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 is often used to model synchronization between coupled oscillators in a network. A particular form of synchronization is phase-locking whereby the oscillators rotate at a common frequency with fixed angle differences. It has been observed that the oscillators will phase-lock if their natural frequencies have small differences, and conversely, that they won't if the natural frequencies have large differences. In [Bronski and Ferguson, SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 128--156] Bronski and the author gave upper and lower bounds for the volume of the set of natural frequencies for which the Kuramoto model exhibits phase-locking. This was done under the assumption that any two oscillators influence each other with equal strength. In this paper the author generalizes these upper and lower bounds by removing this assumption. Similar to [Bronski and Ferguson, SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 128--156], 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 and one of our upper bounds are sums over divergent directed spanning trees. We also relate our results to observations by Poignard, Pereira, Pade in [SIAM J. Appl. Math., 78 (2018), pp. 372--394] and Pade and Pereira in [APS March Meeting Abstracts, 2015, M45.011] on how changes in the network can affect synchronization. [ABSTRACT FROM AUTHOR]