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Small noise and long time phase diffusion in stochastic limit cycle oscillators
- Source :
- Journal of Differential Equations, Journal of Differential Equations, 2017, Journal of Differential Equations, Elsevier, 2017
- Publication Year :
- 2015
- Publisher :
- arXiv, 2015.
-
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.<br />24 pages, 1 figure. Small changes, added four references
- Subjects :
- Dephasing
[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]
Phase (waves)
FOS: Physical sciences
Dynamical Systems (math.DS)
Quantitative Biology - Quantitative Methods
01 natural sciences
Noise (electronics)
Small Noise Limit
010305 fluids & plasmas
Long Time Dynamics
Stochastic differential equation
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]
Limit cycle
0103 physical sciences
FOS: Mathematics
Statistical physics
Limit (mathematics)
Mathematics - Dynamical Systems
010306 general physics
Stable Hyperbolic Limit Cycles
Mathematical Physics
Brownian motion
Quantitative Methods (q-bio.QM)
Mathematics
Applied Mathematics
Probability (math.PR)
Mathematical Physics (math-ph)
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Stochastic Differential Equations
Isochrons
60H10, 34F05, 60F17, 82C31, 92B25
FOS: Biological sciences
Brownian noise
Mathematics - Probability
Analysis
Subjects
Details
- ISSN :
- 00220396 and 10902732
- Database :
- OpenAIRE
- Journal :
- Journal of Differential Equations, Journal of Differential Equations, 2017, Journal of Differential Equations, Elsevier, 2017
- Accession number :
- edsair.doi.dedup.....13967881cd2ab6ac23bb4498b16bfafe
- Full Text :
- https://doi.org/10.48550/arxiv.1512.04436