Deterministic Chaos

Logistic Map Cycles

We have seen that points of an n-cycle of L(x) are fixed points of Lⁿ(x). So we look at graphs of some compositions of the logistic map for various s-values.

Here we see the s = 4 pictures for L²(x), L³(x), and L⁴(x). Click on each picture for an animation of s going from 2 to 4 in steps of 0.25. Click an animation to stop.

L2	L3	L4

A careful comparison of these three animations reveals cycles arise in two different ways.

The 2-cycle appears where one of the fixed points of L(x) becomes unstable.

But the 3-cycle arises from nothing.

Maybe the difference has to do with even and odd cycles.

No, looking at the 4-cycle animation shows two 4-cycles:

the first appears where the 2-cycle becomes unstable,

the second arises from nothing.

In fact, these are the two mechanisms for producing new cycles. The first is called a period-doubling bifurcation, the second a tangent bifurcation.

Here is a brief description of the geneology of cycles.

Return to cycles.