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BASIC THEORY OF ORDINARY DIFFERENTIAL EQUATIONS

Universitext Editorial Board (North America):

S. Axler F.W. Gehring K.A. Ribet

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Universitext Editors (North America): S. Axler, F.W. Gehring, and K.A. Ribet Aksoy/Khamsl: Nonstandard Methods in Fixed Point Theory Anderson: Topics in Complex Analysis Aupetit: A Primer on Spectral Theory Berberian: Fundamentals of Real Analysis BoossBleecker: Topology and Analysis Borkar: Probability Theory: An Advanced Course BSttcher/Sibermann: Introduction to Large Truncated Toeplitt Matrices Carleson/Gamelin: Complex Dynamics Cecil: Lic Sphere Geometry: With Applications to Submanifolds Chae: Lchcsgue Integration (2nd cd.)

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Curtis: Matrix Groups DiBenedetto: Degenerate Parabolic Equations Dimca: Singularities and Topology of Hypersurfaces Edwards: A Formal Background to Mathematics I a/b Edwards: A Formal Background to Mathematics 11 a/b Foulds: Graph Theory Applications Friedman: Algebraic Surfaces and Holomorphic Vector Bundles

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(cnnrinued after index)

Po-Fang Hsieh Yasutaka Sibuya

Basic Theory of Ordinary Differential Equations With 114 Illustrations

Springer

Po-Fang Hsieh Department of Mathematics and Statistics Western Michigan University Kalamazoo, MI 49008

Yasutaka Sibuya School of Mathematics University of Minnesota 206 Church Street SE Minneapolis, MN 55455 USA sibuya 0 math. umn.edu

USA

[email protected]

Editorial Board (North America): S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132

USA

F.W. Gehring

K.A. Ribet

Mathematics Department East Hall

Department of

University of Michigan Ann Arbor. Ml 48109-

University of California at Berkeley Berkeley, CA 94720-3840

1109

USA

Mathematics

USA

Mathematics Subject Classification (1991): 34-01

Library of Congress Cataloging-in-Publication Data Hsieh, Po-Fang.

Basic theory of ordinary differential equations I Po-Fang Hsieh, Yasutaka Sibuya. cm. - (Unrversitext) p. Includes bibliographical references and index. ISBN 0-387-98699-5 (alk. paper) 1. Differential equations. I. Sibuya. Yasutaka. 1930II. Title. III. Series OA372_H84 1999

515'.35-dc2l

99-18392

Printed on acid-free paper.

r 1999 Spnnger-Verlag New York, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc.. 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis.

Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use of general descriptive names, trade names, trademarks. etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act. may accordingly be used freely by anyone. Production managed by MaryAnn Cotton: manufacturing supervised by Jeffrey Taub. Photocomposed copy prepared from the authors' A5-71EX 2.1 files.

Printed and bound by R.R. Donnelley and Sons, Harrisonburg. VA. Printed in the United States of America. 9 8 7 6 5 4 3 2 1

ISBN 0-387-98699-5 Springer-Verlag New York Berlin Heidelberg

SPIN 10707353

To Emmy and Yasuko

PREFACE

This graduate level textbook is developed from courses in ordinary differential

equations taught by the authors in several universities in the past 40 years or so. Prerequisite of this book is a knowledge of elementary linear algebra, real multivariable calculus, and elementary manipulation with power series in several complex variables. It is hoped that this book would provide the reader with the very basic knowledge necessary to begin research on ordinary differential equations. To this purpose, materials are selected so that this book would provide the reader with methods and results which are applicable to many problems in various fields. In order to accomplish this purpose, the book Theory of Differential Equations by E. A. Coddington and Norman Levinson is used as a role model. Also, the teaching of Masuo Hukuhara and Mitio Nagumo can be found either explicitly or in spirit in many chapters. This book is useful for both pure mathematician and user of mathematics. This book may be divided into four parts. The first part consists of Chapters I, II, and III and covers fundamental existence, uniqueness, smoothness with respect to data, and nonuniqueness. The second part consists of Chapters IV, VI, and VII and covers the basic results concerning linear differential equations. The third part consists of Chapters VIII, IX, and X and covers nonlinear differential equations. Finally, Chapters V, XI, XII, and XIII cover the basic results concerning power series solutions. The particular contents of each chapter are as follows. The fundamental existence and uniqueness theorems and smoothness in data of an initial problem are explained in Chapters I and II, whereas the results concerning nonuniqueness are explained in Chapter III. Topics in Chapter III include the Kneser theorem and maximal and minimal solutions. Also, utilizing comparison theorems, some sufficient conditions for uniqueness are studied. In Chapter IV, the basic theorems concerning linear differential equations are explained. In particular, systems with constant or periodic coefficients are treated in detail. In this study, the S-N decomposition of a matrix is used instead of the Jordan canonical form. The S-N decomposition is equivalent to the block-diagonalization separating distinct eigenvalues. Computation of the S-N decomposition is easier than that of the Jordan canonical form. A detailed explanation of linear Hamiltonian systems with constant or periodic coefficients is also given. In Chapter V, formal power series solutions and their convergence are explained. The main topic is singularities of the first kind. The convergence of formal power series solutions is proven for nonlinear systems. Also, the transformation of a linear system to a standard form at a sin-

gular point of the first kind is explained as the S-N decomposition of a linear differential operator. The main idea is originally due to R. Gerard and A. H. M. Levelt. The Gerard-Levelt theorem is presented as the S-N decomposition of a matrix of infinite order. At the end of Chapter V, the classification of the singuvii

viii

PREFACE

larities of linear differential equations is given. In Chapter VI, the main topics are the basic results concerning boundary-value problems of the second-order linear differential equations. The comparison theorems, oscillation and nonoscillation of solutions, eigenvalue problems for the Sturm-Liouville boundary conditions, scattering problems (in the case of reflectionless potentials), and periodic potentials are studied. The authors learned much about the scattering problems from the book by S. Tanaka and E. Date [TD]. In Chapter VII, asymptotic behaviors of solutions of linear systems as the independent variable approaches infinity are treated. Topics include the Liapounoff numbers and the Levinson theorem together with its various improvements. In Chapter VIII, some fundamental theorems concerning stability, asymptotic stability, and perturbations of 2 x 2 linear systems are explained, whereas in Chapter IX, results on autonomous systems which include the LaSalle-Lefschetz theorem concerning behavior of solutions (or orbits) as the independent variable tends to infinity, the basic properties of limit-invariant sets including the Poincar6-Bendixson theorem, and applications of indices of simple closed curves are studied. Those theorems are applied to some nonlinear oscillation problems in Chapter X. In particular, the van der Pot equation is treated as both a problem of regular perturbations and a problem of singular perturbations. In Chapters XII and XIII, asymptotic solutions of nonlinear differential equations as a parameter or the independent variable tends to its singularity are explained. In these chapters, the asymptotic expansions in the sense of Poincare are used most of time. However, asymptotic solutions in the sense of the Gevrey asymptotics are explained briefly. The basic properties of asymptotic expansions in the sense of Poincare as well as of the Gevrey asymptotics are explained in Chapter XI. At the beginning of each chapter, the contents and their history are discussed briefly. Also, at the end of each chapter, many problems are given as exercises. The purposes of the exercises are (i) to help the reader to understand the materials in each chapter, (ii) to encourage the reader to read research papers, and (iii) to help the reader to develop his (or her) ability to do research. Hints and comments for many exercises are provided. The authors are indebted to many colleagues and former students for their valuable suggestions, corrections, and assistance at the various stages of writing this book. In particular, the authors express their sincere gratitude to Mrs. Susan Coddington and Mrs. Zipporah Levinson for allowing the authors to use the materials in the book Theory of Differential Equations by E. A. Coddington and Norman Levinson.

Finally, the authors could not have carried out their work all these years without the support of their wives and children. Their contribution is immeasurable. We thank them wholeheartedly. PFH YS

March, 1999

CONTENTS vii

Preface

Chapter I. Fundamental Theorems of Ordinary Differential Equations I-1. Existence and uniqueness with the Lipschitz condition 1-2. Existence without the Lipschitz condition 1-3. Some global properties of solutions 1-4. Analytic differential equations Exercises I

Chapter II. Dependence on Data II-1. Continuity with respect to initial data and parameters 11-2. Differentiability Exercises II

1

1

8 15

20 23 28 28 32 35

Chapter III. Nonuniqueness III-1. Examples 111-2. The Kneser theorem 111-3. Solution curves on the boundary of R(A) 111-4. Maximal and minimal solutions 111-5. A comparison theorem 111-6. Sufficient conditions for uniqueness Exercises III

41 41

Chapter IV. General Theory of Linear Systems IV-1. Some basic results concerning matrices IV-2. Homogeneous systems of linear differential equations IV-3. Homogeneous systems with constant coefficients IV-4. Systems with periodic coefficients IV-5. Linear Hamiltonian systems with periodic coefficients IV-6. Nonhomogeneous equations IV-7. Higher-order scalar equations Exercises IV

69 69 78

Chapter V. Singularities of the First Kind V-1. Formal solutions of an algebraic differential equation V-2. Convergence of formal solutions of a system of the first kind V-3. The S-N decomposition of a matrix of infinite order V-4. The S-N decomposition of a differential operator V-5. A normal form of a differential operator V-6. Calculation of the normal form of a differential operator V-7. Classification of singularities of homogeneous linear systems Exercises V ix

45 49 52 58 61

66

81

87 90 96 98 102 108 109 113 118 120 121

130 132 137

x

CONTENTS

Chapter VI. Boundary-Value Problems of Linear Differential Equations of the Second-Order VI- 1. Zeros of solutions VI- 2. Sturm-Liouville problems VI- 3. Eigenvalue problems VI- 4. Eigenfunction expansions VI- 5. Jost solutions

VI- 6. Scattering data VI- 7. Reflectionless potentials VI- 8. Construction of a potential for given data VI- 9. Differential equations satisfied by reflectionless potentials VI-10. Periodic potentials Exercises VI Chapter VII. Asymptotic Behavior of Solutions of Linear Systems VII-1. Liapounoff's type numbers VII-2. Liapounoff's type numbers of a homogeneous linear system VII-3. Calculation of Liapounoff's type numbers of solutions VII-4. A diagonalization theorem VII-5. Systems with asymptotically constant coefficients VII-6. An application of the Floquet theorem Exercises VII

Chapter VIII. Stability VIII- 1. Basic definitions VIII- 2. A sufficient condition for asymptotic stability VIII- 3. Stable manifolds VIII- 4. Analytic structure of stable manifolds VIII- 5. Two-dimensional linear systems with constant coefficients VIII- 6. Analytic systems in R2 VIII- 7. Perturbations of an improper node and a saddle point VIII- 8. Perturbations of a proper node VIII- 9. Perturbation of a spiral point VIII-10. Perturbation of a center Exercises VIII Chapter IX. Autonomous Systems IX-1. Limit-invariant sets IX-2. Liapounoff's direct method IX-3. Orbital stability IX-4. The Poincar6-Bendixson theorem IX-5. Indices of Jordan curves Exercises IX

CONTENTS

xi

Chapter X. The Second-Order Differential Equation

-t2 + h(x) dt + g(x) = 0

d X-1. Two-point boundary-value problems X-2. Applications of the Liapounoff functions X-3. Existence and uniqueness of periodic orbits

304 305

309 313

X-4. Multipliers of the periodic orbit of the van der Pol equation X-5. The van der Pol equation for a small e > 0 X-6. The van der Pol equation for a large parameter X-7. A theorem due to M. Nagumo X-8. A singular perturbation problem

318

Exercises X

334

Chapter XI. Asymptotic Expansions XI-1. Asymptotic expansions in the sense of Poincare XI-2. Gevrey asymptotics XI-3. Flat functions in the Gevrey asymptotics XI-4. Basic properties of Gevrey asymptotic expansions XI-5. Proof of Lemma XI-2-6 Exercises XI

342

Chapter XII. Asymptotic Expansions in a Parameter XII-1. An existence theorem XII-2. Basic estimates XII-3. Proof of Theorem XII-1-2 XII-4. A block-diagonalization theorem XII-5. Gevrey asymptotic solutions in a parameter XII-6. Analytic simplification in a parameter

372

Exercises XII

395

Chapter XIII. Singularities of the Second Kind XIII-1. An existence theorem XIII-2. Basic estimates XIII-3. Proof of Theorem XIII-1-2 XIII-4. A block-diagonalization theorem XIII-5. Cyclic vectors (A lemma of P. Deligne) XIII-6. The Hukuhara-T rrittin theorem XIII-7. An n-th-order linear differential equation at a singular point of the second kind XIII-8. Gevrey property of asymptotic solutions at an irregular singular point Exercises XIII

403 403 406 417 420

References

453

Index

462

319 322 327 330

342 353

357 360 363 365

372 374 378 380 385 390

424 428

436 441

443

CHAPTER I

FUNDAMENTAL THEOREMS OF ORDINARY DIFFERENTIAL EQUATIONS In this chapter, we explain the fundamental problems of the existence and uniqueness of the initial-value problem dy dt

(P)

=

y(to) _ 4o

in the case when the entries of f (t, y) are real-valued and continuous in the variable (t, y), where t is a real independent variable and y is an unknown quantity in Rn. Here, R is the real line and R" is the set of all n-column vectors with real entries. In §I-1, we treat the problem when f (t, y) satisfies the Lipschitz condition in W. The main tools are successive approximations and Gronwall's inequality (Lemma

In §1-2, we treat the problem without the Lipschitz condition. In this case, approximating f (t, y) by smooth functions, f-approximate solutions are constructed. In order to find a convergent sequence of approximate solutions, we use Arzeli Ascoli's lemma concerning a bounded and equicontinuous set of functions (Lemma 1-2-3). The existence Theorem 1-2-5 is due to A. L. Cauchy and G. Peano [Peal] and the existence and uniqueness Theorem 1-1-4 is due to E. Picard [Pi1 and E. Lindelof [Lindl, Lind2J. The extension of these local solutions to a larger interval is explained in §1-3, assuming some basic requirement.,, for such an extension. In §I-4, using successive approximations, we explain the power series expansion of a solution in the case when f (t, y) is analytic in (t, y). lit each section, examples and remarks are given for the benefit of the reader. In particular, remarks concerning other methods of proving these fundamental theorems are given at the end of §1-2.

I-1. Existence and uniqueness with the Lipschitz condition We shall use the following notations throughout this book:

y=

Y1,

f,

y2

f2

yn

!y = max{jy,j 1Y21-..

ItYn11.

fn

In §§I-1, 1-2, and 1-3 we consider problem (P) under the following assumption.

Assumption 1. The entries of f (t, y) are real-valued and continuous on a rectangular region:

R = R((to,6),a,b) = {(t,yl: It - tot 0.

To locate a solution in a neighborhood of its initial point, the number or plays an important role as shown in the following lemma.

Lemma I-1-1. If ff = ¢(t) is a solution of problem (P) in an interval it - tol < a < a, then I'(t) - 41 < b in It - tol < a, i.e., (t,5(t)) E R((to,co),a,b) for. It - tol 0. Letting a - 0, we obtain (t} = c"o + f L f (s, 0(s))ds. u

This completes the proof of Theorem 1-2-5. 0 Example 1-2-6. Theorem 1-2-5 applies to the initial-value problem dy (P)

xyl/s,

dx

y(3) = 0.

However, Theorem I-1-4 does not apply to (P). In fact y(x) = 0 is a solution of Problem (P) and also 0

y(x) _

for

x < 3,

for

x > 3

(2(x2_ 9)154 5

is a solution of Problem (P). Note that the right-hand side of the differential equation (P) does not satisfy the Lipschitz condition on any y-interval containing

y=0.

