x

It is evident that iterations which start at

To the left you see parameter plane (A,B) for iterations of the
biquadratic maps. Regions with bounded critical orbits which start at
x and _{1} = 0x are
marked by colors and digits "1" and "2" correspondingly.
_{2} = (-A)^{1/2}From equation x is fixed (i.e.
periodic with period-1) at _{1} = 0B = -A. In a similar way the
points ^{2}x are superstable at
_{2,3} = +-(-A)^{1/2}A = -B. You see these two parabola to the left
(the blue and light-blue wings and tails of swallow).
^{2} |

Let us denote

This is the one parameter quadratic family with

of the 2D biquadratic family (near

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 -> x ^{2} + 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' = x or
^{2} + c_{1} ,
x' = y^{2} + c_{2}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

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