  We consider linear delay differential equations at the verge of Hopf
instability, i.e. a pair of roots of the characteristic equation are on the
imaginary axis of the complex plane and all other roots have negative real
parts. When nonlinear and noise perturbations are present, we show that the
error in approximating the dynamics of the delay system by certain two
dimensional stochastic differential equation without delay is small (in an
appropriately defined sense). Two cases are considered: (i) linear
perturbations and multiplicative noise (ii) cubic perturbations and additive
noise. The two-dimensional system without-delay is related to the projection of
the delay equation onto the space spanned by the eigenfunctions corresponding
to the imaginary roots of the characteristic equation. A part of this article
is an attempt to relax the Lipschitz restriction imposed on the coefficients in
doi:10.1142/S0219493716500131 Also, the multiplicative noise case was not
considered there. Examples without rigorous proofs are worked in
arXiv:1403.3029

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