About non-coincidence of invariant manifolds and intrinsic low dimensional manifolds (ILDM) |
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| Institution: | 1. Institute Pprime, Department of Fluid Flow, Heat Transfer and Combustion, CNRS–University of Poitiers–ENSMA, SP2MI, Bd Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France;2. M2P2 Lab, Aix-Marseille Université–CNRS–Ecole Centrale, 38 rue Frédéric Joliot-Curie, 13451 Marseille Cedex 13, France;3. Department of Applied Mathematics, CNRS–University of Pau and Inria-CAGIRE group, IPRA, avenue de l’université, Pau, France;4. Center for Coastal Physical Oceanography and Ocean, Earth and Atmospheric Sciences, Old Dominion University, Norfolk, VA 23529, USA |
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| Abstract: | The present paper contains an analysis of some aspects of a well known method of Intrinsic Low-Dimensional Manifolds (ILDM), which is regularly used for model reduction purposes in a number of combustion problems. One of these aspects relates to an existence of additional solutions (so-called “ghost”-manifolds), which represent intrinsic low-dimensional manifolds and do NOT represent any slow invariant manifold even for two-dimensional singularly perturbed systems (for a small but finite singular parameter). These “ghost”-manifolds are examples that contradict to the conjecture about the coincidence of ILDM and slow invariant manifolds published previously. Another aspect of the ILDM-method concerns the so-called transition zones (turning manifolds) between different invariant manifolds. It is shown that transition manifolds can not be correctly described by the ILDM-method. This statement is illustrated by an example taken from the mathematical theory of combustion. |
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