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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:264

Article

Giacomin, Giambattista¹  Poquet, Christophe²  Shapira, Assaf¹ 

[1] Univ Paris Diderot, Sorbonne Paris Cite, Lab Probabilites & Modeles Aleatoires, UMR 7599, F-75205 Paris, France

[2] Univ Lyon 1, Inst Camille Jordan, UMR 5208, F-69622 Villeurbanne, France

关键词: Stochastic differential equations;    Stable hyperbolic limit cycles;    Isochrons;    Small noise limit;    Long time dynamics;   

DOI  :  10.1016/j.jde.2017.09.029

来源: Elsevier

【 摘 要 】

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 epsilon SE arrow 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 (c epsilon(-2)), c > 0, and we show both that on the time scale epsilon(-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 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. (C) 2017 Elsevier Inc. All rights reserved.

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