- W.-J. Beyn,
W. Kleß:
Numerical Taylor expansions of invariant
manifolds in large dynamical systems
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In this paper we develop a numerical method for computing
higher order local approximations of invariant manifolds, such as
stable, unstable or center manifolds near steady states of a dynamical
system. The underlying system is assumed to be large in the sense that
a large sparse Jacobian at the equilibrium occurs,
for which only a linear
(black box) solver and a low dimensional invariant subspace is available,
but for which methods like the QR-Algorithm are considered to be too
expensive.
Our method is based on an analysis of the multilinear Sylvester equations
for the higher derivatives which can be solved under certain
nonresonance
conditions. These conditions are weaker than the standard gap conditions
on the spectrum which guarantee the existence of the invariant manifold.
The final algorithm requires the solution of several large linear systems
with a bordered Jacobian. To these systems we apply a block elimination
method recently developed by Govaerts and Pryce (1991, 1993).
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Keywords: Invariant manifolds, numerical methods, higher order
approximations, large dynamical systems.
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Mathematics Subject Classification (1991):
34C35, 58F14, 65F50, 65H17.
preprint_6_96.dvi (124KB, without illustrations)
preprint_6_96.ps (678KB, includes 5 illustrations)
DFG-Project Connecting Orbits (Bielefeld)
DFG-Schwerpunktprogramm Dynamik (Berlin)
Thorsten Pampel, 1998-02-17
Department of Mathematics |
University of Bielefeld