Differential Equations, Dynamical Systems, and Linear Algebra (Pure and Applied Mathematics)

367 Pages·1974·4.562 MB·English

by  Morris W. Hirsch, Stephen Smale

Dvferential Equations,
Dynamical Systems,
and Linear Algebra
MORRIS W. HIRSCH AND STEPHEN SMALE
Uniuersity of Gdijomiu, Berkeley
ACADEMIC PRESS New York Sun Francisco London
A Subsidiary of Harcourr Brace Jovanovich, Publishers
COPYRIGHT 0 1974, BY ACADEMIPCR ESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMIWED IN ANY FORM OR BY ANY MEANS, ELECTRONIC
OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDINQ, OR ANY
INFORMATlON STORAGE AND RETRIEVAL SYSTEM, WITHOUT
PERMlSSlON IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON)L TD.
24/28 Oval Road. London NWl
Library of Congress Cataloging in Publication Data
Hirsch, Moms,D ate
Differential equations, dynamical systems, and
linear algebra.
(Pure and applied mathematics; a series of monographs
and textbooks, v. )
1. Differential equations. 2. Algebras, Linear.
I. Smale, Stephen, Date joint author. 11. Title.
111. Series.
QA3.P8 [QA372] 5 10'.8s [ 5 15'. 35 1 73-1895 1
ISBN 0-12-349550-4
AMS (MOS) 1970 Subject Classifications: 15-01. 34-01
PRINTED IN THE UNlTED STATES OF AMERICA
Cont:ents
Preface ix
CHAPTEIR FIRSETX AMPLES
1. The Simplelt Examples I
2.L ineSyaretema witC.Ohn at&Cnoetffi cients 9
Notes 13
CHAPTE2R NEWTON'SE QUATIOANN DK EPLERL'ASW
1. Harmonic Oscillators 15
2.So me CalcuBlaucskgro und H
3.Co nservatFiovreFc iee lds 17
4.Ce ntraFlo ree Fields 19
5. State8 22
6. Elliptical Planetary Orbits 23
Note, '¥1
CHAPTER' 3 LINEASRY STEMWSI TH CONSTCAONETF FICIENTS
ANDR EALE IGENVALUES
I.B asiLci neAalrg ebra •29 •
2.R eaEli genvalues
3.D iffereEqnutaitailwo intsRh e alD,is tiEnicgetn values 47
4.C omplEeis:g envalues 55
CHAPTE4R LINEASRY STEMS WITH CONSTANT COEFFICIENTS
ANDC OMPLEEXI GENVALUES
1. Complex Veetor Spaces 62
2.R eaOpel rators witCho mplEeigexn values 86
3.A pplicoafCt oimopnlL eixn eAalrg etborD ai fferentF.qiuala tiona 69
CHAPTER 5 LINEASRY STEMASN DE XPONENTIAOFLO SP ERATORS
1. RevioefTw o pologyi nR • 75
2.N ewN ormfsoO rl d 77
3.E xponentoifOa pelrsa tors 82
4.H omogeneouLsi neSayrs tems 89
5.A NonhomogenEeoquusa tion 99
6.H ighOerrd Seyrs tems 102
Notes 108
vi
CONTENTS
CHAPTER 6 LINEAR SYSTEMS AND CANONICAL FORMS OF
OPERATORS
1. The Primary Decomposition 110
2. The S+ N Decomposition 116
3. Nilpotent Canonical Forms 122
4. Jordan and Real Canonical Forms 126
5. Canonical Forms and Differential Equations 133
6. Higher Order Linear Equations 138
7. Operatore on FunC'tion Spaces 142
UIAl'TER i CONTRACTIONS AND GENERIC PROPERTIES OF
OPERATORS
L Sinks and Sources 144
2. Hyperbolic Flows 15-0
3. Generic Properties of Operators 153
4. The Significance of Genericity 158
CHAPTER 8 FUNDAMENTAL THEORY
1. Dynamical S tems and Vector Fields 160
ys
2. The Fundamental Theorem 161
3. Existence and Uniqueness 163
4. Continuity of Solutions in Initial Conditions 169
5. On Extending Solutions 171
6. Global Solutions 173
7. · The Flow of a Differential Equation 174
Notes 178
CHAPTfR 9 STABILITY OF EQUILIBRIA
1. Nonlinear Sinks 180
2. Stability 185
3. Liapunov Functione 192
4. Gradient Systems 199
5. Gradienta and Inner Products 204
Notes 209
CHAPTER 10 DIFFERENTIAL EQUATIONS FOR ELECTRICAL CIRCUITS
1. An RLCC ircuit 211
2. Analysis of the Circuit Equations 215
3. Van der Pol's Equation 217
4-. Hopf Bifurcation 227
5. More General Circuit Equations 228
Notes 238
CHAPTER 11 THE POINCARt-BENDIXSON THEOREM
I. Limit Sets 239
2. Local Sections and Flow Boxes 242
3. Monotone Sequences in Planar Dynamical Systems 244
CONTENTS vii
4. The Poine&re-Bendixson Theorem 2411
5. Applications of the Poine&re-Bendixson Theorem 250
Notes 254
CHAPTER 12 ECOLOGY
l. One Species 255
2. Predator and Prey 258
3. Competing Species 265
Notes 274
CHAPTER 13 PERIODIC A'ITRACTORS
I. ABymptotic Stability of Closed Orbits 276
2. Discrete Dynamical Systems 278
3. Stability and Closed Orbits 281
CHAPTER H CLASSICAL MECHANICS
1. The n-Body Problem 287
2. Hamiltonian Mechanics 290
Notes 2116
CHAPTER 15 NONAlITONOMOUS EQUATIONS AND DIFFERI!NTIAIIILITY
OFFWWS
I. Existence, Uniqueness, and Continuity for Nonaulonomoua
Differential F.quations 296
2. Differentiability of the Flow of Autonomous Equa.tiona 298
CHAPTER 16 PERTURBATION THEORY AND STRUCTURAL STABILITY
l. Persistence of Equilibria 304
2. Persistence of Closed Orbits 309
3. Structural Stability 312
