This is a rendering of the Julia set for $z' = c \sin(z)$ where $c=1+0.2 i$.
This version includes gamma correction for improved contrast ($\gamma = 0.3$), a linear scaling of the colours from darkest to lightest, and $8x8$ sub-pixel sampling to generate a high-quality antialiased image. It might take a long time to complete rendering.
Paul Bourke has some images and explanation of the thorn fractal. This implementation features over sampling, gamma correction, iterative improvement, and it's open source.
The Julia set version of this fractal is generated by computing the escape time for the functions
$$ x_{n+1} = \frac{x_n}{\cos(y_n)} + c_x $$
$$ y_{n+1} = \frac{y_n}{\sin(x_n)}+ c_y $$
I look forward to trying a Mandelbrot version of the fractal soon.
Randomly choose some affine transformations and apply them randomly to a point. Colour the transformed point based on the selected transfomation.
Drawing mostly-transparent shapes to the canvas as fast as possible.
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.
