Iteration is the process of repeatedly taking the results of a
function and feeding them back into the same function.
The function, y=f(x) can represent some process
in motion (called a dynamical system).
Iteration is fundamental in exploring chaos, fractals, and dynamical
systems.
I've put some info about it on my site at
www.hyperscope.net,
but search on Google for "graphical iteration" or "cobweb diagrams"
there's alot of info about the method.
I first saw the method described in the book "Chaos and Fractals
New Frontiers of Science by Heinz-Otto Peitgen, Hartmut Jurgens,
Dietmar Saupe". (its an old book but a new edition came
out a couple of years ago)
I've enjoyed using some of the available programs and Java applets
offered over the internet for doing graphical function iteration,
but I've yet to find the one I wished for.
For example many of the programs only offer a limited number of
predefined functions to iterate on, or they are expensive,
do you really want to purchase Mathematica or Maple just to do
graphical iteration?
Some of the better programs for doing this for some reason are
only available for the Apple computers or they're old DOS programs.
So anyway for these reasons and simply because the iteration of functions
is such an interesting study, and motivated by Donationcoder.com's
programming contest I decided to give it a try with C++ Builder to create
a software for graphical iteration.
So here's how it works;
First the graph of the function y=f(x) and the identity function y=x
are drawn on the same set of axes.
Then to view the iteration enter a starting value in
the text box labeled "x0=".
Click the red and blue button to generate the sequence of
iterates (called an orbit) for the function.
The first time, a vertical line is drawn from x0 on the x-axis to
the y=x line.
From that point we draw a vertical line until we hit the graph of
y=f(x).
Then a horizontal line is drawn to the line y=x .
And we continue this way, this is the feedback process of iteration.
This works because points on the y=x line (the blue line), which is a diagonal
bisector, have the same distance from both axes.
The program could be useful for those starting in the study of chaos, fractals
and dynamics. Iteration even shows up in calculus (Newtons Method) and
differential equations.
One value of this type of display is its intuitive connection to the geometry of
iteration.
I hope this at least explains some of the ideas behind the program.
Regards,
Marc
