## 1 Introduction
This would open up the very large bag of tricks developed for ODE systems. One very interesting recent line of research in this direction is the [PERSON] approximation method of [PERSON] and coworkers [34], but this is not the path we will be following today. Rather, we will return to an old idea [35]: Assuming that the delay is not too large, we can take a Taylor expansion of a DDE in powers of the delay. This immediately reduces the DDE to an ODE. This procedure is not without its risks. Indeed, if we directly integrate such an equation for any order of expansion beyond first order in the delay, we will eventually get spurious \"runaway\" solutions [36; 37; 38]. Since truncation of the Taylor series after the first-order term was known not to generate spurious solutions, this lowest-order approximation of the DDEs has been applied to compute inertial manifolds [32; 39]. However, [PERSON] famous result [36] only applies to directly integrating the approximating ODEs. [PERSON] and coworkers have shown that expanding the delayed term in a series in powers of the delay results in correct approximations of the state-space vector field implied by a DDE [33; 38; 40]. Accordingly, methods for the construction of invariant manifolds that do not rely on integration should generally work with higher-order expansions. Many gene expression models are essentially chemical models with delayed terms. Many excellent reviews of methods for computing invariant manifolds tailored to chemical models have been published [41; 42; 43; 44; 45; 46; 47]. The method used here and in a previous study [39] is based on [PERSON]'s functional iteration method [1; 5]. In favorable cases, this method tends to converge very quickly. Slow convergence can often be accelerated [48], and divergence can also be addressed using straightforward stabilization methods [49; 50]. The approach to constructing inertial manifolds taken here can thus be thought of as combining a perturbative treatment of the delay with an iterative method for computing invariant manifolds. As the foregoing text suggests, computing inertial manifolds of DDEs is a technically challenging undertaking. The widespread availability of DDE integrators and of sophisticated analysis tools such as DDE-BIFTOOL [51] might be thought to make an inertial manifold analysis redundant. However, there is a great deal of insight to be gained from knowing the shape of an inertial manifold, particularly when the manifold is of sufficiently low dimension to be easily visualized. In some cases, readily available experimental methods may not be able to access fast transients. Motion on the inertial manifold then becomes the experimentally observable behavior, and the geometry of the manifold may play a role in determining the stability of a steady state, for example [49]. On occasion, understanding the geometry of the slow manifold in a particular model may lead to the discovery of a class of systems expected to display similar behavior because of similar slow manifold geometries [52]. In other cases,understanding the geometry of slow manifolds may support the design of experiments to distinguish between mechanisms [53]. Additional arguments for the utility of invariant manifold calculations have been presented by [PERSON] [7]. In order to set the stage for the application to DDEs, Section 2 reviews [PERSON]'s iterative method for approximating slow manifolds. The model that will be studied in that section is the zero-delay limit of the DDE model of an inducible enzyme used as our central example. In Section 3, two variations of the DDE model are studied, a three-variable model used to establish some basic techniques (Section 3.1), and a more realistic four-variable model (Sections 3.2 and 3.3). In the latter case, both three- and two-dimensional inertial manifolds are constructed. Whenever possible, insights into model behavior gained from the calculations will be pointed out.