Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8899073 | Journal of Differential Equations | 2018 | 31 Pages |
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 εâ0 - and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times expâ¡(cεâ2), c>0, and we show both that on the time scale εâ2 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 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Giambattista Giacomin, Christophe Poquet, Assaf Shapira,