Two other methods of proving Theorem 1-2-5 are summarized in the following two remarks. Remark 1-2-7. Let every entry of an n-dimensional vector f (t, y-) be a real-valued

continuous function of n + 1 independent variables t and y = (yl, ... , y,,) on a rectangular region R = {(t, y-) : It - tot < a, Iy"-cod < b}. Assume that I f(t,yj 1 < M

I. FUNDAMENTAL THEOREMS OF ODES

14

on R, where M is a positive number. Set a = min (a, M ) . Since f (t, yI is uniformly continuous on R, there exists a positive n umber p(e) for every given positive number a such that I f (tl, y1) - f (t2,1%2)I < E Whenever Itl - t2[ < p(c) and

Iyl - 1%2I 5 p(e) for (xl, WI) E R and (x2, y2) E R. Let 0(e) : to = ro < r1 < < rn(c) = to + a be a subdivision of the interval to < t < to + a such that n(c)-1

m ax (I rj - rj+ll) < nun P(E),

91(t) =

p(E)

M

. Set

co

for

t = to,

1/e(rj-1) + J (r)-1,1/c(rj_1))(t - 7-j-1)

for

rj_1 < t < TI,

(j = 1,2,... ,n(6)). Then, we can show that (i) every entry of y, (t) is piecewise linear and continuous in t and (t, &(t)) E R on the interval to < t < to + a, (ii) the set {yi : E > 0} is bounded and equicontinuous on the interval to < t
to, t < to,

0,

and

[3(t - to), 2

(S.1.3)

y(t) 0,

41

3/2

t > to, t < to

111. NONUNIQUENESS

42

(cf. Figure 1). Actually, the region bounded by two solution curves (S.1.2) and (S.1.3) is covered by solution curves of problem (I11.1.1). Note that, in this case, solution (S.1.1) is the unique solution of problem (P) for t < to. Solutions are not

unique only fort ? to. Example 111-1-2. Consider a curve defined by

y = sin t

(111.1.2)

(-oo < t < +oo)

and translate (111.1.2) along a straight line of slope 1. In other words, consider a family of curves y = sin(t - c) + c,

(111.1.3)

where c is a real parameter. By eliminating c from the relations

dt =

c os(t - c),

y - t = sin(t - c) - (t - c),

we can derive the differential equation for family (111.1.3). In fact, since sin u - u

is strictly decreasing, the relation v = sinu - u can be solved with respect to u to obtain u = G(v) - v, where G(v) is continuous and periodic of period 27r in v, G(2n7r) = 0 for every integer n, and G(v) is differentiable except at v = 2n7r for every integer n. The differential equation for family (111.1.3) is given by dy (II1.1.4)

dt

= 1OS[G(y-t)-(y-t)1.

Since G(2n7r) = 0 and cos(-2n7r) = 1 for every integer n, differential equation (111. 1.4) has singular solutions y = t + 2mr, where n is an arbitrary integer. These lines are envelopes of family (111.1.3) (cf. Figure 2).

FIGURE 1.

FIGURE 2.

Example 111-1-3. The initial-value problem (111.1.5)

dt =

y(to) = 0

has at least two solutions (S.3.1)

y(t) = 0

(-oo < t < +oo),

43

1. EXAMPLES and

4 (t -

(S.3.2)

t,0)2,

t > to

,

y(t) -

t < to

4(t - to)2,

(cf. Figure 3). The region bounded by two solution curves (S.3.1) and (S.3.2) is covered by solution curves of problem (111.1.5).

FIGURE 3.

Consider the following two perturbations of problem dy

dy

=

_

+ E,

y(to) = 0

y2 y2 + C2

y(to) = 0,

lyl

where a is a real positive parameter. Each of these two differential equations satisfies the Lipschitz condition. In particular, the unique solution of problem (111. 1.6) is given by

J4(t-to+2VI -E)2 - E, (S.3.3)

y(t) =

-4(t-to-2f)2

+ E,

t > to t < to

(cf. Figure 4). On the other hand, (S.3.1) is the unique solution of problem (111.1.7). Figure 5 shows shapes of solution curves of differential equation (111.1.7). Note that nontrivial solution of (111.1.7) is an increasing function of t, but it does not reach y = 0 due to the uniqueness.

III. NONUNIQUENESS

44

y

FIGURE 5.

FIGURE 4.

Generally speaking, starting from a differential equation which does not satisfy any uniqueness condition, we can create two drastically different families of curves by utilizing two different smooth perturbations. In other words, a differential equation without uniqueness condition can be regarded as a branch point in the space of differential equations (cf. [KS]). Example 111-1-4. The general solution of the differential equation 2

d) +y2=1

(1II.1.8)

is given by y = sin(t + c),

(111.1.9)

where c is a real arbitrary constant. Also, y = 1 and y = -1 are two singular solutions. Two solution curves (111.1.9) with two different values of c intersect each other. Hence, the uniqueness of solutions is violated (cf. Figure 6). This phenomenon may be explained by observing that (111. 1.8) actually consists of two differential equations: (III.1.10)

dy = dt

1 - yz

and

dy=- l-y2. dt

Each of these two differential equations satisfies the Lipschitz condition for lyI < 1. Figures 6-A and 6-B show solution curves of these two differential equations, respectively. y=I

y=-I FIGURE 6.

FIGURE 6-A.

FIGURE 6-B.

Observe that each of these two pictures gives only a partial information of the complete picture (Figure 6).

2. THE KNESER THEOREM

45

We can regard differential equation (111.1.8) as a differential equation dy

(111.1.11)

dt

=w

on the circle

w2 + y2 = 1.

(111.1.12)

If circle (111.1.12) is parameterized as y = sinu, w = cosu, differential equation (III.1.11) becomes (III.1.13)

d=1

or

cos u = 0 .

Solution curves u = t of (111.1.13) can be regarded as a curve on the cylinder

{(t, y, w) : y =sin u, w = cos u, -oc < u < +oo (mod 2w), -oo < t < +oo} (cf. Figure 7). Figure 6 is the projection of this curve onto (t,y)-plane.

FIGURE 7.

In a case such as this example, a differential equation on a manifold would give a better explanation. To study a differential equation on a manifold, we generally use a covering of the manifold by open sets. We first study the differential equation on each open set (locally). Putting those local informations together, we obtain a global result. Each of Figures 6-A and 6-B is a local picture. If these two pictures are put together, the complete picture (Figure 6) is obtained.

111-2. The Kneser theorem We consider a differential equation

dt = f(t,y-)

(III.2.1)

under the assumption that the R"-valued function f is continuous and bounded on a region (111.2.2)

S2 = {(t, y-)

:

a:5 t < b , Iy1 < +oo }.

Under this assumption, every solution of differential equation (III.2.1) exists on the interval Zfl = {t : a < t < b} if (to, y"(to)) E f2 for some to E Zo (cf. Theorem 1-3-2 and Corollary I-3-4). The main concern in this section is to investigate topological properties of a set which is covered by solution curves of differential equation (111.2.1).

47

2. THE KNESER THEOREM

Theorem 111-2-4 ((Kn]). If A is compact and connected, then SS(A) is also compact and connected for every c E To. Proof.

The compactness of SS(A) was already explained. So, we prove the connectedness only.

Case 1. Suppose that A consists of a point (r, t), where we assume without any loss of generality that r < c. A contradiction will be derived from the assumption that there exist two nonempty compact sets F1 and F2 such that

F1nF2=0.

SS(A)=F1uF2,

(111.2.3)

If t'1 E F1 and 2 E F2, there exist two solutions 1 and 4'2 of (111.2. 1) such that

01(r) _ , 01(c) _ 1, and 42(r) _ ., 02(c) _ 6, (cf. Figure 9). Set

h(µ) _

1(r+11)

{ &T + JJUJ)

for 05µ5c-r, for

-,r)

Let { fk(t, y-) : k = 1, 2,... } be a sequence of R"-valued functions such that (a) the functions fk (k = 1, 2, ...) are continuously differentiable on 0, (b) I fk(t, y1I < M for (t, y) E f2, where M is a positive number independent of (t, yl and k, ra+A = f'uniformly on each compact set in Q (c) k El oo

(cf. Lemma I-2-4). For each k, let &(t,µ) be the unique solution of the initialvalue problem

= fk(t, y-), y(-r+ Iµ4) = h(µ). The solution 15k(t, µ) is continuous

for t E Zo and iµl < c - r. It is easy to show that the family

k=

1, 2,... ; jµj 5 c - r} is bounded and equicontinuous on the interval Io. Note that the functions Z(rk(c, µ) (k = 1, 2, ...) are continuous in µ for 1µI
ahPj P1. (A). h=1

Hence, \jPjPA(A) = \h PiPh(A). Thus, PiPh(A) = 0 whenever j it follows that P1 = P1(A) = P2P}(A).

h. Therefore,

IV. GENERAL THEORY OF LINEAR SYSTEMS

76

Observation IV-1-15. Let A = S + N be the S-N decomposition of an n x n matrix A. Let T be an n x n invertible matrix such that if we set A = T-1ST, then A = diag[A1I1, A2I2, ... , AkIk], Where A1, A2, ... , Ak are distinct eigenvalues

of S (and also of A), I, is the m3 x mj identity matrix, and m3 is the multiplicity of the eigenvalue A,. It is easy to show that

(i) if we set M = T-'NT, then M is nilpotent, MA = AM, and M = diag[M1i M2,... , Mk}, where Mj are mj x m j nilpotent matrices,

(ii) if we set Pj = Tdiag[Ej1iE.,2,... ,E,k]T-1, where E,1 = 0 if j Ejj = Ij, we obtain

(I .

PjPh =0

=P1+P2+...+Pk,

1, while h),

(.1

S = A1P1 + A2P2 + ... + AkPk.

Therefore, P. = P, (S) = P? (A) (j = 1, 2, ... , k) (cf. Observation IV-1-14). The following two remarks concern real diagonalizable matrices.

Remark IV-1-16. Let A be a real nxn diagonalizable matrix and let A1, A2 ... , An be the eigenvalues of A. Then, there exists a real n x n invertible matrix P such

that (1) in the case when all eigenvalues A j (j = 1,2,... , n) are real, then P-1 AP is a real diagonal matrix whose entries on the main diagonal are A1, A2, ... , An, (2) in the case when all eigenvalues are not real, then n is an even integer 2m and P-1A fP = diag[D1, D2, ... , DmJ, where A23_1 = a, + ibl, A2J = a) - ibj, and

Dj

abj

a,

(3) in other cases, P1AP = diag[D1, D2i ... , Dhj, where A23_1 = a)+ib,, A2.1 =

a, - ib,, and D. = I b, a j for j = 1, 2, ... , h - 1 and Dh is a real diagonal matrix whose entries on the main diagonal are A, (j = 2h - 1,... , n). Remark IV-1-17. For any given real n x n matrix A, there exists a sequence {Bk : k = 1, 2,. ..} of real n x n diagonalizable matrices such that lim Bk = A. This can be proved in the following way: (i) let A = S + N be S-N decomposition of A,

(ii) using Remark IV-1-16, assume that S = diag[D1, D2, ... , Dhj, as in (3) of Remark IV-1-16,

(iii) find the form of N by SN = NS, (iv) triangularize N without changing S, (v) use a method similar to the proof of Lemma IV-1-4. Details of proofs of Remark IV-1-16 and IV-1-17 are left to the reader as exercises. Now, we give two examples of calculation of the S-N decomposition.

Example IV-1-18. The matrix A

252

498

4134

698

-234

-465

-3885

-656

15

30

252

42

-10

-20

-166

-25

has two

distinct eigenvalues 3 and 4, and PA(A) = (A - 4)2(A - 3)2,

PA(A)

2

2

1

(A

14)2

A

4 + (A 13)2 + A

3

1. SOME BASIC RESULTS CONCERNING MATRICES

77

Set P2(A) _ (A - 4)2 + 2(A - 3)(A - 4)2.

-2(A-4)(A-3)2'

P1(A) = (A - 3)2

Then,

P1(A) =

-1

-2

134

198

2

2

-134

1

-125

-186

-1

-1

125

186

0

2 0

9

12

-12

0

-6

-8

0 0

-8

0

0 0

6

9

P2(A) =

-198

Therefore,

2

-2

1

5

0

0

12

12

10

0

-6

-5

S = 4P1(A) + 3P2(A) =

250

500

4000

500

-235

-470

-3760

-470

15

30

240

30

-10

-20

-160

-20

N = A - S =

Example IV-1-19. The matrix A =

3

4

3

2

7

4 3

-4 8

A1= 11, A2=1,and

(A+9)

_

1

100(A - 11)

PA(A)

I'

has two distinct eigenvalues

_

1

1 ) 21 1 ) ,

PA(A)

198

134

-125 -186

100(A - 1)2*

Hence, 1

-

(A - 1)2

-

(A + 9)(A - 11)

100

100

Set

P1(A)

_

(A - 1)2

- -

P2(A)

100

(A + 9)(A - 11).

100

Then, 1

0

P1(A) =100- 0 0

56 76

28 38

48

24

1

,

P2(A)

100

1100 0 0

-56 -28 24 -48

Therefore,

56 86

28 38

0

48

34

-16 -16

2

20

-40

32

-4

20

N=A - S= 10

0

10

1

S = 11P1(A) + P2(A) = 10

2

1

-38 76

IV. GENERAL THEORY OF LINEAR SYSTEMS

78

In this case, SN = NS = N and N2 = O. Let Vj = Image (PI(A)) (j = 1,2). Then, by virtue of (1) of Lemma IV-1-10, Pj(A)p" = p" for all IT E Vj (j=1,2). 14

Furthermore, V1 is spanned by

Pa =

14 19 12

1

0

0 0

-2

1

19 12

0

1

and V2 is spanned by

0 0

and

1

-2

Set

. Then,

Po 1 SPo =

11

0

0

1

0 0

0

0

1

Pp 1 NPo = 2

[00

1

-1

0

1

-1

.

It is noteworthy that there is only one linearly independent eigenvector x = for the eigenvalue A2 = 1.

It is not difficult to make a program for calculation of S and N with a computer. For more examples of calculation of S and N, see [HKSI.

IV-2. Homogeneous systems of linear differential equations In this section, we explain the basic results concerning the structure of solutions of a homogeneous system of linear differential equations given by (IV.2.1)

dt = A(t)y',

where the entries of the n x n matrix A(t) are continuous on an interval I = It : a < t < b). Let us prove the following basic theorem. Theorem 1V-2-1. The solutions of (1V2. 1) forms an n-dimensional vector space over C.

We break the entire proof into three observations.

Observation IV-2-2. Any linear combination of a finite number of solutions of (IV.2.1) is also a solution of (IV.2.1). We can prove the existence of n linearly independent solutions of (IV.2.1) on the interval Z by using Theorem I-3-5 with n linearly independent initial conditions at t = to. Notice that each column vector of a solution Y of the differential equation (IV.2.2)

dt = A(t)Y

on an n x n unknown matrix Y is a solution of system (IV.2. 1). Therefore, construct-

ing an invertible solution Y of (IV.2.2), we can construct n linearly independent solutions of (IV.2.1) all at once. If an n x n matrix Y(t) is a solution of equation

(IV.2.2)on an interval I={t:a 0 for some j, some solutions of (IV.3.9) tend to 00 as t - +00, (iii) every solution of (IV.3.9) is bounded for t > 0 if and only if R(AE) < 0 for j = 1, 2, ... , k and NP, (A) = 0 if R(A)) = 0. Now, we illustrate calculation of exp[tA] in two examples. Note that in the case when A has nonreal eigenvalues, we must use complex numbers in our calculation. Nevertheless, if A is a real matrix, then exp[tA] is also real. Hence, at the end of our calculation, we obtain real-valued solutions of (IV.3.9) if A is real.

-2 Example IV-3-3. Consider the matrix A =

0 3

1

0

2

1

-2 0

. The characteristic

polynomial of A is pA(A) _ (A -1)(A+2)2. By using the partial fraction decomposition of

1

PA(A),

and

P2(A)

we derive 1 =

(A + 2)2 - (A + 5)(A - 1)

+ 5)(A - 1),

9

. Setting P1(A) =

(A + 2)2 9

we obtain

9

P1(A) =

0 0

0 0

1

1

0 0, P2(A) = 1

1

0

0

1

0 0

-1

-1

0

Set

S = PI(A) - 2P2(A) =

-2 0 3

0

0

-2 0 3

1

,

N=A-S=

0

0

1

0

0

0.

0

-1

0

3. HOMOGENEOUS SYSTEMS WITH CONSTANT COEFFICIENTS

83

Note that N2 = 0. Hence,

exp[tA] = e'[ 13 + tN ] PI(A) + e-21 [ 13 + tN ] P2(A) e-2t

to-2t

0

e-2c

0

et - (1 +

et - e-2t

0 t)e-2t

et

The solution of the initial-value problem dt = Ay, y(0) = y is y(t) = exp[tA]i). To find a solution satisfying the condition Jim y(0) = 0, we must choose it so that

t +m

P1(A)ij = 0. Such an iJ is given by it = P2(A)c", where c is an arbitrary constant vector in C3.

Example IV-3-4. Next, consider the matrix A =

0

-1

1

0

-1

1

1

-1 . The charac0

teristic polynomial of A is PA(A) = A(A2 + 3) = A(A - if)(A + i\/3-). Using the 1 partial fraction decomposition of , we obtain PA (A)

(A- if)(A+ if) - A(A+ if)- sA(A- if}.

