Bifurcations in Maps with a Quadratic Maximum - Demo 1

Patterns in the Quadratic Map

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.

The superstable cycles are more easily identified in the plot of the Lyapunov exponent. These orbits occur at the values of a where the Lyapunov exponent becomes very large and negative, and are easily picked off with the mouse. Again enlarge the region near a=3.57 to uncover the higher order bifurcations. Precise values of the Lyapunov exponent depend on eliminating the transients (choose a large value of "Transient") and having enough "Points" in the orbit (obviously at least 2n points in the 2n cycle!).

Using the two applets you can construct the table showing the geometric nature of the sequence of bifurcations discussed in the text.

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Last modified Sunday, January 30, 2000
Michael Cross