Applications of Centre Manifold Theory

151 Pages·1981·3.978 MB·English

by  Jack Carr (auth.)

Applied Mathematical Sciences I Volume 35
Jack Carr
Applications of
Centre Manifold Theory
Springer-Verlag
New York Heidelberg Berlin
Arch W. Naylor George R. Sell
University of Michigan University of Minnesota
Department of Electrical Institute for Mathematics
and Computer Engineering and its Applications
Ann Arbor, MI 48104 514 Vincent Hall
USA 206 Church Street, S.E.
Minneapolis, MN 55455
USA
Editors
J. E. Marsden L. Sirovich
Department of Division of
Mathematics Applied Mathematics
University of California Brown University
Berkeley, CA 94720 Providence, RI 02912
USA USA
AMS Subject Classifications: 4601, 4701, 1501, 2801, 34B25, 4001, 4201, 4401, 4501,
54E35,9301
Library of Congress Cataloging in Publication Data
Naylor, Arch W.
Linear operator theory in engineering and science
(Applied mathematical sciences; 40)
Includes index.
1. Linear operators. I. Sell, George R.,
1937- . ll. Title. ill. Series: Applied
mathematical sciences (Springer-Verlag New York
Inc.); 40.
QA1.A647 vol. 40 [ QA329.2] 510s 82-10432
[ 515.7 '246 ]
ISBN-13: 978-0-387-90577-8 e-ISBN-I3: 978-1-4612-5929-9
DOl: 10.1007/978-1-4612-5929-9
(First Springer edition, with a few minor corrections-This title was originally published
in 1971 by Holt, Rinehart and Winston, Inc.)
First softcover printing, 2000.
© 1982 by Springer-Verlag New York Inc.
All rights reserved. No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York
10010, USA.
9 8 7 6 5 432 1
To my parents
PREFACE
These notes are based on a series of lectures given in
the Lefschetz Center for Dynamical Systems in the Division of
Applied Mathematics at Brown University during the academic
year 1978-79.
The purpose of the lectures was to give an introduction
to the applications of centre manifold theory to differential
equations. Most of the material is presented in an informal
fashion, by means of worked examples in the hope that this
clarifies the use of centre manifold theory.
The main application of centre manifold theory given
in these notes is to dynamic bifurcation theory. Dynamic
bifurcation theory is concerned with topological changes in
the nature of the solutions of differential equations as para
meters are varied. Such an example is the creation of periodic
orbits from an equilibrium point as a parameter crosses a
critical value. In certain circumstances, the application of
centre manifold theory reduces the dimension of the system
under investigation. In this respect the centre manifold
theory plays the same role for dynamic problems as the
Liapunov-Schmitt procedure plays for the analysis of static
solutions. Our use of centre manifold theory in bifurcation
problems follows that of Ruelle and Takens [57) and of Marsden
and McCracken [51) .
In order to make these notes more widely accessible,
we give a full account of centre manifold theory for finite
dimensional systems. Indeed, the first five chapters are de
voted to this. Once the finite dimensional case is under
stood, the step up to infinite dimensional problems is
essentially technical. Throughout these notes we give the
simplest such theory, for example our equations are autono
mous. Once the core of an idea has been understood in a
simple setting, generalizations to more complicated situations
are much more readily understood.
In Chapter 1, we state the main results of centre mani
fold theory for finite dimensional systems and we illustrate
their use by a few simple examples. In Chapter 2, we prove
the theorems which were stated in Chapter 1, and Chapter 3
contains further examples. In Section 2 of Chapter 3 we out