1= Setting

P1(A) = 3(A2 + 3),

I A(A + if),

P2(A)

P3(A)

A(A - if),

we obtain 1

1

1

1

Pi(A) = -

1

1

1

3

1

1

1

1-if 1+if -2 1-if 6 1-if 1+iv -2 1

,

-2

P2(A) = -- 1+if

and P3(A) is the complex conjugate of P2(A). If we set

S = (if)P2(A) - (if)P3(A), then S = A. This implies that N = 0. Thus, we obtain

exp(tA] = P1(A) + e,.''P2(A) + e-;J1tP3(A) = P1(A) + 23t (e'3tP2(A))

.

Using

(e'' t(1 + if)) = cos(ft) - \/3-sin(ft), 2 (e'-1t(1 - if))

wa(ft) + f sin(ft),

IV. GENERAL THEORY OF LINEAR SYSTEMS

84

we find

a(t)

c(t)

b(t)

exp[tA] = 1 c(t) a(t) b(t) 3

where

a(t)

c(t)

b(t)

a(t) = I + 2cos(ft), b(t) = 1 - {cos(ft) + f sin(ft)} ,

c(t) = 1 - {wa(ft) - f sin(ft)) Remark IV-3-5. Fhnctions of a matnx In this remark, we explain how to define functions of a matrix A.

I. A particular case: Let A0, I,,, and N be a number, the n x n identity matrix, and an n x n nilpotent matrix, respectively. Also, consider a function f (X) in a neighborhood of A0. Assume that f (A) has the Taylor series expansion (i.e., f is analytic at A0) f(A) = f(A0) + 1

f

(h)

h?Ao)(A

-

Ao)h.

h=1

In this case, define f (,\o1 + N) by

= f(AoI. + N) = f(Ao)I. +

f(h)PLO) .A

h=1

h.t

A

=

f(Ao)II +

n-1 f(h)(AO) NA. h=1

h.

n-1 (h)

Since N is nilpotent, the matrix >2f

is also nilpotent. Therefore, the

h=1

characteristic polynomial pf(A0,+N)(A) of f(AoI + N) is

Pf(aol-N)(A) = (A - 1(A0))". II. The general case: Assume that the characteristic polynomial PA(A) of an n x n matrix A is PA(A) = (A - .l1)m1(A - A 2 ) ' - 2 ... (A - Ak)mr.

where A1,... , Ak are distinct eigenvalues of A. Construct P, (A) (j = 1, ... , k), S, and N as above. Then,

A = (A1In + N)P1(A) + (A21n + N)P2(A) + ... + 0k1n + N)Pk(A). Therefore,

A' = (A11n + N)1P1(A) + (A2In + N)'P2(A) 4- ... + (Ak1n + N)'Pk(A) for every integer P.

3. HOMOGENEOUS SYSTEMS WITH CONSTANT COEFFICIENTS

85

Assuming that a function f (A) has the Taylor series expansion

f(A) = f(A3)

00 0f(h)(A,)(A-A,)'' h!

h=1

at A = A., for every j = 1, ... , k, we define f (A) by (IV.3.10)

f(A) = f (A11, + N)Pk(A) + f (A21n + N)P2(A) + ... + f (Akin + N)Pk(A).

Since P2(S) = P,(A) (cf. Observation IV-1-15), this definition applied to S yields

f(S) = f(AIIn)P1(A) + f(A21.)P2(A) + -' + f(AkII)Pk(A) and f (A) - f (S) has a form N x (a polynomial in S and N). Therefore, f (A) - f (S) is nilpotent. Furthermore, f (S) and f (A) commute. This implies that

f(A) = f(S) + (f(A) - f(S)) is the S-N decomposition of f (A). Thus,

Pf(A)(A) = pf(s)(A) = (A - f(A1))m' ... (A -

f(Ak))m

Example IV-3-6. In the case when f (A) = log(A), define log(A) by

log(A) = log(A1In+N)Pk(A) + log(A21n+N)Pk(A) + ... + log(Akln+N)Pk(A), where we must assume that A is invertible so that A, # 0 for all eigenvalues of A. Let us look at log(AoI,, + N) more closely, assuming that A0 0 0. Since

/

log(Ao + u) = 1000) + log t 1 + -o) = logl o) + \\

+O°

m=1

m+1

(-I)m

Io

we obtain

log(.1oIn + N) = log(Ao)ln +

n-1 (-1)m+1 m=1

m

It is not difficult to show that exp[log(A)J = A. In fact, since (log(A))m = (log(A11n + N))mPk(A) + (log(A21n + N))mp2(A) + ... + (log(Akln + N))mPk(A), it is sufficient to show that exp[1og(Aoln + N)J = A01,, + N. This can be proved by using exp[log(Ao +,u)] =A0 +,u.

IV. GENERAL THEORY OF LINEAR SYSTEMS

86

Observation IV-3-7. In the definition of log(A) in Example N-3-6, we used log(A,). The function log(A) is not single-valued.

Therefore, the definition of

log(A) is not unique.

Observation IV-3-8. Let A = S + N be the S-N decomposition of A. If A is invertible, S is also invertible. Therefore, we can write A as A = S(II + M), where

M = S' N = NS-1. Since S and N commute, two matrices S and M commute. Furthermore, M is nilpotent. Using this form, we can define log(A) by

log(A) = log(S) + log(Ih + M), where

log(S) = Iog(A1)P1(A) + log(A2)P2(A) +

. + log(Ak)Pk(A)

and

(1)m+1

log(IR + Al) = E - m

Mm.

M=1

This definition and the previous definition give the same function log(A) if the same definition of log(A,) is used.

Example N-3-9. Let us calculate sin(A) for A =

3

4

2

7

-4 8

3 4

(cf. Example

3

IV-1-19). The matrix A has two eigenvalues 11 and 1. The corresponding projections are 7 -14 -7 0 14

P1(A) =

25

25

0

L9

L9

0

12 25 25

6 50 25

P2(A) =

0 0

25

25

2s

50

-12

19

5

25

Define S = 11P1(A) + P2(A) and N = A - S. Then N2 = 0. Also, (

t

sin(11 + x) = -0.99999 + 0.0044257x + 0.499995x2 + 0(x3 ), sin(1 + x) = 0.841471+0.540302x-0.420735x 2 + 0(x3).

Therefore,

sin(A) _ (-0.9999913 + 0.0044257N)P1(A) + (0.84147113 + 0.540302N)P2(A) 1.92207 1.0806

-1.8957 -1.42252

-0.407549 -0.591695

-2.1612

0.845065

0.1834

It is known that sin x has the series expansion +00

sin x = 1

(2h +)1)I

T2h+1

4. SYSTEMS WITH PERIODIC COEFFICIENTS

87

Therefore, we can also define sin(A) by sin(A)

-

A2h+1 -- + (2h(-1)h + 1)!

However, this approximation is not quite satisfactory if we notice that

_1h sin(11) = -0.99999

and h=O

112h+1

= -117.147.

(2h + 1)!

IV-4. Systems with periodic coefficients In this section, we explain how to construct a fundamental matrix solution of a system dy dt

(IV.4.1)

= A(t)y

in the case when the n x n matrix A(t) satisfies the following conditions: (1) entries of A(t) are continuous on the entire real line R, (2) entries of A(t) are periodic in t of a (positive) period w, i.e., (IV.4.2)

A(t + w) = A(t)

for

t E R.

Look at the unique n x n fundamental matrix solution 4'(t) defined by the initialvalue problem (IV.4.3)

dY = A(t)Y, .it

Y(0) = In

Since 4i '(t + w) = A(t + w)4S(t + w) = A(t)4'(t + w) and +(0 + w) = 4t(w), the matrix 41(t +w) is also a fundamental matrix solution of (IV.4.3). As mentioned in (1) of Remark IV-2-7, there exists a constant matrix r such that 44(t +w) = 4i(t)r and, consequently, r = 4?(w). Thus, (IV.4.4)

41(t + w) = 4(t)t(w)

for t E R.

Setting B = w-' log(4?(w)J (cf. Example IV-3-6), define an n x n matrix P(t) by (IV.4.5)

P(t) = d'(t) exp(-tBJ.

Then, P(t + w) = ((t + w) exp(-(t + w)BI = 4i(t)0(w) exp(-wB) exp(-tBJ = 4'(t) exp(-tBJ = P(t). This shows that P(t) is periodic in t of period w. Thus, we proved the following theorem.

IV. GENERAL THEORY OF LINEAR SYSTEMS

88

Theorem IV-4-1 (G. Floquet [Fl]). Under assumptions (1) and (2), the fundamental matrix solution 4i(t) of (IV.4.1) defined by the initial-value problem (IV.4.3) has the form

4'(t) = P(t) exp[tB],

(IV.4.6)

where P(t) and B are n x n matrices such that (a) P(t) is invertible, continuous, and periodic of period w in t, (Q) B is a constant matrix such that 4'(w) = exp[wB]. Observation IV-4-2. As was explained in Observation IV-3-8, letting 4'(w) _ S + N = S(I + M) be the S-N decomposition of 4(w), we define log($(w)) by log(4?(w)) = log(S) + log(1 + M), where log(S) = log(A1)P1(4'(w)) + log(a2)P2(4'(w)) + - + log(Ak)Pk('6(w)) -

and

n- 1

l)m+l

log(!! + M) = E (- m

Mm.

M=1

In the case when A(t) is a real matrix, the unique solution 4 (t) of problem (IV.4.3) is also a real matrix. Therefore, the entries of 4'(w) are real. Since S and N are real

matrices, the matrix M = S-'N is real. Therefore, log(I + M) is also real. Let us look at log(S) more closely. If \j is a complex eigenvalue of 4'(w), its complex conjugate A3 is also an eigenvalue of 4'(w). In this case, set A,+1 = A3. It is easy to see that the projection Pj+1(4?(w)) is also the complex conjugate of P,(4'(w)). However, if some eigenvalues of 4'(w) are negative, log[S] is not real. To see this more clearly, rewrite log[s] in the form

log[S] _ E log[A,]Pi(4?(w)) + E log1A3]P'(4'(w))other j

A, 7j } are two constant vectors in C".

Hint. Note that cos(xA) and sin(xA) are not linearly independent when det A = 0. If we define F(u) =

sin u

is

u

then the general solution of the given differential equation

j(x) = cos(xA)c"1 + xF(xA)c2. IV-9. Find explicitly a fundamental matrix solution of the system dy = A(t)y if there exists a constant matrix P E GL(n,C) such that P-"A(t)P is in Jordan canonical form, i.e.,

P-1A(t)P = diag[A, (t)I1 + N1iA2(t)12 + N2,... ,.1k(t)Ik +Nkj, where for each j, Ap(t) is a C-valued continuous function on the interval a S t S 6, I1 is the n, x n j identity matrix, and N, is an nl x n, matrix whose entries are 1 on the superdiagonal and zero everywhere else.

Hint. See [GH]. /r

IV-10. Find log (I

11

]).cos([

\` `

arctan

3

0

1

2

0

1

-1

-1 j).anii 1

698 1).

252

498

4134

-234

-465

-3885

-656

15

30

252

42

-10 -20 -166 -25 IV-11. In the case when two invertible matrices A and B commute, find

log(AB) - log(A) - log B if these three logarithms are defined as in Example IV-3-6.

105

EXERCISES IV

IV-12. Given that A =

0 0

0 0

2

3

1

0

0 0

1

0

-2 0

4

yi

I/2

'

1/3

0

1/4

find a nonzero constant vector ii E C4 in such a way that the solution l(t) of the initial-value problem d9

A9,

9(0)

satisfies the condition lim y"(t) = 0. t -+oo

IV-13. Assume that A(t), B(t), and F(t) are n x n, m x m, and n x m matrices, respectively, whose entries are continuous on the interval a < t < b. Let 4i(t) be

an n x n fundamental matrix of - = A(t)4 and W(t) be an in x in fundamental matrix of Z = B(t)u. Show that the general solution of the differential equation on an ii x m matrix Y dY (E)

= A(t)Y - YB(t)

F(t),

_WT

is given by

Y(t) = 4;(t) C 4,(t)-1

r`

where C is an arbitrary constant n x m matrix.

IV-14. Given the n x n linear system dY dt

= A(t)Y - YB(t),

where A(t) and B(t) are n x n matrices continuous in the interval a < t < b, (1) show that, if Y(to) -I exists at some point to in the interval a < t < b, then Y(t)-i exists for all points of the interval a < t < b, (2) show that Z = Y-1 satisfies the differential equation dZ = B(t)Z - ZA(t). dt

IV-15. Find the multipliers of the periodic system dt = A(t)f, where A(t) _ [cost

simt

-sint

cost

IV. GENERAL THEORY OF LINEAR SYSTEMS

106

Hint. A(t) = exp It 101 0]] IV-16. Let A(t) and B(t) be n x n and n x m matrices whose entries are realvalued and continuous on the interval 0 < t < +oo. Denote by U the set of all R'-valued measurable functions u(t) such that ,u(t)j < 1 for 0 < t < +oo. Fix a

l`

vector { E R". Denote by ¢(t, u) the unique R"-valued function which satisfies the initial-value problem

at

= A(t)i + B(t)u(t), i(0)

where u E U. Also, set

R = {(t, ¢(t, u)) : 0:5 t < +oo, u E R}. Show that R is closed in

Rn+i.

Hint. See ILM2, Theorem 1 of Chapter 2 on pp. 69-72 and Lemma 3A of Appendix of Chapter 2 on pp. 161-163j.

IV-17. Let u(t) be a real-valued, continuous, and periodic of period w > 0 in t on R. Also, for every real r, let ¢(t, r) and t1'(t, r) be two solutions of the differential equation d2y dt2

(Eq)

- u(t) y = 0

such that

o(r, r) = 1,

0'(r, r) = 0,

and

0(7-,T) = 0,

t;J'(T,T) = 1,

where the prime denotes d Set 0(0, r) C(r) = 10'(0.'r)

V(0, r) 1 i,I (0,r) J

,

0(r+w,r) Vl(r+w,r) 0(r+w,r) tG'(r+w,r)

M(r) _ (I) Show that

M(r) = C(r)'1M(O)C(r). (11) Let A+ and A_ be two eigenvalues of the matrix M(r). Let

K _(r)

be eigenvectors of M(r) corresponding to A+ and

I

A_`,

K+(r)J and respectively.

how that (i) K±(r) are two periodic solutions of period w of the differential equation

dK dT

± K2 + u(r) = 0,

(ii) if we set

Qf{t,r) = exp [J t Kt(() d(l r

these two functions satisfy the differential equation (Eq) and the conditions

13f(t+w,r) = A*O*(t,r).

107

EXERCISES IV

Hint for (I). Set 4'(t,r) _ matrix solution of the system

(t

(t'

)

Then, ((t, r) is a fundamental

I JJJ

-

and y = 4'(t, r)c is the solution satisfying the initial condition y(r) = c'. Note that C(r) = 4'(0, r) and M(r) = 4'(r + w, r). Therefore, 4'(t, r) = 4'(t, 0)C(r) and 4'(t + r) = 4'(t, r)M(r). Now, it is not difficult to see 4'(w, r) = 4'(w, 0)C(r) _ 4'(0, r)M(r) = C(r)M(r) and 4'(w, 0) = M(O). Hint for (II). Since eigenvalues of M(r) and M(0) are the same, the eigenvalues of M(r) is independent of r. Solutions 77+ (t) of (Eq) satisfying the condition Y7+ (t +

w) = A+n+(t) are linearly dependent on each other. Hence,

W+(t)

= K+(t) is

independent of any particular choice of such solutions q+(t). In particular K+(t + W) = 17'+ (t + w) = K+(t). Problem (II) claims that the quantity K+(r) can be 17+(t + w)

found by calculating eigenvectors of M(r) corresponding to A+. Furthermore, A+ = exp I

o o

K+(r)drJJ I I.

The same remark applies to A_. The solutions 0±(x, r)

of (Eq) are called the Block solutions.

IV-18. Show that (a) A real 2 x 2 matrix A is symplectic (i.e., A E Sp(2, R)) if and only if det A = 1;

(b) the matrix

01

l

0J

N is symmetric for any 2 x 2 real nilpotent matrix N.

Hint for (b). Note that det(etNJ = I for any real 2 x 2 nilpotent matrix N. IV-19. Let G, H, and J are real (21n ) x (2n) matrices such that G is symplectic,

H is symmetric, and J = I 0n

. Show that G-1JHGJ is symmetric.

IV-20. Suppose that the (2n) x (2n) matrix 4'(t) is the unique solution of the initial-value problem L'P = JH(t)4', 4'(0) = 12n, where J = I 0n n and H(t) l T J is symmetric. Set L(t) = 4'(t)-1JH(t)4'(t)J. Show that d4'dt) = JL(t)4'(t)T, where 4'(t)T is the transpose of 4'(t).

