 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 horizontal coordinate is r, which ranges from -2 to 4, and the vertical coordinate represents x, which ranges from -1 to 2.
• The horizontal line segment in the middle of the picture corresponds to -1 < r < 1, for which x = 0 is an attracting fixed point.
• The curve to the right of it is a portion of a hyperbola for 1 < r < 3, for which x = (r-1)/r is an attracting fixed point.
• Further to the right in the interval 3 < r < 3.449, the curve splits into two branches (hence the term bifurcation). this corresponds to the fact that the orbit in this case converges to a period 2 cycle, i.e., the population eventually oscillates between two different values in alternate years.

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.

• To the right of r = 3.449 each of the two branches splits again and the population oscillates between 4 different values. This is illustrated below in Figure 3.4. This phenomenon is called period doubling. Moving slight further to r = 3.55 (Figure 3.55), the period doubles again to 8.
• Moving still further to the right to r = 3.59 (Figure 3.6), we see that the population takes on many different values, all lying within two different intervals. This is called chaos. Near the right edge at r = 3.99 (Figure 3.7) we see a broader range of chaos.
• Near r = 3.83 (Figure 3.8) the chaos vanishes and we get a period 3 cycle. This interval is called the period 3 window.
• If we zoom in on the uppermost branch of the period 3 window, we see a picture similar to the one above. In particular we see period doubling again. This is an example of self-similarity.

There are some additional features of the Figure 3.2 worth mentioning.

• Changing the initial value of x (or seed population) doe not alter the picture.
• Some other functions besides f[x] = r*x*(1-x) lead tobifurcation diagrams with similar features. In Winfract, make sure that View/Hotkey action is set to Fractint-style prompts, then press z and alter First Function with the right/left arrow keys. Figure 3.3 shoiwns the diagram for when that function is set to tan.

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.