IterLab

Iterlab is a program for viewing graphical iteration graphs.
Iteration is simply the process of repeatedly taking the results of a function and feeding them back into the same function. For this purpose the function, y=f(x) could be representative of some process in motion, called a dynamical system.

A recurrence relation of the form x_n+1 = f( x_n ), is called a one point iteration formula.

The generation of each successive term in the sequence is called a step in the iteration of f(x). The sequence of iterates for a particular starting value x₀ is known as the orbit of x₀. Graphical iteration is a wonderful, simple way to visualize the sequence of iterates by generating them in a purely geometrical way, I don't know who came up with the method but it's very clever indeed.

I first saw this method described in the book Chaos and Fractals by Heinz-Otto Peitgen, Hartmut Jurgens and Dietmar Saupe, The book has been around awhile, It's a great introduction to the topic, a couple of years ago a new edition was published.

It's a big thick book, so it's kinda pricey but you can probably find at the library. They even provide code in BASIC for iterating on a quadratic function, but after awhile you'd like to iterate other functions.

I particularly wondered about iteration of harmonics, well I think Iterlab makes it pretty easy.

For example here's a picture of twenty iterations of the iterator

y=2*sin( (1.0/2.0) * x) + cos(2*x) + (x/2.0)

starting at x₀ = 7.2.




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.

The graphical iteration starts at the initial x₀ value. After that the following steps are repeated:

• the value x_n of the function f(x) is computed.
• from there we draw a vertical line until we hit the graph of y=f(x).
• then a horizontal line is drawn to the identity line y=x.

And we continue this way, this is the feedback process of iteration;
each term in the sequence is defined by its predecessor.

It works because points on the y=x line, which is a diagonal bisector, have the same distance from both axes.

Iterlab is for viewing the one-step feedback process, and represents one class of many types of feedback methods. You could go on and begin iterating functions using complex numbers from there comes the Mandelbrot set and other such beautiful and intriguing objects.

Depending on the function and the starting value, the iterates could converge to a point (sometimes this will happen quickly other times it may take a long time), sometimes they cycle back between two or more points. Sometimes the points repel or take off to infinity, or never come to rest and will keep going all around the function.


written by Marc Alfaro