Stability of Nonautonomous Differential Equations
Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the construction and regularity of topological conjugacies, the
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Luis Barreira Claudia Valls
Stability of Nonautonomous Differential Equations
1926
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1926
Luis Barreira · Claudia Valls
Stability of Nonautonomous Differential Equations
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Authors Luis Barreira Claudia Valls Departamento de Matemática Instituto Superior Técnico Av. Rovisco Pais 1049-001 Lisboa Portugal e-mail: [email protected] [email protected] URL: http://www.math.ist.utl.pt/∼barreira/
Library of Congress Control Number: 2007934028 Mathematics Subject Classification (2000): 34Dxx, 37Dxx ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN 978-3-540-74774-1 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-74775-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2008 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the authors and SPi using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
SPIN: 12114993
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543210
To our parents
Preface
The main theme of this book is the stability of nonautonomous differential equations, with emphasis on the study of the existence and smoothness of invariant manifolds, and the Lyapunov stability of solutions. We always consider a nonuniform exponential behavior of the linear variational equations, given by the existence of a nonuniform exponential contraction or a nonuniform exponential dichotomy. Thus, the results hold for a much larger class of systems than in the “classical” theory of exponential dichotomies. The departure point of the book is our joint work on the construction of invariant manifolds for nonuniformly hyperbolic trajectories of nonautonomous differential equations in Banach spaces. We then consider several related developments, concerning the existence and regularity of topological conjugacies, the construction of center manifolds, the study of reversible and equivariant equations, and so on. The presentation is self-contained and intends to convey the full extent of our approach as well as its unified character. The book contributes towards a rigorous mathematical foundation for the theory
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