A probability based approach on ananlyzing dynamics of oscillators on a   bidirectional ring with propagation delay
  In this paper, we presented a model of pulse-coupled oscillators distributed on a bidirectional ring with propagation delay. In numerical simulations based on this model, we observed phenomena of asynchrony in a certain range of delay factor $\alpha$. To find the cause of these phenomena, we used a new probability based approach of analyzing. In this approach, the mathematical expectation of influence on one oscillator's phase change caused by its neighbor, which is regarded as a random factor, is calculated. By adding this expectation of influence into the firing map $h(\phi)$ introduced by Mirolla and Strogatz, a probability firing map $H$ is invented. By observing the behavior of $H$ 's iteration from $H$ 's graph, we successfully constructed a connection between the asynchrony phenomena and $H$ 's graph. 
Shuishi Yang