Differential equations wiley pdf download

ORDINARY DIFFERENTIAL EQUATIONS JACK K. HALE

KRIEGER PUBLISHING COMPANY MALABAR, FLORIDA

Original Edition 1969 Second Edition 1980 Printed and Published by ROBERT E. KRIEGER PUBLISHING COMPANY, INC. KRIEGER DRIVE MALABAR, FLORIDA 32950

Copyright © 1969 (Original Material) by JOHN WILEY & SONS, INC. Copyright © 1980 (New Material) by ROBERT E. KRIEGER PUBLISHING COMPANY, INC.

All rights reserved. Aro reproduction in any form of this book, in whole or in part (except for brief quotation in critical articles or rcoiews), may be made without written authorization from the publisher. Printed in the United States of America

Library of Congress Cataloging in Publication Data Hale, Jack K. Ordinary differential equations.

Second edition of original published by Wiley-Interscience, New York, which was issued as v. 21 of Pure and applied mathematics. Bibliography: p. Includes index. 1. Differential equations. I. Title. [QA372.H184 19801 515'.352 79-17238 ISBN 0-89874-011-8 10

9

8

7

6

Preface

This book is the outgrowth of a course given for a number of years in the Division of Applied Mathematics at Brown University. Most of the students were in their first and second years of graduate study in applied mathematics, although some were in engineering and pure mathematics. The purpose of the book is threefold. First, it is intended to familiarize the reader with some of

the problems and techniques in ordinary differential equations, with the emphasis on nonlinear problems. Second, it is hoped that the material is presented in a way that will prepare the reader for intelligent study of the current literature and for research in differential equations. Third, in order not to lose sight of the applied side of the subject, considerable space has been devoted to specific analytical methods which are presently widely used in the applications. Since the emphasis throughout is on nonlinear phenomena, the global theory of two-dimensional systems has been presented immediately after the fundamental theory of existence, uniqueness, and continuous dependence. This also has the advantage of giving the student specific examples and concepts which serve to motivate study of later chapters. Since a satisfactory global theory for general n-dimensional systems is not available, we naturally turn to local problems and, in particular, to the behavior of solutions of differential equations near invariant sets. In the applications it is necessary not only to study the effect of variations of the initial data but also in the vector field. These are discussed in detail in Chapters III and IV in which the invariant set is an equilibrium point. In this way many of the basic and powerful methods in differential equations can be examined at an elementary level. The analytical methods developed in these chapters are immediately applicable to the most widely used technique in the practical theory of nonlinear oscillations, the method of averaging, which is treated in Chapter V. When the invariant set corresponds to a periodic orbit and only autonomous perturbations in the vector field are permitted, the discussion is similar to that for an equilibrium point and is given in Chapter VI. On the other hand, when the perturbations in the vector field are nonautonomous or the invari-

ant set is a closed curve with equilibrium points, life is not so simple. In Chapter VII an attempt has been made to present this more complicated ix

x

PREFACE

and important subject in such a way that the theory is a natural generalization of the theory in Chapter IV. Chapter VIII is devoted to a general method for determining when a periodic differential equation containing a small parameter has a periodic solution. The reason for devoting a chapter to this subject is that important conclusions are easily obtained for Hamiltonian systems in this framework and the method can be generalized to apply to problems in other fields such as partial differential, integral, and functional differential equations. The abstract generalization is made in Chapter IX with an application to analytic solutions of linear systems with a singularity,

but space did not permit applications to other fields. The last chapter is devoted to elementary results and applications of the direct method of Lyapunov to stability theory. Except for Chapter I this topic is independent of the remainder of the book and was placed at the end to preserve continuity of ideas.

For the sake of efficiency and to acquaint the student with concrete applications of elementary concepts from functional analysis, I have presented the material with an element of abstraction. Relevant background

material appears in Chapter 0 and in the appendix on almost periodic functions, although I assume that the reader has had a course in advanced calculus. A one-semester course at Brown University usually covers the

saddlepoint property in Chapter III; the second semester is devoted to selections from the remaining chapters. Throughout the book I have made suggestions for further study and have provided exercises, some of which are difficult. The difficulty usually arises because the exercises are introduced when very little technique has been developed. This procedure was followed to permit the student to develop his own ideas and intuition. Plenty of time should be allowed for the exercises and appropriate hints should be given when the student is prepared to receive them. No attempt has been made to cover all aspects of differential equations. Lack of space, however, forced the omission of certain topics that contribute to the overall objective outlined above; for example, the general subject of boundary value problems and Green's functions belong in the vocabulary of every serious student of differential equations. This omission is partly justified by the fact that this topic is usually treated in other courses in applied mathematics and, in addition, excellent presentations are available in the literature. Also, specific applications had to be suppressed, but individuals with special interest can -easily make the correlation with the theoretical results herein. I have received invaluable assistance in many conversations with my colleagues and students at Brown University. Special thanks are due to C. Olech for his direct contribution to the presentation of two-dimensional systems, to M. Jacobs for his thought-provoking criticisms of many parts of

PREFACE

xi

the original manuscript, and to W. S. Hall and D. Sweet for their comments. I am indebted to K. Nolan for her endurance in the excellent preparation of the manuscript. I also wish to thank the staff of Interscience for being so efficient and cooperative during the production process.

Jack K. Hale Providence, Rhode Island September, 1969

Preface to Revised Edition For this revised edition, I am indebted to several colleagues for their assistance in the elimination of misprints and the clarification of the presentation. The section on integral manifolds has been enlarged to include a more detailed discussion of stability. In Chapter VIII, new material is included on Hopf bifurcation, bifurcation with several independent parameters and subharmonic solutions. A new section in Chapter X deals with Wazewski's principle. The Appendix on almost periodic functions has been completely rewirtten using the modern definition of Bochner. Jack K. Hale April1980

Contents

CHAPTER 0.

Mathematical preliminaries 0.1. Banach spaces and examples 0.2. Linear transformations 0.3. Fixed point theorems

1 1

3 4

CHAPTER I.

General properties of differential equations

I.1. Existence 1.2. Continuation of solutions 1.3. Uniqueness and continuity properties 1.4. Continuous dependence and stability 1.5. Extension of the concept of a differential equation 1.6. Differential inequalities 1.7. Autonomous systems-generalities 1.8. 1.9.

Autonomous systems-limit sets, invariant sets Remarks and suggestions for further study

CHAPTER II.

Two dimensional systems 11.1.

Planar two dimensional systems-the Poincare-

11.2.

Bendixson theory Differential systems on a torus Remarks and suggestions for further study

11.3.

51

51

64 76

CHAPTER III.

Linear systems and linearization 111.1.

General linear systems

78 79

xiv

CONTENTS

Stability of linear and perturbed linear systems nth Order scalar equations 111.4. Linear systems with constant coefficients 111.5. Two dimensional linear autonomous systems III.6. The saddle point property 111.7. Linear periodic systems III.8. Hill's equation 111.9. Reciprocal systems III.10. Canonical systems III.11. Remarks and suggestions for further study 111.2.

111.3.

83 89 93 101

106 117 121 131

136 142

CHAPTER IV.

Perturbations of noncritical linear systems IV.1. IV.2. IV.3. IV.4. IV.5 W.6

Nonhomogeneous linear systems Weakly nonlinear equations-noncritical case The general saddle point property More general systems The Duffing equation with large damping and large forcing Remarks and extensions

144 145 154 156 162 168 171

CHAPTER V.

Simple oscillatory phenomena and the method of averaging

V.I. Conservative systems V.2. Nonconservative second order equations-limit cycles V.3. Averaging V.4. The forced van der Pol equation V.5. Duffing's equation with small damping and small harmonic forcing V.6. The subharmonic of order 3 for Duffing's equation V.7. Damped excited pendulum with oscillating support V.8. Exercises V.9. Remarks and suggestions for further study

175 176 184 190 198 199 206 208 210 211

CHAPTER VI.

Behavior near a periodic orbit

213

VI.I. A local coordinate system about an invariant closed curve

214

CONTENTS

VI.2. Stability of a periodic orbit VI.3. Sufficient conditions for orbital stability in two dimensions VI.4. Autonomous perturbations VI.5. Remarks and suggestions for further study

xv 219

'224 226

227

CHAPTER VII.

Integral manifolds of equations with a small parameter

VII.1. Methods of determining integral manifolds VII.2. Statement of results VII.3. A " nonhomogeneous linear " system VII.4. The mapping principle VII.5. Proof of Theorem 2.1 VII.6. Stability of the-perturbed manifold VII.7. Applications VII.8. Exercises VII.9. Remarks and suggestions for further study

229 231

236 239

245 247

248 250 254 256

CHAPTER VIII.

Periodic systems with a small parameter

VIII.!. A special system of equations VIII.2. Almost linear systems VIII.3. Periodic solutions of perturbed autonomous equations VIII.4. Remarks and suggestions for further study

258 259 275 294 296

CHAPTER IX.

Alternative problems for the solution of functional equations

298

IX.!. Equivalent equations IX.2. A generalization

299

IX.3. IX.4. IX.5. IX.6.

303

Alternative problems Alternative.problems for periodic solutions The Perron-Lettenmeyer theorem Remarks and suggestions for further study

302

304 307 309

CONTENTS

xvi

CHAPTER X.

The direct method of Liapunov

311

X.I. Sufficient conditions for stability and instability in autonomous systems X.2. Circuits containing Esaki diodes X.3. Sufficient conditions for stability in nonautonomous systems X.4. The converse theorems for asymptotic stability X.5. Implications of asymptotic stability X.6. Wazewski's principle X.7. Remarks and suggestions for further study

311

320 324 327 331 333 338

APPENDIX

Almost periodic functions References

339 352

Index

360

CHAPTER 0 Mathematical Preliminaries

In this chapter we collect a number of basic facts from analysis which play an important role in the theory of differential equations. 0.1. Banach Spaces and Examples

Set intersection is denoted by n, set union by u, set inclusion bye and x e S denotes x is a member of the set S. R (or C) will denote the real (or complex) field. An abstract linear vector space (or linear space) £' over R (or C) is a collection of elements {x, y, ... } such that for each x, yin X, the sum x + y

is defined, x + y e 27, x + y = y + x and there is an element 0 in E' such that x + 0 = x for all x e X. Also for any number a, b e R (or C), scalar multiplica-

tion ax is defined, ax a E' and 1 x = x, (ab)x = a(bx) = b(ax), (a + b)x = ax + by for all x, y e X. A linear space E is a normed linear space if to each x in E', there corresponds a real number jxj called the norm of x which satisfies (i) (ii) (iii)

jxj >0 for x 0, 101 =0;. Ix + yl < jxj + jyj (triangle inequality); laxl= lai lxlfor all a in R (or C) and x in X.

When confusion may arise, we will write I x for the norm function on X.