CHAPTER V

SINGULARITIES OF THE FIRST KIND In this chapter, we consider a system of differential equations

xfy

(E)

= AX, Y-),

assuming that the entries of the C"-valued function f are convergent power series in complex variables (x, y-) E Cn+I with coefficients in C, where x is a complex inde-

pendent variable and y E C" is an unknown quantity. The main tool is calculation with power series in x. In §I-4, using successive approximations, we constructed power series solutions. However, generally speaking, in order to construct a power 00

series solution y"(x) = E xmd n, this expression is inserted into the given differM=1

ential equation to find relationships among the coefficients d, , and the coefficients d,,, are calculated by using these relationships. In this stage of the calculation, we do not pay any attention to the convergence of the series. This process leads us to the concept of formal power series solutions (cf. §V-1). Having found a formal power series solution, we estimate Ia',,, i to test its convergence. As the function x-I f (x, y) is not analytic at x = 0, Theorem 1-4-1 does not apply to system (E). Furthermore, the existence of formal power series solutions of (E) is not always guaranteed. Nevertheless, it is known that if a formal power series solution of (E) exists, then the series is always convergent. This basic result is explained in §V-2 (cf. [CL, Theorem 3.1, pp. 117-119) and [Wasl, Theorem 5.3, pp. 22-251 for the case of linear differential equations]). In §V-3, we define the S-N decomposition for a lower block-triangular matrix of infinite order. Using such a matrix, we can represent a linear differential operator (LDO)

G[Yj = x

+ f2(x)lj,

where f2(x) is an n x n matrix whose entries are formal power series in x with coefficients in Cn. In this way, we derive the S-N decomposition of L in §V-4 and a normal form of L in §V-5 (cf. [HKS]). The S-N decomposition of L was originally defined in [GerL]. In §V-6, we calculate the normal form of a given operator C by using a method due to M. Hukuhara (cf. [Sill, §3.9, pp. 85-891). We explain the classification of singularities of homogeneous linear differential equations in §V-7. Some basic results concerning linear differential equations given in this chapter are also found in [CL, Chapter 4].

108

1. FORMAL SOLUTIONS OF AN ALGEBRAIC DE

109

V-1. Formal solutions of an algebraic differential equation We denote by C[[x]] the set of all formal power series in x with coefficients in w

00

b,,,x', we define

C. For two formal power series f = E amxm and g = m=0

m=0

f = g by the condition an = b", for all m > 0. Also, the summf + g and the 00

00

(am +b,,,)xm and f g =

product f g are defined by f +g = m=0

(F am-nbn )x"', m=0 n=0

00

respectively. Furthermore, for c E C and f = >2 a,,,xm E C[[x]], we define cf m=0

00

by cf = >camx'. With these three operations, C[[x]] is a eommutattve algebra m=0

over C with the identity element given by E bmxm, where bo = 1 and b,,, = 0 m=0

if m > 1. (For commutative algebra, see, for example, [AM].) Also, we define the

of f with respect to r by dx _ E(m+ 1)a,,,+,x' and the integral

derivative

m=0 rX

(

airi x"' for f = E a,,,xm E C[[xj]. Then, C[[x]]

f (x)dx by f f (x)dx = o

0

00

"0

r

m=0

m=1

is a commutative differential algebra over C with the identity element. There are some subalgebras of C[[x]j that are useful in applications. For example, denote by C{x) the set of all power series in C[]x]] that have nonzero radii of convergence. Also, denote by C[x] the set of all polynomials in x with coefficients in C . Then, C{x} is a subalgebra of C[[x]] and C]x] is a subalgebra of C{x}. Consequently, C[x] is also a subalgebra of C[[x]]. Let F(x, yo, y, , ... , yn) be a polynomial in yo, y,..... yn with coefficients in C[[x]]. Then, a differential equation dy ,...dxn

y,,d"y

F

(V.1.1)

f

=0

is called an algebrutc differential equation, where y is the unknown quantity and x

is the independent variable. If F ` x, f, ... ,

dxn

= 0 for some f in C[[x]], then

such an f is called a formal solution of equation (V.1.1). In this definition, it is not necessary to assume that the coefficients of F are in C{x}. Example V-1-1. To find a formal solution of xLY

(V.1.2)

+y-x=0,

set y = Famx'. Then, it follows from (V.1.2) that ao = 0, 2a, = 1, and m>0

(m + 1)am = 0 ( m > 2 ). Hence, y = 2x is a formal solution of equation (V.1.2).

V. SINGULARITIES OF THE FIRST KIND

110

In general, if f E C{x} is a formal solution of (V.1.1), then the sum of f as a convergent series is an actual solution of (V.1.1) if all coefficients of the polynomial

FareinC{x}. Denote by x°C([x]j the set of formal series x°f (x), where f (x) E C([x]], a is a complex number, and x° = exp[a logx]. For 0 = x°f E x°C[[x]], 0 =

x°g E x°Cj[x]j, and h E C(jx]], define 0 + iP = x(f + g), hO = x°hf, and xL = ox°f + x°x Then, x°C([xfl is a commutative differential module over .

the algebra C((x]]. Similarly, let x°C{x} denote the set of convergent series x°f (x),

where f (x) E C{x}. The set x°C{x} is a commutative differential module over the algebra C{x}. Furthermore, if a is a non-negative integer, then x°C([x]] C C[[x]]. If F(x, yo,... , y,) is a formal power series in (x, yo,... , yn) and if fo(x) E xC[[x]j, ... , fn(x) E xC([xj], then F(x, fo(x),... , fn(x)) E C[[xJj. Also, if F(x, yo,. .. , yn) is a convergent power series in (x, yo,... , yn) and if fa(x) E xC{x}, ... , fn(x) E xC{x}, then F(x, fo(x), ... , f,,(x)) E C{x}. Therefore, using the notation x°C[[x]]n to denote the set of all vectors with n entries in x°C[(x]], we can define a formal solution fi(x) E xC[[x]]n of system (E) by the condition x = Ax, i) in C[[x]j . (Similarly, we define x°C{x}n.) Also, in the case of a homogeneous sys-

tem of linear differential equations to be a formal solution if x theorem.

xfy

A(x)g, a series d(x) E x°C[[xj]n is said

= A(x)e in x°C[[x]jn. Now, let us prove the following

Theorem V-1-2. Suppose that A(x) is an n x n matrix whose entries are formal power series in x and that A is an eigenvalue of A(O). Assume also that A + k are not eigenvalues of A(O) for all positive integers k. Then, the differential equation (V.1.3)

x!Ly

= A(x)g

has a nontrivial formal solution fi(x) = xa f (x) E x"C[[x]]n Proof. 00

Insert g = xa E x'nam into (V.1.3). Setting A(x) _ y2 xmAm, where An E m=0

m=0

Mn(C) and Ao = A(O), we obtain 00

xa

Co

(A + m)xm= xa

LtAm_i.

Therefore, in order to construct a formal solution, the coefficients am must be determined by the equations m-h ,1ao = Aoa"o

and

(A + m)am = Aodd,n + E Am-hah h=0

(m > 1 ).

1. FORMAL SOLUTIONS OF AN ALGEBRAIC DE

111

Hence, ado must be an eigenvector of Ao associated with the eigenvalue A, whereas m-h

Am-hah form? 1. 0

ii',,, = ((A + m)1. - A0)-1 h

For an eigenvalue Ao of A(O), let h be the maximum integer such that Ao + h is also an eigenvalue of A(O). Then, Theorem V-1-2 applies to A = A0 + h. The convergence of the formal solution ¢(x) of Theorem V-1-2 will be proved in §V-2, assuming that the entries of the matrix A(x) are in C(x) (cf. Remark V-2-9). n

Let P = Eah (x)&h (b = x d h=0

/

be a differential operator with the coefficients

ah(x) in C([x]]. Assume that n > 1 and an(x) 54 0. Set ab

P[x'] = x'

fm(s) xm, m=np

where the coefficients f,,, (s) are polynomials in s and no is a non-negative integer 0. Then, we can prove the following theorem. such that

Theorem V-1-3. (i) The degree of f,,,, in s is not greater than n. (ii) If zeros of fno do not differ by integers, then, for each zero r of f,,, there exists a formal series x' (x) E xr+IC[[x]] such that P[xr(1 +O(x))] = 0.

Proof (i) Since 6h(x'] = shx', it is evident that the degree of f,,,, in s is not greater than n.

+o0

(ii) For a formal power series d(x) = E ckxk, we have k=1

+oo

P[x'(I + b(x))] = P(x'] +

L. Ck P[x'+k

J

k=1 +oo

= x'

+oo

{mnof"

(s)xm + > Ckxk k =1

+oo

+ k)xm/ (frn(s m=no }

((,fm(s)

= x' fno(s)x +

+oo

m -no

Ckfm-k(s + k))

+ k=1

xa+no AK( + M=1

x"`m J1

+ E Ckfno+m-k(s + k)) x"'

.

J

k=1

In order that y = x''(1 +b(x)) be a formal solution of P[y] = 0, it is necessary and sufficient that the coefficients cm and r satisfy the equations m

fno (r) = 0,

fno+m (r) + E Ckfno+m-k(r + k) = 0 k=1

(m > 1).

V. SINGULARITIES OF THE FIRST KIND

112

It is assumed that f,,,, (r + m) 54 0 for nonzero integer m if r is a zero of Therefore, r and c,,, are determined by m-1

(r) = 0 and cm = -7,(

1

r + m)

ckfn+m-k(r + k)

(m > 1).

Convergence of the formal solution x'(1+¢(x)) of Theorem V-1-3 will be proved in s is n and the coefficients at the end of §V-7, assuming that the degree of ah(x) of the operator P belong to C{x}. The polynomial f.,, (s) is called the indicial polynomial of the operator P.

Remark V-1-4. Formal solutions of algebraic differential equation (V.1.1) as defined above are not necessarily convergent, even if all coefficients of the polynomial 00

F are in C{x}. For example, y = Y (-1)m(m!)xm+i is a formal solution of the 2y

m=0

differential equation 2d + y - x = 0. Also, the formal solution x' (I + fi(x)) of 2 Theorem V- 1-3 is not necessarily convergent if the degree of f,,. (s) in s is less than

n. In order that f = >a,,,xm be convergent, it is necessary and sufficient that m=0

la,,, I < KA' for all non-negative integers m, where K and A are non-negative num00

bers. Also, it is known that some power series such as

(m!)mxm do not satisfy

any algebraic differential equation. The following result'n--R gives a reasonable necessary condition that a power series be a formal solution of an algebraic differential equation.

Theorem V-1-5 (E. Maillet [Mail). Let F(x, yo, yi, ... , yn) be a nonzero polynomial in yo, yi, ... , y, with coefficients in C{x}, and let f = E amxm E CI[x]] be m=0

a formal solution of the differential equation

F(x, y, Ly, ... , dny dxn = 0.

/

Then,

there exist non-negative numbers K, p, and A such that (V.1.4)

I am I

< K(m!)PA' (m > 0).

We shall return to this result later in §XIII-8. Remark V-1-6. In various applications, including some problems in analytic number theory, sharp estimates of lower and upper bounds of coefficients am of a formal

solution f = F,00axm of an algebraic differential equations are very important. m =9

For those results, details are found, for example, in [Mah], [Pop], [SS1], (SS2], and [SS3]. The book [GerT] contains many informations concerning upper estimates of coefficients lam I.

2. CONVERGENCE OF FORMAL SOLUTIONS

113

V-2. Convergence of formal solutions of a system of the first kind In this section, we prove convergence of formal solutions of a system of differential equations X

(V.2.1)

fy

= AX, Y-),

where y E C' is an unknown quantity and the entries of the C"-valued function f are convergent power series in (x, y7) with coefficients in C. A formal power series 00

E x' , ,,,

(V.2.2)

(c,,, E C" )

E xC([xfl"

rn-t is a formal solution of system (V.2.1) if (V.2.3)

X

dx

= f(x,0)

as a formal power series. To achieve our main goal, we need some preparations.

Observation V-2-1. In order that a formal power series (V.2.2) satisfy condition (V.2.3), it is necessary that f (O, 0) = 0. Therefore, write f in the form

f (x, y-) = o(2-) + A(x)g + F, y~'fv(x), Iel>2

where

(1) p = (pl,... ,p") and the p, are non-negative integers, (2) jpj = pi +

+ pn and yam' = yi'

yn^, where yi,...are the entries

of g,

(3) fo E xC{x}" and f, E C{x}", (4) A(x) is an n x n matrix with the entries in C{x}. Note that and

fo(x) = Ax, 6)

A(x) _ Lf(x,0).

Setting

A(x) = F, x'A"s

Ao =

M=0

LZ

0,0) I

where the coefficients A", are in M"(C), write condition (V.2.3) in the form xd'o dx

= Ao +

f (x, ¢)

- A0

.

Then, (V.2.4)

"t,,

= A0E

+ rym

for m = 1, 2, ... ,

V. SINGULARITIES OF THE FIRST KIND

114

where (J(x))p

f (x, 0) - AoO = o(x) + IA(x) - Ao1 fi(x) +

fp(x) IpI>2

00

1: xm7m M=1

and Im E C n. Note that ry'm is determined when c1 i ... , c,,,_ 1 are determined and

that the matrices mI" - A0 are invertible if positive integers m are sufficiently large. This implies that there exists a positive integers mo such that if 61, ... , C,,,a are determined, then 4. is uniquely determined for all integers m greater than mo. Therefore, the system of a finite number of equations (V.2.5)

mEm

+

= A0

(m = 1,2,... ,mo)

decides whether a formal solution ¢(x) exists. If system (V.2.5) has a solution {c1

i ... , c,,,a }, those mo constants vectors determine a formal solution (x) uniquely.

Observation V-2-2. Supposing that formal power series (V.2.2) is a formal soluN

x'6,. Since

tion of (V.2.1), set ¢N (x) _ m=1

Ax, (x))

- f (x, N(x)) = A(x)(d(x) - N(x)) + F, [d(x)p - N(x)D] ff(x), Ipl>2

it follows that AX, $(x)) - f (X, dN(x)) E xN+C[Ix]]". Also, xd¢N(x)

= xdd(x) dx

dx

Hence, AX, 4V W) (V.2.6)

-

xdoN(x) E xN+1C((x((n. dx

xdON(x) E xN+1C{x}". Set dx

zddN(x) dx

9N,0(x) =

Now, by means of the transformation y" = z1+ y5N(x), change system (V.2.1) to the system

dz

xdx = JN(x,z

(V.2.7)

on i E C", where

9N(x, l = f (x, z + dN(x)) -

xdjN(x)

dx

= 9N,O(x) + f(x,Z+dN(x))f r- f(x,dN(x))

= 9N,0(x) + A(x)z" + L. [(+ $N(x))" - N(x)'] f,,(x) InI?2

115

2. CONVERGENCE OF FORMAL SOLUTIONS

As in Observation V-2-1, write gN in the form 9N(x,

= 9N,O(x) + BN(x)z + L xr 9N,p(x), Ip1?2

where (1) 9N,o E xN+1C{x}" and g""N,p E C{x}",

(2) BN(x) is an n x n matrix with the entries in C{x}, (3) the entries of the matrix BN(X) - Ao are contained in xC{x}. Observation V-2-3. System (V.2.7) has a formal solution 00

(V.2.8)

ON(x) = O(x) - ON(x) _

x'"cn, E xN+1C1jx11 n.

m=N+1

The coefficients cm are determined recursively by

me,, = Aocm + '7m

for m = N + 1, N + 2, ... ,

where

9N(x,ILN(x)) - AO1GN(x) = f(x,$(x)) - Ao1 N(x) -

-

= f (x, fi(x))

AOcN(x) -

dx

xdVx)

= 9N,O(x) + [BN(x) - Ao1 i'N(x) + E (ZGN(x))p 9N,p(x) Ip1>2 ou

=E

xm-'1'm

m=N+1

Note that the matrices min - Ao are invertible for m = N + 1, N + 2, ... , if N is sufficiently large.

Observation V-2-4. Suppose that system (V.2.1) has an actual solution q(x) such that the entries of i(x) are analytic at x = 0 and that ij(0) = 0. Then, the dim (0) of q(x) at x = 0 is a formal solution of Taylor expansion ¢(x) _

j

(V.2.1). Furthermore, is convergent and E xC{x}". Keeping these observations in mind, let us prove the following theorem.

