"Swallows" and "shrimps"

Biquadratic maps and Milnor's swallow

Consider entangling of regular and chaotic dynamics regions on 2D parameter plane. The real biquadratic maps depend on two parameters (A,B)
    xn+1 = (xn2 + 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
    x1 = 0     or
    x2,3 = ±(-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 Fig.2 on the parameter (A,B) plane these three bifurcation curves are shown in red. Note that the critical point x1 = 0 is fixed at B = -A2 and x2 is fixed at A = -B2. All these curves make the "Milnor's swallow" shape [1]. In the applet to the right regions with bounded critical orbits started at x1 = 0 and x2 = (-A)1/2 are marked by different colors and digits "1" and "2" correspondingly.
In the swallow's tail (inside the region bounded by the two red bifurcation curves) can exist two different attractors. E.g. at the two parabola crossing point A = B = -1 there are two superstable fixed points x = 0 and x = -1. Depending on initial conditions iterations go to one or another attractor. Therefore swallows have different tail orientation in the two pictures below, where regions of regular and chaotic dynamics of the critical points x1 (to the left) and x2 (to the right) are shown. Construction algorithm for these images is explained at the bottom of the page.
"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
In the vicinity of the superstable fixed point x1 = 0 (i.e. near the parabola B = -A2) 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 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.
To the left you see in sequence (from the right to the left): tangent bifurcation, period doubling cascade and chaotic region with windows of regular dynamics.

2D bifurcation diagram for nonlinear maps

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

"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 shrimps. Due to slow convergence near the period doubling bifurcations there are (non-chaotic) black strips between zones of different periodicities.

[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 9 Nov 06