[jashkenas]

This equation is an "attractor" because the coordinates remain close to particular points (in this case, the origin).

Not all iterated functions behave this way -- sometimes the coordinates will wander off, and that would be called a "repellor" (http://en.wikipedia.org/wiki/Repellor).

"sin" and "cos" are good primitives for building attractor functions, because they naturally stay bounded close to the origin -- "x²" for example, wouldn't work nearly as well.

As to why the deJong attractor draws particular trajectories ... that's the beauty and mystery of the thing. It's also interesting how it tends to draw particular types of shapes when seeded from different areas -- dusty spheres when seeded close to the origin, skinny loops in the 4th quadrant, folded sheets of cloth in the 1st and 3rd, and stacked diamond circles when seeded in the 2nd.