To the left you see parameter plane (A,B) for iterations of the
biquadratic maps. Regions with bounded critical orbits which start at
x1 = 0 and x2 = (-A)1/2 are
marked by colors and digits "1" and "2" correspondingly.
From equation x = (x2 + A)2 + B it follows that the critical point x1 = 0 is fixed (i.e. periodic with period-1) at B = -A2. In a similar way the points x2,3 = +-(-A)1/2 are superstable at A = -B2. You see these two parabola to the left (the blue and light-blue wings and tails of swallow).
A central feature of a region of periodic stability surrounded by chaotic
behaviour is a point in parameter space at which the map has a superstable
orbit - a periodic orbit which includes a critical point of the map.
Near a superstable period-n orbit, the n-th iterate of the map is generally
well approximated by the quadratic family x -> x2 + c .
In the two parameter case, an orbit will be superstable along a curve in parameter space. In general one expects that along lines transverse to the curve of superstability, the bifurcation diagram will resemble the one-parameter quadratic family.
However, if the map has more than one critical point,
at a point of intersection of two curves of superstability the orbit becomes
"doubly superstable" - to include a second critical point. Near such a point
it is well approximated by the composition of two quadratic map
y' = x2 + c1 , x' = y2 + c2 or
x' = (x2 + c1)2 + c2 ,
and a linear change of coordinates (A,B) leads to canonical two parameter biquadratic family .
To the left you see period-3 window of periodicity (a "swallow" or "shrimp").
 J.Milnor "Remarks on iterated cubic maps"
Exp.Math. 1 (1992), 5.
 B.R.Hunt, J.A.C.Gallas, C.Grebogi, J.A.Yorke, and H.Kocak Bifurcation Rigidity
Physica D 129 (1999), 35.
 Canonical Quartic Map