# 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.

## Abstract

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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