Existence and Exact Multiplicity of Phaselocked Solutions of a Kuramoto Model of Mutually Coupled Oscillators

We investigate existence and exact multiplicity of phase-locked solutions of an integro-differential equation derived from a Kuramoto system of coupled oscillators. Under general assumptions on the form of frequency distribution, we derive new, easily verified criteria which guarantee that either (i) exactly one solution exists, or (ii) exactly two solutions coexist over an entire interval of values of the key parameter γ. We illustrate our results with an example in which each of these possibilities occurs. Problems for future research are suggested.