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We consider linear delay differential equations at the verge of Hopfinstability, i.e. a pair of roots of the characteristic equation are on theimaginary axis of the complex plane and all other roots have negative realparts. When nonlinear and noise perturbations are present, we show that theerror in approximating the dynamics of the delay system by certain twodimensional stochastic differential equation without delay is small (in anappropriately defined sense). Two cases are considered: (i) linearperturbations and multiplicative noise (ii) cubic perturbations and additivenoise. The two-dimensional system without-delay is related to the projection ofthe delay equation onto the space spanned by the eigenfunctions correspondingto the imaginary roots of the characteristic equation. A part of this articleis an attempt to relax the Lipschitz restriction imposed on the coefficients indoi:10.1142/S0219493716500131 Also, the multiplicative noise case was notconsidered there. Examples without rigorous proofs are worked inarXiv:1403.302

Nonlinear and additive white noise perturbations of linear delay differential equations at the verge of instability: An averaging approach