2015/12/14 by Giambattista Giacomin, Giacomin, Giambattista, Christophe Poquet +3 · 1 citation
Computer Science · Environmental Science · Physics and Astronomy · #Nonlinear Dynamics and Pattern Formation #Ecosystem dynamics and resilience #Advanced Thermodynamics and Statistical Mechanics
paper · pdf · doi:10.48550/arxiv.1512.04436
We study the effect of additive Brownian noise on an ODE system that has a\nstable hyperbolic limit cycle, for initial data that are attracted to the limit\ncycle. The analysis is performed in the limit of small noise - that is, we\nmodulate the noise by a factor \ε searrow 0 - and on a long time\nhorizon. We prove explicit estimates on the proximity of the noisy trajectory\nand the limit cycle up to times \exp\(c \ε-2\), c>0,\nand we show both that on the time scale \ε-2 the "'dephasing"\n(i.e., the difference between noiseless and noisy system measured in a natural\ncoordinate system that involves a phase) is close to a Brownian motion with\nconstant drift, and that on longer time scales the dephasing dynamics is\ndominated, to leading order, by the drift. The natural choice of coordinates,\nthat reduces the dynamics in a neighborhood of the cycle to a rotation, plays a\ncentral role and makes the connection with the applied science literature in\nwhich noisy limit cycle dynamics are often reduced to a diffusion model for the\nphase of the limit cycle.\n