Hah! I bet I have an explanation of Saturn's rings, including the
ripples and braids. It's just a four-body resonance strange
attractor. There's Saturn. Then there's two very small moons close
together in resonance (7:8 or such). Those moons close together form
a gravitational basin, sort of like the Lagrange points set up by
Sun+Jupiter. And there is a bunch of pebble like stuff orbiting the
center of that basin, like marbles in a bowl. Like the Trojan
asteroids in the Sun+Jupiter's lagrange point. Fancier stuff, like
ring particles interacting or something special about Saturn's
oblateness, wouldn't be relevant.

If that's right, I should have no trouble throwing together a
simulation of it with my orbit applet. I'll try that tonight.