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. The rabbit is a parabolic Julia set for internal angle 1/3 (parameter c is a root point between period 1 and period 3 components of Mandelbrot set).
There are two planes associated with . One of these, the (or ) plane, will be called the mapping plane, since sends this plane into itself. The other, the (or ) plane, will be called the control plane.
The nature of what happens in the mapping plane under repeated application of depends on where (or ) 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 . 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 is the control parameter, and Figure 2 shows the Mandelbrot set when is the control parameter. Since and are affine transformations of one another (a linear transformation plus a translation), the filled Julia sets look much the same in either the or planes.
Quaternion julia set with parameters c = −0,123 + 0.745i and with a cross-section in the XY plane. The "Douady Rabbit" julia set is visible in the cross section
Representation of the dynamics inside the 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 is within one of these four sprouts, the associated filled Julia set in the mapping plane is a Douady rabbit. For these values of , it can be shown that has and one other point as unstable (repelling) fixed points, and as an attracting fixed point. Moreover, the map has three attracting fixed points. Douady's rabbit consists of the three attracting fixed points , , and and their basins of attraction.
For example, Figure 3 shows Douady's rabbit in the plane when , a point in the five-o'clock sprout of the right disk. For this value of , the map has the repelling fixed points and . The three attracting fixed points of (also called period-three fixed points) have the locations
The red, green, and yellow points lie in the basins , , and of , respectively. The white points lie in the basin of .
The action of 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.
Figure 3: Douady's rabbit for or .
As a second example, Figure 4 shows a Douady rabbit when , a point in the eleven-o'clock sprout on the left disk. (As noted earlier, 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 itself are located at and . The three major lobes on the left, which contain the period-three fixed points ,, and , meet at the fixed point , and their counterparts on the right meet at the point . It can be shown that the effect of on points near the origin consists of a counterclockwise rotation about the origin of , or very nearly , followed by scaling (dilation) by a factor of .