 In these notes I will use mathematical notation similar to the syntax of Mathematica.
4. The logistic Mandelbrot set, January 28.

The logistic Mandelbrot set (Figure 4.1) is another way to get an overview of the long term behavior of the logistic equation. As usual we define f[x_] := r*x*(1-x), but no we allow x and r to be complex numbers. each point in the picture corresponds to a complex value of r = r1 + r2*I, where r1 and r2 are the horizontal and vertical coordinates. the region shown is -3 < r1 < 5 and -3 < r2 < 3.

For each value of r we consider the orbit for x = 1/2. This is known as the critical orbit because the derivative there, f'[1/2] is zero. For each value of r this orbit may or may not go to infinity. The logistic Mandelbrot set L is the set of values of r for which the criticial orbit does not go to infinity. It is shown in the picture as the dark blue region. (Note that it includes the real values 0 < r < 4 that we studied above.) Points outside of L are colored according to how fast the corresponding orbit goes to infinity. Figure 4.1. The logistic Mandelbrot set, drawn by Fractint using Fractals/Fractal_Formula/mandellambda.

We will now account for the most obvious feature of the picture, the two large circles, which each have radius 1 and are centered at 0 and 2. We will refer to them collectviely as the main body of L.We will see that for each value of r within it, the function f[x] has an attracting fixed point. Recall that its fixed points are at 0 and (r-1)/r, and that the derivatives f'[x] at these points are r and 2-r respectively. The two large circles are the regions where

|r| < 1 and |2-r| < 1

respectively.

Next we will account for second largest circular regions in the picture, each having radius roughly .2 and centered are roughly 3.2 and -1.2. These are the regions where the function f2[x] (2.8) has an attracting fixed point. We will refer to them collectiuvely as the period 2 body of L. We saw above (2.10) that f2 has four fixed points, namely

0, (r - 1)/r, r + r^2 - (-3 - 2*r + r^2)^(1/2))/(2*r), and r + r^2 + (-3 - 2*r + r^2)^(1/2))/(2*r).

The first two of these are also fixed points of f[x] and have already been considered. The derivative of f2 (2.11) at each of the second two is 4 + 2*r - r^2. We need to determine when this quantity has absolute value less than 1. If we denote the derivative by u, then

Solve[u == 4 + 2*r - r^2,r]

gives

{{r -> 1 - Sqrt[5 - u]}, {r -> 1 + Sqrt[5 - u]}}.

For |u| < 1 these regions are roughly but not precisely circular. We can find their 'centers' with

NSolve[{u == 4 + 2*r - r^2, u == 0}, r]

and get

{{r -> -1.23607}, {r -> 3.23607}},

which corresponds to what we see in the picture. We can make similar calculations with u = 1 and u = -1.

What can we say about the period 3 body? Making calculations like those above with the function

f3[x_] := f[f[f[x]]]

is not easy, even with Mathematica. However it is easy to find out for which values of r the criticial point x = 1/2 is fixed by f3. The code

NSolve[f3[1/2]==1/2, r]

gives 7 solutions,

{{r -> -1.83187}, {r -> -0.552675 - 0.959456 I}, {r -> -0.552675 + 0.959456 I},

{r -> 2.}, {r -> 2.55268 - 0.959456 I},

{r -> 2.55268 + 0.959456 I}, {r -> 3.83187}}.                             (4.2)

The first and last of these are the most interesting. We have already looked at the cobweb diagram for r = 3.83 (Figure 3.8), and found that it leads to a period 3 orbit. Here is a close up of the corresponding region of L. (One can use Fractint to get a picture with a specified center and magnification by typing z F6 F4 after switching the Hotkey Action to Fractint syle. The default magnification is .333, so the scale below is really 150 times smaller than in the Figure 4.1.) Figure 4.3. The logistic Mandelbrot set L near r = 3.83187, magnified 50 times.

The blue region in this picture is a replica of the classical Mandelbrot set (Figure 5.2 below), which we will discuss later. The function f3 has an attracting fixed point for each r in the cardioid shaped main body. The circle to its right is the region where f6 has an attarcting fixed point. The corresponding picture for r = -1.83187 is a mirror image of this one. Each of the four other fixed points (4.2) besides r = 2 is in the center of a circulr region attached to the main body of L. One of them is shown below. Figure 4.4. The logistic Mandelbrot set L near r = 2.55268 + 0.959456 I, magnified 5 times.