AFTERWORD
3)9
APPENDIX I ELEMENTARY FACTS
1. Set Theoretic C.Onventions 322
2. Complex Numbers 323
3. Determinants 324
4. Two Propositions on Linear Algebra 325
APPENDIX II POLYNOMIALS
I. The Fundamental Theorem of Algebra 328
APPENDIX III ON CANONICAL FORMS
l. A Decomposition Theorem 331-
2. Uniqueness of Sand N 333
3. Canonical Forms for Nilpotent O raton 334
pe
viii CONTENTS
APPENDIIVX T HEI NVERflJSNECT IONT HEOREM 337
REFERENCES 340
ANSWERTSO SELECPTREODB LEMS 343
SubjIencdte x 355
Preface
This book is about dynamical aspects of ordinary differential equations and the
relations between dynamical systems arid certain fields outside pure mathematics.
A prominent role is played by the structure theory of linear operators on finite-
dimensional vector spaces; we have included a self-contained treatment of that
.
subject
The background material needed to understand this book is differential calculus
of several variables. For example, Serge Lang’s Calculus of Several Variables, up to
the chapter on integration, contains more than is needed to understand much of our
text. On the other hand, after Chapter 7 we do use several results from elementary
analysis such as theorems on uniform convergence; these are stated but not proved.
This mathematics is contained in Lang’s Analysis I, for instance. Our treatment of
linear algebra is systematic and self-contained, although the most elementary parts
have the character of a review; in any case, Lang’s Calculus of Several Variables
develops this elementary linear algebra at a leisurely pace.
While this book can be used as early as the sophomore year by students with a
strong first year of calculus, it is oriented mainly toward upper division mathematics
and science students. It can also be used for a graduate course, especially if the later
chapters are emphasized.
It has been said that the subject of ordinary differential equations is a collection
of tricks and hints for finding solutions, and that it is important because it can
solve problems in physics, engineering, etc. Our view is that the subject can be
developed with considerable unity and coherence; we have attempted such a de-
velopment with this book. The importance of ordinary differential equations
vis d vis other areas of science lies in its power to motivate, unify, and give force to
those areas. Our four chapters on “applications” have been written to do exactly
this, and not merely to provide examples. Moreover, an understanding of the ways
that differential equations relates to other subjects is a primary source of insight
and inspiration for the student and working mathematician alike.
Our goal in this book is to develop nonlinear ordinaty differential equations in
open subsets of real Cartesian space, R”,i n such a way that the extension to
manifolds is simple and natural. We treat chiefly autonomous systems, emphasizing
qualitative behavior of solution curves. The related themes of stability and physical
significance pervade much of the material. Many topics have been omitted, such as
Laplace transforms, series solutions, Sturm theory, and special functions.
The level of rigor is high, and almost everything is proved. More important,
however, is that ad hoc methods have been rejected. We have tried to develop
X PREFACE
proofs that add insight to the theorems and that are important methods in their
own right.
We have avoided the introduction of manifolds in order to make the book more
widely readable; but the main ideas can easily be transferred to dynamical systems
on manifolds.
The first six chapters, especially Chapters 3-6, give a rather intensive and com-
plete study of linear differential equations with constant coefficients. This subject
matter can almost be identified with linear algebra; hence those chapters constitute
a short course in linear algebra as well. The algebraic emphasis is on eigenvectors and
+
how to find them. We go far beyond this, however, to the “semisimple nilpotent”
decomposition of an arbitrary operator, and then on to the Jordan form and its real
analogue. Those proofs that are far removed from our use of the theorems are
relegated to appendices. While complex spaces are used freely, our primary concern
is to obtain results for real spaces. This point of view, so important for differential
equations, is not commonly found in textbooks on linear algebra or on differential
equations.
