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We study the effect of additive Brownian noise on an ODE system that has astable hyperbolic limit cycle, for initial data that are attracted to the limitcycle. The analysis is performed in the limit of small noise - that is, wemodulate the noise by a factor $\varepsilon \searrow 0$ - and on a long timehorizon. We prove explicit estimates on the proximity of the noisy trajectoryand the limit cycle up to times $\exp\left(c \varepsilon^{-2}\right)$, $c>0$,and we show both that on the time scale $\varepsilon^{-2}$ the "'dephasing"(i.e., the difference between noiseless and noisy system measured in a naturalcoordinate system that involves a phase) is close to a Brownian motion withconstant drift, and that on longer time scales the dephasing dynamics isdominated, to leading order, by the drift. The natural choice of coordinates,that reduces the dynamics in a neighborhood of the cycle to a rotation, plays acentral role and makes the connection with the applied science literature inwhich noisy limit cycle dynamics are often reduced to a diffusion model for thephase of the limit cycle.

Small noise and long time phase diffusion in stochastic limit cycle oscillators