Intuitively, I understand that with an even number of stages, the output of the last would be the same logic level as the input of the first, so that the output eventually latches to a certain logic level. Whereas, with odd number of inverters, the outputs switch and the resulting frequency is determined by the gate delay.
However I also read that Barkhaussen's criterion does not apply to non-linear circuits, and cannot be used to evaluate the behaviour of a ring oscillator. If that is the case, how do you explain the sustenance of oscillations in the oscillator? More precisely, what would the transient behaviour be like, that leads to oscillations?
Also, is there a necessary DC operating point that either oscillator converges to in the steady state? Is the operating point "stable" in the case of even and "unstable" in the case of odd stages?