line Hopf bifurcation theory for Z-dimensional systems. In
Section 3 of Chapter 3 we apply this theory to a singular per
turbation problem which arises in biology. In Example 3 of
Chapter 6 we apply the same theory to a system of partial dif
ferential equations. In Chapter 4 we study a dynamic bifurca
tion problem in the plane with two parameters. Some of the
results in this chapter are new and, in particular, they con
firm a conjecture of Takens [64). Chapter 4 can be read in
dependently of the rest of the notes. In Chapter 5, we apply
the theory of Chapter 4 to a 4-dimensional system. In Chap
ter 6, we extend the centre manifold theory given in Chapter
2 to a simple class of infinite dimensional problems. Fin
ally, we illustrate their use in partial differential equa
tions by means of some simple examples.
first became interested in centre manifold theory
through reading Dan Henry's Lecture Notes (34). My debt to
these notes is enormous. would like to thank Jack K. Hale.
Dan Henry and John Mallet-Paret for many valuable discussions
during the gestation period of these notes.
This work was done with the financial support of the
United States Army, Durham, under AROD DAAG 29-76-G0294.
Jack Carr
December 1980
TABLE OF CONTENTS
Page
CHAPTER 1. INTRODUCTION TO CENTRE MANIFOLD THEORY 1
1.1. Introduction . . • I
1. 2. Motivation •... 1
1. 3. Centre Manifolds . 3
1.4. Examples . . . . . 5
1. S. Bifurcation Theory . . • . 11
1.6. Comments on the Literature 13
CHAPTER 2. PROOFS OF THEOREMS 14
2.1. Introduction . . . l4
2.2. A Simple Example . . . • . . . 14
2.3. Existence of Centre Manifolds. 16
2.4. Reduction Principle . ....... . 19
2. S • Approximation of the Centre Manifold 2S
2.6. . Properties of Centre Manifolds . . . . 28
2.7. Global Invariant Manifolds for Singular
Perturbation Problems. 30
2.8. Centre Manifold Theorems for Maps. 33
CHAPTER 3. EXAMPLES . . . . . . . . 37
3.1. Rate of Decay Estimates in Critical Cases. 37
3.2. Hopf Bifurcation . . . .• ...•. • 39
3.3. Hopf Bifurcation in a Singular Perturbation
Problem .... .. ... . . . 44
3.4. Bifurcation of Maps ...... . 50
CHAPTER 4. BIFURCATIONS WITH TWO PARAMETERS IN TWO
SPACE DIMENSIONS S4
4.1. Introduction . 54
4.2. Preliminaries. 57
4.3. Scaling. . . 64
4.4. The Case £1 > 0 64
4.5. The Case £ < 0 77
4.6. More Seal in! . . . . . . • • . . . 78
4.7. Completion of the Phase Portraits. 80
4.8. Remarks and Exercises ..... 81
4.9. Quadratic Nonlinearities • .. 83
CHAPTER 5. APPLICATION TO A PANEL FLUTTER PROBLEM 88
5.1. Introduction . 88
5.2. Reduction to a Second Order Equation 89
5.3. Calculation of Linear Terms. . 93
5.4. Calculation of the Nonlinear Terms . 95
Page
CHAPTER 6. INFINITE DIMENSIONAL PROBLEMS. 97
6.1. Introduction . . 97
6.2. Semigroup Theory 97
6.3. Centre Manifolds 117
6.4. Examples 120
REFERENCES 136
INDEX ... 141
CHAPTER 1
INTRODUCTION TO CENTRE MANIFOLD THEORY
1.1. Introduction
In this chapter we state the main results of centre
manifold theory for finite dimensional systems and give some
simple examples to illustrate their application.
1.2. Motivation
To motivate the study of centre manifolds we first
look at a simple example. Consider the system
3
x • ax (1.2.1)
J
where a is a constant. Since the equations are uncoupled
we can easily show that the zero solution of (1.2.1) is
asymptotically stable if and only if a < O. Suppose now that
(1.2.2)
Since the equations are coupled we cannot immediately decide
if the zero solution of (1.2.2) is asymptotically stable, but
we might suspect that it is if a < o. The key to understand
ing the relation of equation (1.2.2) to equation (1.2.1) is
1