draft

"Swallows" and "shrimps"

Milnor's swallow

The real biquadratic maps   xn+1 = (xn2 + A)2 + B   have the critical points
    x1 = 0 ,
    x2,3 = +-(-A)1/2,   for   A < 0 .
It is evident that iterations which start at +-(-A)1/2 coincide.
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).

In the vicinity of the superstable fixed point x1 = 0 for large |A| we can neglect the x4 term
    xn+1 = (xn2 + A)2 + B ~ 2Axn2 + A2 + B .
Let us denote t = 2Ax then
    tn+1 = tn2 + 2A(A2 + B) .
This is the one parameter quadratic family with C = 2A(A2 + 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) - A2
of the 2D biquadratic family (near B = -A2). One can get similar formula for the second critical point.

"Shrimps hunter" applet

 For each parameter pair (A,B) the map is iterated 500 times (starting at one of the critical points) and then the orbit is examined for periodic behaviour. If the orbit is becoming unbounded a light-grey dot is plotted. If the orbit is found to approach an orbit with low period the dot is colored according to the period. If the orbit has period greater then 64 a black point is plotted to enhance the visibility of smaller shrimp. Due to slow convergence near the period doubling bifurcations there are (non-chaotic) black strips between zones of different periodicities.

Bifurcation diagram for the real biquadratic family

"Shrimps Hunter" controls Click mouse in window to find period p of the point. Click mouse + <Alt>(<Ctrl>) to Zoom In(Out) 2 times. Hold <Shift> to modify Zoom In/Out x4

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 [2].

To the left you see period-3 window of periodicity (a "swallow" or "shrimp").

The real cubic family

[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

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updated 1 September 2002