A sequence {xn} in a normed linear space E' converges to x in X if lim, I xn - xi = 0. We shall write this as lim xn = x. A sequence {xn} in X'is a Cauchy sequence if. for every e > 0, there is an N(s) > 0 such that jxn - x,nl < e if n, m >_ N(s). The space 2' is complete if every Cauchy sequence in X converges to an element of X. A complete normed linear space is a Banach space. The s-neighborhood of an element x of a normed linear space E' is {y in X: y - xj < e}. A set S in ° ' is open if for every x e S, an e-neighborhood of x is also contained in X. An element x is a limit point of a set S if each e-neighborhood of x contains points of S. A set S is closed if it contains its limit points. The closure of a set S is the union of S and its limit points. A set S is dense in E' if the closure of S is X. If S is a subset of E', I

ORDINARY DIFFERENTIAL EQUATIONS

2

A is a subset of R and Va, a e A is a collection of open sets of X such that Ua E A Va S. then the collection Va is called an open covering of S. A set S in . is compact if every open covering of S contains a finite number of open '

sets which also cover S. For Banach spaces, this is equivalent to the following: a set S in a Banach space is compact if every sequence {xn}, xn E S, contains a subsequence which converges to an element of S. A set S in . 1 ' is bounded if there exists an r > 0 such that S c {x e 2C: IxI < r}. Example 1.1. Let Rn(Cn) be the space of real (complex) n-dimensional column vectors. For a particular coordinate system, elements x in Rn(Cn) will

be written as x = (xi, ... , xn) where each xj is in R(C). If x = (xl, ... , xn), y = (yl, ..., yn) are in Rn(Cn), then ax + by for a, b in R(C) is defined to be (axl + by,, ..., axn + byn). The space Rn(Cn) is clearly a linear space. It is a Banach space if we choose IxI, x = col(xl, ..., xn), to be either supilxil, Yi Ixil or [Ei IxiI2]4. Each of these norms is equivalent in the sense that a sequence converging in one norm converges in any of the other norms. Rn(Cn)

is complete because convergence implies coordinate wise convergence and R(C) is complete. A set S in Rn(Cn) is compact if and only if it is closed and bounded. EXERCISE 1.1. If E is a finite dimensional linear vector space and I I, are two norms on E, prove there are positive constants m, M such that

m I xI < jjxjj < M I xI for all x in E.

Example 1.2. Let D be a compact subset of Rm [or Cm] and %(D, Rn) [or '(D, Cn)] be the linear space of continuous functions which take D into Rn [or Cn]. A sequence of functions (On, n =1, 2, ... } in W(D, Rn) is said to converge uniformly on D if there exists a function 0 taking D into Rn

such that for every e > 0 there is an N(e) (independent of n) such that n(x) - O(x)l < e for all n >_ N(e) and x in D. A sequence Jon) is said to be uniformly bounded if there exists an M > 0 such that 10n(x)I 0, there is a 8 > 0 such that l

n =1 , 2, ... , - gn(y)I < e, if Ix - yi < 8, x, y in D. A function f in '(D, Rn) is said to be Lipschitzian in D if there is a constant K such that I f (x) - f (y)I < KI x - yI for all I

x, y, in D. The most frequently encountered equicontinuous sequences in '(D, Rn) are sequences {tbn} which are Lipschitzian with a Lipschitz constant independent of n. LEMMA 1.1. (Ascoli-Arzela). Any uniformly bounded equicontinuous sequence of functions in r(D. Rn) has a subsequence which converges uniformly on D.

MATHEMATICAL PRELIMINARIES

3

LEMMA 1.2. If a sequence in '(D, Rn) converges uniformly on D, then the limit function is in '(D, Rn).

If we define 101 =maxIO(x)I, 2ED

then one easily shows this is a norm on W(D, Rn) and the above lemmas show

that '(D, Rn) is a Banach space with this norm. The same remarks apply to

'(D, Cn). EXERCISE 1.2.

Suppose m = n = 1. Show that le (D, R) is a normed

linear space with the norm defined by III II = f IO(x)I dx.

Give an example to show why this space is not complete. What is the completion of this space? 0.2. Linear Transformations

A function taking a set A of some space into a set B of some space will be referred to as a transformation or mapping of A into B. A will be called the domain of the mapping and the set of values of the mapping will be called the range of the mapping. If f is a mapping of A into B, we simply write f : A -* B and denote the range off by f (A). If f : A -* B is one to one and continuous together with its inverse, then we say f is a homeomorphism of A onto B. If .s, GJ are real (or complex) Banach spaces and f: , ' -* ON is such that f (alxl + a2 x2) = al f (xi) + a2 f (x2) for all xl, x2 in . and all real (or complex) numbers al, a2, then f is called a linear mapping. A linear mapping f of . ' into °J is said to be bounded if there is a constant K such that if (x)I u < KI xI, for

all x in .. LEMMA 2.1.

f:

Suppose

', 9 are Banach spaces. A linear mapping

-->9 is bounded if and only if it is continuous. EXERCISE 2.1.

Prove this lemma.

EXERCISE 2.2.

Show that each linear mapping of Rn (or Cn) into

R"n (or Cm) can be represented by an m x n real (or complex) matrix and is therefore necessarily continuous. The norm I f I of a continuous linear mapping f: '-*OJ is defined as

IfI =sup{Ifxiu: IxIX =1}. It is easy to show that I f I defined in this way satisfies the properties (i)-(iii)

4

ORDINARY DIFFERENTIAL EQUATIONS

in the definition of a norm and also that

for alix in T.

IfxIy -. The fact that f is bounded in a neighborhood of (w, y) implies x is uniformly continuous on [a, w) and x(t) -*y as t - c o-. Thus, there is an extension of x to the interval [a, co + a]. Since w + a > co, this is a contradiction and shows there is a tU such that (t,x(t)) is not in U for tp < t < w. Since Uis ap arbitrary compact set, this proves (t,x(t)) tends to the boundary of D. The proof of the theorem is complete. EXERCISE 2.1. For t, x scalars, give an example of a function f (t, x) which is defined and continuous on an open bounded connected set D and

18

ORDINARY DIFFERENTIAL EQUATIONS

yet not every noncontinuable solution 0 of (1.1) defined on (a, b) has 0(a + 0),

0(b - 0) existing. The above continuation theorem can be used in specific examples to verify that a solution is defined on a large time interval. For example, if it is desired to.show that a solution is defined on an interval [to, 00), it is sufficient

to proceed as follows. If the function f (t, x) is continuous for t in (t1, 00),

tl_ to and y such that P < y < a and define the rectangle Dl as Dl ={(t, x): to < t < T, jxj < y}. Then f (t, x) is bounded on Dl and the continuation theorem implies that the solution x(t) can be continued to the boundary of Dl. But y > S implies that x(t) must reach this boundary by reaching the face of the rectangle defined by t = T. Therefore x(t) exists for to < t < T. Since T is arbitrary, this proves the assertion.

1.3. Uniqueness and Continuity Properties

A function f (t, x) defined on a domain D in Rn+I is said to be locall li schitzian in x if for any closed bounded set U in D there is a k = kU such that If (t, x) -f (t, y)j < k Ix - y for (t, x), (t, y) in U. If f (t, x) has continuous first partial derivatives with respect to x in D, then f (t, x) is locally lipschitzian in x.

If f (t, x) is continuous in a domain D, then the fundamental existence theorem implies the existence of at least one solution of (1.1) passing through a given point (to, xo) in D. Suppose, in addition, there is only one such solution x(t, to, xo) through a given (to, xo) in D. For any (to, xo) e D, let (a(to, xo), b(to, xo)) be the maximal interval of existence of x(t, to, xo) and let E c Rn+2 be defined by E = {(t, to, xo) : a(to, xo) < t < b(to, xo), (to, xo) a D}. The trajectory through (to, xo) is the set of points in Rn+I given by (t, x(t, to, xe))

for t varying over all possible values for which (t, to, xo) belongs to E. The set E is called the domain of definition of x(t, to, xo). The basic existence and uniqueness theorem under the hypothesis that f (t, x) is locally lipschitzian in x is usually referred to as the Picard-Lindeld f theorem. This result as well as additional information is contained in THEOREM 3.1. If f (t, x) is continuous in D and locally lipschitzian with respect to x in D, then for any (to, xo) in D, there exists a unique solution x(t, to, xo), x(to, to, xo) = xo, of (1.1) passing through (to, xo). Furthermore,

GENERAL PROPERTIES OF DIFFERENTIAL EQUATIONS

19

the domain E in Rn+2 of definition of the function x(t, to, xo) is open and x(t, to, xo) is continuous in E. PROOF.

Define Ia = Ia(to) and B(a, f, to, xo) as in the proof of Theorem

1.1. For any given closed bounded subset U of D choose positive a, P so that B(a, S, to, xo) belongs to D for each (to, xo) in U and if V = u {B(a, P, to, xo) ; (to, xo) in U),

then the closure of V is in D. Let M = sup{I f (t, x) I , (t, x) in V} and let k be the lipschitz constant of f (t, x) with respect to x on V. Choose &, P so

that 0 < & < a, 0 <

P, M& _ to. The case s < to is treated in the same manner. This implies the closed set U = {(t, x(t, to, x0), to < t< s } belongs to E. Therefore, we can apply the previous results to see that x(t, , , )

20

ORDINARY DIFFERENTIAL EQUATIONS

is a continuous function of (t, , q) for It - fI < a, (6, -q) in U. There exists an integer k such that to + k& > s >_ to + (k - 1)&. From uniqueness, we have x(t + to + &, to, xo) = x(t + to + &, to + &, x(to + a, to, xo)) for any t. But the

previous remarks imply this function is continuous for Itl < a. Therefore, x(e, to, xo) is continuous for 16 -toI 0 there is a 81 > 0 such that Ix(t, s,

E

A) -x(t, t0, x0, Ao) I 0 and any to >_ 0, there is a 8 = 8(e, to) such that I xoI < 8 implies

jx(t, to, xo) I < e for t e [to, oo). The solution x = 0 is uniformly stable if it is stable and 8 can be chosen independent of to >_ 0. The solution x = 0 is called

asl/mvtotically stable if it is stable and there exists a b = b(to) such that ixol < b implies- l x(t, to, xo) I -*0 as t -moo. The solution x = 0 is 4niformly, asymvtotically stable if it is uniformly stable, b in the definition of s,srmptotic stability can be chosen independent of to >_ 0, and for every rl > 0 there is a

T(i) > 0 such that Ixol < b implies jx(t, to, xo)j _ to + T(-q). The solution x = 0 is unstable if it is not stable.

GENERAL PROPERTIES OF DIFFERENTIAL EQUATIONS

27

Pictorially, stability is the same as in the above diagram except the solution must remain in the infinite cylinder of radius e for t ? to.

We can discuss the stability and asymptotic stability of any other solution x(t) of the equation by replacing x by x + y and discussing the zero solution of the equation y =f (t, x + y) -f (t, x). The definitions of stability of an arbitrary solution t(t) are the same as above except with x replaced by x - x(t).

LEMMA 4.1. 1 If f is either independent of t or periodie in t, then the solution x = 0 of (1.1) being stable (asymptotically stable) implies the solution x =0 of (1.1) is uniformly stable (uniformly asymptotically stable). EXERCISE 4.1.

Prove Lemma 4.1.

EXERCISE 4.2.

Discuss the stability and asymptotic stability of every

solution of the equations z = -x(1 - x), x + x = 0, and .x + 2-1[x2 + (x4 + 4x2)'/2]x = 0. The latter equation has the family of solutions x = c sin(ct + d) where c, d are arbitrary constants. Does stability defined in the above way depend on to in the sense that a

solution x = 0 may be stable at one value of to and not at another? The answer is no! For tl 0 such that ixol < 8(to, e) implies jx(t, to, xo) < e, t >_ to. Continuity with respect to initial data implies the existence of a 81 = 81(t1, e, to, S) > 0 so small that 1xl < 81(t1, e) implies Ix(t, ti, xl) < 8(to, e), t1 < t< to. Then ix(t, ti, xl)l < e fort >_ t1, provided that 1xii _ to, it is not quite so obvious. Let V(ti, e) _ {x in Rn: x = x(ti, to, xo) for xo in the open ball of radius 8(to, s) centered at zero}. Since the mapping x(tl, to, ) is a homeomorphism, there exists a 81(t1, e) such that {x: Ixj ti and 1xiI < 8(t1, e); that is, stability at t1. EXERCISE 4.3. In the above definition of asymptotic stability of the solution x = 0, we have supposed that x = 0 is stable and solutions with

initial. values

neighborhood of zero approach zero as t ---> oo. Is it possible

to have the latter property and also have the solution x = 0 unstable? Show this cannot happen if x is a scalar. Give an example in two dimensions where all solutions approach zero as t --> oo and yet the solution x = 0 is unstable. Is it possible to give- such an example in two dimensions for an equation whose right hand sides are independent of t?

