Structure of long-term average frequencies for Kuramoto oscillator systems

We study the long-term average frequency as a function of the natural frequency for Kuramoto oscillators with periodic coefficients. Unlike the case for more general periodically forced oscillators, this function is never a "devil's staircase"; it may have plateaus at integer multiples of the forcing frequency, but we prove it is strictly increasing between these plateaus. The proof uses the fact that the flow maps for Kuramoto oscillators extend to Möbius transformations on the complex plane, and that Möbius transformations have particularly simple dynamics that rule out p:q mode locking except in the case of fixed points (q=1). We also give a criterion for the degeneration of an integer plateau to a single point and use it to explain the absence of plateaus at even multiples of the collective frequency for a Kuramoto system with a bimodal frequency distribution.