Deterministic Chaos

The Univrsal Function

The next picture shows the 128th composition for s values corresponding to a superstable 4-cycle.

As we expect, the graph has 8 fixed points:

two for the (unstable) fixed points of L,

two for the (unstable) 2-cycle of L, and

four for the (superstable) 4-cycle of L.

Graphical iteration of L⁴ shows that almost every point in the interval iterates to one of the points of the 4-cycle.

So the graph of the universla function is mostly horizontal segments, of height one of the values of the 4-cycle.

(Almost every because of course the fixed points and 2-cycle of L are fixed points of L⁴, so they remain fixed under iteration of L⁴. Similarly, points that iterate exactly to these fixed points do not iterate to the 4-cycle points.)

Return to the universal Function.