Gaps in the Saturn Rings A spectacular example of the instability of rational KAM-tori are the gaps in the rings of the planet Saturn.
The time a particle orbiting in Cassini's gap (between the A-ring and the B-ring, r=118,000km) would need around Saturn is exactly half the time the "shepherd-moon" Mimas needs to orbit Saturn. The quotient of the revolving frequencies is 2:1. Any particle orbiting in Cassini's gap is therefore unstable against the perturbation caused by Mimas and it is consequently thrown out of its orbit.
Here are some other references I ran across while looking: