Noise and slow-fast dynamics in a
three-wave resonance problem
Physical Review E 47, 3122-3127 (1993)
G.D. Lythe and M.R.E. Proctor
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 9EW,
United Kingdom.
Abstract
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
=
where
is the
small timescale ratio and
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.
PACS: 02.50-r, 64.60Ht, 05.70Fh, 47.54.+r
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