Coherence Stability and Effect of Random Natural Frequencies in Populations of Coupled Oscillators

We consider the (noisy) Kuramoto model, that is a population of oscillators, or rotators, with mean-field interaction. Each oscillator has its own randomly chosen natural frequency (quenched disorder) and it is stirred by Brownian motion. In the limit this model is accurately described by a (deterministic) Fokker-Planck equation. We study this equation and obtain quantitatively sharp results in the limit of weak disorder. We show that, in general, even when the natural frequencies have zero mean the oscillators synchronize (for sufficiently strong interaction) around a common rotating phase, whose frequency is sharply estimated. We also establish the stability properties of these solutions (in fact, limit cycles). These results are obtained by identifying the stable hyperbolic manifold of stationary solutions of an associated non disordered model and by exploiting the robustness of hyperbolic structures under suitable perturbations. When the disorder distribution is symmetric the speed vanishes and there is a one parameter family of stationary solutions, as pointed out by Sakaguchi (Prog Theor Phys 79:39-46, 1988): in this case we provide more precise stability estimates. The methods we use apply beyond the Kuramoto model and we develop here the case of active rotator models, that is the case in which the dynamics of each rotator in absence of interaction and noise is not simply a rotation.