## Complex Dynamics software: comdyn

This program is designed to display, using color graphics, the main ideas in the study of iterations of (complex-valued) rational functions. See e.g. Carleson and Gamelin, Complex Dynamics. The software was written by my student Felix Huang and is partly based on a program written by an earlier student Mike Stark.

If you do not have an account on our system, you can obtain a copy of the C code written for X Window Systems, by sending me a mail message, as described below. If you are logged on to one of our department machines and want to run the program, just type the following command (or use your mouse to cut and paste):

~marshall/PUBLIC/xcomdyn

Then move the mouse cursor to one of the windows. The "identity" window is a standard coloring of the plane. A spectrum of pure color determines the argument of a point, and shading determines the modulus. The point w=1 is pure red and the other cube roots of 1 are green and blue. A darker shade means smaller absolute value. So w=0 is black and infinity is white. A function f is represented in the "complex plane" window by coloring each point z with the color of w=f(z) in the identity window. Click on "Iterate" in the main window, then type in your favorite rational function in the xterm window. For example:
(2z^3 + 5)/(3z^2 - 2)
(Then move the mouse cursor to the complex plane window). The program will then compose this function with itself many times, displaying the successive iterates. Historical note: Newton studied the iterates of this rational function.

Another example is to Click on Show, then enter in the xterm window:
f=z^4 + 3z + 3; f[1]
This will show just the map f. Can you see where the zeros of f are located? Newton's method to accurately find the zeros of f is to iterate Nf= z-f(z)/f'(z). Click on Nf, which will iterate the Newton function. The iterates of Nf will converge to the zeros of f in the regions where the color is eventually constant. You can stop the iteration by clicking on "Stop". If you hold down the second mouse button and move the mouse cursor to various regions in the picture, the "position" window will show the values of z and the current iterate of f. Clicking on "Cont" will continue the iterations Notice that there are regions which "cycle" instead of converging.

Felix's Masters thesis which describes the program and gives many examples, can be found at: Thesis . The examples are also located in Examples . As you read the text, you can cut and paste the analytic form of the examples from this file.

The C-code can be obtained by sending me a request at the address below. If you have netscape, just click on the address and send the message.

Questions? Send them to me at:

marshall@math.washington.edu