# plotting1

<b><u>Orbits of f(z) = z^2 + c,</u> </b>
where c= r*e^(i*2*pi*alpha)/2 - r^2*e^(i*4*pi*alpha)/4.<br/>
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 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.
<applet code="plotting1.class" name="plotting1" width="440" height="280">
</applet>

Some hints and exercises
The interesting dynamics take place for r near 1.0. When 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.