Strange attractors

Not to trump my colleague at chunk 34 of the Processing Book we are busily writing, but how does one explain an attractor, exactly?

In an iterated function system, particularly the random iteration algorithm, you pick a point, any point (nothing up the sleeve) and watch as it magically gets sucked into this attractor. Then the point hops around the attractor forever, apparently randomly moving about, but otherwise very nicely confined and constrained.

We don't tend to plot the first 10 iterations or so. This seems rather arbitrary, but I've not yet found a good explanation for how we decide when to start plotting.

What's the simplest attractor you could have? f(x) = 0? Ah, so f(0) = 0 is a fixed point of this system. And, f(x) = x has all points as fixed points. So we can explain that for some systems the set of fixed points has an interesting shape. Also that after a while we can remain on that shape, if we start out close enough to it to get sucked in by its 'gravitational pull'. What would that look like?

Pendulum? Not very interesting - its fixed point must be its bottom point. But it's a helpful analogy for thinking about attractors as fixed points, which is what IFSs are, except that they are sets of points. I don't want to get too mathematical.

Could do the Cantor set, I suppose, or an IFS to draw a line or a square or some such thing. The former is probably too abstract. The latter is visually uninteresting, but at least you can plainly see the attractor.

Hmm... more reading required.