Theorem V-2-5. Suppose that fo(x) = f (x, 0) E xN+1C{x}n and that the matrices mIn - A0 (m > N + 1) are invertible, where Ao = (V.2.1) has a unique formal solution

69

00

(V.2.9)

t(x) =

E X'n4n E x^'+1C((x}]n. m=N+1

Furthermore, ¢ E xN+1C{x}n.

(0, 0). Then, system

V. SINGULARITIES OF THE FIRST KIND

116

Remark V-2-6. Under the assumptions of Theorem V-2-5, system (V.2.1) possibly has many formal solutions in xC[[x)J". However, Theorem V-2-5 states that there is only one formal solution in xN+1C[[xJJn

Proof of Theorem V-2-5.

We prove this theorem in six steps.

Step 1. Using the argument of Observation V-2-1, we can prove the existence and uniqueness of formal solution (V.2.9). In fact,

f (x, 4) - Ao¢ = o(x) + JA(x) - Ao}fi(x) +

E (_)P_)

IpI?2 00

xmrym.

m=N+1

(m > N + 1) are

This implies that rym = 0 for m = 1, 2, ... , N. Hence, uniquely determined by

for m=N+1,N+2,....

mc,,, = Ao6,,, + rym

Step 2. Suppose that system (V.2.1) has an actual solution q(x) satisfying the following conditions:

(i) the entries of rl(x) are analytic at x = 0, (ii) there exist two positive numbers K and 6 such that

jy(x)I < KIxjN+1 00

xm

Then, the Taylor expansion E m=N+1

ml

a.,.7m'1

IxI < 6.

for

(0) of #(x) at x = 0 is a formal solution

of (V.2.1) (cf. Observation V-2-4). Since such a formal solution is unique, it follows

that $(x) = E

xm d1ij -

(0). Because the Taylor expansion of if(x) at x = 0 is

m=N+1

convergent, the formal solution ¢ is convergent and

E

xN+1C{x}".

Step 3. Hereafter, we shall construct an actual solution f(x) of (V.2.1) that satisfies conditions (i) and (ii) of Step 2. To do this, first notice that there exist three positive numbers H, b, and p such that (I)

If(x,o)I 5

HIxIN+1

for

IxI < 5

and

(II)

If(x,91) - f(x,y2)I 5 (IAoI + 1) 1111 - y2I for

Ix) < b and

Iy, I

2).

Am =

Form > 2, we write Am - Am-! [

B.

... Am,

0 Amm J

Set Nm =

m

l-1

nt. Then, A. is an

N", x Nm matrix, while Bm is an nm x N,"_ 1 matrix. Also, let Amm = Smm +Nmm

3. THE S-N DECOMPOSITION OF A MATRIX OF INFINITE ORDER 119

and Am = Sm + Aim be the S-N decompositions of Amm and Am, respectively, where Smm is an nm x nm diagonalizable matrix, Sm is an Nm x Nm diagonalizable matrix, Nmm is an nm x nm nilpotent matrix, Aim is an N. x Nn nilpotent matrix, SmmNmm = NmmSmm, and SmAm = NmSm. The following lemma shows how the two matrices Sm and Aim look. Lemma V-3-1. The matrices Sm and Aim have the following forms: S"'

S1 = s11,

J N, = Ni1,

0l

rI Sm_1

=

L Cm

Aim =

(m > 2),

Smm

[Aim_i

fm

(m > 2), O 1 Nmm

where Cm and fm are am x N,_1 matrices. Proof Consider the case m > 2. Since the matrices Sm and Aim are polynomials in Am with constant coefficients, it follows that

Sm = [B m1

and

Aim =

I

fm1 01,

ILM0

where 13m-1 and Dm-1 are Nm-1 x Nm_1 matrices, Cm and fm are nm x Nm_1 matrices, and µm and vm are nm x nm matrices. Furthermore, Dm_1 and Vm are nilpotent. Also, Bm-1Dm-1 = Dm-1Bm-i and tlmvm = 1.m µm. Hence, it suffices to show that Bm-1 and µm are diagonalizable. Note that since Sm is diagonalizable, Sm has Nm linearly independent eigen vectors. An eigenvector of Sm has one of two forms k

PJ

and [?.] , where p is an

eigenvector of Bm_1i whereas q is an eigenvector of µm. Therefore, if we count those independent eigenvectors, it can be shown that Bm-1 has N,, -I linearly independent eigenvectors, while Pm has nm linearly independent eigenvectors. Note that Nm = Nm_ 1 + nm. This completes the proof of Lemma V-3-1. 0 Lemma V-3-1 implies that the matrices Sm and Nn have the following forms:

S1 = SI1, S11

0

C21

S22

0 ... 0 ...

Cm2

...

...

Nil

0

D

...

Y21

JN22

0

Fm 1

Fm2

... ...

Sm = Cmi and

(m>2)

N1 = N11, Aim =

0 0 Smm

0 0 Nmm

(m>2),

V. SINGULARITIES OF THE FIRST KIND

120

where C,k and Fjk are n3 x nk matrices. Set S11

0

C21

S22

0 0

Cml

Cm2

"'

S =

0 N22

0 0

"'

Fm2

...

.Wmm

N11

f f21

1m 1

Smm 0

N= 0

Then,

A=S+N

(V.3.1)

and

SN=NS.

We call (V.3.1) the S-N decomposition of the matrix A.

V-4. The S-N decomposition of a differential operator Consider a differential operator

C(yl = x

(V.4.1)

+ f2(x)Y,

w

where f2(x) _ > xt1 and 1

Let us identify a formal power series

E

t=o ao

xt dt with the vector p =

a1

a2

E C. Then, the operator C is represented

t>o

by the matrix flo

0

f2l

In + f2o

0 0

f12

l1

21, + f2o

11m

f4.-l

fZn-2

0 ... 0 ... 0 ...

A =

...

f21

mI, + fZa

0

.

5. A NORMAL FORM OF A DIFFERENTIAL OPERATOR

121

where I is the n x n identity matrix. Let A = S + N be the S-N decomposition

j)

of A. Since A =

11

R2

A1, where (i =

and I,> is the oo x oo identity

L

matrix, the matrices S and N have the forms al

(V.4.2)

O 0 0

0 0

0

So a2

I. + So al

21n + So

am

am-l

am-2

...

...

...

...

...

...

at

min + SO

0

No 0

...

S=

and

O No

O

V1

V2

V1

No

Vm

Vm-1

Vm-2

No

(V.4.3)

0

N=

...

VI

respectively, where the a. and v, are n x n matrices and S2o = So +No is the S-N decomposition of the matrix S2o. 00

00

Set a(x) = So + E xtat and v(x) = No +

trot.

Then, S represents a

tvl

t_1

+ a(x)y, while N represents multiplication by differential operator Lo[y1 = x v(x) (i.e., the operator: y" -+ v(x)y-). Since A = S + N and SN = NS, it follows that

(V.4.4)

L[gf = Lo[Yj + v(x)y"

and

Lo[v(x)y1 = v(x)Lo[yl-

We call (V.4.4) the S-N decomposition of operator (V.4.1). We shall show in th next section that Lo[yj is diagonalizable and v(x)n = O.

V-5. A normal form of a differential operator Let us again consider the differential operator

L[yj = X

(V.5.1)

f

+ Q(x)U,

00

where f2(x) = Ex1fle and Sgt E WC). In §V-4, we derived the S-N decomposition (V.5.2)

t=o

L[y-] = Lo[yl + v(x)y"

and

Co[v(x)yl = v(x)Lo[yj.

V. SINGULARITIES OF THE FIRST KIND

122

where (V.5.3)

+ a(x)y

Co[YI = x

and

Co

a(x) = So +

00 xtat

and

v(x) = No +

xtvt t_1

t=1

(cf. (V.4.4)). Notice that N = So + No is the S-N decomposition of S2o and that the operator Go[ul and multiplication by v(x) are represented respectively by matrices (V.4.2) and (V.4.3). Set

0 I. +,%

0 0

0 0

al

21 + So

0

am_ 1

am-2

Sm =

Since So is diagonalizable, there exist an invertible n x n matrix P0 and a diagonal matrix Ao such that

SoPo = PoAo,

NO = diag[al, A2, ... , . \n)-

Note that A, (j = 1, 2,... , n) are eigenvalues of So. Hence, A. (j = 1, 2,... , n) are also eigenvalues of flo. For every positive integer m, Sm is diagonalizable. Hence, Po

there exists an (m + 1)n x n matrix P.. =

Pi

such that SmPm = PmA0. If

Pm

m is sufficiently large, we can further determine matrices Pt for t > m + 1 by the equations t (V.5.4)

(tl + S0)Pt + E ahP1_h = PtAo. h=1

Equation (V.5.4) can be solved with respect to Pt, since the linear operator Pt -+ (tl + So)P1 - PtAo is invertible if m is sufficiently large. In this way, we can find

A P

1

an oo x n matrix P =

I

such that SP = PAa. Set P(x) = >x'P,. Then,

Pm

t=D

entries of P(x)-1 are also formal power series in x and (V.5.5)

Co[P(x)j = P(x)Ao.

5. A NORMAL FORM OF A DIFFERENTIAL OPERATOR

123

Define two differential operators and an n x n matrix by X [U1

{

= P(x)-iC(P(x)uZ,

/Co(u1 = P(x)-'Co(P(x)Uj,

vo(x) = P(x)-'v(x)P(x),

respectively. Then 1C(u'j = X 0(u'] + vo(x)u'. Observe that

.- (V.5.6)

_.. _, f _.

_.

=x du

,

l du' \

_

. _.

1

+Aou".

This shows that the operator Co(y1 is diagonalizable. Furthermore, dvo

(V.5.7)

x

dxx) + Aovo(x) =

P(x)-'Lo(P(x)vo(x)1

= P(x)`Co[v(x)P(x)1

= P(x)-'v(x)Lo(P(x)1 = vo(x)P(x)-'Co(P(x)1 = vo(x)Ao

This shows that the entry i,k(x) on the j-th row and k-th column of the matrix vo(x) must have the form v,k(x) = 7jkxAk-a', where ryjk is a constant. Since Lt(x) is a formal power series in x, it follows that (V.5.8)

vjk(x) = 0

if

Ak - )j is not a non-negative integer.

Observe further that the matrix &. (x) can be written in the form

vo(x) = x-^°rx^°,

where

'711

'Yin

'7ni

Inn

r=

Hence, for any non-negative integer p, we have vo(x)P = x-A0rPxAo, where

z^O = exp ((log x)Ao] = diag(za' , Z112'... , za^ I. On the other hand, since No is nilpotent, vo(x)P can be written in a form i (x)P =

X'pQP(x), where mp is a non-negative integer such that lim mP = +oo and P+oo the entries of the matrix Qp(x) are power series in x with constant coefficients. Therefore, L0(x)P = 0 if p is sufficiently large. This implies that the matrix r is nilpotent. Since v(x) = P(x)vo(x)P(x)-1, we obtain v(x)^ = O. Thus, we arrive at the following conclusion.

Theorem V-5-1. For a given differential operator (V.5.1), let 0o = So +No be the S-N decomposition of the matrix N. Then, there exists an n x n matrix P(x) such that (1) the entries of P(x) are formal power series in x with constant coefficients, (2) P(O) is invertible and SoP(0) = P(0)Ao, where A0 is a diagonal matrix whose diagonal entries are eigenvalues of Q0r

V. SINGULARITIES OF THE FIRST KIND

124

(5) the transformation

P(x)u

(V.5.9)

changes the differential operator (V.5.1) to another differential operator (V.5.10)

P(x)-1G(P(x)ui = KOK + vo(x)u,

where

Ko(61 = x

dil

+ Aou,

vo(x) = x-^Orx^0,

and

r= Ynl

Inn

with constants "ljk such that (V.5.11)

Vj k = 0

if

Ak - Aj is not a non-negative integer.

Furthermore, the matrix r as nilpotent.

Remark V-5-2. It is easily verified that the matrix P(x) is a formal solution of the system

xd)

= P(x)(Ao + vo(x)) - Sl(x)P(x).

Since the entries of vo(x) are polynomials in x, the power series P(x) is convergent if f2(x) is convergent (cf. Theorem V-2-7). Therefore, in such a case, v(x) is convergent and, hence, o(x) is convergent.

Observation V-5-3. Choose integers l1, ... , to so that A.,+f j = Ak +fk if Aj -Ak is an integer. Then, vo(x) = x4rx-L, where L = diag(e1, f2, ... , en]. If we set 7t{V1 = x'! 1C(xf and ho{v") = x-LICo(x' i , it follows that dv' ?{o(vl = x'L fXLTf + zLLv + Aoz' 7} = xdx + (Ao + L)v

Hence,

(V.5.12)

R{v = x

+ (Ao + L + r)v.

Note that Ao + L = diag(A1 + P1, A2 + e2, ... , An + enj

and (V.5.13)

(Ao + L)r = r(A0 + L).

It was already shower in §V-4 that r is nilpotent. Thus, the following theorem is proved.

5. A NORMAL FORM OF A DIFFERENTIAL OPERATOR

125

Theorem V-5-4. The transformation y = P(x)xLiT changes the system

G[yi = xdy + fl(x)yl = 0

(V.5.14) to

VV! = x

(V.5.15)

d1U

+ (Ao + L + r)u = 0.

Observation V-5-5. The matrix n-1

o{x) = x-Ao-L-r = x-Ao-L I

+ h-1

h

(-1)h[logx[ rh h!

is a fundamental matrix solution of system (V.5.15). Note that if (A3+tj)-(Ak+tk) is an integer, then A, + ej = Ak + tk in Ao + L.

Remark V-5-6. A fundamental matrix solution 4D(x) of system (V.5.14) is given by 4>(x) = P(x)x4o(x), which can be written in the form n-1

45(x) = P(x)xLx-M-L In +

F {-1)h (lo x h rh h!

h=1

IL

Since L - A0 - L = -A0, the matrix 44(x) can be also written in the form n-1

4i(x) =

P(x)x-ne

In +

h

(-1)h (lohx) rh

h=1

I However, x-Ao and I

n-I

and

In + E(-1)h h=1

n-i

nr >(-1)h [lohx[

h

rh do not commute, whereas

x-A°-L

h=1

[log x[ h h!

rh commute.

I

Remark V-5-7. The methods used in §§V-3, V-4, and V-5 are based on the

original idea given in [GerL[.

Example V-5-8. In order to illustrate the results of this section, consider the differential equation of the Bessel functions (V.5.16)

za (zT)

+ (z2 - a2)y = 0,

where a is a non-negative integer. If we change the independent variable z by x = z2, (V.5.16) becomes x

d

lxdx

2y=0.

V. SINGULARITIES OF THE FIRST KIND

126

This equation is equivalent to the system

-1

0

where 11(x) =

y=

+ Q(x)y = 0,

x

(V.5.17)

0

=10+x11i c o =

x - a2 0

4

a2 4

-1

0

and 0i = 0

0

1

4

0

To begin with, let us remark that the S-N decomposition of the matrix 11 is given by S2o = So + No, where 80 =

O

if

{Q

if

Also, set Ao =

a = 0, a > 0,

a

0

2

a

N

and

if

N0 - to

if

a = 0,

a>0.

Note that two eigenvalues of Q are

0

2

Now, we calculate in the following steps:

Step 1. Fix a non-negative integer m so that m > 2 -

(_a)

= a.

Step 2. Find three 2(m + 1) x 2(m + 1) matrices A,,,, Sm, and Nm (cf. §§V-4 and V-5). Po P1

Step 3. Find a 2(m + 1) x 2 matrix Pm =

, where the Pr are 2 x 2 matrices

Pm

such that SmPm = Pn&. in

m

Step 4. Find two 2 x 2 matrices Mm(x) = No + >2xtvt and Qm(x) _ >xtPt t-o

t=1

(cf. §§V-4 and V-5).

Step 5. We must obtain Qm(x)-1Mm(x)Qm(x) = vo(x) +O(xm+i), where Po InoP0 vo(x)

r0 0

a(a)xn 0

if

a = 0,

if

a>0,

J

where a(a) is a real constant depending on a (cf. Theorem V-5-1). After these calculations have been completed, we come to the following conclusion.

5. A NORMAL FORM OF A DIFFERENTIAL OPERATOR

127

+00

Conclusion V-5-9. There exists a unique 2 x 2 matrix P(x) = >xtPe such that 1=o

(1) (2) (8)

the matrices Pl for t = 0,1, ... , m are given in Step 8, the power series P(x) converges for every x, the transformation y = P(x)u" changes (V.5.17) to

dii xaj + (Ao + vo(x)) i = 0'.

