Asymptotic analysis of two reduction methods for systems of chemical reactions

This paper 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 [Combust. Flame 88 (1992) 239] and an iterative method due to Fraser [J. Chem. Phys. 88 (1988) 4732] and further developed by Roussel and Fraser [J. Chem. Phys. 93 (1990) 1072]. 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 asymptotic expansions of these manifolds (,epsilon down arrow 0, where E measures the ratio of the reaction time scales) are compared with the asymptotic expansion of M-epsilon,, the slow manifold given by geometric singular perturbation theory. It is shown that the expansions of the ILDM and M-epsilon agree up to and including terms of O(epsilon); the former has an error at O(epsilon(2)) that is proportional to the local curvature of M-0. The error vanishes if and only if the curvature is zero everywhere. The iterative method generates, term by term, the asymptotic expansion of Mepsilon,. Starting from M-0, the ith application of the algorithm yields the correct expansion coefficient at O(epsilon'), while leaving the lower-order coefficients invariant. Thus, after e applications, the expansion is accurate up to and including the terms of O(epsilon(l)). The analytical results are illustrated on a planar system from enzyme kinetics (Michaelis-Menten-Henri) and a model planar system due to Davis and Skodje. (C) 2002 Published by Elsevier Science B.V.