As mentioned earlier, no matter what the value of the complex parameter c is, in the iteration of the complex quadratic map there is a unique trapping set T_c and a corresponding escape set E_c. The Julia set (J_c) is the boundary between the set T_c and the set E_c. The Mandelbrot set is an answer to the following kind of enquiry. Of the infinite number of possible Julia sets that exists, is there any organizing principle that classifies these Julia sets.

The key results for this classification of Julia sets were already there in the works of Julia and Fatou who knew about the topological dichotomy in the Julia set. The result states that for any choice of the complex parameter c the associated Julia set J_c and the trapping set T_c are either topologically connected (severely deformed circles) or totally disconnected (generalized Cantor dust like).

This was indeed the key result that clued Mandelbrot, in 1979, to visualize a set in the complex parameter space c which is called the Mandelbrot set. The Mandelbrot set consists of all values of c that have connected Julia sets. Picking value of c that is outside the Mandelbrot set, and iterating the equation to obtain the J_c for this particular choice of c gives a disconnected Julia set.

Note important , as it is, the classification of Julia set in terms of disconnected sets, this still doesn't allow one to visualize the shape of the set of points, in the parameter space, for which the Julia set is connected. The genius is in the realization of the interrelation between the above mentioned dichotomy and in the long term behavior of the critical point.

The computer graphical renderings of Mandelbrot set is made possible by this important fact which states -- The trapping set T_c is connected if and only if the critical orbit is bounded. This definition makes it possible to draw a portrait of the Mandelbrot set.

For each complex number c, a sequence of iterates Z_n is defined by 3. The complex number c is a member of the Mandelbrot set if and only if |Z_n| is finite for all values of n. The bars indicate the magnitude of Z_n given by Z_n = Ö(X_n² + Y_n²) where X_n is the real component and Y_n the imaginary component of Z_n. The point, in the complex parameter space, is colored white if the orbit is unbounded for that particular value of c and is colored black if the orbits are bounded.

The figure shown below is the Mandelbrot set (in black). It extends from the cusp of the cardoid at Re c = 0.25 to the tip of the tail at Re c = -2 along the real axis and from Im c = -1.25 to Im c = 1.25 along the imaginary axis.

The basic algorithm to generate the Mandelbrot set is as follows. For each pixel c, start with Z = 0. Iterate the above equation up to N times, exiting if |Z| gets large. If you finish the loop, the point is probably inside the Mandelbrot set. If you exit, the point is outside and can be colored according to how many iterations were completed. You can exit if |Z| > 2, since if Z gets this big it will go to infinity. The maximum number of iterations, N, can be selected as desired, for instance 200. Larger values of N will give sharper detail but take longer.

A note about why we start from Z₀ = 0. Zero is the critical point of Mandelbrot equation given by  2. That is, a point where d/dz (Z² + c) = 0. Critical points are important because by a result of Fatou: every attracting cycle (T_c) for a polynomial or rational function attracts at least one critical point. Thus, testing the critical point shows if there is any stable attractive cycle. For the case of equation with multiple critical points, all the critical points must be tested.

DETAILS

For the sake of clarity the largest cardoid (heart) shaped central region of the Mandelbrot set will be referred to as the main body of the Mandelbrot set (M₁ -- the region labeled 1 in the figure(3) below). All other pieces that are attached to the main body will be referred to as the buds. The largest bud that is attached to the main body (along the real axis) will be called the M₂ bud (bud labeled 2 in figure(3) below). The main body of the Mandelbrot set intersects the real axis at Âc = 0.25 and Âc = -0.75. Extending the stability analysis criteria discussed for the case logistic equation, it is easy to see that the fixed point of the complex quadratic iterator is stable along the real axis for precisely the interval mentioned above.

The determination of the boundary of the main body of the Mandelbrot set relies on the realization that any value of the complex parameter picked from within the main body of the Mandelbrot set the corresponding Julia set is a boundary between the Escape set and the trapping set of the stable fixed point of the quadratic map. The boundary of the main body defines the locus of points (in the parameter space) for which the fixed point is indifferent, that is, the modulus of the derivative of the map about the fixed point is exactly equal to 1. Using this fact one can determine the explicit expression for the outline of the M-set's main body.

If z is the fixed point of complex quadratic map, it follows that z satisfies the equation z² - z + c = 0. The derivative of the map about the fixed point z is given by 2z which in polar coordinates can be expressed as 2z = re^if. Combining these two equations, and solving for c, we obtain

c = 1 2 r eif - 1 4 r2 e2if (1)

Note, for the value of r < 1 the above equation determines the points inside the main body of the Mandelbrot set and r = 1 gives the bondary of M₁. The above equation is the parametrization of the curve in the complex plane for 0 £ f < 2p. Thus, is explicitly seen as an equation of cardoid when expressed as

Âc = cos(f)/2 - cos(2f)/4 Ác = sin(f)/2 - sin(2f)/4 (2)

by equating the real and imaginary parts of the equation.

It turns out, that at the parameter values, f = 2p/k, where k = 2, 3, 4, 5 ¼, one of the main buds of the Mandelbrot set is attached to M₁ set. Moreover, the period of the attractive cycles that belong to these buds is given by the number k in 2p/k. Also, there is another amazing fact about the arrangement of the buds. Two given buds of periods p and q at the cardoid detemine the period of the largest bud in between them as p+q. (This is illustrated for the case of p = 2 and q = 3 in figure(3) below). Similar rules are true for buds on buds.

Figure 3: The buds of the Mandelbrot set corresponding to Julia sets that bound the basins of attraction (trapping sets) of periodic orbits. The numbers in the figure indicate the periods of these orbits.

Figure 4: The plot of equation (2) which defines the boundary of the main body (M1) of the Mandelbrot set and the numbers indicate the periodicity of the buds that attach to the main body of the Mandelbrot set and the point where they attach to the main body of the Mandelbrot set.

The above two remarkable property corresponding to the periodicity of the bud was the reason for indexing the buds attached to the main body of the M-set as M_n. Thus, from the above argument the period 2 bud is attached at an angle p (setting k = 2 in f = 2p/k), similarly period 3 is the attached at f = 120 and so on. Figure(4) above shows the buds of the Mandelbrot set corresponding to Julia sets that bound basins of attraction of periodic orbits. The numbers in the figure indicate the periods of these orbits.