It is not appropriate at this time to have a detailed discussion of stability, but we will continually bring out more of the properties of this concept.

28

ORDINARY DIFFERENTIAL EQUATIONS

1.5. Extension of the Concept of a Differential Equation

In Section 1.1, a differential equation was defined for continuous vector fields f. As an immediate consequence, the initial value problem for (1.1) is equivalent to the integral equation c

x(t) = xo +

(5.1)

f

f (s, x(s)) ds.

to

For f continuous, any solution of this equation automatically possesses a continuous first derivative. On the other hand, it is clear that (5.1) will be meaningful for a more general class of functions f if it is not required that x have a continuous first derivative. The purpose of this section is to make these notions precise for a class of functions f. Suppose D is an open set in Rn+1 and f : D - . Rn is not necessarily continuous. Our problem is to find an absolutely continuous function x defined on a real interval I such that (t, x(t)) e D for t, in I and x(t) =f (t, x(t))

(5.1)

for all t in I except on a set of Lebesgue measure zero. If such a function x and interval I exist, we say x is a solution of (5.1). A solution of (5.1) through (to, xo) is a solution x of (5.1) with x(to) = xo . We will not repeat the phrase "except on a set of Lebesgue measure zero" since it will always be clear that this is understood. Suppose D is an open set in Rn+1. We say that f: D - Rn satisfies the Caratheodory conditions on D if f is measurable in t for each fixed x, continuous in x for each fixed t and for each compact set U of D, there is an integrable function mu(t) such that I f (t, x)I < 'mv(t),

(5.2)

(t, x) e U.

For functions f which satisfy the Caratheodory conditions on a domain D, the conclusions of Sections 1 and 2 carry over without .change. If the function f (t, x) is also locally Lipschitzian in x with a measurable Lipschitz function, then the uniqueness property of the solution remains valid. These results are stated below, but only the details of the proof of the existence theorem are given, since the other proofs are essentially the same. THEOREM.5.1. (Caratheodory). If D is an open set in Rn+1 and f satisfies the Caratheodory conditions on D, then, for any (to, xo) i1f D, there is a solution of (5.1) through (to, xo).

Suppose a, 9 are positive numbers chosen so that the rectangle {(t, x): It - to a, I x - xol 0 and f A(s) ds = + oo, then the solution x = 0 is asymptotically stable. EXERCISE 6.1.

If A(t) > 0 and f ooA(s) ds = + co for all to , is the` solution to

x = 0 of the previous discussion uniformly asymptotically stable? Discuss the case where A(t) is not of fixed sign.

GENERAL PROPERTIES OF DIFFERENTIAL EQUATIONS

35

EXERCISE 6.2. Suppose f: Rn+1--> Rn is continuous and there exists a positive definite matrix B such that x Bf (t, x) < -A(t)x x for all t, x where A(t) is continuous for tin (- oo, oo). Prove that any solution of the equation z =f (t, x), x(to) = xo, exists on [to, oo) and give sufficient conditions for stability and asymptotic stability. (Hint: Find the derivative of the function V(x) = x Bx along solutions and use the fact that there is a positive constant a such that x Bx >_ zx - x for all x.)

Consider the equation x =f (t, x), If (t, x)I < 0(t) Ix) for all t, :c in R X R, f - 0(t) dt < oo. (a) Prove that every solution approaches a constant as t - oo. (b) If, in addition, EXERCISE 6.3.

f(t, x) -f (t, y)I < 4(t) IX - yl for all x, y, prove there is a one to one correspondence between the initial values and the limit values of the solution. (c) Does the above result imply anything for the equation

x = -x + a(t)x,

f o Ia(t)I dt < oo?

(Hint: Consider the transformation x = e-ty.) (d) Does this imply anything about the system xl = X2,

x2 = - xl + a(t)xl,

f 00 Ia(t)j dt < co,

where xj, x2 are scalars? EXERCISE 6.4.

Consider the initial value problem

z + a(z, z)z + P(z) = u(t),

z(0) _ 6,

z(0) = 71,

with a(z, w), g,(z) continuous together with their first partial derivatives for all z, w, u continuous and bounded on (- oo, co), a > 0, zf(z) >_ 0. Show there is one and only one solution to this problem and the solution can be defined on [0, oo). Hint: Write the equation as a system by letting z = x, z = y, define V (X, y.) = y2/2 + f o f(s) ds and study the rate of change of V(x(t), y(t)) along the solutions of the two dimensional system. COROLLARY 6.5. Let w(t, u) satisfy the conditions of Theorem 6.1 and in addition be nondecreasing in u. If u(t) is the same function as in Theorem

36

ORDINARY DIFFERENTIAL EQUATIONS

6.1 and v(t) is continuous and satisfies v(t) < va + fa w(s, v(s)) &,

(6.6)

a 0, then integrating by parts in Lemma 6.2 gives

Js) +

`p(t) 0, r > 0 such that the mapping T is a continuously differentiable homeomorphism (or diffeomorphism) of It x S, , I= = {t: Itl - oo and v < Tp.k < tp.k - v. Therefore, there is a

subsequence which we label the same as before such that Tp.k -*TO as k -+ oo and 0 < To < tp - v/2. But this clearly implies that the path yp described by 0(t, p) satisfies 0(7-o, p) = p. This is a contradiction since pq was assumed to be an arc. The path cylinder C is obtained as the union of the arcs of the trajectories p'q' with p' in Ep-1. It remains only to show that this is homeomorphic to a Ep-1 closed cylinder. For I = [0, 1], define the mapping G: x I -> Rn by G(p', s) = 0(stp , p'), where tp is defined above. It is clear that this mapping is a homeomorphism and therefore C is a closed path cylinder. This proves the lemma. Now suppose y is a closed-path. Lemma 7.2 implies y is the orbit of a nonconstant periodic solution O(t,p) of (7.1) of least period t > 0. Take a p,En-i C E"-1 transversal En-1 at p. There1 is another transversalpE"- at p p p such that, for any q1E Ep- , there is a tq > 0, continuously differentiable

in q,x(ltq,q) in En- ,x(t,q) not in En--' for 0 < t < tq, and the mapping F: E"pp`

set F(Ep

X1 [0,1) -+ R' defined by F(q,s) = x(stq,q) is a diffeomorphism. The X [0,1)) is called a path ring enclosing y. We have proved the

following result.

46

ORDINARY DIFFERENTIAL EQUATIONS

LEMMA 7.5.

If y is a closed path, there is a path ring enclosing y.

It may be that a solution of an autonomous equation is not defined for all t in R as the example x = x2 shows. In the applications, one is usually only interested in studying the behavior of the solutions in some bounded set G and it is very awkward to have to continually speak of the domain of defini-

tion of a solution. We can avoid this situation by replacing the original differential equation by another one for which all solutions are defined on (- oo, oo) and the paths defined by the solutions of the two coincide inside G. When the paths of two autonomous differential equations coincide on a set G, we say the differential equations are equivalent on G. LEMMA 7.6.

If f in (7.1) is defined on Rn and G c Rn is open and

bounded, there exists a function g: Rn -* Rn such that z = g(x) is equivalent to (7.1) on G and the solutions of this latter equation are defined on (- co, co). PROOF. If f = (fl, ... , fn), we may suppose without loss of generality that G c {x: I f j(x)I < 1 , j =1, 2, ... , n}. Define g = (gj, ... , gn) by gf = fj oi , where each Oj is defined by 1 1

qj(x) =

fj(x) 1

f1(x)

if I fj(x)I < 1, if fj(x) > 1,

if fj(x)< -1.

Corollary 6.3 implies that g satisfies the conditions of the lemma since lg(x)I is bounded in B.

1.8. Autonomous Systems-Limit Sets, Invariant Sets

In this section we consider system (7.1) and suppose f satisfies enough conditions on Rn to ensure that the solution 0(t, p), (O, p) = p, is defined for all tin B and all p in Rn and satisfies the conditions (i)-(iii) listed at the beginning of Section 1.7.

The orbit y(p) of (7.1) through p is defined by y(p) = {x: x = 0(t,p), -oo _ 0} and the negative semiorbit through p is y -(p) = {x: x = q(t, p), t < 0). If we do not wish to distinguish a particular point on an orbit, we will write y, y+, y for the orbit, positive semiorbit, negative semiorbit, respectively. The positive or w-limit set of an orbit y of (7.1) is the set of points in

GENERAL PROPERTIES OF DIFFERENTIAL EQUATIONS

47

Rn which are approached along y with increasing time. More precisely, a point q belongs to the w-limit set or positive limit set co(y) of an orbit. y if there exists a sequence of real numbers {tk}, tk -a oo as k -->- oo such that 0(tk, p) -*q as k - oo. Similarly, a point q belongs to the ce-limit set or negative limit set a(y) if there is a sequence of real numbers {tk}, tk - - - 00 as k -* oo such that 4,(tk, p) -* q as k -* oo.

It is easy to prove that equivalent definitions of the w-limit set and a-limit set are w(Y) = n Y+(p) = pEV

«(Y) = n Y -(P) = PEY

n

u c(t, p)

n

u o(t, p)

7 E(-00,00)tZT

T E (- 00, 00) t:9 T

where the bar denotes closure. A set M in Rn is called an invariant set f (7.1) if, for any p in M, the solution (t, p) of (7.1 through belongs to M for tin - oo, oo .Any orbit of (7.1) is obviously an invariant set of (7.1). A set M is called positively (negatively) invariant if for each p in M, 0(t, p) belongs to M for t > 0 (t < 0). THEOREM 8.1. The a- and w-limit sets of an orbit y are closed and invariant. Furthermore, if y+(y-) is bounded, then the w-(a-) limit set is nonempty compact and connected, dist(4(t, p), w(y(p))) --0 as t -> oo and dist( (t, p), a(( ))) -* 0 as t--> -oo. PROOF.

The closure is obvious from the definition. We now prove the

positive limit sets are invariant. If q is in w(y), there is a sequence {tn}, to - . ao as n -> oo such that q(tn , p) -* q as n -* oo. Consequently, for any fixed tin (- oo, 00), c(t + to , p) = 0(t, On, p)) -' 0(t, q) as n co from the continuity of 0. This shows that the orbit through q belongs to w(y) or w(y) is invariant. A similar proof shows that a(y) is invariant. If y+ (y) is bounded, then the co- (a-) limit set is obviously nonempty and bounded. The closure therefore implies compactness. It is easy to see that dist(q(t, p), w(y(p))) -->0 as t * oo, dist(o(t, p), a(y(p))) -a0 as t --> - oo. This

latter property clearly implies that w(y) and a(y) are connected and the theorem is proved. COROLLARY 8.1.

The limit sets of an orbit must contain only complete

paths.

A sit M in Rn is called a minimal set of (7.1) if it is nonempty, closed and invariant and has no proper subset which possesses these three properties. LEMMA 8.1.

