The splitting of the stable fixed point 
into a
stable 2-cycle as 
becomes greater than 3 is called a
bifurcation (a fancy Latin word for splitting). As
increases further, more bifurcations happen. In particular, we expect
another bifurcation just beyond 
. Trying
, we obtain the web diagram in this
figure. 100 iterations are plotted in
this figure, and it appears the forward orbit settles into a cycle of
4 points.
Figure 68: The web diagram for : a 4-cycle appears
Therefore, apparently our stable 2-cycle has now bifurcated into a stable 4-cycle.
The algebra of these bifurcations becomes increasingly vexing. To
find a 4-cycle, we must solve the equation 
. Unfortunately,
this is a degree 
polynomial equation, and that's tough even
on a computer. But wait! We know some of the roots come from 
.
So we may restrict ourselves to the polynomial
Alas, this is still a degree 16-4=12 polynomial, which is wee bit out of our range.
Moreover, the attractive 4-cycles also satisfy 
. Perhaps this can help. At the beginning and end of the range of
's where there is an attracting 4-cycle, we must have
. These are degree 15 equations. It is possible to
combine two different polynomial equations in two variables x and
into a single equation in one of the variables, particularly
. Even this is not terribly pleasant to do. Fortunately, there is
a dynamical approach that we know very well.