In these notes I will use mathematical notation similar to the syntax of Mathematica.
3. The logistic bifurcation diagram, January 26.
We have been studying what happens when we start with an intial value of x and repeatedly apply the logistic function f[x] = r*x*(1-x) of (1.2) for various values of the growth constant r between 0 and 4. The functions
f[x], f[f[x]], f[f[f[x]]], ...
are called iterates of f. The sequence of numbers produced in this way is called the orbit of x. This kind of problem is called a discrete dynamical system.
Winfract has a tool called a bifurcation diagram for getting an overview of this situation. To get it, select biflambda from the Fractals/Fractal Formula menu. For a faster rendition, go to the View menu and turn off Pixel-by-Pixel Update. Figure 3.1. Bifurcation diagram for -2 <= r <= 4 and -1 <= x <= 2.
This picture illustrates the long term behavior of the system for various values of r, beginning with the initial value x = .66.
The picture below is a closeup of the upper right hand portion of the picture above. (One can alter the region shown by press z in Winfract and then editing.)
Figure 3.2. Bifurcation diagram for 2.8 < r < 4 and 0 < x < 1.
This picture shows several interesting features.
There are some additional features of the Figure 3.2 worth mentioning.
Figure 3.3. Bifurcation diagram for the function r*Tan[x]*(1-Tan[x]).
Figure 3.4. Cobweb diagram for r = 3.48 showing a period 4 cycle.
Figure 3.5. Cobweb diagram for r = 3.55 showing a period 8 cycle.
Figure 3.6. Cobweb diagram for r = 3.59 showing chaos within two different intervals.
Figure 3.7. Cobweb diagram for r = 3.99 showing a broad range of chaos.
Figure 3.8. Cobweb diagram for r = 3.83 showing a period 3 cycle.
Figure 3.9. Bifurcation diagram showing the uppermost branch of the period 3 window, with period doubling and other features similar to Figure 3.2. The region shown is 3.81 < r < 3.86 and .94 < x < .98.
This page was last revised on February 2, 1998.