## 4 Discussion  
Throughout the examples considered in this paper, we consistently obtained rapid convergence of the iterative schemes at fixed \(m\). On the other hand, the expansion of the delayed nonlinearity in powers of \(\delta\) was more delicate. Sometimes, the best results were obtained for \(m=1\), and sometimes for \(m=2\). It seems likely, given the results of [PERSON] [33; 38; 40] that there is no fundamental limitation on the order of expansion in the delay, and that the \"best\" order of expansion therefore depends on the model and parameters. In [PERSON]'s work, it was assumed that the delay was sufficiently small for the Taylor series to converge. Here, relatively large values of the delays have been considered,  
Figure 10: Trajectories of the DDE model (21) and of the system reduced to the inertial manifold for \(m=2\). The parameters for the first two curves are those of Figure 9. In particular, \(\delta=3\). For the third curve, the DDE model was integrated with \(\delta\) increased to 7.  
and convergence is not assured, but useful results can still be obtained provided we pick the \"correct\" order of expansion. The situation appears to be (at least) analogous to asymptotic series, where we get sensible approximations provided we cut off the expansion as soon as it starts to diverge [93]. Now that the method has been shown to give reasonable results in a number of challenging cases, the stage is set for a more deliberate analysis of its mathematical properties. Despite the relatively large values of the delay used in this study, it was possible to calculate manifolds on which the dynamics are at least qualitatively correct. Note that the calculations were carried out to either side of the Andronov-Hopf bifurcation, even though results were not reported for every case tried. Thus, the method proposed here is able to represent inertial manifolds of DDEs over a range of parameters. This raises the question of whether there is some other development of the delay term that could be used. The [PERSON] approximation method of [PERSON] and coworkers [34] has already been mentioned. As a general point, it would be useful to have a variety of approximate transformations of delay equations valid in different parameter regimes. Scaling played an important role throughout the work reported here. For every model variation analyzed, values of the singular perturbation parameters \(\varepsilon_{1}\), \(\varepsilon_{2}\) and \(\beta\) would inform our choice of representations of the manifold. When the correct representation was chosen, rapid convergence of the iterative process was always obtained. In some calculations not reported here, choosing the \"wrong\" representation for the parameter range considered always led to difficulties with the iterative method, essentially because the manifolds were nearly vertical in parts of the state space. When analyzing smaller models for which scaling is practical, especially if we have some idea of the relevant parameter values, it is therefore well worth investing some time into a scaling analysis. This is, of course, not an original recommendation [63; 65]. Implicitly, in selecting the representations of the manifolds in the delay model, we assumed that a version of [PERSON]'s geometric singular perturbation theory [12] and, in particular, of his persistence and smoothness results [11], extended to inertial manifolds. Others have made the same assumption [94]. [PERSON] has shown that similar persistence and smoothness results hold using the delay as the small parameter in the singular perturbation expansion [33]. In [PERSON]'s approach, the singular limit is an ODE. The singular limit of systems such as (21) can be a set of DDEs with an algebraic constraint, or a set of ODEs with an algebraic functional constraint (e.g., \(\rho(s(\theta-\delta))=r(\theta)\) if \(\varepsilon_{2}\to 0\)). To the author's knowledge, persistence and smoothness of inertial manifolds for such systems has not been systematically studied. This is clearly an important area for future work.