Douady rabbit
Encyclopedia
The Douady rabbit, named for the French mathematician Adrien Douady
Adrien Douady
Adrien Douady was a French mathematician.He was a student of Henri Cartan at the Ecole Normale Supérieure, and initially worked in homological algebra. His thesis concerned deformations of complex analytic spaces...
, is any of various particular filled Julia sets
Filled Julia set
The filled-in Julia set \ K of a polynomial \ f is a Julia set and its interior.-Formal definition:The filled-in Julia set \ K of a polynomial \ f is defined as the set of all points z\, of dynamical plane that have bounded orbit with respect to \ f...
associated with the c near the center period 3 buds of Mandelbrot set for complex quadratic map
Complex quadratic polynomial
A complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers.-Forms:When the quadratic polynomial has only one variable , one can distinguish its 4 main forms:...
.
Forms of the complex quadratic map
There are two common forms for the complex quadratic map
. The first, also called the complex logistic mapLogistic map
The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
, is written as
where
is a complex variable and 
is a complex parameter. The second common form is
Here
is a complex variable and 
is a complex parameter. The variables 
and 
are related by the equation
and the parameters
and 
are related by the equations
Note that
is invariant under the substitution 
.Mandelbrot and filled Julia sets
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 setFilled Julia set
The filled-in Julia set \ K of a polynomial \ f is a Julia set and its interior.-Formal definition:The filled-in Julia set \ K of a polynomial \ f is defined as the set of all points z\, of dynamical plane that have bounded orbit with respect to \ f...
consists of all points in the mapping plane whose images remain bounded under indefinitely repeated applications of
. The Mandelbrot setMandelbrot set
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
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 transformationsAffine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
of one another (a linear transformation plus a translation), the filled Julia sets look much the same in either the
or 
planes. |  |
Figure 1: The Mandelbrot set in the  plane. | Figure 2: The Mandelbrot set in the  plane. |
The Douady rabbit
The Douady rabbit is most easily described in terms of the Mandelbrot set as shown in Figure 1. 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 
. |
Figure 4: Douady's rabbit for  or  . |
External links
- http://mathworld.wolfram.com/DouadysRabbitFractal.html