## 3 Computation of Inertial Manifolds for DDEs
### Computing a Two-Dimensional Inertial Manifold
The reduced model obtained from Equation (21) for a computed manifold \((\mathcal{C}(e,s),\mathcal{R}(e,s))\) is the planar system  
\[\epsilon_{1}s =\kappa-es+\mathcal{C}(e,s)\left(s+\alpha\right), \tag{39a}\] \[\dot{e} =\mathcal{R}(e,s)-e. \tag{39b}\]  
Given explicit functional equations, we could try to carry out the iterations analytically in a symbolic algebra system, but again the expressions quickly grow to the point where they cannot practically be handled, so numerical iteration was used. The second iterate was obtained analytically for each \(m\) and used to verify reasonable accuracy of the numerical scheme. These differences (not shown) are generally very small (\(O(10^{-6})\) or less), except at the \(s=0\) edge. Note that the boundary of the computational domain is often a source of instability in this class of iterative methods [50; 86]. Now consider the first iterate of Equation (38) starting from the naive initial function:  
\[\mathcal{R}^{(1)}_{1}=\rho(s)-\frac{\delta}{\epsilon_{1}}\frac{d\rho}{ds} \left[\kappa-es+\mathcal{C}^{(1)}_{1}(s+\alpha)\right]. \tag{40}\]  
For \(n>1\), \(d\rho/ds\to 0\) as \(s\to 0\). Thus, \(\mathcal{R}^{(1)}_{1}\) itself tends to zero as \(s\to 0\). Estimating the derivative in this region therefore involves taking differences of small quantities, which is difficult to do accurately,resulting in larger errors in the numerical iteration process in this region. Fortunately, these errors do not destabilize the iterative scheme, at least if we stop iteration as soon as reasonable convergence is achieved. Convergence is (again) very fast, so three or four iterates are sufficient to reduce the iteration error below the threshold where it has any observable effect on the dynamics. Because the approximated manifolds are now paired surfaces \(\mathcal{C}(e,s)\) and \(\mathcal{R}(e,s)\), they are readily visualized. Figure 9 shows the computed inertial manifolds at various levels of truncation, \(m\). Despite its coupling to \(\mathcal{R}\) (Equation (37)), \(\mathcal{C}\) depends very little on \(m\), so a representative approximation is shown. On the other hand, the shapes of the \(\mathcal{R}^{(m)}\) surfaces are dramatically different. This is a possible sign of a divergent series in \(\delta\). (Note that \(\delta\) is not particularly small here.) However, which representation of the inertial manifold is the best? A simple way to answer the latter question is to look at trajectories. For the parameters of Figure 9, the DDE model has a limit cycle (Figure 10). The system reduced to the \(m=1\) approximate inertial manifold also has a limit cycle (not shown), but this limit cycle wanders out of the positive quadrant of the \((e,s)\) plane. We might accept unphysical excursions of trajectories started from some initial conditions, but an unphysical attractor is not acceptable in any potential application. The \(m=2\) attractor is, on the other hand, an entirely sensible-looking limit-cycle entirely contained in the positive quadrant, although it is too large and of a different shape than the limit cycle of the DDE model (Figure 10). The significantly larger size of the limit cycle in the \(m=2\) reduced model is associated with an Andronov-Hopf bifurcation that occurs at a smaller value of \(\delta\) than in the DDE model. For \(m=3\), all we get is a strongly damped oscillation (not shown). It thus seems that the \(m=2\)approximation is the best representation of the inertial manifold available in the context of this method for these parameters. The observation that the Andronov-Hopf bifurcation is advanced in the \(m=2\) reduced model relative to the original DDE model raises an interesting question, particularly since the limit cycle in the reduced model has a different shape than that in the DDE model: are the dynamics of the reduced model simply shifted along the \(\delta\) axis in the parameter space? Along with the two limit cycles already discussed, Figure 10 shows the attractor of the DDE model at a larger value of \(\delta\).