If A is a nonempty compact, invariant set of (7.1), there is

a minimal set M C A.

48

ORDINARY DIFFERENTIAL EOTTATIONS

PROOF. Let F be a family of nonempty subsets of Rn defined by F = {B: B c A, B .compact, invariant}. For any B1, B2 in F, we say B2 < B1 if B2 c B1. For any F1 c F totally ordered by " < ", let C = nB E F.B. The family F1 has the finite intersection property. Indeed, if B1; B2 are in F1, then either B1 < B2 or B2 < B1 and, in either case, B1 o B2 is nonempty

and invariant or thus belongs to Fl. The same holds true for any finite collection of elements in Fl. Thus, C is not empty, compact and invariant and for each B in F1, C < B. Now suppose an element D of F is such that D < B for each B in Fl. Then D c B for each B in F1 which implies D C C or D < C. Therefore C is the minimum of Fl. Since each totally ordered subfamily of F admits a minimum, it follows from Zorn's lemma that there

is a minimal element of F. It is easy to see that a minimal element is a minimal set of (7.1) and the proof is complete. Let us return to the examples considered in Section 1.7 to help clarify the above concepts. In example 7.1, the co-limit set of every orbit except the orbit consisting of the critical point {0} is empty. The cc-limit set of every orbit is {0}. The only minimal set is {0}. In example 7.2, the w-limit set of the orbits {0 < x < 1}, {x < 0}, is {0}, the cc-limit set of {x > 1}, {0 < x < 1} is {0} and {0} and {1} are both minimal sets. In example 7.3, the w- and x-limit set of any orbit is itself, every orbit is a minimal set and any circular disk about the origin is invariant. In example 7.4, the circle {r =1} and the point {r = 0} are minimal sets, the circle {r =1} is the w-limit set of every orbit except {r = 0}, while the point {r = 01 is the x-limit set of every orbit inside

the unit circle. In example 7.6, the torus r = 1 is a minimal set as well as the circle r = 0, the w-limit set of every orbit except r = 0 is the torus r = 1 and the x-limit set of every orbit inside the torus r = 1 is the circle r = 0. Let us give one other artificial example to show that the w-limit sets do not always need to be minimal sets. Consider r and 0 as polar coordinates which satisfy the equations

sin20+(1-r)3, r(1 - r). The w-limit set of all orbits which do not lie on the sets {r =1} and {r = 01 is the circle r =1. The circle r =1 is invariant but the orbits of the equation on r =1 consist of the points {O = 0}, {0 = rr} and the arcs of the circle {0 < 0 < 7r}, {7r < 0 < 2rr}, The minimal sets on this circle are just the two points {0 = 0}, {0 = 7T}. EXERCISE 8.1. Give an example of a two dimensional system which has an orbit whose w-limit set is not empty and disconnected. THEOREM 8.2.

If K is a positively invariant set of system (7.1) and

K is homeomorphic to the closed unit ball in Rn, there is at least one equilibrium point of system (7.1) in K.

GENERAL PROPERTIES OF DIFFERENTIAL EQUATIONS PROOF.

49

For anyrI > 0, consider the mappingtaking p in K into 0(rI, p)

in K. From Brouwer's fixed point theorem, there is a pI in K such that 0(rI, pi) =pz, and, thus, a periodic orbit of (7.1) of period ri. Choose a sequence rm > 0, rm -*0 as m -)- oo and corresponding points pin such that

q(rm, pm) = pm . We may assume this sequence converges to a p* in K as m -* oo since there is always a subsequence of the pm which converge. For any t and any integer m, there is an integer km(t) such that km(t)rm 5 t < km(t)rm + rm and 0(km(t)rm, pm) =pm for all t since 0(t, pm) is periodic of period rm in t. Furthermore,

l0(t, p*) -p*I < I0(t, p*) - #(t, pm)I + I,(t, Pm) -Pm! + Ipm -p*I = 10(t, p*) - 0(t, pm)I ± 10(t - km(t)rm, pm) -pm1 + I pm -P*I, and the right hand side approaches zero as m oo for all t. Therefore, p* is an equilibrium point of (7.1) and the theorem is proved. Some of the most basic problems in differential equations deal with the characterization of the minimal sets and the behavior of the solutions of the equations near minimal sets. Of course, one would also like to be able to describe the manner in which the w-limit set of any trajectory can be built up from minimal sets and orbits connecting the various minimal sets. In the case of two dimensional systems, these questions have been satisfactorily answered. For higher dimensional systems, the minimal sets have not been completely classified and the local behavior of solutions has been thoroughly discussed only for minimal sets which are very simple. Our main goal in the following chapters is to discuss some approaches to these questions.

1.9. Remarks and Suggestions for Further Study

For a detailed proof of Peano's theorem without using the Schauder theorem, see Coddington and Levinson [1], Hartman [1]. When uniqueness

of trajectories of a differential equation is not assumed, the union of all trajectories through a given point forms a type of funnel. For a discussion of the topological properties of such funnels, see Hartman [1]. There are many other ways to generalize the concept of a differential equation. For example, one could permit the vector field f (t, x) to be continuous in t, but discontinuous in x. Also, f (t, x) could be a set valued function.

In spite of the fact that such equations are extremely important in some applications to control theory, they are not considered in this book. The interested reader may consult Flugge-Lotz [1], Andre and Seibert [1], Fillipov [1], Lee and Marcus [1]. The results on differential inequalities in Section 6 are valid in a much

more general setting. In fact, one can use upper right hand derivatives in

50

ORDINARY DIFFERENTIAL EQUATIONS

place of right hand derivatives, the assumption of uniqueness can be eliminated by considering maximal solutions of the majorizing equation and even some types of vector inequalities can be used. Differential inequalities are also very useful for obtaining uniqueness theorems for vector fields which are not Lipschitzian. See Coppel [1], Hartman [1], Szarski [1], Laksmikantham

and Leela [1]. Sections 7 and 8 belong to the geometric theory of differential equations begun by Poincare [1] and advanced so much by the books of Birkhoff [1],

Lefschetz [1], Nemitskii and Stepanov [1], Auslander and Gottschalk [1]. The presentation in Section 7 relies heavily upon the book of Lefschetz [1].

A function : R X Rn into Rn which satisfies properties (i-iii) listed at the beginning of Section 7 is called a dynamical system. Dynamical systems can

and have been studied in great detail without any reference to differential equations (see Gottschalk and Hedlund [1], Nemitskii and Stepanov [1]). All results in Section 7 remain valid for dynamical systems. However, the proofs are more difficult since the implicit function theorem cannot be invoked. The concepts of Section 8 are essentially due to Birkhoff [1]. The definitions of stability given in Section 4 are due to Liapunov [1]. For other types of stability see Cesari [1], Yoshizawa [2].

CHAPTER II Two Dimensional Systems

The purpose of this chapter is to discuss the global behavior of solutions

of differential equations in the plane and differential equations without critical points on a torus. In particular, in Section 1, the w-limit set of any bounded orbit in the plane is completely characterized, resulting in the famous

Poincare-Bendixson theorem. Then this theorem is applied to obtain the existence and stability of limit cycles for some special types of equations. In Section 2, all possible w-limit sets of orbits of smooth differential equations without singular points on a torus are characterized, yielding the result that the w-limit set of an orbit is either a periodic orbit or the torus itself. Differential equations on the plane are by far the more important of the two types discussed since any system with one degree of freedom is described

by such equations. On the other hand, in the restricted problem of three bodies in celestial mechanics, the interesting invariant sets are torii and, thus, the theory must be developed. Also, as will be seen in a later chapter, invariant torii arise in many other applications.

M. Planar Two Dimensional Systems-The Poincare-Bendixson Theory In this section, we consider the two dimensional system (1.1)

z =f (x)

where x is in R2, f : R2 -a R2 is continuous with its first partial derivatives and

such that the solution 0(t, p), 0(0, p) =p, of (1.1) exists for -oo R2 is a homeomorphism.

The beautiful results for 2-dimensional planar systems are made possible

because of the Jordan curve theorem which is now stated without proof. Recall that a Jordan curve is the homeomorphic image of a circle.

52

ORDINARY DIFFERENTIAL EQUATIONS

JORDAN CURVE THEOREM. I Any Jordan curve J in R2 separates the plane; more precisely, R2\J = Se u Si where Se and Si are disjoint open sets, Se is unbounded and called the exterior of J, Si is bounded and called the interior of J and both sets are arcwise connected. A set B is arcwise connected if p, q in B implies there is an arc pq joining p and q which lies entirely in B. Let p be a regular point, L be a closed transversal containing p, L° be its interior,

V = {p in L°: there is a tp > 0 with 7!(t, p) in L° and 0(t, p) in R2\L for 0 < t < tp}, and let W =h-1(V) where h: [-1, 1] -3 L is a homeomorphism. Also, let g: W -* (-1, 1) be defined by g(w) = h-1c6(th(,,,) , h(w)). See Fig. 1.1.

Figure II.I.I LEMMA 1.1. The set W is open, g is continuous and increasing on W and the sequence {gk(w)}, k = 0, 1, ... , n < oo is monotone, where gk(w) _ g(gk-1(w)), k = 1, 2, ... , g°(w) = w. PROOF. For any p in V c L° let q = 0(tp, p) in L°. From Section 1.7, we have proved that the are pq of the path through p can be enclosed in an open path cylinder with pq as axis and the bases of the cylinder lying in the interior L° of the transversal L. This proves W is open. From continuity with respect to initial data, tp is continuous and we get continuity of g. To prove the last part of the lemma, consider the Jordan curve J given by C = {x: x = 0(t, p), 0 oo. But from Section 1.7, there must be a path cylinder containing po such that any orbit passing sufficiently near po must contain an are which crosses the transversal L at some point. Therefore, there exist points qk = 0(tk, p) in Lo, tk -> co as k -*. oo such that qk -> po as k --> oo. But Lemma 1.1 implies that the qk approach po monotonically in the sense that h-I(qk) is a monotone sequence. Suppose now po is any other point in w(y) n Lo. Then the same argument holds to get a sequence qk -->p0 monotonically. Lemma 1.1 then clearly imples that po = po and the corollary is proved. COROLLARY 1.2.

If y+ and w(y+) have a regular point in common, then

y+ is a periodic orbit. PROOF. If po in y+ n w(y+) is regular, there is a transversal of (1.1) containing po in its interior. From Corollary 1.1, if w(y+) 0 y+, there is a

sequence qk = 0A, p) -->po monotonically. Since po is in y+, this contradicts Lemma 1.1. Corollary 1.1 therefore implies the result. THEOREM 1.1. If M is a bounded minimal set of (1.1), then M is either a critical point or a periodic orbit. PROOF. If y is an orbit in M, then a(y) and w(y) are not empty and belong to M. Since a(y) and w(y) are closed and invariant we have a(y) = w(y) = M. If M contains a critical point, then it must be the point itself

ORI)INARY DIFFERENTIAL EQUATIONS

54

since, it is equal to w(y) for some y. If M = w(y) does not contain a critical point, then y c w(y) implies y and w(y) have a .regular point in common which implies by Corollary 1.2 that y is periodic. Therefore y = w(y) = M and this proves Theorem 1.1. LEMMA 1.2. If w(y+) contains regular points and also a periodic orbit yo, then w(y+) = yo. PROOF.

