Fast and slow invariant manifolds in chemical kinetics

In the framework of the classical theory of Singularly Perturbed Systems (SPS) there exist two main types of invariant manifolds, i.e. fast and slow manifolds with their own intrinsic dynamics. There are many different methods for dimension and complexity reduction for mechanisms of chemical kinetics, based on the concept of slow invariant manifolds. For instance, most known and popular techniques such as Quasi-Steady State (QSSA) and Partial Equilibrium Approaches (PEA) as well as recently developed Intrinsic Low Dimensional Manifolds (ILDM) and Computational Singular Perturbation (CSP) methods have shown a potential to treat automatically very complex mechanisms comprising hundreds of chemically reacting species. In this respect, the importance of slow invariant manifolds has become more or less clear. However, the importance of fast invariant manifolds is much less transparent. Though without an accurate knowledge about fast manifolds (fast fibers in other terminology) slow manifolds cannot be appropriately identified. In this paper we shall discuss a concept of Singularly Perturbed Vector Fields (SPVF) that is a coordinate free theory of singularly perturbed systems with the main emphasis made on fast invariant manifolds and on the method which allows identification and evaluation of these objects. The importance of these objects in model reduction of mechanisms of chemical kinetics will be addressed. Motivation and application of the method will be demonstrated by a simple model example and by analysis of conventional QSSA and PEA methods.