Invariant manifolds as pullback attractors of nonautonomous differential equations

Bernd Aulbach; Martin Rasmussen; Stefan Siegmund

doi:10.3934/dcds.2006.15.579

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2006, Volume 15, Issue 2: 579-596. Doi: 10.3934/dcds.2006.15.579

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Invariant manifolds as pullback attractors of nonautonomous differential equations

1.
Department of Mathematics, University of Augsburg, D-86135 Augsburg
2.
Department of Mathematics, University of Frankfurt, D-60325 Frankfurt

Received: December 2004

Revised: October 2005

Published: April 2006

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Abstract

We discuss the relationship between invariant manifolds of nonautonomous differential equations and pullback attractors. This relationship is essential, e.g., for the numerical approximation of these manifolds. In the first step, we show that the unstable manifold is the pullback attractor of the differential equation. The main result says that every (hyperbolic or nonhyperbolic) invariant manifold is the pullback attractor of a related system which we construct explicitly using spectral transformations. To illustrate our theorem, we present an application to the Lorenz system and approximate numerically the stable as well as the strong stable manifold of the origin.

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Mathematics Subject Classification: 34C30, 34D45, 37B55, 37D10, 65L20.

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