Gd. Lythe et Mre. Proctor, NOISE AND SLOW-FAST DYNAMICS IN A 3-WAVE RESONANCE PROBLEM, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 47(5), 1993, pp. 3122-3127
Recent research on the dynamics of certain fluid-dynamical instabiliti
es shows that when there is a slow invariant manifold subject to fast
time-scale instability the dynamics are extremely sensitive to noise.
The behavior of such systems can be described in terms of a one-dimens
ional map, and previous work has shown how the effect of noise can be
modeled by a simple adjustment to the map. Here we undertake an in-dep
th investigation of a particular set of equations, using the methods o
f stochastic integration. We confirm the prediction of the earlier stu
dies that the noise becomes important when mu/lnepsilon/ = O(1), where
mu is the small time-scale ratio and epsilon is the noise level. In a
ddition, we present detailed information about the statistics of the s
olution when the noise is a dominant effect; the analytical results sh
ow excellent agreement with numerical simulations.