An Invariant-Manifold Approach to Lumping

Abstract

Differential equation models of chemical or biochemical systems usually display multiple, widely varying time scales, i.e. they are stiff. After the decay of transients, trajectories of these systems approach low-dimensional invariant manifolds on which the eventual attractor (an equilibrium point in a closed system) is approached, and in which this attractor is embedded. Computing one of these slow invariant manifolds (SIMs) results in a reduced model of dimension equal to the dimension of the SIM. Another approach to model reduction involves lumping, the formulation of a reduced set of variables that combine the original model variables and in terms of which the reduced model is framed. In this study, we combine lumping with a constructive method for SIMs based on the iterative solution of the invariance equation. We illustrate these methods using a simple model of a linear metabolic pathway, and a model for hydrogen oxidation. The former is treated with a linear lumping function, while a nonlinear lumping function based on a Lyapunov function is used in the latter.