A Douady rabbit is a fractal derived from the Julia set of the function ${\textstyle f_{c}(z)=z^{2}+c}$, when parameter ${\displaystyle c}$ is near the center of one of the period three bulbs of the Mandelbrot set for a complex quadratic map.

It is named after French mathematician Adrien Douady.

The Douady rabbit is generated by iterating the Mandelbrot set map ${\displaystyle z_{n+1}=z_{n}^{2}+c}$ on the complex plane, where parameter ${\displaystyle c}$ is fixed to lie in one of the two period three bulb off the main cardioid and ${\displaystyle z}$ ranging over the plane. The resulting image can be colored by corresponding each pixel with a starting value ${\displaystyle z_{0}}$ and calculating the amount of iterations required before the value of ${\displaystyle z_{n}}$ escapes a bounded region, after which it will diverge toward infinity.

It can also be described using the logistic map form of the complex quadratic map, specifically

which is equivalent to

${\displaystyle w_{n+1}=w_{n}^{2}+c}$.

Irrespective of the specific iteration used, the filled Julia set associated with a given value of ${\displaystyle \gamma }$ (or ${\displaystyle \mu }$) consists of all starting points ${\displaystyle z_{0}}$ (or ${\displaystyle w_{0}}$) for which the iteration remains bounded. Then, the Mandelbrot set consists of those values of ${\displaystyle \gamma }$ (or ${\displaystyle \mu }$) for which the associated filled Julia set is connected. The Mandelbrot set can be viewed with respect to either ${\displaystyle \gamma }$ or ${\displaystyle \mu }$.

Noting that ${\displaystyle \mu }$ is invariant under the substitution ${\displaystyle \gamma \to 2-\gamma }$, the Mandelbrot set with respect to ${\displaystyle \gamma }$ has additional horizontal symmetry. Since ${\displaystyle z}$ and ${\displaystyle w}$ are affine transformations of one another, or more specifically a similarity transformation, consisting of only scaling, rotation and translation, the filled Julia sets look similar for either form of the iteration given above.