Orbits of f(z) = z^2 + c, where c= r*e^(i*2*pi*alpha)/2 - r^2*e^(i*4*pi*alpha)/4.
The black square is a region in the complex plane containing the attractor (if there is one). First enter a value of r (near 1.0) and integers p and q, p < q. Set the number of colors equal to q (or to a divisor of q if q > 20). Then click on any point in the black square to see the orbit of that point. To see the approximate location in the complex plane put a check next to the Axes button. Recall that if there is an attractor, the point (0,0) will be attracted to it.

Some hints and exercises
The interesting dynamics take place for r near 1.0. When r < 1 there is an attracting fixed point which becomes parabolic at r = 1 and then for r > 1 an attracting cycle of period q is born.
1. For r = 1.012, p = 5, q = 13 see if you can locate the repelling fixed point.
2. For "large" q the hyperbolic component with period q is very small. for example to see a period 19 attracting cycle, choose p = 11, q = 19 and r = 1.005.
3. To see orbits when c is in the 1/2 bulb off the 1/3 bulb try r = 1.22, p = 1 and q = 3 and number of colors = 6.
4.a. To get an idea of the dynamics at a parabolic fixed point, let r =1 and let p and q be ratios of successive Fibonacci numbers. (Recall that the limit of these ratios is (1-sqrt(5))/2 which is known to admit a Siegel disk. (Try r = 1, p = 610, q = 987 and click on several places.)
b. To see a kind of "phyllotaxis", keep the same p and q, but let r = 1.001 to get a spiral out, or r = .999 to get a spiral in. Set the number of points to 10000.