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 timescale instability the dynamics are extremely sensitive to noise. The behaviour 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 modelled 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 mu|log(epsilon)| = O(1), where mu is the small timescale ratio and \epsilon 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.