  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.
