Continuous and Discrete Dynamics near Manifolds of Equilibria
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Bernd Aulbach
Continuous and Discrete Dynamics near Manifolds of Equilibria
Spri nger-Verlag Berlin Heidelberg New York Tokyo 1984
Author Bernd Aulbach Mathematisches Institut der Universitat Am Hubland, 8700 Wurzburg, FRG
Sponsored by the Volkswagenwerk Foundation.
AMS Subject Classification (1980): 34-02, 34C05, 34C20, 34C30, 34C35, 34C45, 34D05, 34D20, 39AlO, 39A11, 58FlO, 58F15, 58F19, 92AlO ISBN 3-540-13329-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13329-1 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Dedicated to the memory of the first mathematician I met, my father
Preface
In the qualitative theory of dynamical systems non isolated equilibria are rarely studied objects in comparison with isolated equilibria. This research monograph is devoted to the analysis of solutions of nonautonomous ordinary differential equations or autonomous ordinary difference equations near manifolds of stationary solutions. This study is not only motivated from a theoretical but also from a practical point of view as an application to the fundamental selection model of population genetics shows.
The principal results of this book can be comprehended with a basic knowledge of ordinary differential equations. A full understanding and appreciation of this text, however, requires some familiarity with several concepts of the qualitative theory of dynamical systems such as
w limit sets, invariance, Ljapunov functions, stability, asymptotic behavior and invariant manifolds. Apart from this, the presentation is self contained. All quoted results are stated in full detail.
I am most grateful to Prof. Dr. H. W. Knobloch for his helpful criticism and suggestions for improvement. Many thanks are also due to Dr. D. Flockerzi for his careful reading of the manuscript. Anyhow, all remaining faults are mine. Finally I would like to thank the Volkswagenwerk foundation for financial support during the preparation of the final version of this book.
Wurzburg, February 1984
Bernd Aulbach
Table of Contents
Preface
V
Basic Notation 1. Introduction
VIII .
I. DIFFERENTIAL EQUATIONS 2. Basic Results
9
3. Asymptotic Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • .
24
4. Saddle Point Property . . . . . . . . . . . . • . . . . . . . . . . . . . . • . . .
35
5. Equations with a Nonstationary Integral Manifold ....
38
6. The Basic Selection Model from Population Genetics for Overlapping Generations
45
II.
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