Characterization of stable manifolds for nonuniform exponential dichotomies

Luis Barreira; Claudia Valls

doi:10.3934/dcds.2008.21.1025

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2008, Volume 21, Issue 4: 1025-1046. Doi: 10.3934/dcds.2008.21.1025

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Characterization of stable manifolds for nonuniform exponential dichotomies

Luis Barreira^1, and
Claudia Valls^2,

1.
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa
2.
Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received: July 31, 2007

Revised: February 29, 2008

Published: September 2008

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Abstract

We establish the existence of smooth stable manifolds for nonautonomous differential equations $v'=A(t)v+f(t,v)$ in a Banach space, obtained from sufficiently small perturbations of a linear equation $v'=A(t)v$ admitting a nonuniform exponential dichotomy. In addition to the exponential decay of the flow on the stable manifold we also obtain the exponential decay of its derivative with respect to the initial condition. Furthermore, we give a characterization of the stable manifold in terms of the exponential growth rate of the solutions.

Keywords:

Nonuniform exponential dichotomies,
stable manifolds.

Mathematics Subject Classification: Primary: 37D10, 34D99.

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