Phase noise spectrum of oscillators described by Itô stochastic differential equations

We present a description in terms of phase and amplitude fluctuations for nonlinear oscillators subject to white Gaussian noise, described by Itô stochastic differential equations. The equations derived for the amplitude and the phase are rigorous, and their validity is not limited to the weak noise limit. We show that using Floquet´s 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. The resulting phase reduced model is analyzed, and the asymptotic values of probability density function, auto-correlation matrix and power spectral density are found.