The iPhone’s panorama mode is pretty impressive. Dallas Road, Victoria BC

The standard map is the first chaotic thing I remember seeing, probably in an AfterDark screen saver. It has an interesting physical interpretation involved rotors, but that isn't important for the pretty pictures.

$$ x_{n+1} = x_n + K \sin(y_n) \mod 2\pi $$

$$ y_{n+1} = y_n + x_n + K \sin(y_n) \mod 2\pi $$

Clifford Attractors are attributed to Clifford Pickover. They're reminiscent of smoke gently wafting in still air. paulbourke.net has some more images, and generates colour in a different way.

The formula has four parameters, each chosen randomly on $[-2,2]$. Not all random combinations generate a strange attractor. In some they fall into a fixed cycle or even a single fixed point. If that happens click the Restart button to get a new set of parameters.

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.

My notes on setting up a new mac, probably only useful for me.