On the computation of the third order terms of the series defining the center manifold for systems of delay differential equation

When studying a general system of delay differential equation with a single constant delay, we encounter a certain lack of uniqueness in determining the coefficient of one of the third order terms of the series defining the center manifold. We solve this problem by considering a perturbation of the considered problem, perturbation that allows us to remove the singularity. The result generalizes a similar result obtained for scalar differential equations (J. Dyn. Diff. Eqns., 24/2012).