Some basic remarks on eigenmode expansions of time-delay dynamics

We describe in elementary terms how eigenmode expansions can be used to deal with differential-difference equations. As particular applications we present the full analytical solution of linear stochastic time-delay systems and the weakly nonlinear analysis of nonlinear differential-difference equations in the limit of large time delay. Our exposition is essentially based on an explicit analytical expression for the linear spectrum in terms of the Lambert W-function and on the explicit formula for the eigenfunctions of the adjoint equation.