## 3 Computation of Inertial Manifolds for DDEs
### The \"Fast RNA\" Case
The expansion contains powers of \(c\), so in general it is not convenient (if it is even possible) to write down explicitly the functional equation. However, for \(m=1\), we have, after labeling for iteration,  
\[\mathcal{C}_{k+1}^{(1)}=\frac{es-\beta\varepsilon_{1}(\rho(s)-e)\frac{ \partial\mathcal{C}_{k}^{(1)}}{\partial e}+\beta(\kappa-es)\left(\delta\frac {\partial\mathcal{C}_{k}^{(1)}}{\partial e}\frac{d\rho}{ds}-\frac{\partial \mathcal{C}_{k}^{(1)}}{\partial s}\right)}{s+\gamma+\beta\varepsilon_{1}- \beta(s+\alpha)\left(\delta\frac{\partial\mathcal{C}_{k}^{(1)}}{\partial e} \frac{d\rho}{ds}-\frac{\partial\mathcal{C}_{k}^{(1)}}{\partial s}\right)}. \tag{27}\]  
The \(m=0\) case is recovered by setting \(\delta=0\) in this equation. If we start iteration from the initial function \(\mathcal{C}_{0}^{(1)}=0\), we get the following as the first iterate:  
\[\mathcal{C}_{1}^{(1)}=\frac{es}{s+\gamma+\beta\varepsilon_{1}}. \tag{28}\]  
A zero initial function is often a good choice for initializing this style of iteration. In the case of ordinary chemical models, this naive initial function generally returns the steady-state approximation as the first iterate [1; 5; 80], which is a useful property of this class of iterative schemes. Note that Equation (28) is readily invertible with respect to \(e\) except when \(s=0\). There is therefore some hope that more accurate expressions for the inertial manifold generated by iteration will also be locally invertible, and therefore that the model reduction scheme described above will work. Equation (27) can be iterated analytically using a symbolic algebra language. The \(m=2\) functional equation gives a quadratic equation in \(c\). It could also be iterated analytically, but numerical iteration was used for \(m=2\) and \(m=3\) instead. Iteration of Equation (27) and of its implicit counterparts for \(m>1\) converges very quickly. The differences between the manifolds obtained for different values of \(m\) are essentially negligible (Figure 4). We can draw a few lessons from these calculations:  
1. Because the manifolds computed for different values of \(m\) are essentially identical, it appears that the higher-order corrections to the vector field are unnecessary to capture the dynamics, at least in the parameter regime of Figure 4.
2. The calculation for \(m=1\) (even if carried out numerically) is much better behaved than for larger values of \(m\). In the numerical calculations with larger \(m\), localized imperfections appear in the manifolds. A simple routine was developed for removing these imperfections, essentially by interpolating through the region where they appear, but we can see from Figure 4b that there are small residual artifacts in the surfaces. (\(\mathcal{C}_{3}^{(1)}\), on the other hand, is perfectly smooth.) 3. If we integrate the model reduced to the inertial manifold, the \(m=1\) reduced model best reproduces the trajectories of the three-variable DDE model (Figure 5). After an unphysical excursion to negative concentrations, the \(m=1\) model reproduces the trajectory of the three-variable model reasonably well. On the other hand, the \(m=2\) model has a limit cycle, which does not appear in the three-variable model until \(\delta\) is above 0.54, and the \(m=3\) model is excessively damped. The negative concentrations in the reduced model may be due to a poor representation of the inertial manifold near the \(c=0\) axis, and would be an interesting subject for future investigations. The more serious misbehavior of the \(m=2\) and \(m=3\) models is likely due to a compounding of numerical issues, mostly the difficulties in solving the functional equations accurately in certain parts of the surface, numerical error during the inversion of the relation \(c=\mathcal{C}_{k}^{(m)}(e,s)\), and of course the interaction of these two factors. Finally, note that the \(\delta\) dependence of the inertial manifold is dynamically significant.