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 tex2html_wrap_inline523 = tex2html_wrap_inline523 where tex2html_wrap_inline523 is the small timescale ratio and tex2html_wrap_inline523 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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