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Stability of Nonautonomous Differential Equations

  • Book
  • © 2008

Overview

Authors:
  1. Luis Barreira

    1. Instituto Superior Técnico, 1049-001, Lisboa, Portugal

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    You can also search for this author in PubMed   Google Scholar

  2. Claudia Valls

    1. Instituto Superior Técnico, 1049-001, Lisboa, Portugal

    View author publications

    You can also search for this author in PubMed   Google Scholar

Part of the book series: Lecture Notes in Mathematics (LNM, volume 1926)

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Table of contents (12 chapters)

  1. Front Matter

    Pages I-XIV
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  2. Exponential dichotomies

    1. Front Matter

      Pages 17-17
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    2. Introduction

      Pages 1-16

    3. Exponential dichotomies and basic properties

      Pages 19-25

    4. Robustness of nonuniform exponential dichotomies

      Pages 27-51

  3. Stable manifolds and topological conjugacies

    1. Front Matter

      Pages 53-53
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    2. Lipschitz stable manifolds

      Pages 55-73

    3. Smooth stable manifolds in Rn

      Pages 75-117

    4. Smooth stable manifolds in Banach spaces

      Pages 119-143

    5. A nonautonomous Grobman–Hartman theorem

      Pages 145-167

  4. Center manifolds, symmetry and reversibility

    1. Front Matter

      Pages 169-169
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    2. Center manifolds in Banach spaces

      Pages 171-196

    3. Reversibility and equivariance in center manifolds

      Pages 197-215

  5. Lyapunov regularity and stability theory

    1. Front Matter

      Pages 217-217
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    2. Lyapunov regularity and exponential dichotomies

      Pages 219-248

    3. Lyapunov regularity in Hilbert spaces

      Pages 249-263

    4. Stability of nonautonomous equations in Hilbert spaces

      Pages 265-276

  6. Back Matter

    Pages 277-290
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Keywords

About this book

The main theme of this book is the stability of nonautonomous di?erential equations, with emphasis on the study of the existence and smoothness of invariant manifolds, and the Lyapunov stability of solutions. We always c- sider a nonuniform exponential behavior of the linear variational equations, given by the existence of a nonuniform exponential contraction or a nonu- form exponential dichotomy. Thus, the results hold for a much larger class of systems than in the “classical” theory of exponential dichotomies. Thedeparturepointofthebookisourjointworkontheconstructionof- variant manifolds for nonuniformly hyperbolic trajectories of nonautonomous di?erential equations in Banach spaces. We then consider several related - velopments,concerningtheexistenceandregularityoftopologicalconjugacies, the construction of center manifolds, the study of reversible and equivariant equations, and so on. The presentation is self-contained and intends to c- vey the full extent of our approach as well as its uni?ed character. The book contributes towards a rigorous mathematical foundation for the theory in the in?nite-dimensional setting, also with the hope that it may lead to further developments in the ?eld. The exposition is directed to researchers as well as graduate students interested in di?erential equations and dynamical systems, particularly in stability theory.

Reviews

From the reviews: “In this book, the authors give a unified presentation of a substantial body of work which they have carried out and which revolves around the concept of nonuniform exponential dichotomy. … This is a well-written book which contains many interesting results. The reader will find significant generalizations of the standard invariant manifold theories, of the Hartman-Grobman theorem … . Anyone interested in these topics will profit from reading this book.” (Russell A. Johnson, Mathematical Reviews, Issue 2010 b)

Authors and Affiliations

  • Instituto Superior Técnico, 1049-001, Lisboa, Portugal

    Luis Barreira, Claudia Valls

About the authors

Luis Barreira is a Full Professor of Mathematics at Instituto Superior Técnico, Lisbon and a member of the Center for Mathematical Analysis, Geometry, and Dynamical Systems. He obtained his PhD from the Pennsylvania State University in 1996. In 2007 he has been awarded the Gulbenkian Science Prize.  

Claudia Valls is an Invited Assistant Professor at Instituto Superior Técnico, Lisbon and a Postdoctoral Fellow at the Center for Mathematical Analysis, Geometry, and Dynamical Systems, of which she is also a member. She obtained her PhD from the Universitat de Barcelona in 1999. 

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