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Volume 20,     Number 2,     Summer 2012

 

FURTHER STUDIES OF THE FUNCTIONAL
EQUATION TRUNCATION APPROXIMATION
MARC R. ROUSSEL

Abstract. The approximation of invariant manifolds has important applications for understanding dynamical systems, as well as for model reduction. The intrinsic low-dimensional manifold (ILDM) method constructs an approximation to a slow invariant manifold of a set of autonomous ordinary di erential equations by finding the locus of points at which the rate vector lies in the subspace spanned by the slow eigenvectors. We had previously conjectured that an alternative method based on explicit neglect of the curvature of the slow manifold, the functional equation truncation approximation (FETA), is exactly equivalent to the ILDM method. The proof of this equivalence is provided here. The proof hinges on the demonstration that one of the FETA equations computes a d-dimensional direct sum of eigenspaces. Numerical examples treated include the Michaelis-Menten model of enzyme kinetics and the Lorenz equations. We conclude that ILDM-class methods are broadly applicable to the approximation of attracting manifolds.

 

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