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.