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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)

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(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.
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Another example is to Click on Show, then enter in the xterm window:
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f=z^4 + 3z + 3; f[1]

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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.
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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.
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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.
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Questions? Send them to me at:

marshall@math.washington.edu
Return to Don Marshall's home page.

* Last revised on: July, 1997 *