If not, then the connectedness of w(y+) implies the existence

p in w(y+)\yo and a po in yo such that p. ->po as n --,. oo. of a sequence Since po is regular, there is a closed transversal L such that po is in the interior

Lo of L. From Corollary 1.1, w(y+) r Lo = {po}. From the existence of a path cylinder in Section 1.7, there is neighborhood N of po such that any orbit entering N must intersect Lo. In particular, y(p.n) for n sufficiently large must

intersect Lo. But we know this occurs at po. Thus p,a belongs to yo for n sufficiently large which is a contradiction. THEOREM 1.2

(Poincare-Bendixson Theorem).

If y+ is a bounded

positive semiorbit and w(y+) does not contain a critical point, then either (i)

Y+ = w(Y+),

or

(ii)

w(Y+) =

Y+\Y+,

In either case, the w-limit set is a periodic orbit. The same result is valid for a negative semiorbit.

PROOF. By assumption and Theorem 1.8.1, w(y+) is nonempty, compact invariant and contains regular points only. Therefore, by Lemma 1.8.1, there is a bounded minimal set M in w(y+) and M contains only regular points. Theorem 1.1 implies M is a periodic orbit yo. Lemma 1.2 now implies the theorem. An invariant set M of (1.1) is said to be stable if for every e-neighborhood U, of M there is a 8-neighborhood U6 of M such that p in U6 implies y+(p) in U,. M is said to be asymptotically stable if it is stable and in addition there is a b > 0 such that p in Ub implies dist(q,(t, p), M) --)- 0 as t -goo. If M is a periodic orbit, one can also define stability from the inside and outside of M in an obvious manner. COROLLARY 1.3. For a periodic orbit yo to be asymptotically stable it is necessary and sufficient that there is a neighborhood 0 of yo such that w(y(p)) =yo for any p in G.

PROOF. We first prove sufficiency. Clearly dist(o(t, p), yo) --)-0 as t ->oo

for every p in G. Suppose L is a transversal at po in yo and suppose p is in

TWO DIMENSIONAL SYSTEMS

55

G n Se , q is in G n S$ , where Se and Si are the exterior and interior of yo, respectively. From Corollary 1.1, there are sequences p), qk= 4 (tk , q) in L approaching po as k -* oo. Consider the neighborhood Uk of Yo which lies between the curves given by the are pkpk+l of y(p) and the segment of L between pk and Pk+1 and the are gkgk+l of y(p) and the segment of L between qk and qk+1. Uk is a neighborhood of yo. The sequences {tk}, {tk} satisfy tk+l - tk --* a, tk+1 -t; --> a as k-* oo where a is the period of yo. This follows from the existence of a path ring around yo. Continuity with respect to initial data then implies for any given e-neighborhood Ue of yo, there is a k sufficiently large so that p in Uk implies q(t, p) in Ue for t >_ 0 and yo is stable. To prove the converse, suppose yo is asymptotically stable. Then there must exist a neighborhood G of yo which contains no equilibrium points and G\yo contains no periodic orbits. The Poincare-Bendixson theorem implies the w-limit set of every orbit is a periodic orbit. Since yo is the only such orbit in G, this proves the corollary. COROLLARY 1.4. Suppose yl, Y2 are two periodic orbits with Y2 in the interior of yl and no periodic orbits or critical points lie between yl and Then both orbits cannot be asymptotically stable on the sides facing one another. Y2.

PROOF. Suppose yl, y2 are stable on the sides facing one another. Then there exist positive orbits yl, y2 in the region between yl, Y2 such that yl = Yi\Yl, Y2 = 2\ Ys For any pl in yl, P2 in Y2 construct transversals L1, L2 .

There exist pj 0 pi in yl n L1, p2 0 p2 in y2 n L2. Consider the region S bounded by the Jordan curve consisting of the arc pip" of y, and the segment of the transversal L1 between pi and pi and the curve consisting of the are

peps of y2 and the segment of the transversal L2 between p2 and p2 (see Fig. 1.2). The region S contains a negative semiorbit. Thus, the PoincareBendixson Theorem implies the existence of a periodic orbit in this region. This contradiction proves the corollary. THEOREM 1.3. Let y+ be a positive semiorbit in a closed bounded subset K of R2 and suppose K has only a finite number of critical points. Then one of the following is satisfied: (i) w(y+) is a critical point; (ii) w(y+) is a periodic orbit; (iii) w(y+) contains a finite number of critical points and a set of orbits yi with a(Vi) and w(ya) consisting of a critical point for each orbit y{ . See Fig. 1.3. PROOF. co(y+) contains at most a finite number of critical points. If co(y+) contains no regular points, then it must be just one point since it is

ORDINARY DIFFERENTIAL EQUATIONS

56

Figure 11.1.2

(i)

(ii)

Figure 11.1.3

connected. This is case (i). Suppose w(y+) has regular points and also contains a periodic orbit yo. Tlien w(y+) = yo from Lemma 1.2. Now suppose w(y+) contains regular points and no periodic orbits. Let yo be an orbit in w(y+). Then w(yo) c w(y+). If po in w(yo) is a regular point and L is a closed transversal to po with interior Lo, then Corollary 1.1 implies w(y+) r Lo = w(yo) n Lo = {p0} and yo must meet Lo at some qo. Since yo belongs to w(y+) we have qo = po which implies by Corollary 1.2 that yo is periodic. This contradiction implies w(yo) has no regular points. But, w(yo)

is connected and therefore consists of exactly one point, a critical point. A similar argument applies to the a-limit sets and the theorem is proved. COROLLARY 1.5.

If y+ is a positive semiorbit contained in a compact set

in S2 and w(y+) contains regular points and exactly one critical point po, then there is an orbit in w(y+) whose a- and w-limit sets are {po}. We now discuss the possible behavior of orbits in a neighborhood of a periodic orbit. Let yo be a periodic orbit and Lo be a transversal at po in yo,

TWO DIMENSIONAL SYSTEMS

57

h: (-1, 1) -. Lo be a homeomorphism with h(0) = po. If g is the function defined in Lemma 1.1, then g(O) =0 since yo is periodic. Since the domain W of definition of g is open, 0 is in W, g is continuous and increasing, there is an E > 0 such that g is defined and g(w) > 0 for w in (0, e) and g(w) < 0 for w

in (- e, 0). We discuss in detail the case g(w) > 0 on (0, e) and the case g(w) < 0 on (- e, 0) is treated in a similar manner. Three possibilities present themselves. There is an rI, 0 < eI < e, such that (i) (ii) (iii)

g(w) < w for w in (0, El); g(w) > w for w in (0, el); g(w) =w for a sequence wn >0, W n -* 0 as n-* oo.

In case (i), gk(w) is defined for each k > 0, is monotone decreasing and gk(w) -* 0 as k -* oo. In fact, it is clear that gk(w) is defined for k > 0. Lemma 1.1 states that gk(w) is-monotone and the hypothesis implies this sequence is decreasing. Therefore, gk(w) -awo > 0 as k -* oo. But, this implies g(wo) = wo and therefore wo = 0. Similarly, in case (ii), if we define g-k(w) to be the inverse of gk(w) then g-k(w), is defined for each k > 0, is decreasing and g-k(w) --> 0 as k --> oo.

If we interpret these three cases in terms of orbits and limit sets, we have THEOREM 1.4. If yo is a periodic orbit and G is an open set containing yo, Ge = G n Se , Gi = G n Si where Se and Si are the interior and exterior of yo, then one of the following situations occur: (i) there is a G such that either yo = cu(y(p)) for every p in Ge or yo = a(y(p)) for every p in Ge; (ii) for each G, there is a p in Ge, p not in yo, such that oc(y(p)) = y(p) is a periodic orbit. Similar statements hold for Gi.

We call yo a limit cycle if there is a neighborhood G of yo such that either w(y(p)) = ,yo for every p E G or a(y(p) = ,yo for every p E G.

The Poincare-Bendixson theorem suggests a way to determine the existence of a nonconstant periodic solution of an autonomous differential equation in the plane. More specifically, one attempts to construct a domain D in R2 which is equilibrium point free and is positively invariant; that is, any solution of (1.1) with initial value in D remains in D for t z 0. In such a case, we are assured that D contains a positive semiorbit + and thus a periodic solution from the Poincare-Bendixson theorem Furthermore, if we can ascertain that there is only one periodic orbit in D, it will be asymptotically stable from Theorem 1.4 and Corollary 1.3. These ideas are illustrated for the Lienard type equation (1.2)

ii + g(a)it + u = 0

ORDINARY DIFFERENTIAL EQUATIONS

58

where g(u) is continuous and the following conditions are satisfied: (1.3)

G(u) = def fog(s) ds is odd in u, oo as Jul --± oo and there is a > 0 such that G(u) > 0 (b) G(u) for u> S and is monotone increasing. (c) There is an a > 0 such that G(u) G(u), decreasing if v < G(u) and the function v = v(t) is decreasing if u > 0, increasing if u < 0. Also, the slopes of the paths v = v(u) described by (1.5) are horizontal on the v-axis and vertical on the curve v = G(u). These facts and hypothesis (1.3b) on G(u) imply that a solution of (1.4) with initial value A = (0, vo) for vo sufficiently large describes an orbit with an arc of the general shape shown in Fig. 1.4.

Figure 11.1.4

59

TWO DIMENSIONAL SYSTEMS

Observe that (u, v).a solution of (1.4) implies (-u, -v) is also a solution from hypothesis (1.3a). Therefore, if we know a path ABECD exists as in Fig. 1.4, then the reflection of this path through the origin is another path. In particular, if A = (0, vo), D = (0, -vi), vi < vo, then the complete positive semiorbit of the path through any point A' = (0, vo), 0 < vo < vo must be bounded. In fact, it must lie in the region bounded by the arc ABECD, its reflection through the origin and segments on the v-axis connecting these arcs to form a Jordan curve. The above symmetry property also implies that (1.4) can have a periodic orbit if and only if vI =vo. We show there exists a vo > 0 sufficiently large so that a solution as in Fig. 1.4 exists with A = (0, vo), D = (0, -vi), vi < vo. Consider the function V(u, v) _ (u2 + v2)/2. If u, v are solutions of (1.4) and (1.5), then

(a) W _ -uG(u),

(1.6)

dV _ (b)

(c)

du

dV

uG(u) v - G(u)

= G(u).

Using these expressions, we have

V(D) - V(A) = f

dV = (fAB + f l -uG(u) du + f CDJ V - G(u)

ABECD

G(u) dv

BEC

along the orbits of (1.4). It is clear that this first expression approaches zero monotonically as vo -a oo. If F is any point on the u-axis in Fig. 1.4 between (P, 0) and E, and #(vo) = f 0(u) dv, then BEC

- 0(vo) = - fBEG G(u) dv = f CE B

G(u) dv > f ER

G(u) dv > FJ x FK

where FJ, FK are the lengths of the line segments indicated in Fig. 1.4. For fixed F, FK oo as vo -* oo and this proves q(vo) -- - oo as vo -± oo. Thus, there is a vo such that V(D) < V(A). But this implies vI _ 0 along solutions of (1.4) if Jul < a. Finally, the PoincareBendixson Theorem implies the existence of a periodic solution of (1.4) and we have THEOREM 1.5. If G satisfies the conditions (1.3), then equation (1.2) has a nonconstant periodic solution.

60

ORDINARY DIFFERENTIAL EQUATIONS

Figure 11.1.5

If further hypotheses are made on G, then the above method of proof will yield the existence of exactly one nonconstant periodic solution. In fact, we can prove THEOREM 1.6. If 0 satisfies the conditions (1.3) with a = fl, then equation (1.2) has exactly one periodic orbit and it is asymptotically stable. PROOF. With the stronger hypotheses on 0, every solution with initial value A = (0, vo), vo > 0, has an are of an orbit as shown in Fig. 1.5.

