Volume Bounds for the Phase-locking Region in the Kuramoto Model with Asymmetric Coupling

The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among them. It has been observed that if the natural frequencies of the oscillators are similar they will phase-lock, meaning that they oscillate at a common frequency with fixed phase differences. Conversely, we do not observe this behavior when the natural frequencies are very dissimilar. Previously, Bronski and the author gave upper and lower bounds for the volume of the set of frequencies exhibiting phase-locking behavior. This was done under the assumption that any two oscillators affect each other with equal strength. In this paper the author generalizes these upper and lower bounds by removing this assumption. Similar to the previous work with Bronski, where the upper and lower bounds are sums over spanning trees of the network, our generalized upper and lower bounds are sums over certain directed subgraphs of the network. In particular, our lower bound is a sum over directed spanning trees. Finally, we numerically simulate the dependence of the number of directed spanning trees on the presence of certain two edge motifs in the network and compare this dependence with that of the synchronization of oscillator models found by Nykamp, Zhao, and collaborators.