Attractivity and Bifurcation for Nonautonomous Dynamical Systems
Although, bifurcation theory of equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this book, two diffe
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Martin Rasmussen
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1907
Martin Rasmussen
Attractivity and Bifurcation for Nonautonomous Dynamical Systems
Author Martin Rasmussen Institut für Mathematik Lehrstuhl für Angewandte Analysis Universität Augsburg 86135 Augsburg Germany e-mail: [email protected]
Library of Congress Control Number: 2007925370 Mathematics Subject Classification (2000): 34D05, 37B25, 37B55, 37D10, 37G35 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-71224-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-71224-4 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-71225-1 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 2007 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: 12027767
VA41/3100/SPi
543210
To Professor Bernd Aulbach and my parents
Preface
This book has been developed from my dissertation, which I wrote at the University of Augsburg from 2002 to 2005. I first became acquainted with several definitions of attractor for nonautonomous dynamical systems when I was preparing my diploma thesis, and the question arose whether a nonautonomous bifurcation theory can be founded based on suitable notions of nonautonomous attractor (and repeller). At the beginning of my time as a Ph. D. student, I developed local notions of attractor and repeller for several time domains (the past, the future, the whole time and finite time intervals), and I distinguished between two bifurcation scenarios. The first scenario describes the loss of attractivity and repulsivity, and the second one deals with transitions of attractors and repellers. All definitions are introduced in Chapter 2 of this book. As a test for the new definitions, I then considered asymptotically autonomous differential equations; these are systems whose behavior becomes autonomous when time tends to the past or the future. I found conditions for the occurrence of a nonautonomous bifurcation in case the underlying autonomous
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