Linear Web

This applet is designed to show you three interrelated concepts:

The orbit of a seed under iteration of the linear function Ax + B.
The corresponding time-series graph.
The web diagram or graphical iteration of this function.

See the book Chaos for details about what these terms mean.

When you open this applet, you see four "windows". In the upper left you will view the graph of Ax + B together with its graphical iteration or web diagram. You will click in this window to choose the appropriate seed x₀. In the upper right are two sliders to control the values of A and B. In the lower left is displayed the corresponding time series of the chosen seed x₀. And is the lower right is the list of the orbit of x₀.

To use the applet, first select a value of A and B using the sliders. Note how you see the graph of y = Ax + B moving as you vary these "parameters." Be sure to change these values of A and/or B, for otherwise you will be viewing the default function y = x, which,to put it mildly, is not very interesting! You also see the diagonal line y = x. Now click on a point in the window that contains the graph. The x-coordinate of this point yields the seed for the orbit x₀. Click the iterate button to see the first point on the orbit of this seed. This is displayed graphically in the web-diagram window and in the time series window, and numerically in the orbit list. Don't know what you are seeing? Go to the book Chaos for details. Click iterate again for the next point on the orbit. Keep clicking the Iterate button to see subsequent points on the orbit.

Now change the parameters A or B with the sliders. Note how the orbit of your original seed changes.

Questions:

For which values of A do you see an attracting fixed point?
For which values of A do you see a repelling fixed point?
For which values of A do you see neither an attracting nor a repelling fixed point?
For which values of A do you see a cycle of period 2?
How does changing the value of B alter the answers to the above questions?

Go to Linear Web Applet.

Created by Adrian Vajiac and Robert L. Devaney.

For comments and suggestions write to Robert L. Devaney at bob@bu.edu