x_{n+1} = (x_{n}^{2} + A)^{2}
+ B.
You see in Fig.1 that it can have one or two attracting fixed points. Each of them attract nearest critical points x_{1} = 0 or x_{2,3} = ±(-A)^{1/2}, for A < 0 (it is evident that orbits starting at ±(-A)^{1/2} coincide). Let A = -1 then for B ~ 1 we see the first tangent bifurcation and two fixed points stable and unstable appear (the highest curve - a). Under decreasing B the second tangent bifurcation takes palce (curve - b). At last in reverse tangent bifurcation stable and unstable fixed points merge together and disappear (the lower curve - c). |
In the vicinity of the superstable fixed point x_{1} = 0
(i.e. near the parabola B = -A^{2})
for large |A| we can neglect the x^{4} term
x_{n+1} = (x_{n}^{2} + A)^{2} + B ~ 2Ax_{n}^{2} + A^{2} + B. Let us denote t = 2Ax then t_{n+1} = t_{n}^{2} + 2A(A^{2} + B). This is quadratic family with C = 2A(A^{2} + B). Therefore for large |A| any bifurcation value C_{*} of the quadratic maps (e.g. tangent, period doubling or crisis bifurcation) corresponds to bifurcation curve B_{*} = C_{*} / (2A) - A^{2} of the 2D biquadratic family (near B = -A^{2}). One can get similar formula for the second critical point. |
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' = x^{2} + c_{1} , x' = y^{2} + c_{2} or x' = (x^{2} + c_{1})^{2} + c_{2} , and a linear change of coordinates (A,B) leads to canonical two parameter biquadratic family [2]. To the left you see period-3 window of periodicity (a "swallow" or "shrimp"). |
[1] J.Milnor "Remarks on iterated cubic maps"
Exp.Math. 1 (1992), 5.
[2] B.R.Hunt, J.A.C.Gallas, C.Grebogi, J.A.Yorke, and
H.Kocak Bifurcation
Rigidity
Physica D 129 (1999), 35.
[3] Canonical Quartic Map