As s increases from 0 to 4, we observe 
* Each stable cycle goes through an infinite sequence of perioddoublings. 
 For example, the stable fixed point gives rise to a stable 2cycle just as the fixed point becomes
unstable. 

 This stable 2cycle gives rise to a stable 4cycle just as the 2cycle becomes unstable. 

 This stable 4cycle gives rise to a stable 8cycle just as the 4cycle becomes unstable. 

In general, any stable ncycle gives rise to a family 
2ncycle →
4ncycle → 8ncycle → 16ncycle → ... 
* Each tangent bifurcation produces a stable cycle, that then gives rise to a
cascade of perioddoubling bifurcations. 
* For each svalue, there is at most one stable cycle, and for many there are none. 
* If there is a stable cycle, the iterates of 1/2 converge to the cycle. This was
proved by Fatou and Julia. (What's special about x = 1/2 is that it is the
critical point, the point at which the logistic map's derivative is 0.) 
* Call the range of svalues of a stable cycle and all its stable
perioddoubling descendants a periodic window. 
 The order in which these periodic windows arise is a bit complicated, but
completely understood. 


 They showed every periodic window
contains an svalue for which x = 1/2 belongs to the cycle. These
cycles are called superstable. 

 With each superstable
cycle, Metropolis, Stein, and Stein associated a sequence of symbols L and R, denoting
whether a point of the cycle falls to the Left or Right of 1/2. 

 The two 4cycles
shown below have sequences RLR and RLL. (Fourcycles containing
x = 1/2 have only three symbols, because the cycle is always understood to
start with x = 1/2.) 


 The heart of understanding the ordering of the periodic windows is finding how the symbol sequences
of any two superstable cycles are related to the order of the svalues at which they occur. 

* For some svalues there is no stable cycle at all, and the dynamics are chaotic.
Jakobsen proved the chaotic svalues are a nonnegligible portion of the range
3.6 ≤ s ≤ 4, but the amount of chaos is difficult to estimate. 