(V.5.18)

We illustrate the scheme given above in the case when a = 0. First we fix m = 2. Then, 0

-1

0

0

0

0

0

0

0

0

00

0

0

0

0

0

0

0

0

0

0

00

0

0

1

-1

0

0

1

0

0

0

0

1

0

0

2

0

0

2

A2 =

S2 = 1

4 0

0

0

0

0

0 0

4

1

0 0

0

0

-1

2

0

-

2

1

1

4

2

1

1

4

4

3

3

1

1

32

32

4

2

1

3

1

1

16

32

4

4

and

N2=

0

-1

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

-1

0

0

1

1

4

2

0

1

3

3

1

1

32

32

4

2

16

T2-

0

9

64

-320 -128

-16 -32

16 48

1192

Calculating eigenvectors of S2, we find a 6 x 2 matrix P2 =

that S2P2 = 0. Note that, in this case, Ao = 0.

0

1

1

0

such

V. SINGULARITIES OF THE FIRST KIND

128

Set 1

1

3

3

4

2

32

32

0

1

1

4

16

3

32-

M2 (x) = 1 o + xvl + x2v2 and

-320

192

(V.5.19)

-16

P1

-1 8]

PO = [ 64

16 ]

= [ -32 48

'

P2 ,

[0

1

=

01 '

Q2(r) = P° + xPl + x2P2. ]

Then, Qz-' 11f2(x)Q2(x)

-1

+ O(x3). Note that

5

1

1,°

2

12

634

64

[0

[192 64

01]

-128] =

4

[--21

2].

64

Thus, we arrived at the following conclusion. 00

Conclusion V-5-10. There exists a unique 2 x 2 matrix P(x)

Zx'Pr such e=o

that (1) (2)

(3)

the matrices Pt for i = 0, 1, 2 are given by (V.5.19). the power series P(x) conueryes for every x, the transformation y" = P(x)u" changes the system dy

xdx +

(V.5.20)

0

-1

X

0

Y = 0'

4 to

d6

x3i + [_1 2, u" = 0.

(V.5.21)

-

Furthermore, the transformation y" = P(x)Po iv changes (V.5.20) to (V.5.22)

x

+ 10

0

" = 0.

5. A NORMAL FORM OF A DIFFERENTIAL OPERATOR

In the case when a=1, wefixm=2. Then 1 0 -1 0 0 0 0

0

-1

0

0

0

0

0

0

0

0

0

1

-1

0

0

1

0

0

2

-1

1

0

0

0

0

0

0

1

-1

0

0

0

-4

1

0

0

0

0

0

2

-1

129

4

0

A2 = 0 0

1

132=

0

0

-4

2

4

8 3

1

1

16

8

4

1

1

16

16

3

1

3

1

1

64

16

16

8

4

J

1

1

4

2

4

and

Ar2

=

0

0

0

0

0

01

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

8

4

1

1

16

8

16 -3

1

1

1

16

8

4

1

1

1

64

16

16

8

1

00 0

0 384 192

Calculating eigenvectors of S2, we find a 6 x 2 matrix P2 =

_72

32

-16

-124 such

2

0

5

1

1

that S2P2 = P2Ao. Note that, in this case, A0 =

0 1

2

L0 vl =

. Set

1 1

2

1

1

1

1

8

4

16

16

1

1

v2 =

1

3 16

8

64

16

M2(x) = xl/1 + x2v2, and

(V.5.23)

Po

84

62

[19 3 J' Pl -

Q2(x) = Po + xP1 + T2 p2.

-48 -12 [-72 -14]

,

1'2 = [2 1j

,

V. SINGULARITIES OF THE FIRST KIND

130

x 0

Then, Q2-'M2(X)Q2(X) = 0

48 0

+ Q(x3). Thus, we arrived at the following

conclusion. +oo

Conclusion V-5-11. There exists a unique 2 x 2 matrix P(x) = >xtPt such that (1) (2) (3)

e=o

the matrices Pt for P = 0, 1, 2 are given by (V.5.23), the power series P(x) convet es for every x, the transformation y = P(x)u changes the system -1

xdx +

(V.5.24)

V = 0 0

to

x-

(V.5.25)

+

2 0

48I u = U. 1

2

For a further discussion, see IHKS[. A computer might help the reader to calculate S2i N2, and P2 in the cases when a = 0 and a = 1. Such a calculation is not difficult in these cases since eigenvalues of A2 are found easily (cf. §IV-1).

V-6. Calculation of the normal form of a differential operator In this section, we present another proof of Theorem V-5-1. The main idea is to construct a power series P(x) as a formal solution of the system dP(x)

= P(x)(Ao + vo(x)) - Q(x)P(x)

(cf. Remark V-5-2). Another proof of Theorem V-5-1.

To simplify the presentation, we assume that So = A0 = diag[,ulI,, µ2I2, ... , Aklk[, where pi, p2, ... , µk are distinct eigenvalues of no with multiplicities m1, respectively, and the matrix I, is m, x m j identity matrix. Since AOA(o = NoAo, the matrix No must have the form No = diag(N01,No2, ... ,Nok[, Where Nor is an 00

mi x m1 nilpotent matrix. Let us determine two matrices P(x) = I,a + E xmP,,, M=1

6. CALCULATION OF THE NORMAL FORM

131

00

and B(x) = Sto + E xmBm by the equation m=1

(in+mm) (Oo

xad

+

*n=1 xmBm)

(In + m 'nPm)

Cep + m=1

This equation is equivalent to m-1

mPm = PmS2o - S20Pm + Bm - 1m +

(m > 1).

( PhBm-h - fl.-hPh) [=1

Therefore, it suffices to solve the equation

mX + (A0 +N4)X - X(Ao+No) - Y = H,

(V.6.1)

where X and Y are n x n unknown matrices, whereas the matrix H is given. If we write X, Y, and H in the block-form X11

Xkl

...

Xlk

[Y11

...

Xlk

H11

...

Xlk

Xkk

Ykl

...

Ykk

Hkl

...

Hkk

where XXh, YYh, and H,h are m, x mh matrices, equation (V.6.1) becomes

(m + P) - µh)X)h + NO,Xjh - X,hNOh - Yjh = Hjh,

where j, h = 1,... , k. We can determine XJh and Y,h by setting Y,;h = 0 if

m + u, - Ah 4 0, and X,,h = 0 if m + p, - µh = 0. More precisely speaking, if m + p - Ph 96 0, we determine X3h uniquely by solving

(m + p, - µh)Xlh + NojXjh - X3hNoh = H)h. If M + Aj - Ah = 0, we set Y,h = H,h. In this way, we can determine P(z) and B(x). In particular, go + B(x) has the form x ^°I'xA0, where r is a constant d n x n nilpotent matrix. Furthermore, the operator x +A0 and the multiplication ds operator by No + B(x) commute. D The idea of this proof is due to M. Hukuhara (cf. [Si17, §3.9, pp. 85-891).

Remark V-6-1. In the case of a second-order linear homogeneous differential equation at a regular singular point x = a, there exists a solution of the form +00

01(z) = (x - a)aE c, (x - a)n, where the coefficients c,, are constants, co 0 0, n=O

V. SINGULARITIES OF THE FIRST KIND

132

and the power series is convergent. If there is no other linearly independent solution of this form, a second solution can be constructed by using the idea explained in Remark IV-7-3. This second solution contains a logarithmic term. Similarly, a third-order linear homogeneous differential equation has a solution of the form +00

c, (x - a)" at a regular singular point x = a. Using this solu-

01(x) _ (x - a)° n=0

tion, the given equation can be reduced to a second-order equation. In particular, if there exists another solution ¢(x) of this kind such that ¢1 and ¢2 are linearly independent, then the idea given in Remark IV-7-4 can be used to find a fundamental set of solutions. In general, a fundamental matrix solution of system (V.5.14) can be constructed if the transformation y" = P(x)xLV of Theorem V-5-4 is found. In fact, if the P(x)xL2-no-L-r is a fundamental definition of fo(x) of Observation V-5-5 is used, matrix solution of (V.5.14). The matrix P(x) can be calculated by using the method of Hukuhara, which was explained earlier.

V-7. Classification of singularities of homogeneous linear systems In this chapter, so far we have studied a system (V.7.1)

x dt = A(x)y,

y E C",

where the entries of the n x n matrix A(r) are convergent power series in x with complex coefficients. In this case, the singularity at x = 0 is said to be of the first kind. If a system has the form (V.7.2)

xk+1

= A(x)y",

y" E C",

where k is a positive integer and the entries of the n x n matrix A(x) are convergent power series in x with complex coefficients, then the singularity at x = 0 is said to be of the second kind In §V-5, we proved the following theorem (cf. Theorem V-5-4).

Theorem V-7-1. For system (V.7.1), there exist a constant n x n matrix A0 and an n x n invertible matrix P(x) such that (i) the entries of P(x) and P(x)-1 are analytic and single-valued in a domain 0 < jxi < r and have, at worst, a pole at x = 0, where r is a positive number, (ii) the transformation (V.7.3)

y' = P(x)i7

changes (V.7.1) to a system (V.7.4)

du = Aou".

Theorem V-7-1 can be generalized to system (V.7.2) as follows.

7. CLASSIFICATION OF SINGULARITIES OF LINEAR SYSTEMS

133

Theorem V-7-2. For system (V.7.2), there exist a constant n x n matrix A0 and an n x n invertible matrix P(x) such that (i) the entries of P(x) and P(x)-1 are analytic and single-valued in a domain V = {x : 0 < lxi < r}, where r is a positive number, (ii) the transformation

y = P(x)i

(V.7.5) changes (V.7.2) to (V.7.4).

Proof.

Let 4i(x) be a fundamental matrix of (V.7.2) in D. Since A(x) is analytic and single-valued in D, fi(x) _ I (xe2"t) is also a fundamental matrix of (V.7.2). Therefore, there exists an invertible constant matrix C such that 46(x) = 4i(x)C (cf. (1) of Remark IV-2-7). Choose a constant matrix Ao so that C. = exp[2rriAo] (cf. Ex-

ample IV-3-6) and let P(x) = 4(x)exp[-(logx)Ao]. Then, P(x) and P(x)-1 are analytic and single-valued in D. Furthermore,

dP(x) = dam) exp(_(logx)Ao] - P(x)(x-'Ao)

= x-(k+1)A(x)P(x) - P(x)(x-'Ao). This completes the proof of the theorem. 0 An important difference between Theorems V-7-1 and V-7-2 is the fact that the matrix P(x) in Theorem V-7-2 possibly has an essential singularity at x = 0. The proof of Theorem V-7-2 immediately suggests that Theorem V-7-2 can be extended to a system (V.7.6)

dg = F(x)y, dx

where every entry of the n x n matrix F(x) is analytic and single-valued on the domain V even if such an entry of F(x) possibly has an essential singularity at x = 0. More precisely speaking, for system (V.7.6), there exist a constant n x n matrix Ao and an n x n invertible matrix P(x) satisfying conditions (i) and (ii) of Theorem V-7-2 such that transformation (V.7.5) changes (V.7.6) to (V.7.4). Now, we state a definition of regular singularity of (V.7.6) at x = 0 as follows.

Definition V-7-3. Let P(x) be a matrix satisfying conditions (i) and (ii) such that transformation (V.7.5) changes (V.7.6) to (V.7.4). Then, the singularity of (V.7.6) at x = 0 is said to be regular if every entry of P(x) has, at worst, a pole

atx=0.

Remark V-7-4. Theorem V-7-1 implies that a singularity of the first kind is a regular singularity. The converse is not true. However, it can be proved easily that a regular singularity is, at worst, a singularity of the second kind. Furthermore, if (V.7.2) has a regular singularity at x = 0, then the matrix A(0) is nilpotent. This is a consequence of the following theorem.

V. SINGULARITIES OF THE FIRST KIND

134

Theorem V-7-5. Let A(x) and B(x) be two n x n matrices whose entries are formal power series in x with constant coefficients. Also, let r and s be two positive integers. Suppose that there exists an n x n matrix P(x) such that (a) the entries of P(x) are formal power series in x with constant coefficients, (b) det(P(x)] ,E 0 as a formal power series in x, (c) the transformation y = P(x)ii changes the system x'

16 -1

X

= A(x)y to x' 2i _

B(x)il. Suppose also that s > r. Then, the matrix B(O) must be nilpotent.

Proof

Step 1. Applying to the matrix P(x) suitable elementary row and column operations successively, we can prove the following lemma.

Lemma V-7-6. There exist two n x n matrices +oo

+oo

T(x) _

xmTm

and

S(x) _

x"'Sm, m=o

m=0

and n integers 1\1,,\2, ... , an such that (i) the entries of n x n matrices T,,, and Sm are constants. (ii) det To 0 0 and det So 0 0, (iii)

T(x)P(x)S(x) = A(z) = diag(x''',zA2,...

(iv)A1 I and of the operator P as in §V-1, a,(x) 0 0. Defining the indicial polynomial we prove the following theorem.

Theorem V-7-10. (i) In the case when the degree of f b in s is equal to n, if we change the equation

P[y1 = 0 to a system by setting yt = y and yj = 61-1 [y) (j = 2, ... , n), the system has a singularity of the first kind at x = 0.

137

EXERCISES V

(ii) In the case when the degree of fn ins is less than n, if we change the equation P[y] = 0 by setting y1 = y and y, = b'-1 [y] (j = 2,... , n), the system has an irregular singularity at x = 0.

Proof n

00

Ltahi

Look at P(xd] _ >ah(x)bh[x°] = xe= x"

fm(S)xm and set Tn=r+o

ah(x) = xnobh(x). Then, the functions bh(x) are analytic at x = 0. Furthermore, is n, we must have bn(0) # 0. Therefore, in this case we can if the degree of n-1

write the equation P(y) = 0 in the form bo[y] = -bn(Ebh(x)bh(yJ. Claim (i) h=0

follows immediately from this form of the equation.

If the degree of fn0 is less than n, we must have bn(0) = 0 and b,(0) - 0 for some j such that 0 < j < n. To show (ii), change Yh further by zh = x(h-1)ayh with a suitable positive rational number a so that the system for (z1, ... , zn) has a singularity of the second kind of a positive order (cf. the arguments given right after Definition V-7-8, and also §XIII-7).

From Theorem V-7-10, we conclude that the formal solution x''(1 + O(x)) of Theorem V-1-3 is convergent if the degree of f,, (s) in s is n. The following corollary of Theorem V-7-10 is a basic result due to L. Fuchs [Fu].

Corollary V-7-11. The differential equation Ply] = 0 has, at worst, a regular singularity at x = 0 if and only if the functions ah(x) (h = 0, ... , n - 1) are x

analytic at x = 0. Some of the results of this section are also found in [CL, Chapter 4, pp. 108-137].

EXERCISES V

V-1. Show that if A is a nonzero constant, H is a constant n x m matrix, and N1 and N2 are n x n and m x in nilpotent matrices, respectively, then there exists one

and only one n x m matrix X satisfying the equation AX + NIX - XN2 = H. V-2. Show that the convergent power series

y = F(a,x) =

1+a

(a+1)...(a+m-1Q(Q+1)

m=1

satisfies the differential equation

x(1-x)!f 2 + where a, /3, and -y are complex constants.

+1)J

Y

- c3y = 0,

Q+m-1) x,,

V. SINGULARITIES OF THE FIRST KIND

138

Comment. The series F(a, Q, ry, x) is called the hypergeometric series (see, for example, [CL; p. 1351, 101; p. 159], and [IKSYJ).

V-3. For each of the following differential equations, find all formal solutions of c

the form x'' I 1 + > cmxm] . Examine also if they are convergent. L

m=1

(i)

xb2y + aby + /3y = 0,

(ii)

b2y + aby + $y = rxmy,

where b = xa, the quantities a, 3, and 7 are nonzero complex constants and m is a positive integer.

V-4. Given the system (E)

dt = (N + R(t))il,

N = [0

O]

,

and R(t) = t-3 I 0 0J

,

show that (i) (E) has two linearly independent solutions ,

where

(t)

o m!(m+1)!'

and

)1 J2(t) = 1612 (t (t)

where

,

02(t) = t +

+00

bmt-'n - ¢1(t)logt,

m-1

with the constants bm determined by b_1 = 1, bo = 0, and 2m + 1 (m = 1,2,... ), m(m + 1)bm - b,,,-, = m!(m+ 1)! IN) = O for the fundamental matrix solution Y(t) = [if1(t)12(t)J, (ii) 1 lim t-1(Y(t)-e

(iii) the limit of e-'NY(t) as t -+ +oo does not exists.

