Julia Sets and the Mandelbrot set
In this section, we will explore the logistic map in the complex plane. This topic may not be as applied as the real logistic map. But the stunning beauty and the striking complexity of the Julia sets and the Madelbrot set that arise make this topic a feature attraction in the garden of mathematics. Here mathematics and art are mixed, and their distinctions blurred.

For the convenience of our discussions, we first rewrite the logistic map as
ƒc(z) = z 2 + c (7.1)
where c is a complex parameter. Actually, it is easy to see that this quadratic map and the logistic map g(w) = λw(1 - w) are topologically conjugate under the transformation figure 32 and figure 33.
Next we study the long-time behaviors of the orbits in the complex plane under the map (7.1). With the memory of the real logistic map afresh, you may be less inclined to predict simple dynamics in this map, even though it looks indeed simple. This attitude turns out to be shrewd. In fact, what you will see next may be beyond your wildest dreams.

First we introduce some notations. For a given value of c and a starting point z, either figure 34 or remains bounded for all n.
Let kc = {z : figure 33}.
It is called the filled-in Julia set.
The set Jc = δkc is called the Julia set.
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Section 7: Julia Sets and the Mandelbrot Set