This shows a combination of an IFS (iterated function system) and a particle system. In an IFS, points are attracted towards a fractal shape by iterating the positions over a set of affine transformations. The original algorithm starts with just one random point and plots its current position over a series of randomized transformations. In this example, I instead start out with a number of randomly distributed particles, which are then given target positions by running the IFS transformations.

When the general shape has settled, one additional iteration is performed every few seconds. It's instructive to see how particles are warped from one place on the fractal shape to a completely different position.

The demo is implemented in Processing using processing.js. With 5000 particles moving around, performance depends on your browser and hardware. Unfortunately, PShape support is not yet complete in processing.js, which would probably help to speed this up.