"Swallows" and "shrimps"

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 leads to canonical two parameter biquadratic family [1].

"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

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 [2].

"Shrimps hunter"

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. Note that due to slow convergence near the period doubling bifurcations there are (non-chaotic) black strips between zones of different periodicities.

[1] B.R.Hunt, J.A.C.Gallas, C.Grebogi, J.A.Yorke, and H.Kocak Bifurcation Rigidity
Physica D 129 (1999), 35.
[2] J.Milnor "Remarks on iterated cubic maps" Exp.Math. 1 (1992), 5.

Contents   Previous: Chaotic transient   Next: Circle maps
updated 9 Nov 2006