Noise and resonant mode interactions

Ann. New York Acad. Sci 706 42-53 (1993)

M.R.E. Proctor and G.D. Lythe

Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 9EW, United Kingdom.


A set of ordinary differential equations describing the resonant interaction of two or three oscillators is shown to be highly sensitive to noise when there are two timescales present in the dynamics. A quantitative theory using stochastic calculus gives detailed information on the statistics of the noise-controlled dynamics. In this paper we have given a necessarily brief description of the dynamics of noise-sensitivity in the three-wave resonance model. The methods can readily be generalised to more complicated problems, and it is indeed a matter of some interest to extend the analysis to more complicated slow manifolds (with higher dimension, more elaborate stability properties, etc). We emphasise that the noise-sensitivity of these systems is greatest when the timescale ratio is small, and that this happens naturally near a point of bifurcation for the primary wave. Indeed, sufficiently close to the bifurcation point the dynamics must be controlled by the noise, no matter how small the latter may be, and thus the observed behaviour might seem to an experimenter to be a Hopf bifurcation, but one giving `oscillations' (actually noisily periodic orbits) whose amplitude depends entirely on ambient noise levels. The above resonant interaction seems fairly specialised since the coupling term in the $y$ equation is linear in $y$. It is this that permits the manifold to become unstable. But in fact resonances with the same property can be found for any pair of integers $m,n$ if there exists another integer $l$ such $ln=2m$, although if $l>1$ the resonant terms are of at least the same order as the non resonant cubic interaction terms, so their inclusion is a more delicate matter. We hope to explore these generalisations more fully in future work.
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