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

Sofia Borok, Igor Goldfarb, Vladimir Gol'dshtein

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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.

Original languageEnglish
Pages (from-to)1029-1038
Number of pages10
JournalCommunications in Nonlinear Science and Numerical Simulation
Issue number6
StatePublished - 1 Aug 2008


  • Fast-slow systems
  • Intrinsic low-dimensional manifold method (ILDM)
  • Invariant slow manifolds
  • Reduction methods
  • Singularly perturbed systems

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Applied Mathematics


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