Dear Tumeda,

I am afraid that Barkhausen Criteria still holds for oscillator.
Although osciallator has no stable o.p, it is instructive to
check B.C. first. Please have a look at Razavi's paper.

My question is, how about your gain? I have a look on you
schematic, you have 4 stages, and one of them is cross- connected. And then it is sure that there will be a positive
feedback, if the gain condition is met.