Figure 1.13 shows the impact map of the strange attractor discovered at the end of the period doubling route to chaos.
Figure 1.13: Strange attractor in the bouncing ball system arising at the end of
a period doubling cascade. (Generated by the Bouncing Ball program.)
This strange attractor looks almost like a simple curve (segment of an upside-down parabola) with gaps. Parts of this curve look chopped out or eaten away. However, on magnification, this curve appears not so simple after all. Rather, it seems to resemble an intricate web of points spread out on a narrow curved strip. Since this chaotic solution is not periodic (and hence, never exactly repeats itself) it must consist of an infinite collection of discrete points in the impact (velocity vs. phase) space.
This strange set is generated by an orbit of the bouncing ball system, and it is chaotic in that orbits in this set exhibit sensitive dependence on initial conditions. This sensitive dependence on initial conditions is easy to see in the bouncing ball system when we solve for the impact phases and velocities for the exact model with the numerical procedure described in Appendix A. First consider two slightly different trajectories that converge to the same period one orbit. As shown in Table 1.2, these orbits initially differ in phase by 0.00001. This phase difference increases a little over the next few impacts, but by the eleventh impact the orbits are indistinguishable from each other, and by the eighteenth impact they are indistinguishable from the period one orbit. Thus the difference between the two orbits decreases as the system evolves.
Table 1.2: Convergence of two different initial conditions to a period one
orbit. The digits in bold are where the orbits differ. At the zeroth hit the
orbits differ in phase by 0.00001. Note that the
difference between the orbits decreases so that after 18 impacts both
orbits are indistinguishable from the period one orbit. The operating
parameters are: A = 0.01 cm, frequency = 60 Hz, 
, and the initial
ball velocity is 8.17001 cm/s. The impact phase is presented as 
so that it is normalized to
be between zero and one.
Table 1.3: Divergence of initial conditions on a strange attractor
illustrating sensitive dependence on initial conditions. The parameter
values are the same as in Table 1.2 except for A = 0.012 cm. The bold
digits show where the impact phases differ. The orbits differ in phase
by 0.00001
at the zeroth hit, but by the twenty-third impact they differ at every
digit.
An attracting periodic orbit has both long-term and short-term predictability. As the last example indicates, we can predict, from an initial condition of limited resolution, where the ball will be after a few bounces (short-term) and after many bounces (long-term).
The situation is dramatically different for motion on a strange attractor. Chaotic motions may still possess short-term predictability, but they lack long-term predictability. In Table 1.3 we show two different trajectories that again differ in phase by 0.00001 at the zeroth impact. However, in the chaotic case the difference increases at a remarkable rate with the evolution of the system. That is, given a small difference in the initial conditions, the orbits diverge rapidly. By the twelfth impact the error is greater than 0.001, by the twentieth impact 0.01, and by the twenty-fourth impact the orbits show no resemblance. Chaotic motion thus exhibits sensitive dependence on initial conditions. Even if we increase our precision, we still cannot predict the orbit's future position exactly.
As a practical matter we have no exact long-term predictive power for chaotic motions. It does not really help to double the resolution of our initial measurement as this will just postpone the problem. The bouncing ball system is both deterministic and unpredictable. Chaotic motions of the bouncing ball system are unpredictable in the practical sense that initial measurements are always of limited accuracy, and any initial measurement error grows rapidly with the evolution of the system.
Strange attractors are common to a wide variety of nonlinear systems. We will develop a way to name and dissect these critters in Chapters 4 and 5.