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 lowdimensional
manifold (ILDM) method constructs an approximation
to a slow invariant manifold of a set of autonomous ordinary
dierential 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 ddimensional
direct sum of eigenspaces. Numerical examples treated
include the MichaelisMenten model of enzyme kinetics and the
Lorenz equations. We conclude that ILDMclass methods are
broadly applicable to the approximation of attracting manifolds.
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