A method to determine the dimension of long-time dynamics in multi-scale systems
by
Sybille Handrock-Meyer
Technische Universität Chemnitz, Fakultät für Mathematik, Chemnitz, Germany
Coauthors: L.V. Kalachev (University of Montana, Department of Mathematical Sciences, Missoula, MT 59812, USA), K.R. Schneider (Weierstrass--Institut für Angewandte Analysis und Stochastik, Berlin, Germany)

Modeling reaction kinetics in a homogeneous medium usually leads to stiff systems of ordinary differential equations the dimension of which can be large. The problem of determination of the minimal number of phase variables needed to describe the characteristic behavior of large scale systems is extensively addressed in current chemical kinetics literature from different point of views. Only for a few of these approaches there exists a mathematical justification.

A well-known approach to reduce the dimension of such systems without loosing essential information on the system behavior is the quasi-steady state assumption (QSSA): the derivative of fast variables is assumed to be zero. This procedure eliminates redundant variables whereas at the same time their influence on the system's behavior is taken into account. The application of the QSSA requires some knowledge of the underlying chemistry, or the explicit representation of the corresponding differential system in the form of a singularly perturbed system.

In this talk we describe and justify a procedure allowing directly to determine how many and which state variables are essential in a neighborhood of a given point of the extended phase space. This method exploits the wide range of characteristic time-scales in a chemical system and its mathematical justification is based on the theory of invariant manifolds. The procedure helps to get chemical insight into the intrinsic dynamics of a complex chemical process.

Date received: August 11, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caia-01.