We have seen that points of an
ncycle of L(x) are fixed points of L^{n}(x). So we
look at graphs of some compositions of the logistic map for various
svalues. 
Here we see the s = 4 pictures for
L^{2}(x), L^{3}(x), and L^{4}(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. 

A careful comparison of these three animations reveals cycles
arise in two different ways, illustrated by these observations. 
The 2cycle appears where one of the
fixed points of L(x) becomes unstable, 
but the 3cycle seemss to appear from nothing. 

Maybe the difference has to do with even and odd cycles. 
No, looking at the 4cycle animation shows two 4cycles: 
the first appears where the 2cycle becomes unstable, 
the second arises from nothing. 

In fact, these are the two mechanisms for producing
new cycles. The first is called a
perioddoubling bifurcation, the second a
tangent bifurcation. 
Here is a brief description of the geneology
of cycles. 