Up: Iteration

# 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.

• Use the Java Applet Iteration of the Logistic Map to see how the dynamics close to the (positive) fixed points depends on the parameter . For each , which point is attracting? Are there the same number of attracting points for every ?

• For this question you can either use Devaney's Chaos Lab #2, Mathematica or the Java Applet The Period Doubling Route to Chaos. In the latter case, follow the abstracted instructions.

Fix the starting point x0 = 0.1. For several choices of between 0 and 3.5, investigate the eventual behavior of x0 = 0.1. In Devaney's Chaos Lab #2 you can push the Transient button to see what the eventual behavior is; in the other cases, the eventual behavior is shown in red.

Draw a diagram with on the horizontal axis and x on the vertical axis. For each value that you checked, place a dot at the x values that are part of the eventual behavior of x0 = 0.1. If you checked the eventual behavior for enough values, you should be able to sketch the curves connecting the dots, showing the continuous dependence on . Do not go beyond = 3.5.

Up: Iteration