initial conditions. It is common in a nonlinear system for many solutions to coexist . That is, it is possible to see several different periodic and chaotic motions for the same parameter values. These coexisting orbits are only distinguished by their initial conditions.
The ``period doubling route to chaos'' we saw above is common to a wide variety of nonlinear systems and will be discussed in depth in section 2.8.
