One-Dimensional Dynamical Systems
Part 3: Iteration
Another, generally more preferred, way of visualizing orbits is based
on the graph of the function that represents the dynamical system. It
starts by showing the graph of a function y = f(x)
and the graph of the identity function y = x. Suppose we have
We can visualize this orbit in the (x, y)-plane by drawing lines between points on the graph of f and points on the line y = x. The following sequence represents the same orbit as above:
Points of the form (xn, xn) are on the graph of the identity function, that is, on the line y = x. The points (xn, xn+1) = (xn, f(xn)) are on the graph of f.
Visualizing an orbit with graphical iteration for the Logistic map.
Starting with (x0, x0) on the diagonal, draw a vertical line to the graph of f. Recall that f(x0) is the next point in the orbit of x0. Next, draw a horizontal line from (x0, f(x0)) to the diagonal. This way, we have repositioned our point so that a vertical line from this new position will give us f(f(x0)). This procedure, called graphical iteration, makes visualization easier and provides a geometric shortcut; you can even sketch an orbit instead of computing it.
Graphical iteration on the computer can be done with the following software. If you are using a Macintosh computer, follow the instructions for the software Chaos and Dynamics. On the other machines, you should follow the instructions for using Mathematica.
Written by Hinke Osinga
Comments to: firstname.lastname@example.org
Created: Apr 1 1998 --- Last modified: Thu Apr 9 13:45:03 1998