We focus on the sequence of period doubling bifurcations in the quadratic map for 3.0<a<3.57 - see chapter 4 for the full range of a. (You will probably want to increase the "Speed" of the iteration.)
Successively enlarge the bifurcations near x=0.5 by tracing a rectangle around the desired region dragging the mouse on the stopped plot. You can get a precise estimate for the value of a at various phenomena by clicking on the point on the running plot. As finer scales are investigated you will need to increase the number of points over which transients are eliminated by increasing "Transient", and the number of points plotted in the orbit by increasing "Points".
The discussion in the text focuses on the values of a for which x=0.5 is a point on the orbit i.e. where the lines in the bifurcation diagram cross x=0.5, and the distance to the closest point on the orbit. You can find these values using the mouse to discover the geometric scaling, although it is easier on the bifurcation diagram to identify the bifurcation points rather than the "superstable" points.
Using the two applets you can construct the table showing the geometric nature of the sequence of bifurcations discussed in the text.