V-5. Let y be a column vector with n entries and let Ax, y-) be a vector with n entries which are formal power series in n + 1 variables (x, yj with coefficients in C. Also, let u' be a vector with n entries and let P(x, u) be a vector with n entries which are formal power series in n + 1 variables (x, u") with coefficients in C. Find u + xP(x, u7 changes the most general P(x, u") such that the transformation the differential equation

dy"

ds

=

f (X, y-) to

du"

= 0.

Hint. Expand xP(x, u) and f (x, u+xP(x, u)) as power series in V. Identify coefficients of

d[xP(x, V-)]

with those of Ax, u'+zP(x, iX)) to derive differential equations

which are satisfied by coefficients of xP(x, u).

139

EXERCISES V

V-6. Suppose that three n x n matrices A, B, and P(x) satisfy the following conditions:

(a) the entries of A and B are constants, (b) the entries of P(x) are analytic and single-valued in 0 < [xj < r for some positive number r,

dii (c) the transformation y' = P(x)u changes the system xfy = Ay" to xjj = Bu. Show that there exists an integer p such that the entries of xPP(x) are polynomials in x.

Hint. The three matrices P(x), A, and B satisfy the equation

xd) = AP(x) -

+"0

P(x)B. Setting P(x) = >2 xmPm, we must have mPm = APm - PmB for all m=-oo

integers m. Hence, there exists a large positive integer p such that Pm = 0 for ImI ? P. and let A(ye) be an n x n V-7. Let ff be a column vector with n entries {y,... , matrix whose entries are convergent power series in {yj, ... , yn} with coefficients in C. Assume that A(0) has an eigenvalue A such that mA is not an eigenvalue of A(0 for any positive integer m. Show that there exists a nontrivial vector O(x)

with n entries in C{x} such that y" = b(exp[At]) satisfies the system L = A(y-)y.

Hint. Calculate the derivative of (exp(At]) to derive the system Axe = for

.

N

V-8. Consider a nonzero differential operator P = >2 ak(x)Dk with coefficients k=O

ak(x) E C([x]], where D = dx. Regarding C([x]] as a vector space over C, define a homomorphism P : C[[x]] -. C[[x]j. Show that C((xjj/P(C[[x]j] is a finitedimensional vector space over C.

Hint. Show that the equation P(y) = x"`'4(x) has a solution in C([xjj for any O(x) E C[[x]] if a positive integer N is sufficiently large. To do this, use the indicial polynomial of P. N

V-9. Consider a nonzero differential operator P = >2 ak(x)dk with coefficients k=0

ak(x) E CI[x]j, where b = x2j, n > 1, and

54 0. Define the indicial poly-

nomial as in Theorem V-1-3. Show that if the degree of is n and if n zeros {A1, ... , A, } of do not differ by integers, we can factor P in the following form: P = a+,(x)(b - &1(x))(b -

42(x))...(6

- 0n(x)),

V. SINGULARITIES OF THE FIRST KIND

140

where all the functions ¢, (x) (j = 1, 2, ... , n) are convergent power series in x and 4f(O) = Aj.

Hint. Without any loss of generality, we can assume that an(x) = 1. Then, An + n-1

Eak(O)Ak = (A - A1)... (A - An). Define constants {yo... , In-2} by A' ' + k=0 n-2 ,YkAk

= (A - A2) ... (A - An). For v] (X) E xC[[xI] and ch(x) E xC([x]l (h =

k=O

0,. . . , n - 2), solve the equation (

(C)

n-2

P = (6 - Al -

2(7h+Ch(x))6h

h=0

If we eliminate O(x) by Al + V(x) = 1`n-2 + ct-2(x) - an-1(x), condition (C) becomes the differential equations 6(CO(x)1 = a0(x) + (-Yn-2 + Cn-2(x) -a. -1(x))(1'0 +40(x)), 1 b[ch(x)] = ah(x) + (1'n-2 + Cn-2(x) - an-1(x))(7h +Ch(x)) - (1h-1 + Ch-1(x)),

where h=I,...,n-2. n

V-10. Consider a linear differential operator P = E atbt, where ao,... , an are t=o

complex numbers and 6 = x. The differential equation (P)

P[y1= 0

is called the Cauchy-Euler differential equation. Find a fundamental set of solution of equation (P).

Hint. If we set t = log x, then b = dt V-11. Find a fundamental set of solutions of the differential equation

(6-06-8)(6-a-8)[yl =xn'Y' where 6 = xjj and m is a positive integer, whereas a and /3 are complex numbers such that they are not integers and R[al < 0 < 8t[01. V-12. Find the fundamental set of solutions of the differential equation (LGE)

f(1-x2)LJ +a(a+1)y=0

at x = 0, where a is a complex parameter.

141

EXERCISES V

Remark. Differential equation (LGE) is called the Legendre equation (ef. [AS, pp. 331-338] or [01, pp. 161-189]). 1 V-13. Show that for a non-negative integer n, the polynomial Pn(x) = 2nn!

x do [(x2 - 1)'] satisfies the Legendre equation d

]

[(1-x2)

+n(n+1)Pn=0.

Show also that these polynomials satisfy the following conditions:

(1) Pn(-x) = (-1)nPn(x),

(2) Pn(1) = 1, (3) JPn(t)j ? 1 for {xj > 1,

+00

(4)

1 - 2xt + t =

E Pn(x)tn, and (5) lPn(x)I < 1 for Jxj < 1. n=o

Hint. Set g(x) = (1 - x2)n. Then, (1 - x2)g'(x) + 2nxg(x) = 0. Differentiate this relation (n + 1) times with respect to x to obtain

(1,-

x2)g(n+2)(x)

- 2(n + 1)xg(n+1)(x) - n(n + 1)g(n)(x)

+ 2nxg(n+1) (x) + 2n(n + 1)g(")(x) = 0 or

(1 -X 2)g(n+2)(x) - 2ng(n+1)(x) + n(n + 1)g(n)(x) = 0.

Statements (1), (2), (3), and (4) can be proved with straight forward calculations. Statement (5) also can be proved similarly by using (4) (cf. [WhW, Chapter XV, Example 2 on p. 303]). However, the following proof is shorter. To begin with, set F(x) = Pn(x)2

x2)P''(x))2. n(n + 1){1 -x2)((1 2

Then, F(±1) = Pn(±1)2 = 1, p(X) _

,and F(x) = Pn(x)2 if (x) (P"(x)) n(n + 1)

0. Therefore, for 0 < x < 1, local maximal values of Pn(x)2 are less than F(1) = 1, whereas, for -1 < x < 0, local maximal values of Pn(x)2 are less than F(-1) = 1. Hence, Pn(x)I < 1 for JxI < 1 (cf. [Sz, §§7.2-7.3, pp. 161-172; in particular, Theorems 7.2 and 7.3, pp. 161-162]. See also (NU, pp. 45-46].) The polynomials Pn(x) are called the Legendne polynomials.

V-14. Find a fundamental set of solutions of (LGE) at x = oo. In particular, show what happends when a is a non-negative integer.

V-15. Show that the differential equation b"y = xy has a fundamental set of solutions consisting of n solutions of the following form: (logx)k-1-A

A-o

(k - 1 - h)!

OA(x)

(k = 1, ... , n),

where the functions Oh(x) (h = 0,... , n -1) are entire in x, 00(0) = 1, and Oh(O) _ 0 (h = 1, ... , n -1). Also, find O0 (x).

142

V. SINGULARITIES OF THE FIRST KIND

V-16. Find the order of singularity at x = 0 of each of the following three equations. (i) x5{66y - 362y + 4y} = y, (ii) x5{56y - 3b2y + 4y} + x7{b3y - 55y} = y, (iii) x5y"' + 5x2y" + dy + 20y = 0.

V-17. Find a fundamental matrix solution of the system x2 x2

dbi = xyi + y2, dx dy2

dx

= 2xy2 + 2y3,

x2 dx3 = x3y! + 3y3.

V-18. Let A(x) be an n x n matrix whose entries are holomorphic at x = 0. Also, let A be an n x n diagonal matrix whose entries are non-negative integers. Show that for every sufficiently large positive integer N and every C"-valued function fi(x) whose entries are polynomials in x such that those of xA¢(x) are of degree N, there exists a C"-valued function f (x; A, ¢, N) such that (a) the entries off are polynomials in x of degree N-1 with coefficients depending on A, N, and ¢, (b) f is linear and homogeneous in tb, (c) the linear system xAdd9 = A(x)3l + z has a solution #(x) whose entries are holomorphic at x = 0 and il(x) is linear and homogeneous in ', fi(x)) = O(xN+i) as x 0. (d) Hint. See [HSW].

V-19. Let A(x) and A be the same as in Exercise V-18. Assuming that n > trace(A), show that the system xA dz = A(x)y has at least n - trace(A) linearly independent solutions holomorphic at x = 0. Hint. This result is due to F. Lettenmeyer [Let]. To solve Exercise V-19, calculate f (x; A, ¢, N) of Exercise V-18 and solve f(x; A, 4, N) = 0 to determine a suitable function 0. 00

V-20. Suppose that for a formal power series ¢(x) _ > cx' E C[]x]], there exm=0 /dl d ist two nonzero differential operators P = E ak (x) ` and Q = > bk (x) (d) k I

k=0

k

rk=10

= 0. Show

with coefficients ak(x) and bk(x) in C{x} such that P[0] = 0 and Q I

that 0 is convergent.

l

J

EXERCISES V

143

Hint. The main ideas are (a) derive two algebraic (nonlinear) ordinary differential equations (E)

F(x,v,v',... ,v(")) = 0

for v =

and

G(x,v,v',... ,v(' ) = 0

from the given equations P[¢) = 0 and Q ['01 = 0.

(b) eliminate all derivatives of v from (E) to derive a nontrivial purely algebraic equation H(x, v) = 0 on v. See [HaS1] and [HaS2[ for details.

CHAPTER VI

BOUNDARY-VALUE PROBLEMS OF LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND-ORDER

In this chapter, we explain (1) oscillation of solutions of a homogeneous secondorder linear differential equation (§VI-1), (2) the Sturm-Liouville problems (§§VI2-VI-4, topics including Green's functions, self-adjointness, distribution of eigenvalues, and eigenfunction expansion), (3) scattering problems (§§VI-5-VI-9, mostly focusing on reflectionless potentials), and (4) periodic potentials (§VI-10). The materials concerning these topics are also found in [CL, Chapters 7, 8, and 11], [Hart Chapter XI], [Copl, Chapter 1], [Be12, Chapter 61, and [TD]. Singular selfadjoint boundary-value problems (in particular continuous spectrum, limit-point and limit-circle cases) are not explained in this book. For these topics, see [CL, Chapter 9].

VI-1. Zeros of solutions It is known that real-valued solutions of the differential equation L2 + y = 0 are linear combinations of sin x and cos x. These solutions have infinitely many zeros on the real line P. It is also known that real-valued solutions of the differential equation

2 - y = 0 are linear combination of e= and a-x. Therefore, nontrivial solutions has at most one zero on the real line R. Furthermore, solutions of the differential

equation2 + 2y = 0 have more zeros than solutions of L2 + y = 0.

In this

section, keeping these examples in mind, we explain the basic results concerning zeros of solutions of the second-order homogeneous linear differential equations. In §§VI-1-VI-4, every quantity is supposed to take real values only. We start with the most well-known comparison theorem concerning a homogeneous second-order linear differential equation (VI.1.1)

dx2

+ 9(x)y = 0.

Theorem VI-1-1. Suppose that (i) g, (x) and g2(x) are continuous and g2(x) > 91(x) on an interval a < x < b, (ii)

d 2 01 (x)

+ 91(x)o1(x) = 0 and

d2622x)

+ g2(x)02(x) = 0 on a < x < b,

(iii) S1 and 6 are successive zeros of 01(x) on a < x < b. Then. q2(x) must vanish at some point £3 between t:l and C2. 144

145

1. ZEROS OF SOLUTIONS Proof.

Assume without any loss of generality that Sl < 1;2 and 01(x) > 0 on 1;1 < x < £2.

(1:2) < 0. A contradiction will be derived from the assumption that 02(x) > 0 on S1 < z < 1;2. In fact, assumption (ii) implies 1 (l:l) > 0, 01(S2) = 0, and

Notice that 41(1::1) = 0,

02(x)

VI .1. 2

_ 01 ( x ) d202 (x) =

d 20, (7)

[ 92

(x) - 91 (x)10 (x)0 2 (x) 1

and, hence, (VI - 1 - 3 )

0202)

1L(

6)

-

02 V

1

)!!LV ) =

rE2

(x) - 91 (x)j¢ (x)02 (x)dx . 1

1

E1

The left-hand side of (VI.1.3) is nonpositive, but the right-hand side of (VI.1.3) is

positive. This is a contradiction. 0 A similar argument yields the following theorem.

Theorem VI-1-2. Suppose that (i) g(x) is continuous on an interval a < x < b, (ii) 0i(x) and 02(x) are two linearly independent solutions of (VI.1.1), (iii) 1;1 and 6 are successive zeros of 01 (x) on a < x < b. Then, 02(x) must vanish at some point 3 between t;1 and 1;2. Proof.

Assume without any loss of generality that Sl < £2 and .01 (x) > 0 on S1 < x < 1;2.

Notice that 01(11) = 0,

1 (S1) > 0, 01(12) = 0, and X1(1;2) < 0. Note also that

if 02(1) = 0 or 02(6) = 0, then 01 and 02 are linearly dependent. Now, a assumption that 02(x) > 0 on fi < x < 2 contradiction will be derived from the assumption a- (x) - p1(z) d2 02 (x) = 0 and, hence, In fact, assumption (u) implies 02(x) dx2 dS2 (VI.1.4)

02(2)

1(6) - 4'201)

1(W =0.

Since the left-hand side of (VI.1.4) is positive, this is a contradiction. 0 The following result is a simple consequence of Theorem VI.1.2.

Corollary VI-1-3. Let g(x) be a real-valued and continuous function on the interval Zo = {x : 0 < x < +oo}. Then, (a) if a nontrivial solution of the differential equation (VI.1.1) has infinitely many zeros on I , then every solution of (VI.1.1) has an infinitely many zeros on Zo,

(b) if a solution of (VI.1.1) has m zeros on an open subinterval Z = {x : a < x < 3} of Zo, then every nontrivial solution of (VI.1.1) has at most m + 1 zeros on Z.

Denote by W(x) = W(x;01,02) =

I

the Wronskian of the set of

02(x) functions (01(x), 02(x)}. For further discussion, we need the following lemma.

VI. BOUNDARY-VALUE PROBLEMS

146

Lemma VI-1-4. Let g(x) be a real-valued and continuous function on the interval To = {x : 0 < x < +oo}, and let 771(x) and M(x) be two solutions of differential equation (VI 1.1). Then, (a) W (x; 772 , ri2 ), the Wronskian of {rli (x), 712(x)}, is independent of x, 771(x),

(b) if we set t;(x) =

then

772(x)

4(x) = dx

c

, where c is a constant.

772(x)2

Also, if 77(x) is a nontrivial solution of (VI.1.1) and if we set w(x) =

then

we obtain

dw(x)

(VI.1.5)

+ w(x)2 + g(x) = 0 .

Proof.

(a) It can be easily shown that d

771(x)

772(x)

dx I77i(x)

772(x)

(b) Note that

=-

dl;(x}

771(x)

=

dx

77i'(x)

772(x) 772(x)

I rl' (x)

712(x) !

712(x)2 771(x)

772(x)1

1

-g(x) !

712(x) 712(x)

771(x)

and that

dw(x) dx

=0

if'(x) :l(x)

1 = -g(x) - w(x)2. 12 ((xx))

The following theorem shows the structure of solutions in the case when every nontrivial solution of differential equation (VI.1.1) has only a finite number of zeros

on To= {x:0 xo > 0. Using (b) of Lemma VI-1-4, we can derive the

147

1. ZEROS OF SOLUTIONS

(1(x) following three possibilities (1) =limo(2(x) = 0, (ii) =U Urn

( 1( x )>0

x-+oo (2 (x)

.

Set

(i(x) 711(x) _

+oo, and (iii)

(2(x)

in case (i), in case (ii),

(i(x) - 7'(2(x)

in case (iii),

(2(x)

(i(x) 1(2(x)

772(x) =

in case (i), in cage (ii), in case (iii).

Then, (a) follows immediately. (b) Conclusion (b) of Lemma VI-1-4 implies that

_Id(Y12(x)

1

ql (x)2 ^ c dx . q1(x))

1

and

_-1d

>)1(x)

c dx \ 2(x)

712(x)2

,

where c is the Wronskian of q1 and rte. Hence, (VI.1.6) follows. +oc

(c) If q2 is bounded, we must have j()2 -= +00. 0 The following theorem gives a simple sufficient condition that solutions of (V1.1.1) have infinitely many zeros on the interval Zo.

