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

TL;DR

This paper analyzes how small additive noise affects the phase dynamics of systems with stable limit cycles over long times, showing that noise induces a Brownian motion-like phase diffusion with drift.

Contribution

It provides explicit estimates on the proximity of noisy trajectories to the limit cycle and characterizes the long-term phase diffusion as a Brownian motion with drift.

Findings

Phase diffusion approximates a Brownian motion with drift over long times.

Proximity estimates hold up to exponentially long times in the inverse noise squared.

The natural phase coordinate simplifies the dynamics to a rotation, linking to applied science models.

Abstract

We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise - that is, we modulate the noise by a factor $\varepsilon \searrow 0$ - and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and 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 natural coordinate system that involves a phase) is close to a Brownian motion with constant drift, and that on longer time scales the dephasing dynamics is dominated, 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 a central role and makes the connection with the applied science literature in which noisy limit cycle dynamics are often reduced to a diffusion model for the phase of the limit cycle.