## 3 Computation of Inertial Manifolds for DDEs
### Elimination of the Fastest Relaxation Mode

Table 1: Sizes of expressions generated during the iterative solution of Equation (30) for different iteration levels (\(k\)) and orders of expansion (\(m\)) with respect to \(\delta\). Values are given in characters, with SI prefixes used to represent larger values. this function at, say, different values of \(c\), but these are not very enlightening. Instead, we compute trajectories of the reduced system using \(r=\mathcal{R}_{k}^{(m)}(e,s,c)\) for different values of \(k\), and then calculate the scaled difference between iterates, the iteration error (IE):  
\[\mathrm{IE}_{k}^{(m)}(\theta)=\frac{2\left(\mathcal{R}_{k}^{(m)}\left(e(\theta ),s(\theta),c(\theta)\right)_{k}^{(m)}-\mathcal{R}_{k-1}^{(m)}\left(e(\theta ),s(\theta),c(\theta)\right)_{k-1}^{(m)}\right)}{\left|\mathcal{R}_{k}^{(m)} \left(e(\theta),s(\theta),c(\theta)\right)_{k}^{(m)}\right|+\left|\mathcal{R}_ {k-1}^{(m)}\left(e(\theta),s(\theta),c(\theta)\right)_{k-1}^{(m)}\right|}. \tag{35}\]  
In this equation, \(\left(e(\theta),s(\theta),c(\theta)\right)_{k}^{(m)}\) refers to a trajectory of the reduced model. This is a very stringent test because differences in frequency can cause large differences in the trajectories even if the overall dynamics are very similar. Figure 7 shows these iteration errors for a parameter set in the regime where we expect to be able to carry out this reduction. Near the initial condition, it is difficult to measure \(\mathrm{IE}_{k}^{(m)}\) reliably because of the small denominator in Equation (35), so values of this statistic near \(\theta=0\) should be ignored. For \(k=2\), we see large-amplitude oscillations, indicating that the two manifolds are sufficiently different to cause a loss of coherence. When we calculate the next iterate, the iteration error becomes much smaller and loses any obvious periodicity, which is evidence of rapid convergence of the iterations. The same behavior is observed for higher values of \(m\). While iteration converges rapidly, it turns out that the reduced models are only qualitatively correct. Figure 8 shows trajectories of the DDE model as well as of the state-space ODE models. For the parameters chosen, the DDE model has a stable focus. On the other hand, the reduced models for \(m=1\) and 2 have limit-cycle attractors, i.e., they undergo an Andronov-Hopf bifurcation at a smaller value of \(\delta\) (keeping the other parameters constant) than the DDE model, near \(\delta=0.84\) for the \(m=1\) reduced model vs. \(\delta=1.04\) for the DDE model. As for the \(m=3\) reduced model, it is much more strongly damped than the DDE model, and it fails completely if we try to increase \(\delta\) too much. Note that the value of \(\varepsilon_{2}\) chosen for the calculations of Figures 7 and 8 is not particularly small compared to \(\varepsilon_{1}\). The \(m=1\) and \(m=2\) state-space models are qualitatively correct, even if there are some quantitative discrepancies such as the exact parameter values at the Andronov-Hopf bifurcation. The agreement improves if we decrease \(\varepsilon_{2}\). Figure 7: Scaled iteration differences calculated using Equation (35) for \(m=1\) for the parameters \(\varepsilon_{1}=0.1\), \(\varepsilon_{2}=0.02\), \(\alpha=1\), \(\beta=1\), \(\gamma=1.5\), \(\kappa=0.1\), \(n=4\) and \(\delta=0.9\). Trajectories used in these calculations were started from \((e,s,c)=(0,0,0)\).