The Feigenbaum diagram
The Feigenbaum diagram plots the stable orbits of the logistic iteration map , where and : Given any initial value , the iterations of the map give the sequence , where for . After an initial transient phase, for sufficiently large and for suitable values of the orbit stabilises: the applet shows for any value of (on the abscissa axis) the values of (on the ordinate axis).
You can zoom in and out the diagram by setting the values , , , and . It is possible to measure values of and by means of the corresponding sliders, in order to select correctly portions of the diagram. Verify the self-similarity properties of the figure. Try to explain the various sections of the diagram. In particular, find a meaning to the visible bifurcations.