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
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.