Julia Sets and the Mandelbrot Set

Background

Complex numbers. The Mandelbrot set and Julia sets are generated using complex numbers, z. Addition and multiplication of complex numbers can be reformulated as operations on pairs (x, y) of real numbers.

The Quadratic Mandelbrot set.

The iteration scheme is

z0 -> z1 = z02 + c -> z2 = z12 + c -> ...

where z and c are complex numers.

Here is the definition of Julia sets, along with a brief dsicussion of computational issues: finite resolution, run away to infinity, and accuracy vs time.

Here is the definition of the Mandelbrot set.

Where they live: Julia sets. Each point in the plane is the start of an iteration, so we say Julia sets live in the Dynamical Plane.

Where they live: the Mandelbrot set. Each point in the plane is a c-value for which an iteration starting with z0 = 0 is performed. We say the Mandelbrot set lives in the Parameter Plane.

A Generalized Mandelbrot set. One way to generalize the (quadratic) Mandelbrot set and Julia sets is with the iteration scheme zn -> zn+1 = znk + c. Note k= 2 gives the familiar quadratic Mandelbrot set. The software can handle k = 3, 4, and 5.

For iterations of the form zn+1 = znk + c, the iterations always begin at z0 = 0 because this is the critical point of zk + c.

A function with two critical points. Here the iteration scheme is zn -> zn+1 = zn2/2 + zn3/3 + c.

The function z2/2 + z3/3 + c has two critical points, z = 0 and z = -1.

Consequently, we have three plausible definitions for the Mandelbrot set: all c for which

the iterates of z0 = 0 remain bounded,

the iterates of z0 = -1 remain bounded, and

the iterates of both z0 = 0 and z0 = -1 remain bounded.

Here are pictures of each, and a brief sketch of the relation of critical points and the structure of the Julia sets.

The general cubic function. Here the iteration scheme is zn -> zn+1 = zn3 + A*zn + C.

Why did we choose this cubic polynomial? For that matter, why did we choose z2 + c for the quadratic Mandelbrot polynomial? These choices are justified by a change of variables.

Note the parameter space has four real dimensions, axes labeled a, b, c, and d if we write A = a + i*b and C = c + i*d. That is, the general cubic analog of the Mandelbrot set is four-dimensional. Certainly, we cannot view the entire space. Instead, in the software we present cross-sections: hold two of a, b, c and d fixed, plot the picture in the other two coordinates.

What do we iterate? For this function, (0,0) is not a critical point. This cubic polynomial has two critical points, sqrt(-A/3) and -sqrt(-A/3).

For a point (a,b,c,d) to belong to this Mandelbrot set, the iterates of both critical points must remain bounded.

Return to Mandelbrot set and Julia Set Lab.