IJPAM: Volume 44, No. 1 (2008)

ON A MODIFIED VERSION OF ILDM METHOD
AND ITS ASYMPTOTIC ANALYSIS

S. Borok$^1$, I. Goldfarb$^2$, V. Gol'dshtein$^3$
$^1$Division of Mathematical Sciences, SPMS
Nanyang Technological University
No. 1, Nanyang Walk, Blk 5, 637616, SINGAPORE
e-mail: borok@ntu.edu.sg
$^{2,3}$Department of Mathematics
Ben-Gurion University of the Negev
P.O. Box 653, Beer-Sheva, ISRAEL
$^3$e-mail: vladimir@bgu.ac.il


Abstract.It is known that processes which take place in complex chemical kinetics and combustion systems have very different time scales. It is often desirable to decouple such systems into fast and slow sub-systems for the reduction of their complexity. One of such reduction methods is Intrinsic Low-Dimensional Manifolds (ILDM) method proposed by Maas and Pope. This method successfully locates invariant manifolds of a considered system, but also has a number of disadvantages. One of the main problems of ILDM numerical realization is the existence of so-called ``ghost"-manifolds that do not have any connection to the dynamics of the system. It is shown that even for two-dimensional singularly perturbed system, for which the fast-slow decomposition is explicit, the ``ghost"-manifolds can appear. In the present paper a modified version of the ILDM-method is discussed. This modification, which we call TILDM approach, has a much better performance in the context of ``ghost"-manifolds problem. The asymptotic analysis of the TILDM method explains why one of the main reasons for ``ghost"-manifolds appearance is absent for TILDM.

Received: March 19, 2008

AMS Subject Classification: 34E13, 34E05, 80A30, 80A25

Key Words and Phrases: intrinsic low-dimensional manifold method (ILDM), invariant slow manifolds, reduction methods, fast-slow systems, singularly perturbed systems, asymptotic analysis

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 44
Issue: 1