Causes for “ghost” manifolds

Abstract

The paper concerns intrinsic low-dimensional manifold (ILDM) method suggested in [Maas U, Pope SB. Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, combustion and flame 1992;88:239–64] for dimension reduction of models describing kinetic processes. It has been shown in a number of publications [Goldfarb I, Gol’dshtein V, Maas U. Comparative analysis of two asymptotic approaches based on integral manifolds. IMA J Appl Math 2004;69:353–74; Kaper HG, Kaper TJ, Asymptotic analysis of two reduction methods for systems of chemical reactions. Phys D 2002;165(1–2):66–93; Rhodes C, Morari M, Wiggins S. Identification of the low order manifolds: validating the algorithm of Maas and Pope. Chaos 1999;9(1):108–23] that the ILDM-method works successfully and the intrinsic low-dimensional manifolds belong to a small vicinity of invariant slow manifolds. The ILDM-method has a number of disadvantages. One of them is appearance of so-called “ghost”-manifolds, which do not have connection to the system dynamics [Borok S, Goldfarb I, Gol’dshtein V. “Ghost” ILDM – manifolds and their discrimination. In: Twentieth Annual Symposium of the Israel Section of the Combustion Institute, Beer-Sheva, Israel; 2004. p. 55–7; Borok S, Goldfarb I, Gol’dshtein V. About non-coincidence of invariant manifolds and intrinsic low-dimensional manifolds (ILDM). CNSNS 2008;71:1029–38; Borok S, Goldfarb I, Gol’dshtein V, Maas U. In: Gorban AN, Kazantzis N, Kevrekidis YG, Ottinger HC, Theodoropoulos C, editors. “Ghost” ILDM-manifolds and their identification: model reduction and coarse-graining approaches for multiscale phenomena. Berlin–Heidelberg–New York: Springer; 2006. p. 55–80; Borok S, Goldfarb I, Gol’dshtein V. On a modified version of ILDM method and its asymptotic analysis. IJPAM 2008; 44(1): 125–50; Bykov V, Goldfarb I, Gol’dshtein V, Maas U. On a modified version of ILDM approach: asymptotic analysis based on integral manifolds. IMA J Appl Math 2006;71:359–82; Flockerzi D. Tutorial: intrinsic low-dimensional manifolds and slow attractors. Magdeburg: Max-Planck-Institut; 2001–2005. <www.mpi-magdeburg.mpg.de/people/flocke/tutorial-ildm.pdf>; Flockerzi D, Heineken W. Comment on “Identification of low order manifolds: validating the algorithm of Maas and Pope”. Chaos 1999;9:108–23; Flockerzi D, Heineken W. Comment on “Identification of low order manifolds: validating the algorithm of Maas and Pope”. Chaos 2006;16:048101]. The present work studies the causes for the “ghost” manifolds appearance for the case of a two-dimensional singularly perturbed system.