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IMA Journal of Numerical Analysis Advance Access published online on December 2, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn059
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Computation of integral manifolds for Carathéodory differential equations

Christian Pötzsche

Centre for Mathematical Sciences, Munich University of Technology, 85748 Garching, Germany

Martin Rasmussen{dagger}

Department of Mathematics, Imperial College, London SW7 2AZ, UK

{dagger} Corresponding author. Email: m.rasmussen{at}imperial.ac.uk

Received on 30 June 2008. Accepted for publication 2 September 2008.


  Abstract

We derive two numerical approximation schemes for local invariant manifolds of nonautonomous ordinary differential equations (ODEs) that can be measurable in time and Lipschitzian in the spatial variable. Our approach is inspired by the previous work of Jolly & Rosa (2005 IMA J. Numer. Anal., 25, 698–725) on autonomous ODEs and based on the truncated Lyapunov–Perron operators. Both of our methods are applicable to the full hierarchy of strongly stable, stable, centre stable and the corresponding unstable manifolds, and exponential refinement strategies yield exponential convergence. Several examples illustrate our approach.

Key Words: invariant manifolds; integral manifolds; Lyapunov–Perron operator; Carathéodory condition


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