With the notations the same as in the proof of Theorem 1.5 and with E = (uo, 0), we have

V(D) - V(A) = f

G(u) dv > 0,

ABECD

if uo < a. This implies'no periodic orbit can have uo < a. For uo > a, if we introduce new variables x = G(u), y = v to the right of line BC in Fig. 1.5 (this is legitimate since G(u) is monotone increasing in this region), then the are BEC goes into an arc B*E*C* with end points on the y-axis and the second expression 0(vo) = L G(u) dv = fB*E*C* x dy is the EC negative of the area bounded by the curve B*E*C* and the y-axis. Therefore, 0(vo) is a monotone decreasing function of vo. It is easy to check that fAB + fBCG(u)du is decreasing in vo and so V(D) - V(A) is decreasing in vo. Also, in the proof of Theorem 1.5, it was shown that V(D) - V(A) approaches

-- as vo - °°. Therefore, there is a unique vo for which V(D) = V(4) and thus a unique nonconstant periodic solution. Theorem 1.4 and Corollary 1.3 imply the stability properties of the orbit and the proof is complete. An important special case of Theorem 1.6 is the van der Pol equation (1.7)

ii-k(1-u2)u+u=0, k>0.

TWO DIMENSIONAL SYSTEMS

61

In the above crude analysis, we obtained very little information concerning the location of the unique limit cycle given in Theorem 1.6. When a differential equation contains a parameter, one can sometimes discuss the

precise limiting behavior as the parameter tends to some value. This is illustrated with van der Pol's equation. (1.7). Suppose k is very large; more specifically, suppose k = e-1 and let us determine the behavior of the periodic solution as a -->0+. Oscillations of this type are called relaxation oscillations System (1.7) is equivalent to eu = v - G(u), (1.8)

v= - eu, where G(u) = u3/3 - u. From Theorem 1.6, equation (1.8) has a unique asymptotically stable limit cycle P(e) for every e > 0. From (1.8), if a is small and the orbit is away from the curve v = G(u) in Fig. 1.6, then the u

Figure 11.1.6

coordinate has a large velocity and the v coordinate is moving slowly. Therefore, the orbit-has a tendency to jump in horizontal directions except when

it is very close to the curve v = G(u). These intuitive remarks are made precise in THEOREM 1.7. As s -> 0, the limit cycle of (1.8) approaches the Jordan curve J shown in Fig. 1.6 consisting of arcs of the curve v = G(u) and horizontal line segments.

To prove this, we construct a closed annular region U containing J such that dist(U, J) is any preassigned constant and yet for a sufficiently small, all paths cross the boundary of U inward. U will thus contain (from the Poincare-Bendixson theorem) the limit cycle r(e). The construction of

62

ORDINARY DIFFERENTIAL EQUATIONS

U is shown in Fig. 1.7 where h is a positive constant. The straight lines 81 and 45 are tangent to v = G(u) + h, v = G(u) - h respectively and the lines 56, 12, 9-10, 13-14, are horizontal while 23, 67, 11-12, 15-16 are vertical. The remainder of the construction should be clear. The inner and outer

Figure 11.1.7

boundaries are chosen to be symmetrical about the origin. Also marked on the figure are arrows designating the direction segments of the boundaries

crossed. These are obtained directly from the differential equation and are independent of e > 0. It is necessary to show that the other segments of the boundary are also crossed inward by orbits if a is small. By symmetry, it is only necessary to discuss 34, 45 and 10-11. At any point (u, G(u) - h) on 34, along the orbits of (1.8), we have dv

du

- e2

v -G(u)

_ e2u h <

e2u(3)

h

where u(3) is the value of u at point 3. Hence for s small enough, this is less than g(4) < g(u) which is the slope of the curve G(u) - h. Thus, v < 0 on this arc implies the orbits enter the region along this arc. Along the are 45, we have Iv - G(u) I > h and, hence, the absolute value of the slope of the path ldv/dul = I -e2u/[v - G(u)]l < e2u(4)/h approaches zero as s -> 0. For s small enough this can be made < g(4) which is the slope

of the line 45. Thus, v < 0 on. this are implies the orbits enter into U if e is small enough.

TWO DIMENSIONAL SYSTEMS

63

LetK be the length of the arc 11-12. ForK small enough, Iv - G(u) I > K

along the arc 10-11. Hence, Idv/dul along orbits of (1.8) is less than e2,./K < E2u(11)/K, which approaches zero as s --* 0. Thus, for a small, the orbits enter U since is > 0 on this arc. This shows that given a region U of the above type, one can always choose a small enough to ensure that the orbits cross the boundary of U inward. This proves the desired result since it is clear that U can be made to approximate J as well as desired by appropriately choosing the parameters used in the construction. EXERCISE 1.1. Prove the following Theorem. Any open disk in R2 which contains a bounded semiorbit of (1.1) must contain an equilibrium point. Hint: Use the Poincare-Bendixson Theorem and Theorem I.8.2.'

EXERCISE 1.2. Give a generalization of Exercise 1.1 which remains valid in R3? Give an example. EXERCISE 1.3. Prove the following Theorem. If div f has a fixed sign (excluding zero) in a closed two cell 1, then S2 has no periodic orbits. Hint: Prove by contradiction using Green's theorem over the region bounded by a periodic orbit in 0.

EXERCISE 1.4.

Consider the two dimensional system z =f (t,

x),

f (t + 1, x) = f (t, x), where f has continuous first derivatives with respect to x. Suppose L is a subset of R2 which is homeomorphic to the closed unit disk. Also, for any solution x(t, xo), x(0, xo) = xo, suppose there is a T(xo) such that x(t, xo) is in 0 for all t >_ T(xo). Prove by Brouwer's fixed point theorem

that there is an integer m such that the equation has a periodic solution of period m. Does there exist a periodic solution of period I? EXERCISE 1.5. Suppose f as in exercise (1.4) and there is a A > 0 such that x f (t, x) < -A I X12 for all t, x. If g(t) = g(t + 1) is a continuous function, prove the equation t = f (t, x) + g(t) has a periodic solution of period 1.

EXERCISE 1.6. Suppose yo

is a periodic orbit of a two dimensional

system and let- G, and Gi be the sets defined in Theorem 1.4. Is it possible for

an equation to have a(y(p)) = yo for all noncritical points p in Gi and co (y(q)) = yo for all q in Ge? Explain. EXERCISE 1.7. For Lienard's equation, must there always be a periodic orbit which is stable from the outside? Must there be one stable from the

inside? Explain.

64

ORDINARY DIFFERENTIAL EQUATIONS

EXERCISE 1.8 Is it possible to have a two dimensional system such that each orbit in a bounded annulus is a periodic orbit? Can this happen for

analytic systems? Explain.

11.2. Differential Systems on a Torus

In this section, we discuss the behavior of solutions of the pair of first order equations (2.1)

0),

where (2.2)

4(q + 1, 0) _ t(c6, 0 + 1) _ 0(0, 0), O(0 + 1, 0) = 0(0, 0 + 1) = 0(0, 0).

We suppose (D, 0 are continuous and there is a unique solution of (2.1) through any given point in the 0, 0 plane. Since (D, 0 are bounded, the solutions will exist on (- oo, oo). If opposite sides of the unit square in the (0, 0)-plane are identified, then this identification yields a torus g- and equations (2.1) can be interpreted as a differential equation on a torus. An orbit of (2.1) in the (0, 0)-plane when interpreted on the torus may appear as in Fig. 2.1.

Figure 11.2.1

We also suppose that (2.1) has no equilibrium points and, in particular, that b(q, 0) 0 0 for all 0, 0. The phase portrait for (2.1) is then determined by (2.3)

= A(0, 0), TO

TWO DIMENSIONAL SYSTEMS

65

A(0 +1, 0)=A(c6, 0+1)=A(O, 0), where A(0, 0) is continuous for all 0, 0. The discussion will center around the sohitions of (2.3). The torus 9- can be embedded in R3 by the relations x = (R + r cos 27rO) cos 27rq.

y = (R + r cos 21rO) sin 27r9, z = r sin 21r0,

0< 0. Therefore L(rm, sm) is above L which implies rm/sm = r/s belongs to Rl. Similarly, if n/m is not in Rl and s/r < m/n, then s/r is in R0 . Thus, all rational numbers .with possibly one exception are included in R0 or in Rl and Ro and R1 define a real number p. It remains to show that p is the rotation number defined in the theorem. Suppose m is a given integer and let n be the largest integer such that n/m is in Ro. Then n < pm 0 such that 0(m, C) - C - k >_ a > 0, 6 in [0, 1). For any C

in (- oo, oo), there are an integer p and a e in [0, 1) such that C = p + C. Relation (2.5) then implies 0(m, C) - C - k ? a for all in (-co, oo). A repeated application of this inequality yields 0(rm, 6) - >_ r(k + a) for any integer- r. Dividing by rm and letting r - oo we have p > k/m + a/m which is a contradiction. This completes the proof of the theorem. COROLLARY 2.1.

Among the class of functions A(o, 0) which are

Lipschitzian in 0, the rotation number p = p(A) of (2.3) varies continuously with A; that is, for any e > 0 and A there is a 8 > 0 such that I p(A) - p(B) I < e if max050,esi IA(0, 0) - B(o, 0)I < S. PROOF. If OA(s6, 0) and 0B(¢, 0) designate the solutions of (2.3) for A and B, respectively, z(O) = OA(¢, 0) - No, 0), and L is the Lipschitz

constant for A, then dz

do

= [A(0, z(o) + No, 0)) -A (0, OB(4', 0))] [B(o, OB(o, 0)) - A(0, No, 0))J,

and

DrJzl<

dz

< L Izl + sup I B(0, 0) - A(#, 0)I 0;9010;91

70

ORDINARY DIFFERENTIAL EQUATIONS

for all 0. Thus, 0A(0, 0) - BB(/, 0)I < L-18L0 sup I B(t, B- A(0, 0'! osO, 0;51

for all ¢ > 0.

In the proof of part (ii) of Theorem 2.1, an estimate on the rate of approach of the sequence OA (MI 0)/m to the rotation number p(A) was obtained; namely, I0A(m, 0)/m - p(A)l < 1/Imj for all m. Therefore, JP(A) - P(B) I < I P(A) -

m

OB(m 0)

OA(m, 0) - OB(m, 0) m I

+

- p(B)

M

OA( M, 0) - OB(m, 0)

m

for all integers m. For any s > 0, choose Iml so large that 1/Imi < e/3. For any such given but fixed m, choose 8 > 0 such that I BA(m, 0) - OB(m, 0)1 < 1 if maxose,msi J A(0, 0) - B(¢, 0)I < 8. This fact and the preceding. inequality prove the result. The conclusion of Corollary 2.1 actually is true without assuming A(0, 0) is Lipschitzian in 0. The proof would use a strengthened version of Theorem 1.3.4 on the continuous dependence of solutions of differential equations on the vector field when uniqueness of solutions is assumed. It is an interesting exercise to prove these assertions. THEOREM 2.2.

If the rotation number p is rational, then every tra-

jectory of (2.3) on the torus is either a closed curve or approaches a closed

curve.

