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, illustrated by these observations.
 The 2-cycle appears where one of the fixed points of L(x) becomes unstable, but the 3-cycle seemss to appear 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.