2015/12/14 by Giambattista Giacomin, Giacomin, Giambattista, Christophe Poquet +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Environmental Science · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Ecosystem dynamics and resilience #Nonlinear Dynamics and Pattern Formation #math-ph #math.DS #math.MP #math.PR #msc:34F05 #msc:60F17 #msc:60H10 #msc:82C31 #msc:92B25 #q-bio.QM
paper · pdf · doi:10.48550/arxiv.1512.04436
24 pages, 1 figure. Small changes, added four references
arxiv created 2016/01/18 · arxiv updated 2016/01/19
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 ε \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(c ε-2), c>0, and we show both that on the time scale ε-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.