(Peixoto). Since p is rational, there exists a closed trajectory which intersects every meridian of .l. Therefore, \y is topologically equivalent to an annulus F. The differential equation (2.3) on J\y is equivalent to a planar differential equation on F. Since there are no equilibrium points, the Poincare-Bendixson Theorem, Theorem 1.2, yields the conclusion of the theorem. The remainder of this chapter is devoted to a discussion of the behavior of the orbits of (2.3) when the rotation number p is irrational. Let T : C -> C be the mapping induced by (2.3) which takes the meridian C of into itself. For any P in C, let PROOF.

y on

D(P) = {TnP, n =0, ±1, +2, ...}, and D'(P) be the set of limit points of D(P). Also, let 0 be the empty set.

TWO DIMENSIONAL SYSTEMS

LEMMA 2.1.

71

Suppose p is irrational, m, n are given integers, P is a

given point in C and a, fi are the closed arcs of C with a n fl = {TmP, TnP},

a u $ = C. Then D(Q) n 0 ° ¢ o, D(Q) n #° 0 0 for every Q in C, where a°, 90 are the interiors of a, P respectively. PROOF.

The set UkTk(m-n)ao covers C. For, if not, the sequence

{Tk(m-n)(TnP)} would approach a limit Po and T(m-n)Po = P0 which, from

Theorem 2.1, contradicts the fact that p is irrational. Consequently, for any TP(n-m)Q Q in C, there is an integer p such that Q is in TP(m-n)ao; that is, is in a° and D(Q) n 0 ° o. The same argument applies to S. THEOREM 2.3.

If p is irrational, D'(P) = F is the same for all P,

TF = F and either (i)

F = C (the ergodic case)

or

(ii)

F is a nowhere dense perfect set.

PROOF.

If S belongs to D'(P), there is a sequence {Pk} c D(P) ap-

proaching S as k -*co. For any pointsPk, Pk+I of this sequence and Q in C, it follows from Lemma 2.1 that there is an integer nk such that TnkQ belongs to the shortest of the arcs cc, 9 on C connecting these two points. Therefore TnkQ ->S as k --> oo and D'(P) c D'(Q). The argument is clearly the same to obtain D'(Q) c D'(P) which proves the first statement of the theorem. If Q is in F, then there is a sequence nk and a P such that Tnk P _*Q as n oo. This clearly implies TQ belongs to F and T-1Q belongs to F. Therefore T F = F. If R is an arbitrary element of F, then the fact that F = D'(Q) for every Q implies for any Q e F there is a sequence of integers nk such that TnkQ -. R. Therefore, the set of limit points of F is F itself and F is perfect. Suppose F contains a closed arc y of C. Then y contains a closed subarc a with endpoints TnP, TmP for some integers n, m and P in C. Therefore, by Lemma 2.1, Uk Tka covers C and since Ta, T2a, ... belong to F we have F = C. This proves the theorem. Our next objective is to obtain sufficient conditions which will ensure that T is ergodc; that is, the limit set F of the iterates of T is C.

Let Pn = TnP, n = 0, ±1, ±2, .... If p is irrational and a is any closed are of C with P as an endpoint, Lemma 2.1 implies there is an integer n such that either Pn or P_n is the only point Pk in the interior a° of a for Ikj < n. Since no power of T has a fixed point, for any N > 0, a can be chosen

so small that n >_ N. For definiteness, suppose P_n is in 0°. Let P0 P_n denote the are of C with endpoints P°, P_n and which also belongs to cc. We

associate an orientation to this are which is the same as the orientation of

ORDINARY DIFFERENTIAL EQUATIONS

72

C. Also, let Pk Pk-n , k = 0, 1, ... , n - 1, designate the are of C joining Pk, Pk-n which has the same orientation as C. LEMMA 2.2. PROOF.

The arcs Pk Pk-n, k = 0, 1, ... , n -1, are disjoint.

If the assertion is not true, then there exists an /' from the

set {-n, -n + 1, ..., n-1} and a k from {0, 1, ..., n -1 } such that Pe belongs to the interior Pk Pk _ n of Pk Pk _ n . Therefore, Pe _ k is in Po P' from the orientation preserving nature of powers of T. This is impossible in

case -n < l - k < n from the choice of n. Suppose -2n + 1 < 1- k < -n. Since Pe belongs to Pk Pk_n , it follows that Pe+n Pe and Pk Pk_n intersect and, in particular, Pk is in Pe+n P° . Thus P._n_ e is in PO P° n which is impossible since 0 < k -,f - n < n. This proves the lemma.

Let sJ(e) = 0(1, 6), 0 < 6 < 1. If p is irrational and

THEOREM 2.4.

possesses a continuous first derivative 0' > 0 which is of bounded variation, then T is prgodic. PROOF. Let 6k = cok(e), k =0, +1, be recursively deThen fined by choosing /-1(e) as the unique solution of 0(e) = TkP = (0,.:/ik(e)), P = (0, e). From the product rule for differentiation, we

have dw-k( ) _ h 0'(eJ-k)J de de L i=o where :I'(e) = do(e)/de. Suppose P and n are chosen as prior to Lemma 2.2. Since Pk Pk-n , k = 0, 1, ..., n - 1, are disjoint we have dY k( )

log (don(s)

=

1

11 0'(es), J=o

ds/i-n(e)

de

log(f '(ef)) -log\

de

n-1

)

=o

[log '(61) -log

2 Se-vyz

and, therefore, Sk + S-k does not approach zero as k -- oo.

TWO DIMENSIONAL SYSTEMS

73

If C\F is not empty (that is, T is not ergodic), then take an open are a in C\F with end points in F. This can be done since F is nowhere dense

and perfect. Since TF = F and T preserves orientation, all of the arcs Tka, k =0, +1, ... are in C\F. Also Tka o Vex = 0, k =A j, since the end points of these arcs are in F and if one coincided with another the end points would correspond to a fixed point of a power of T. Therefore, compactness of C yields 8k + S_k -. 0 as k --> oo. This contradiction implies C\F is empty and proves the theorem. Remark. The smoothness assumptions on 0 in Theorem 2.4 are satisfied if A (0, 8) in (2.3) has continuous first and second partial derivatives with respect to 0. In fact, Theorem 1.3.3 and exercise 1.3.2. imply that 0'(6), 1"(C) are continuous and, in particular, &'(C) is of bounded variation for 0 < 6 < 1.

Also, this same theorem states that 80(¢, C)/8e is a solution of the scalar equation dy 8A(#, 0) do 80 with initial value 1 at = 0. Thus, O'(C) = 80(1, e)/8e > 0, 0 _< C < 1. Denjoy [1] has shown by means of an example that Theorem 2.4 is false if the smoothness conditions on 0 are relaxed. There is no known way to determine the explicit dependence of the rotation number p of (2.3) on the function A(q, 0), and, thus, in particular, to assert whether or not p is irrational. However, the result of Denjoy was the first striking example of the importance of smoothness in differential equations to eliminate unwanted pathological behavior. Suppose the notation is the same as in Theorem 2.4 and the proof of Theorem 2.4. LEMMA 2.3.

If p is irrational and a is a fixed real number, then the

function g(Cn + m) = np + m, en = On (e), n, m integers, is an increasing function on the sequence of real numbers {en + m}. PROOF. Throughout this proof,. n, m, r, s will denote integers. The order of the elements in {C + m} does not depend upon e; that is, en + m < $r + s

implies 4. + m < + s for any C. This is equivalent to saying that fn - Cr < s - m implies n - r < s - m for any C. If this were not true, there would be an 71 such that '']n -'fir is an integer which in turn implies some power of T has a fixed point, contradicting the fact that p is irrational. It suffices therefore to choose e = 0. Recall that 0m(0) = 0(m, 0). If p < 0(m, 0) < r, then a repeated application of (2.6) yields

0(m, 0) + (k -1)p < 0(km, 0) < 0(m, 0) + (k - 1)r,

74

ORDINARY DIFFERENTIAL EQUATIONS

for any k > 0. Thus, 0)

+ (1

- k) P:5 m

e(k Taking the limit as k

0(kk 0) '

< e(k

0)

+ (1

-k

I r.

oo, we obtain p < mp < r. Since p is irrational,

p f

exp(-s sin log s) ds

t

0

>

ftAea

exp(s cos a) ds tR

> tn(ea - 1) exp(tn cos oz).

Choose c2 = 0, c1 =1. Since sin log(tn en) = 1, we have Ix2(tnen)1 > tn(ea - 1) exp(btn),

where b = (1 - 2a)en + cos «. If we choose a so that b > 0, then I x2(tn en)) - 00 as n -* oo and the system is unstable. EXERCISE 2.1. Suppose there is a constant K such that a fundamental matrix solution X of the real system (1.3) satisfies JX(t)j < K, t >_ P and

LINEAR SYSTEMS AND LINEARIZATION

89

t

lim inf f tr A(s) ds > -co. .

t

oo $

Prove that X-1 is bounded on [fl, oo) and no nontrivial solution of (1.3) approaches zero as t -> oo.

Suppose A satisfies the conditions in Exercise 2.1 and B(t) is a continuous real n x n matrix for t >_ P with f IA(t) - B(t)I < oo. Prove that every solution of B(t)y is bounded on [f, oo). For any solution x of (1.3), prove there is a unique solution y of B(t)y such that y(t) - x(t) 0 as t oo. EXERCISE 2.2.

EXERCISE 2.3. Suppose system (1.3) is uniformly asymptotically stable, f satisfies the conditions of Theorem 2.4 and b(t) --0 as t -> oo. Prove there is

a T > 9 such that any solution x(t) of x = A(t)x +. f (t, x) + b(t) approaches zero as t -->- oo if I x(T) I is small enough. EXERCISE 2.4. Generalize the result of Exercise 2.3 with b(t) replaced by g(t, x) where g(t, x) -*0 as t -* oo uniformly for x in compact sets.

Suppose there exists a continuous function c(t) such that c(s) ds < y, t >_ 8, for some constant y = y(fl) and f: Rn+1 -->Rn is continuous with If (t, x) I < c(t)IxI. Prove there is a constant r > 0 such that the solution x = 0 of (2.11) is uniformly asymptotically stable if y < r.

ft+1

EXERCISE 2.5.

EXERCISE 2.6.

Generalize Exercises 2.3 and 2.4 with f satisfying the

conditions of Exercise 2.5.

III.3. n1h Order Scalar Equations

Due to the frequency of occurrence of nth order scalar equations in the

applications, it is worthwhile to transform the information obtained in Section 1 to equations of this type. Suppose y is a scalar, al, ... , an and g are continuous real or complex valued functions on (- oo, + oo) and consider the equation (3.1)

Dny +

al(t)Dn-ly + ... + an(t)y = g(t),

where D represents the operation of differentiation with respect to t. The function D2y is the second derivative of y with respect to t, and so forth.

ORDINARY DIFFERENTIAL EQUATIONS

90

Equation (3.1) is equivalent to (3.2)

(z ='Ax + h y

0

1

0

0

Dy

0

0

1

0

Dn-2y

0

0

0

1

-an -an-1 -an-2 ... -al

Dn-ly

0

h=

A=

X

0

0

-9

From this representation of (3.1), a solution of (3.2) is a column vector of dimension n, but the (j + 1)th component of the solution vector is obtained

by differentiation of the first component j times with respect to t and this first component must be a solution of (3.1). Consequently, any n x n matrix solution [61, ..., en], fj an n-vector, of (3.2) must satisfy cj = col(4)j, D4)j, Dn-l4j),where OJ, j =1, 2, ..., n, is a solution of (3.1). If 01, ..., On are n-scalar functions which are (n -1)-times continuously differentiable, the Wronskian A(01, ... , On) of 01, ..., On is defined by

(3.3)

0(4)1, ... , on) = det

01

02

Dot

D02

Dn-14)1 Dn102

"' ... ...