Our approach to linear algebra is a fairly intrinsic one; we avoid coordinates
where feasible, while not hesitating to use them as a tool for computations or proofs.
On the other hand, instead of developing abstract vector spaces, we work with
linear subspaces of Rno r Cn,a small concession which perhaps makes the abstraction
more digestible.
Using our algebraic theory, we give explicit methods of writing down solutions
to arbitrary constant coefficient linear differential equations. Examples are included.
+
In particular, the S N decomposition is used to compute the exponential of an
arbitrary square matrix.
Chapter 2 is independent from the others and includes an elementary account
of the Keplerian planetary orbits.
The fundamental theorems on existence, uniqueness, and continuity of solutions
of ordinary differential equations are developed in Chapters 8 and 16. Chapter 8 is
restricted to the autonomous case, in line with our basic orientation toward dynami-
cal systems.
Chapters 10, 12, and 14 are devoted to systematic introductions to mathematical
models of electrical circuits, population theory, and classical mechanics, respectively.
The Brayton-Moser circuit theory is presented as a special case of the more general
theory recently developed on manifolds. The Volterra-Lotka equations of competing
species are analyzed, along with some generalizations. In mechanics we develop
the Hamiltonian formalism for conservative systems whose configuration space is
an open subset of a vector space.
The remaining five chapters contain a substantial introduction to the phase
portrait analysis of nonlinear autonomous systems. They include a discussion of
“generic” properties of linear flows, Liapunov and structural stability, Poincarb
Bendixson theory, periodic attractors, and perturbations. We conclude with an
Afterword which points the way toward manifolds.
PREFACE xi
The following remarks should help the reader decide on which chaptcrs to rrad
and in what order.
Chapters 1 and 2 are elementary, but they present many ideas that recur through-
out the book.
Chapters 3-7 form a sequence that develops linear theory rather thoroughly.
Chapters 3, 4, and 5 make a good introduction to linear operators and linear differ-
ential equations. The canonical form theory of Chaptcr 6 is thc basis of the stability
results proved in Chapters 7, 9, and 13; howcvrr, this hravy algebra might be post-
poned at a first exposure to this material and the results takcn on faith.
The existence, uniqueness, and continuity of solutions, proved in Chapter 8, are
used (often implicitly) throughout the rest of the book. Depending on the reader’s
taste, proofs could be omitted.
A reader intcrrstcd in the nonlinear material, who has some background in linear
theory, might start with the stability theory of Chapter 9. Chapters 12 (rcology),
13 (periodic attractors), and 16 (perturbations) depend strongly on Chaptor 9, whilr
the section on dual vector spaccs and gradicnts will mnkc Chapters 10 (clcctrical
circuits) and 14 (mechanics) easier to understand.
Chaptcr 12 also depends on Chapter 11 (PoincarbBcndixson) ; and the material
in Section 2 of Chapter I1 on local sections is used again in Chapters 13 and 16.
Chapter 15 (nonautonomous equations) is a continuation of Chapter 8 and is
used in Chapters 11, 13, and 16; however it can be omitted at a first reading.
The logical dependence of the later chapters is summarized in the following chart :
The book owes much to many people. We only mention four of them here. Ikuko
Workman and Ruth Suzuki did an excellent job of typing the manuscript. Dick
Palais made a number of useful comments. Special thanks are due to Jacob Palis,
who read the manuscript thoroughly, found many minor errors, and suggested
several substantial improvements. Professor Hirsch is grateful to the Miller Institute
for its support during part of the writing of the book.
1
Chapter
First Examples
The purpose of this short chapter is to develop some simple examples of differen-
tial equations. This development motivates the linear algebra treated subsequently
and moreover gives in an elementary context some of the basic ideas of ordinary
differential equations. Later these ideas will be put into a more systematic exposi-
tion. In particular, the examples themselves are special cases of the class of differen-
tial equations considered in Chapter 3. We regard this chapter as important since
some of the most basic ideas of differential equations are seen in simple form.
11. The Simplest Examples
The differential equation
is the simplest differential equation. It is also one of the most important. First,
what does it mean? Here z = z(t) is an unknown real-valued function of a real
.
variable t and dx/dt is its derivative (we will also use z’ or z’ (t) for this derivative)
The equation tells us that for every value of t the equality
x’(t) = w(t)
is true. Here a denotes a constant.
The solutions to (1) are obtained from calculus: if K is any constant (real num-
ber), the function f(t) = Ksti s a solution since
f(t) = aK@ = aj(t).