On the Alternative Characterization of Asymptotic Phase‐locking in the Delayed Kuramoto Model

We present an alternative geometric characterization for asymptotic phase‐locking in terms of the cardinality of the set of collision times for the Kuramoto ensemble under the effect of time‐delays. More precisely, for a sufficiently small time‐delay, we show that the emergence of asymptotic phase‐locking is equivalent to the finiteness of collisions. For a two‐oscillator system, we derive the explicit relation in terms of coupling strength and time‐delay for the equivalence relation between asymptotic phase‐locking and finiteness of collisions. This generalizes earlier corresponding results in the absence of time‐delays. We also provide several simulation results and compare them with analytical results.