Noise and slow-fast dynamics in a three-wave resonance problem

Recent research on the dynamics of certain fluid-dynamical instabilities 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-dimensional 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-depth investigation of a particular set of equations, using the methods of stochastic integration. We confirm the prediction of the earlier studies that the noise becomes important when {μ}{}ln{ε}{}=O(1), where {μ} is the small time-scale ratio and {ε} is the noise level. In addition, we present detailed information about the statistics of the solution when the noise is a dominant effect; the analytical results show excellent agreement with numerical simulations.