Figure 1.16: Bouncing ball bifurcation diagram. (Generated by the Bouncing Ball program.)
A bifurcation diagram provides a nice summary for the transition between different types of motion that can occur as one parameter of the system is varied. A bifurcation diagram plots a system parameter on the horizontal axis and a representation of an attractor on the vertical axis. For instance, for the bouncing ball system, a bifurcation diagram can show the table's forcing amplitude on the horizontal axis and the asymptotic value of the ball's impact phase on the vertical axis, as illustrated in Figure 1.16. At a bifurcation point, the attracting orbit undergoes a qualitative change. For instance, the attractor literally splits in two (in the bifurcation diagram) when the attractor changes from a period one orbit to a period two orbit.
This bouncing ball bifurcation diagram (Fig. 1.16) shows the classic period doubling route to chaos. For table amplitudes between 0.01 cm and 0.0106 cm a stable period one orbit exists; the ball impacts with the table at a single phase. For amplitudes between 0.0106 cm and 0.0115 cm, a period two orbit exists. The ball hits the table at two distinct phases. At higher table amplitudes, the ball impacts at more and more phases. The ball hits at an infinity of distinct points (phases) when the motion is chaotic.