Computation of nonautonomous invariant and inertial manifolds

  • We derive a numerical scheme to compute invariant manifolds for time-variant discrete dynamical systems, i.e., nonautonomous difference equations. Our universally applicable method is based on a truncated Lyapunov-Perron operator and computes invariant manifolds using a system of nonlinear algebraic equations which can be solved both locally using (nonsmooth) inexact Newton, and globally using continuation algorithms. Compared to other algorithms, our approach is quite flexible, since it captures time-dependent, nonsmooth, noninvertible or implicit equations and enables us to tackle the full hierarchy of strongly stable, stable and center-stable manifolds, as well as their unstable counterparts. Our results are illustrated using a test example and are applied to a population dynamical model and the Hénon map. Finally, we discuss a linearly implicit Euler-Bubnov-Galerkin discretization of a reaction diffusion equation in order to approximate its inertial manifold.

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Metadaten
Author:Christian PötzscheGND, Martin RasmussenORCiDGND
URN:urn:nbn:de:bvb:384-opus4-4793
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/602
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2007-41)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Release Date:2007/12/11
Tag:Bubnov-Galerkin-Approximation
inertial manifold; Bubnov Galerkin approximation
GND-Keyword:Dynamisches System; Invariante Mannigfaltigkeit; Inertialmannigfaltigkeit; Newton-Verfahren; Fortsetzungsmethode
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik