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

For each value of ** r** we consider
the orbit for

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| < 1** and

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

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

The first two of these are also fixed points of ** f[x]**
and have already been considered. The derivative of

`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

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

`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

Figure 4.3. The logistic Mandelbrot set ** L**
near

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

Figure 4.4. The logistic Mandelbrot set ** L**
near

*This page was last revised on February 2, 1998.*