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Journal of Dynamics and Differential Equations

, Volume 24, Issue 3, pp 427–481 | Cite as

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

  • José M. ArrietaEmail author
  • Alexandre N. Carvalho
  • José A. Langa
  • Aníbal Rodriguez-Bernal
Article

Abstract

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

Keywords

Nonautonomous dynamical systems Hyperbolic global bounded solutions Unstable manifolds Dichotomy Singular perturbations Attractors Lower semicontinuity 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • José M. Arrieta
    • 1
    Email author
  • Alexandre N. Carvalho
    • 2
  • José A. Langa
    • 3
  • Aníbal Rodriguez-Bernal
    • 1
  1. 1.Departamento de Matemática Aplicada, Facultad de MatemáticasUniversidad Complutense de MadridMadridSpain
  2. 2.Instituto de Ciências Matemáticas e de ComputaçaoUniversidade de São PauloSão CarlosBrazil
  3. 3.Departamento de Ecuaciones Diferenciales y Análisis NuméricoUniversidad de SevillaSevillaSpain

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