Asymptotic analysis of two reduction methods for systems of chemical reactions

This article concerns two methods for reducing large systems of chemical kinetics equations, namely, the method of intrinsic low-dimensional manifolds (ILDMs) due to Maas and Pope and an iterative method due to Fraser and further developed by Roussel and Fraser. Both methods exploit the separation of fast and slow reaction time scales to find low-dimensional manifolds in the space of species concentrations where the long-term dynamics are played out. The analysis is carried out in the context of systems of ordinary differential equations with multiple time scales and geometric singular perturbation theory. A small parameter $\eps$ measures the separation of time scales. The underlying assumptions are that the system of equations has an asymptotically stable slow manifold $\mathcal{M}_0$ in the limit as $\eps \downarrow 0$ and that there exists a slow manifold $\mathcal{M}_\eps$ for all sufficiently small positive $\eps$, which is asymptotically close to $\mathcal{M}_0$. It is shown that the ILDM method yields a low-dimensional manifold whose asymptotic expansion agrees with the asymptotic expansion of $\mathcal{M}_\eps$ up to and including terms of $\mathcal{O} (\eps)$. The error at $\mathcal{O} (\eps^2)$ is proportional to the local curvature of $\mathcal{M}_0$. The iterative method generates, term by term, the asymptotic expansion of the slow manifold $\mathcal{M}_\eps$. The analytical results are illustrated in two examples.