Nonlinear Web

• The logistic function Cx(1 - x)

• The quadratic function x² + C

• Cubic functions C(x - x³/3)

• Quartic functions Cx(1 - x)(1 - 2x)

• Sine functions C sin(x)

• Cosine functions C cos(x)

• Tent functions Cx if 0 <= x <= 0.5; C(1 - x) if 0.5 < x <= 1.

• The Doubling function 2x mod 1

After opening nonlinear web, you will see the graph of the logistic function F(x) = 3x(1 - x) displayed. You may change this graph to that of F(x) = Cx(1 - x) for 0 < C < 4 by using the scroll bar on the right.

You may select other functions from the list above using the menu in the upper left hand corner. You may also view the graph of the nth iterate (1 <= n<= 6) of the function using the lower left menu. Remember to click Clear in order to remove the old graph from the window.

Moving the cursor over the graph window, you will see displayed in the lower left hand corner the current x-value. Click on a point in the window to display the web diagram corresponding to the chosen x-value. The first twenty five iterations are displayed in black; the subsequent 175 iterations are displayed in red. This allows you to see the "fate of the orbit" without viewing the transient behavior. Click on Delete Transients to remove the first part of the orbit. Click on Add Transients. to restore the transient behavior.

You may see the orbit one step at a time by using the Iterate button. Be sure that the Del Tans button is showing (otherwise you will not see the first few iterates). Each click of this the Iterate button gives a successive point on the orbit. Clicking a new x-value changes to the orbit of that point. The seed corresponding to the chosen orbit is displayed in the lower left.

Go to Nonlinear Web Applet.

For more information about the web diagram, consult the book Chaos

Created by Rodin Enchev. Modified by Adrian Vajiac and Robert L. Devaney.

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