About non-coincidence of invariant manifolds and intrinsic low dimensional manifolds (ILDM)

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.