Amplitude and phase equations for nonlinear oscillators with noisy interactions

We give a description in terms of phase and amplitude deviation for networks of nonlinear oscillators with noise. The case of white Gaussian noise is considered. The equations for the amplitude and the phase are rigorous, and their validity is not limited to the weak noise limit. We show that using Floquet theory, a partial decoupling between the amplitude and the phase is obtained. The decoupling can be exploited to describe the oscillator's dynamics solely by the phase variable. We discuss to what extent the reduced model is appropriate and some implications on the role of noise on the frequency and the synchronization of the oscillators.