The Tinkerbell map is a chaotic dynamical system with an interesting strange attractor. The iterated function is

$$

x_{n+1} = x_n^2 - y_n^2 + a x_n + b y_n

$$

$$

y_{n+1} = 2 x_n y_n + c x_n + d x_n

$$

In this example, the parameters are $a=0.9$, $b=-0.6013$, and $c=2.0$. $d$ is randomly chosen on $(0.35, 0.55)$, and the colour is based on $d$.

See also Wikipedia.