Six lectures on dynamical systems

316 Pages·1996·172.289 MB·English

by  Aulbach B., Colonius F. (eds.)

SIX LECTURES ON
DYNAMICAL
SYSTEMS
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SIX LECTURES ON
DYNAMICAL
SYSTEMS
editors
B. Aulbach & F. Colonius
Institut fur Mathematik
Universitat Augsburg
Germany
World Scientific
Singapore* New Jersey •London •Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
SIX LECTURES ON DYNAMICAL SYSTEMS
Copyright © 1996 by World Scientific Publishing Co. Pte. Ltd.
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Preface
This collection of articles grew out of a tutorial workshop which has been conducted
at the Department of Mathematics at the University of Augsburg in June 1994.
This workshop, organized by the editors of this volume, was part of a particular
graduate student program ("Graduiertenkolleg Mathematik: Analyse, Optimierung
und Steuerung komplexer Systeme") financed by the DFG and the Bavarian Ministry
of Education, Culture, Science and Art.
The six articles in this volume cover different facets of the mathematical theory of
dynamical systems. The topics range from the topological foundations through invari­
ant manifolds, decoupling, linearization, perturbations and computations to control
theory. In any case, emphasis is put on motivation and on guiding ideas, and the
theoretical explanations are accompanied and illustrated by many examples. Each
contribution is self-contained and provides an in-depth insight into some topic of
current interest.
We hope that this volume will stimulate further research in the field of dynamical
systems.
Augsburg, December 1995 Bernd Aulbach and Fritz Colonius
v
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Contents
Preface v
Dynamical Systems: The Topological Foundations
by Ethan Akin
1 Introduction: Discrete Time Dynamics and Closed Relations 1
2 Recurrence for Closed Relations 7
3 Invariant Sets and Lyapunov Functions 14
4 Attractors and Basic Sets 24
5 Maps Between Dynamical Systems 33
Bibliography 42
Integral Manifolds for Caratheodory Type
Differential Equations in Banach Spaces
by Bernd Aulbach and Thomas Wanner
1 Prologue 45
2 Basic Definitions and Results 48
3 Quasibounded Solutions 60
4 The Fundamental Existence Theorem for Integral Manifolds 76
5 Hierarchies of Integral Manifolds 90
6 The Classical Five Integral Manifolds 97
7 The Missing Sixth Integral Manifold 106
8 Epilogue 109
Appendices 110
A Strong Measurability and the Bochner Integral 110
B Banach's Fixed Point Theorem and Related Results 114
References 117
vii
viii Contents
Control Theory and Dynamical Systems
by Fritz Colonius and Wolfgang Kliemann
1 Introduction 121
2 Control Systems as Dynamical Systems 125
3 Limit Behavior of Controlled Trajectories 131
4 Two Perturbation Theorems for Ordinary Differential Equations 134
5 Two Examples 139
6 Analysis of Linearized Control Systems 144
7 Spectral Theory of Nonlinear Control Systems 150
References 156
Shadowing in Discrete Dynamical Systems
by Brian A. Coomes, Huseyin Kogak and Kenneth J. Palmer
1 Introduction 163
2 The Infinite-time Shadowing Theorem 165
3 Finite-time Shadowing 175
3.1 The Finite-time Shadowing Theorem 175
3.2 Implementation of the Theorem in Dimension One 177
3.3 Example: The Logistic Map 181
3.4 Implementation of the Theorem in Dimension Two 182
3.5 Example: Tinkerbell Map 188
4 Periodic Shadowing 190
4.1 Periodic Shadowing Theorem 190
4.2 Necessary and Sufficient Condition that L~l Exists 193
4.3 Implementation of the Theorem in Dimension One 194
4.4 Example: The Logistic Map 196
4.5 Implementation of the Theorem in Dimension Two 196
4.6 Example: The Henon Map 201
4.7 Lyapunov Exponents for Periodic Orbits of Maps 201
4.8 Example: The Henon Map 208
References 208
Contents ix
Perturbations of Invariant Manifolds of
Ordinary Differential Equations
by George Osipenko and Eugene Ershov
1 Introduction 213
2 Basic Notions and Motivational Examples 216
3 Perturbation of Equilibria 223
4 Invariant Manifolds of Pseudo-Hyperbolic Linear Maps 231
5 Perturbation of Invariant Manifolds Through an Equilibrium 245
6 Perturbation of Periodic Orbits and Invariant Manifolds Through Them . 251
7 Sacker-Neimark and Mane Theorems 254
8 Locally Non-unique Invariant Manifolds 258
References 263
The Reduction of Discrete Dynamical and
Semidynamical Systems in Metric Spaces
by Andrejs Reinfelds
1 Introduction 267
2 Preliminaries 268
3 Auxiliary Lemmas 270
4 Fixed Point 273
5 Invariant Sets 274
6 Conjugacy of Homeomorphisms 1 281
7 Conjugacy of Noninvertible Mappings 290
8 Conjugacy of Homeomorphisms 2 293
9 Application to Stability Theory 304
10 Shadowing Lemma 305
References 308
Dynamical Systems: The Topological
Foundations
Ethan Akin
Mathematics Department
The City College
137 Street and Convent Ave.
New York, N. Y. 10031 USA
[email protected]
Abstract
Of importance for the study of smooth dynamical systems, the
concepts of Lyapunov function, attractor and chain recurrence are
entirely topological. The theory connecting these ideas provides tools
with which to begin the study of a wide variety of dynamical systems.
1 Introduction: Discrete Time Dynamics and
Closed Relations
A dynamical system is a mathematical attempt to describe the phenomenon
of change. The label is applied to a wide variety of models: differential
equations in continuous time, iterations in discrete time, and stochastic
processes. The first examples which students meet usually occur in calculus
and differential equations courses: falling bodies, radioactive decay and
the simple harmonic motions of a spring. These are differential equations
with which you are probably familiar. Let us use them to introduce the
conceptually simpler, but less familiar, iteration processes that will be the
focus of our attention.
1