This is a local copy. The original can be found at
http://www.lut.ac.uk/departments/ma/gallery/doubling
used with permission from Andy Burbanks

Period Doubling Interactive JAVA Applet Produced by Andy Burbanks

This Applet simulates graphical iteration of the Logistic Map f(x) = ax(1-x).

This apparently simple function can display a whole range of different behaviours when iterated (applied repeatedly to some initial value of x). The type of behaviour observed depends on the value of the parameter a.

The horizontal slider controls the initial point x0. The vertical slider controls the parameter value a. (Sliders can be operated by clicking and/or dragging.) The final few iterations are shown in red to help detect periodic orbits (cycles).

Shown below are some examples of the behaviour you should find.

[APPLET] [APPLET] [APPLET] [APPLET]

The four plots show a 2-cycle, 4-cycle, 8-cycle, and ``chaos'', respectively.