Multiscale Analysis of Stochastic Delay Differential Equations

We apply multiscale analysis to stochastic delay differential equations, deriving approximate stochastic equations for the amplitudes of oscillatory solutions near critical delays of deterministic systems. Such models are particularly sensitive to noise when the system is near a critical point, which marks a transition to sustained oscillatory behavior in the deterministic system. In particular, we are interested in the case when the combined effects of the noise and the proximity to criticality amplify oscillations which would otherwise decay in the deterministic system. The derivation of reduced equations for the envelope of the oscillations provides an efficient analysis of the dynamics by separating the influence of the noise from the intrinsic oscillations over long time scales. We focus on two well-known problems: the linear stochastic delay differential equation and the logistic equation with delay. In addition to the envelope equations, the analysis identifies scaling relationships between small noise and the proximity of the bifurcation due to the delay which enhances the resonance of the noise with the intrinsic oscillations of the systems.