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Journal of Dynamics and Differential Equations

, Volume 25, Issue 4, pp 1139–1158 | Cite as

Generalized Nonuniform Dichotomies and Local Stable Manifolds

  • António J. G. Bento
  • César M. Silva
Article

Abstract

We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform dichotomy for the evolution operator that contains the nonuniform exponential and polynomial dichotomies as a very particular case. The family of dichotomies considered allow situations for which the classical Lyapunov exponents are zero. Additionally, we give new examples of application of our stable manifold theorem and study the behavior of the dynamics under perturbations.

Keywords

Invariant manifolds Nonautonomous differential equations Nonuniform generalized dichotomies 

Mathematics Subject Classification (2000)

37D10 34D09 37D25 

Notes

Acknowledgments

This work was partially supported by FCT though Centro de Matemática da Universidade da Beira Interior (project PEst-OE/MAT/UI0212/2011).

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade da Beira InteriorCovilhãPortugal

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