0. Don Dn-lon

A set of scalar functions 01, ... , On defined on a 0, a > 0 such that (6.2)

(a)

JeAt7r+xl < Keatllr+xl,

(b)

IeAt7r_xl < Ke-atln._xI,

t < 0, t Z 0,

for all x in C". These relations are immediate from the observation that there exists a nonsingular matrix U such that U-'A U = diag(A+, A-) where A+ is a k x k matrix whose eigenvalues have positive real parts and 'A_ is an (n - k) x (n - k) matrix whose eigenvalues have negative real parts. From

LINEAR SYSTEMS AND LINEARIZATION

107

Theorem 4.2, there are constants K1 > 0, a > 0 such that IeA+tl 0 such that the matrix A + 8f (x)/8x as a function of x has k eigenvalues with positive real parts, n - k with negative real parts for IxI < S. From the implicit function theorem, the equation Ax +f (x) = 0 has a unique solution xo in the region IxI < S. The transformation x = xo + y yields the equation

- )] y +f (xo + y) -f (xo) - - - )y

[A+-_ax

Ox

def

= By + g(y),

where B has k eigenvalues with positive real parts, n - k with negative real parts and g(y) =o(Iyl) as IyI-p.0.

109

LINEAR SYSTEMS AND LINEARIZATION

On the strength of this remark, we consider the preservation of the saddle point property for equation (6.3) for families of continuous functions f which at least satisfy f (x) =o(IxI) as Ixl -* 0. LEMMA 6.1.

If f : Cn -± Cn is continuous, x = 0 is a saddle point of type

(k) of (4.1), or+, ,r_ are the projection operators defined in (6.1), then, for any solution x(t) of (6.3) which exists and is bounded on [0, oo), there is an x_ in 7r-Cn such that x(t) satisfies (6.4)

x(t) = eAtx-

+ f eA(t-s)ir-f (x(s)) ds + f :a-As r+f Wt + s)) cts, o

for t >_ 0. For any solution x(t) of (6.3) which exists and is bounded on (-oo, 0], there is an x+ in 7r+Cn such that t

(6.5)

0

x(t) = eAtx+ + f eA(t-8)7r+ f (x(s)) ds + f

e-As r_ f (x(t + s)) ds

o

for for t_ 0. There is a constant L such that Iir+xl _ 0. For any a in [0, oo), the solution

x(t) satisfies t

it+x(t) = CA(t-a)ir+x((7) +

f eA(t-s)ir+ f (x(s)) ds,

tin [0, cc),

since Aor+ = or+ A, Air_ = 7r_ A.

Since the matrix A satisfies (6.2), IeA(t-a),r+x(a)I _ 0,

0

0

or the inequality 0

(6.7)

0

u(t) < Keat + L f 00-8)u(s) ds + M f

evsu(t + s) ds,

t _< 0.

-OD

t

If

def L M

+ - < 1,

(6.8)

y

then, in either case, (6.9)

u(t) < (1- ) IKe [a-(I-a) 'Lnti

PROOF. We only need to prove the lemma for u satisfying (6.6) since the transformation t - . -t, s -* -s reduces the discussion of (6.7) to (6.6). We first show that u(t) - 0 as t -± oo. If 8 = limt_, u(t), then u bounded

implies 8 is finite. If 0 satisfies f 0 implies there is a t1 >_ 0 such that u(t) < 0-18 for t >_ t1. From (6.6), for t >_ t1, we have (6.10)

u(t) < Ke-at + Le-at f easu(s) ds + o

\a

+ Y J

Since the lim sup of the right hand side of (6.10) as t -* oo is < 8, this is a contradiction. Therefore, 8 = 0 and u(t) --0 as t -- oo. If v(t) = supszt u(s), then u(t) -* 0 as t - . oo implies for any t in [0, oo), there is a t1 >_ t such that v(t) = v(s) = u(t1) for t < s < t1, v(s) < v(t1) for

111

LINEAR SYSTEMS AND LINEARIZATION

s > t1. From (6.6), this implies

v(t) = u(ti) < Ke-ate + L ft e-01-0v(s) ds 0

+ L f e-a(t,-s)v(s) ds + M ti

t

00 e-vsv(t'+ s) ds

fo

t

Ke-ate + L f e-a(4-8)V(s) ds + flv(t), 0

where P = L/a + M/y < 1. If z(t) = eatv(t), then t1 >_ t implies t

z(t) < (1 - f)-1K + (1 - p)-1L f z(s) ds. 0

From Gronwall's inequality, we obtain z(t) < (1 - fl)-1K exp(1 -8)-1Lt and, thus, the estimate (6.9) in the lemma for u(t). EXERCISE 6.1. Suppose a, b, c are nonnegative continuous functions on [0, oo), u is a nonnegative bounded continuous solution of the inequality oo

u(t) < a(t) + f t b(t - s)u(s) ds + f c(s)u(t + s) ds, 0

t >_ 0,

0

and a(t) -->0, b(t) -*0 as t --> oo, f o'* b(s) ds < ee, f ."o c(s) ds < oo. Prove that u(t) --> 0 as t -* oo if

f, b(s)ds+ fc(s)ds_ 0, is the function given in (6.11). Choose 8 so that (6.13)

4KK1,q(8) < a,

8K2Kirl(8) < 3a-

With this choice of 8 and for any x_ in 7r_Cn with Ix_1 < 8/2K, define I(x_, 8) as the set of continuous functions x: [0, oo)--Cn such theit JxJ _ supost_ 0. Since x is in 9(x_, 8), it is easy to see that Tx is defined and continuous for t ? 0 with [a_ Tx](0) = x-. From (6.2), (6.11), (6.13), we obtain I (Tx)(t)I < Ke-atIx_I + j:Ke.n(ts)IlT_f(x,(8))I ds +

Ke-a8,r+,f (x(t + s))I 0

< Ke-atJ x-I +

KKl 71(8)IxI [2 - e-at] a

2KK1

_ 0. Thus T is a contraction on 5(x_, 6) and there is a unique fixed point x_) in 9(x-, 6) and this fixed point satisfies (6.4). Using the same estimates as above, one shows that the function x*( , x_)

is continuous in x_ and

0) = 0. However, more precise estimates of the dependence of x_) on x_ are needed. If we let x* = x*( , x_), then, from (6.4), x* = x*( , t

Ix*(t) -i*(t)I < Ke-011x_ -z-)I +KK1'h(8) f e-a(t_8)Ix*(s) -x*(s)I ds 0

+KK1 (8) f , e-a8I x*(t+s) -x*(t-+-8))I ds 0

for t

0. We may now apply Lemma 6.2 to this relation. In Lemma 6.2, let y = a, M = L = KK177(8). If u(t) = I x*(t) - x*(t)I , 8 satisfies (6.13) and appropriate identification of constants are made in Lemma 6.2, then (6.15)

I x*(t, x-) - x*(t, x_)I < 2K(exp

- 2)

Ix_ -:9-1,

t >_ 0.

Since x*( , 0) = 0, relation (6.15) implies these solutions satisfy a relation of the form (6.12a) and approach zero exponentially at t - . oo. Let B612K denote the open ball of radius 812K in C'" with center at the origin. Let S,7,-,t designate the initial values of all those solutions of (6.3) which

114

ORDINARY DIFFERENTIAL EQUATIONS

remain inside B6 for t > 0 and have ir_ x(0) in B6/2K . From the above proof,

S.*-k =1x: x = x*(0, x_), x- in (7r-C") n

Let 9(x_) = x*(0, x-), xin (a_Cn) n B612K The function g is a continuous map of (ir_Cn) n B612K onto Sn_k and is given by B812K}.

o

(6.16)

9(x-) = x- + J

e-A87r+ f (x* (s, x-)) d s.

From (6.2), (6.11), (6.13), (6.15), we have

I9(x-) - 9(x-)I > Ix- -x"-I - J

x*(s, x-) - x*(s, z-))I ds 0

>_Ix_-x_I(1-

4K2K17](8)1

3«

J

> 1 Ix- -x-I, for all x-, x"_ in (1r_Cn) n B6/2K . Therefore g is one-to-one. Since g-1 = ir-

is continuous it follows that g is a homeomorphism. This shows that Sn_k is homeomorphic to the open unit ball in C. and, in particular, has dimension n - k. However, S*_k may not be positively invariant. If we' extend S*-k to a set Sn_k by adding to it all of the positive orbits of solutions with initial values in Sn-k , then Sn-k is positively invariant and also homeomorphic to the open unit ball in C,4-k from the uniqueness of solutions of the equation.' The set Sn_k coincides with Sn-k when x in Sn_k implies I7r_ xI < 812K.

From (6.14), (6.15) and the fact that

0) = 0, we also obtain

Iir+x*(0, x-)I < KK1 f, e-a8.q(Ix*(s, x-)I )Ix*(s, x-)I ds 0

< KKl

_

,

J0

e-a8n(2KIx-I )2KI x_I ds

2K2K1 a

I(2KIx-I)Ix I

Consequently Iir+x*(0, x_)I1Ix-I -->.0 as Ix-1 0 in Sn_k which shows that Sn_k is tangent to it-Cn at x = 0. Using relation (6.5), one constructs the set Uk in a completely analogous manner. This completes the proof of the theorem. In the proof of Theorem 6.1, it was actually shown that the mapping g taking it-Cn n B812K into Sn_k is Lipschitz continuous [see relatiofs (6.15) and (6.16)]. Since the solutions of (6.3) also depend Lipschitz continuously on the initial data if (6.11) is satisfied, it follows that the stable manifold

LINEAR SYSTEMS AND LINEARIZATION

115

Sn_k and also the unstable manifold Uk are Lipschitz continuous; that is, Sn_k(Uk) is homeomorphic to the unit ball in Cn-k(Ck) by a mapping which is Lipschitz continuous. It is also clear from the proof of Theorem 6.1 that the Lipschitz condition of the type specified in (6.11b) was unnecessary. One could have assumed only that

If (x) -f (A 1 for a < a0 . If a0 were a double root of B(a) = 1, then Lemma 8.6 would imply it is a maximum, which is impossible. This proves the lemma. By combining the information in the above lemmas we obtain

ORDINARY DIFFERENTIAL EQUATIONS

128

THEOREM 8.1. There exist two sequences {ao < a1 < a2 < oo as k -> co, a2 < a3 < ..} of real numbers, ak ,

}, {al

ac 0, in the complex plane of radius s

and center po and a SI > 0 such that (9.2) has exactly one characteristic multiplier pp(B) in DE(po) for all B in Rsaf, I A - BI < S1. Since (9.2) is reciprocal, p 1(B), is also a characteristic multiplier. But, po 1(B) _ po(B)II po(B)I2 po(B) unless Ipo(B)I = 1. On the other hand, the hypothesis Ipo(A)I =1 implies po 1(A) = p0(A) and by continuity of p0(A) in A, we can find a So < S1 such that po 1(B), p0(B) belong to DE(p0) if I A - BI < So , B in 9si. This implies Ipo(B)I = 1 for I A - BI < So, B in gtsad, and proves the lemma.

If A is in 3tsad and all of the characteristic multipliers

THEOREM 9.1.

of (9.1) are distinct and have unit moduhi, thenA is strongly stable relative to 3tsa1. PROOF. This is immediate from Lemma 9.1 and the Floquet representation of the solutions of a periodic system.