Nonautonomous continuation of bounded solutions

Christian Pötzsche

doi:10.3934/cpaa.2011.10.937

Article Contents

2011, Volume 10, Issue 3: 937-961. Doi: 10.3934/cpaa.2011.10.937

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Nonautonomous continuation of bounded solutions

Christian Pötzsche^1,

1.
Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, D-85758 Garching

Received: March 2009

Revised: September 2009

Published: May 2011

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Abstract

We show the persistence of hyperbolic bounded solutions to nonautonomous difference and retarded functional differential equations under parameter perturbation, where hyperbolicity is given in terms of an exponential dichotomy in variation. Our functional-analytical approach is based on a formulation of dynamical systems as operator equations in ambient sequence or function spaces. This yields short proofs, in particular of the stable manifold theorem.
As an ad hoc application, the behavior of hyperbolic solutions and stable manifolds for ODEs under numerical discretization with varying step-sizes is studied.

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Mathematics Subject Classification: 39A11, 37B55, 37C60, 37G10, 37G15.

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