  This article revisits the approximation problem of systems of nonlinear delay
differential equations (DDEs) by a set of ordinary differential equations
(ODEs). We work in Hilbert spaces endowed with a natural inner product
including a point mass, and introduce polynomials orthogonal with respect to
such an inner product that live in the domain of the linear operator associated
with the underlying DDE. These polynomials are then used to design a general
Galerkin scheme for which we derive rigorous convergence results and show that
it can be numerically implemented via simple analytic formulas. The scheme so
obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i)
a simple DDE with distributed delays whose solutions recall Brownian motion;
(ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics;
and (iii) a periodically forced DDE with two discrete delays arising in climate
dynamics. In all three cases, the Galerkin scheme introduced in this article
provides a good approximation by low-dimensional ODE systems of the DDE's
strange attractor, as well as of the statistical features that characterize its
nonlinear dynamics.
