Various researchers have proposed using self-timed networks to generate and distribute clocks and other timing signals. We consider one of the simplest self-timed networks, a ring, and note that for timing applications, self-timed rings should maintain uniform spacing of events. In practice, all previous designs of which we are aware cluster events into bursts. In this paper, we describe a dynamical systems approach to verify the temporal properties of self-timed rings. With these methods, we can verify that a new design has the desired uniform spacing of events. The key to our methods is developing an appropriate model of the timing behaviour of our circuits. Our model is more accurate than the simplistic interval bounds of timed-automata techniques, while providing a higher level of abstraction than non-linear differential equation models such as SPICE. Evenly spaced and clustered event behaviours are distinguished by simple geometric features of our model.