The Aharonov-Bohm effect is associated with cyclic motion. It is one of a number of anholonomic effects, and this means that the dynamical description depends on the current position of the system and on the path by which it reached that position.
An example of an anholonomic effect is Foucault's famous pendulum, which simply demonstrates the Earth's rotation. The Sagnac effect - a light beam passing around a rotated system of mirrors - is another example. Modern dynamical developments such as Hannay's angle and Berry's phase are further useful examples.