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One-Dimensional Dynamical Systems

Part 3: Iteration

Graphical 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 an orbit:

x0, f(x0), f(f(x0)), ... = x0, x1, x2, ...

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:

(x0, x0), (x0, x1), (x1, x1), (x1, x2), (x2, x2), ...

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.

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Written by Hinke Osinga
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Created: Apr 1 1998 --- Last modified: Thu Apr 9 13:45:03 1998