Douady rabbit

The Douady rabbit is any of various particular filled Julia sets associated with the parameter near the center period 3 buds of Mandelbrot set for complex quadratic map.
Name.
Douady's rabbit or the Douady rabbit is named for the French mathematician Adrien Douady.
The fat rabbit or chubby rabbit has c at the root of 1/3-limb of the Mandelbrot set. It has a parabolic fixed point with 3 .
Forms of the complex quadratic map.
There are two common forms for the complex quadratic map formula_1. The first, also called the "complex logistic map", is written as
where formula_3 is a complex variable and formula_4 is a complex parameter. The second common form is
Here formula_6 is a complex variable and formula_7 is a complex parameter. The variables formula_3 and formula_6 are related by the equation
and the parameters formula_4 and formula_7 are related by the equations
Note that formula_7 is invariant under the substitution formula_15.
Mandelbrot and filled Julia sets.
There are two planes associated with formula_1. One of these, the formula_3 (or formula_6) plane, will be called the "mapping plane", since formula_1 sends this plane into itself. The other, the formula_4 (or formula_7) plane, will be called the "control plane."
The nature of what happens in the mapping plane under repeated application of formula_1 depends on where formula_4 (or formula_7) is in the control plane. The "filled Julia set" consists of all points in the mapping plane whose images remain bounded under indefinitely repeated applications of formula_1. The "Mandelbrot set" consists of those points in the control plane such that the associated filled Julia set in the mapping plane is connected.
Figure 1 shows the Mandelbrot set when formula_4 is the control parameter, and Figure 2 shows the Mandelbrot set when formula_7 is the control parameter. Since formula_3 and formula_6 are affine transformations of one another (a linear transformation plus a translation), the filled Julia sets look much the same in either the formula_3 or formula_6 planes.
The Douady rabbit.
The Douady rabbit is most easily described in terms of the Mandelbrot set as shown in Figure 1 (above). In this figure, the Mandelbrot set, at least when viewed from a distance, appears as two back-to-back unit discs with sprouts. Consider the sprouts at the one- and five-o'clock positions on the right disk or the sprouts at the seven- and eleven-o'clock positions on the left disk. When formula_4 is within one of these four sprouts, the associated filled Julia set in the mapping plane is a Douady rabbit. For these values of formula_4, it can be shown that formula_1 has formula_35 and one other point as unstable (repelling) fixed points, and formula_36 as an attracting fixed point. Moreover, the map formula_37 has three attracting fixed points. Douady's rabbit consists of the three attracting fixed points formula_38, formula_39, and formula_40 and their basins of attraction.
For example, Figure 3 shows Douady's rabbit in the formula_3 plane when formula_42, a point in the five-o'clock sprout of the right disk.
For this value of formula_4, the map formula_1 has the repelling fixed points formula_35 and formula_46. The three attracting fixed points of formula_47 (also called period-three fixed points) have the locations
The red, green, and yellow points lie in the basins formula_51, formula_52, and formula_53 of formula_47, respectively. The white points lie in the basin formula_55 of formula_1.
The action of formula_1 on these fixed points is given by the relations
Corresponding to these relations there are the results
Note the marvelous fractal structure at the basin boundaries.
As a second example, Figure 4 shows a Douady rabbit when formula_64, a point in the eleven-o'clock sprout on the left disk. (As noted earlier, formula_7 is invariant under this transformation.) The rabbit now sits more symmetrically on the page. The period-three fixed points are located at
The repelling fixed points of formula_1 itself are located at formula_35 and
formula_71. The three major lobes on the left, which contain the period-three fixed points formula_72,formula_73, and formula_74, meet at the fixed point formula_35, and their counterparts on the right meet at the point formula_76. It can be shown that the effect of formula_77 on points near the origin consists of a counterclockwise rotation about the origin of formula_78, or very nearly formula_79, followed by scaling (dilation) by a factor of formula_80.