Theorem VI-1-6. Let g(x) be a real-valued and continuous function on the in-

terval lo. If f

g(x)dx = +oo, then every solution of the differential equation

0

(VI.1.1) has infinitely many zeros on Zo. Proof.

Suppose that a solution 17(x) satisfies the condition that 77(x) > 0 for x > (x) Then, w(x) satisfies differential equation (VI.1.5). xo > 0. Set w(x) = !L. 77(x) Hence, lien w(x) = -oo. This implies that lim 17(x) = 0 since 17(x) = r7(xo) :-+co :-+oo

x exp Vo

This contradicts (c) of Theorem VI-1-5. O J

The converse of Theorem VI-1-6 is not true, as shown by the following example.

Example VI-1-7. Solutions of the differential equation

(VI.1.7)2 +

A

= 0,

1 z2

where A is a constant, have infinitely many zeros on the interval -oo < x < +oo if and only if A > 4 Proof

Look at (VI.1.7) near x = oo. To do this, set t = r

(VL1.8) L

462 + 26 + 1 + t,

x 0,

to change (VI.1.7) to

VI. BOUNDARY-VALUE PROBLEMS

148

where b = t. Note that t = 0 is a regular singular point of (VI.1.8) and the indicial equation is 4s2 + 2s + A = 0 whose roots are s = - 2 f

4 - A. These

roots are real if and only if A < 4 This verifies the claim. 0 +00

Note that J 0

1 + x2

dx

2*

The following theorem shows some structure of solutions in the case when +00

ig(x)ldx < +oo. 0

Theorem VI-1-8. Let g(x) be a real-valued and continuous function on the interf+C0

valA. If 1

jg(x)ldx < +oo, then there exist unbounded solutions of differential

0

equation (VI. 1. 1) on Ia. Proof.

The assumption of this theorem and (VI.1.1) imply that lim dq(x) = -Y exists :-+oo dx for any bounded solution n(x) of (VI.1.1). If 7 0, then i(x) is unbounded. Hence, z

lim odd) = 0. Therefore, calculating the Wronskian of two bounded solutions

of (VI.1.1), we find that those bounded solutions are linearly dependent on each other. This implies that there must be unbounded solutions. 0

Remark VI-1-9. Theorems VI-1-1 and VI-1-2 are also explained in ]CL, §1 of Chapter 8, pp. 208-211] and [Hart, Chapter XI, pp. 322-403]. For details concerning other results in this section, see also [Cop2, Chapter 1, pp. 4-33] and [Be12, Chapter 6, pp. 107-142].

VI-2. Sturm-Liouville problems A Sturm-Liouville problem is a boundary-value problem dd-x (p(x) Ly) + u(x)y = f (.T),

(BP)

y(a) cos a

-I p(a)y'(a) sin or = 0,

y(b) cos Q - P(b)y(b) sin )3 = 0,

under the assumptions: (i) the quantities a, b, a, and $ are real numbers such that a < b, (ii) two functions u(x) and f (x) are real-valued and continuous on the interval

Z(a,b) = {x: a < x < b}, (iii) the function p(x) is real-valued and continuously differentiable, and p(x) > 0 on 1(a, b). In this section, we explain some basic results concerning problem (BP).

149

2. STRUM-LIOUVILLE PROBLEMS

Let 4(x) and O(z) be two solutions of the homogeneous linear differential equation (VI-2.1)

-dx

(p(x)A) + u()y = 0

such that (VI.2.2)

m(a) = sins,

p(a)d'(a) = cosa,

,p(b) = sin f3,

p(b)t'(b) = cosfi,

respectively. Then, these two solutions satisfy the boundary conditions (VI.2.3)

¢(a) cos a - p(a)4,'(a) sin a = 0,

i (b) cos /3 - p(b)0'(b) sin /3 = 0.

The two solutions 4(x) and /;(x) are linearly independent if and only if (VI.2.4)

4(b) cos Q - p(b)©'(b) sin /3 # 0 or

'(a) cos a - p(a)rb'(a) sin a # 0.

The first basic result of this section is the following theorem, which concerns the existence and uniqueness of solution of (BP).

Theorem VI-2-1. If the two solutions 4(x) and ip(x) of (VI.2.1) are linearly independent, then problem (BP) has one and only one solutwn on the interval I(a, b). Proof

Using the method of variation of parameters (cf. Remark IV-7-2), write the general solution y(x) of the differential equation of (BP) and its derivative y'(x) respectively in the following form: (VI.2.5) 6

VW = CIOW + C20 W + 4(x)

J

'j'(_)f (_) 4

P(f)WW

Y' (X) = Clo'(x) + C21k'(x) + 0'(x) I s P(A)W

+ i,&(x)1= o(f)f a

4

(x) J

JJJu P(A)W V)

A,

where cl and c2 are arbitrary constants and W (x) denotes the Wronskian of Now, using (VI.2.3) and (VI.2.4), it can be shown that solution (VI.2.5) satisfies the boundary conditions of (BP) if and only if cl = 0 and c2 =

0. 0

Observation VI-2-2. It can be shown easily that p(x)W(x) is independent of x. Observation VI-2-3. Under the assumption that the two solutions 40(x) and tfi(x) of (VI.2.1) are linearly independent, the unique solution of (BP) is given by rb

y(x) = d(x) (VI.2.6)

Jx P(A)W (O

Y 'W = O(x)

rb owfwdC 1r

W

10(x) I

i

(- ))- W(f )

+ V,'(x) r= 4(W W 4. Ja P(OW (O

VI. BOUNDARY-VALUE PROBLEMS

150

Setting

4(x)1() G(x, ) =

(VI.2.7)

p(OW (c) O WOW

m - P( bn,)z - a

n = 1, 2,....

Observation VI-3-13. If u(x) < p and p(x) > p > 0 on I(a.b), determine 81(x, A) by the initial-value problem

p dx = 1 + (p(p

sin2(81),

81(a, A) _ ?r.

Then, since u(x) - A < p - A and p(x) > p > 0 for x E T(a, b), -oo < A < +oo, and a < s, it follows from Lemma VI-3-8 that

0(c,,\) < Oi(c,A)

for

a 3, eigenvalues An satisfy the following estimates:

< Iml (i)

if

n2

An < n2

2

P

Cb

- a)

'

and 2

n2 +

< IµI +

P(b- a)

n2

rr

2

4PCb-a) C1

if

_An> n2

1

1

n

n

-P Wa

where in, p, P and p are real numbers such that

P > p(x) > p > 0

and

p > u(x) > m

for

x E Z(a, b).

VI. BOUNDARY-VALUE PROBLEMS

162

Remark VI-3-15. Lemma VI-3-8, Lemma VI-10, and Theorem VI-3-11 are also found in (CL, §§1 and 2 of Chapter 8] and [Hart, Chapter XI].

VI-4. Eigenfunction expansions Let us define a differential operator L[y] = dx

(p(z)) + u(x)y and the vector !LV

space V(a, b) over the real number field R in the same way as in §VI-2. Also, as in §VI-2, define the inner product (f, g) for two real-valued continuous functions f b

and g on Z(a, b) _ {x : a < x < b} by (f, g) = operator L has the following property:

(f,L[g]) = (L[f],g)

for

It is known that the

J

f E V(a,b) and g E V(a,b)

(cf. Theorem VI-2-5). Define a norm of a continuous function on X(a, b) by 11f 11 =

(f, f). Then, 1(f, g)f < Ilf II llgll. The following theorem is a basic result in the theory of self-adjoint boundary-value problems. Theorem VI-4-1. Let Al and A2 be two distinct eigenvalues of the problem L[y] = Ay,

(EP)

y E V(a, b).

Let r11(x) and r}2(x) be eigenfunctions corresponding to Al and A2, respectively.

Then, (rll,rh) = 0. Proof.

Note that (r71,L[r12]) = A2(rh,g2) and (Ljrll1,rh) = Al(rl1,rn). This implies that A2(r11, r12) = AI (r11, r12). Therefore, (r1i, r12) = 0 since Al 0 A2 0

It is known that problem (EP) has real eigenvalues ( lien

Al > A2 > ... > An > ...

Let r)1(x),-. *2(x),

.

n-++oo

An = -co).

be the eigenfunctions corresponding to Al, A2,...

,

such that 11,11 = 1 (n = 1, 2, ... ). Theorem VI-4-1 implies that (' h,''m) = 0 if h # k. From these properties of the eigenfunctions rjn, we obtain k

k 11f

-

>(f,vh)T1hII2 = Ilfll2 h=1

and hence

-

E(f.rlh)2 > 0 h=1

+oo E(f,7,lh)2

< IIffl2

(the Bessel inequality)

h=1

for any continuous function f on Z(a, b). In this section, we explain the generalized Fourier expansion of a function f (x) in terms of the orthonormal sequence {rl (x) : n = 1,2,... }. As a preparation, we prove the following theorem.

163

4. EIGENFUNCTION EXPANSIONS

Theorem VI-4-2. Let µo not be an eigenvalue of (EP) and let f (x) be a continuous function on Z(a, b). Then, the series (f, 1h)t1h(x) h=1

Ah - Jb

is uniformly convergent on the interval Z(a, b).

Proof Define a linear differential operator Go by £o[yl = G[yl - poy. Then, there exists Green's function of the boundary-value problem (BP)

y E V (a, b)

Go [y1 = f (x),

such that the unique solution of problem (BP) is given by b

y(x) =

J.

(cf. §VI-2; in particular, Observation VI-2-3). Define the operator G[f[ by b

G(f 1 = (G(x, ), f) =

J

G(x, Of WA

for any continuous function f(x) on 1(a, b). Since Go[g[f]l = f, we obtain (I,G[.1) = (4[9[f)), 9191) = (9[f], 4191911) = (9[fl,9)

for any continuous functions f (z) and g(x) on Z(a, b). Also, since Go[rlh1 = c[nh1 Ahr1hUO . Hence, A01% = (Ah - l'o)llh, we obtain 9['7h] =

(f, rlh)ghI lz

< K

lz

(f, '1017h

=K

h=l,

(f, ih)2, h=11

where K is a positive constant such that J!G(x, ) Il < K on 1(a, b). I

the Bessel inequality implies that T(a, b).

0

Jim

Finally,

12

li,ls-+oo h=l ,j

(f' nh)nh(x) l = 0 uniformly on - 110

Now, we claim that the limit of the uniformly convergent series of Theorem VI-4-2 is equal to 9[f].

164

VI. BOUNDARY-VALUE PROBLEMS

Theorem VI-4-3. For any continuous function f (x) on Z(a, b), the sum +00 (h,nh) is equal to 9[f]. Proof Set 91 (f ] = 9 If, - 1: (f, TO 17h . Then, it suffices to show Ah - PO h=1=1 (VIA. 1)

lim IIGe[f111 = 0

for every continuous function f (x) on Z(a, b). We prove (VIA.1) in four steps. Without any loss of generality, we assume that Ah - uo < 0 for h > t . Also, note that (VI.4.2)

(f,9e[9]) = (94f1,9)

for any real-valued continuous functions f and 9 on Z(a, b). Step 1. It can be shown easily that Gr[>jh] = chrjh, where

Ch =

for

h = 1,2,... ,e,

for

h > t.

Let y be an eigenvalue of 9t and let ¢(r) be an eigenfunction of 9t associated with

y. Then, ¢ E V(a,b), 11011 # 0, and y¢ = 9[41 -

(0, 17017h

h=1 An

- go

.

Also, (VI.4.2)

implies that (0,9471h]) = (91'101,170. Hence, ch(¢, nh) = y(¢, nh) Consequently,

(¢, rjh) = 0 if y 0 ch. Therefore, (¢, rjh) = 0 for h = 1, ... , I and y¢ = G[¢] if y 96 0. Hence, either y = O or y = ch (h > P). Step 2. In this step, we prove that (VI.4.3)

sup 1190JI1 = sup

nfa=1

uni=1

1(911f1.f)I-

In fact, the inequality I(9e[f),f)1

5 11Ge[f]IIIIfII implies sup 1I9t[f111 >ufu=1 sup 1(91[f],f)I Also, since (Ge[f ±9],f ±9) = (911f],f)+(9t[9],9)±2(9t[f],9),

Of H=1

we obtain

4(91[f],9) = (91 [f+9) ,f+9) - (9e[f-9],f-9)
I(Gt[ne+I],ne+i)I = Ice+II > Ill 11=1

0.

Also, note that c = sup (911f], f) or -c = inf (Gt[f], f). Suppose that [if U=1

Iif11=1

c = sup (9t If], f ). Then, there exists a sequence f (j = 1, 2.... ) of continuous 11111=1

functions on T(a, b) such that lim (Gt[ fi], fi) = c and IIf, II = 1(j = 1, 2, . . . ) Since {Gt[ fj] : j = 1,2.... } is bounded and equicontinuous on T(a, b), assume without any loss of generality that lim Gt[ f)] = g E G(a, b) uniformly on T(a, b). -.+00

Let us look at (VI-4.4)

IIGt[fjI - cf,

II2

= I1Gt[fj]II2 + c2 - 2c(Gt[f,], f,) 0. Therefore, c must be an eigenvalue of Gt. However, since all eigenvalues of Gt are nonpositive, this is a contradiction. Therefore, -c = inf (91 [fit f ). We can now prove in a similar way

)+oo 11 9t [f, I

Illil=l

that 1

-c = ct+1 =

),ttl - µ0

Step 4. Since sup IIGt[f]II =

, we conclude that lien sup I191[f]II = 0 1 !Lo - t+1 t-'+0°1!11!=l and the proof of (VI.4.1) is completed. The following theorem is the basic result concerning eigenfunction expansion. 11111=1

+00

Theorem VI-4-4. For every f E V(a, b), the series f = 1: (f, nh)nh converges to h=1

f uniformly on T(a, b). Proof.

Set g = .Co[f], then f = g[g]. Therefore,

f=

+00

h=1 +00

_ h=1

(9+17 h)17h

+C0 = 1: (GO[J]+nh)nh

Ah - {b (1 £ofrlh])r)h

ah _

h=1

Ah - 110 +00

_ >(f, rlh)rlh h=1

VI. BOUNDARY-VALUE PROBLEMS

166

For every continuous function f on 1(a, b) and a positive number e, there exists an f, E V(a, b) such that 11f - fe II (f, rth)2

(the Parseval equality).

h=1

Furthermore, this, in turn, implies that if f (x) and g(x) are continuous on 1(a, b), then h=1

= Ay, y'(0) = 0, y(a) = 0, Example VI-4-7. Using the eigenvalue problem dX2 we construct an orthogonal sequence {cos(nx) : n = 0, 1, 2.... } of eigenfunctions. 00

This yields the Fourier cosine series a-° 2 + Ean cos(nx) of a function f (x), where n=1

an =

2

r*

a oJ

f(x) cos(nx)ds. By virtue of Theorem V1-4-4, this series converges

uniformly to f (x) on 1(0, 7r) if f (x) is twice continuously differentiable on 1(0, zr)

and f'(0) = 0 and f'(Tr) = 0.

Example VI-4-8. Using the eigenvalue problem d = Ay,

y(O) = 0, y(7r) = 0, we construct an orthogonal sequence {sin(nx) : n = 1,2,...l of eigenfunc00

tions. This yields the Fourier sine series >bn sin(nx) of a function f (x), where n=1

bn =

f/ f (x) sin(nx)ds. By virtue of Theorem VI-4-4, this series converges

ao

uniformly to f (x) on 1(0, ir) if f (x) is twice continuously differentiable on 1(0, ir) and f (0) = 0 and f (a) = 0. Example VI-4-9. If f (x) is twice continuously differentiable on Z(-ir, ir), f (-vr) _

f (ir), and f'(1r) = f'(-7r), set fe(x) = 2 if (x)+f (-x)] and fo(x)

[f (x)-f (-x))

167

4. EIGENFUNCTION EXPANSIONS

Then, fe satisfies the conditions of Example VI-4-7, while fo(x) satisfies the conditions of Example VI-4-8. Thus, we obtain the Fourier series cc

+

ao

[an cos(nx) + bn sin(nx)], n=1

where an =

7r

J

* f (x) oos(nx)ds and bn =

!

r J

f (x) sin(nx)ds. The Fourier

series converges uniformly to f (x) on Z(-7r, 7r).

Remark VI-4-10. The Fourier series of Example VI-4-9 can be constructed also from the eigenvalue problem L2 = Ay, y(-7r) = y(7r), y'(-7r) = y'(ir). Including this eigenvalue problem, more general cases are systematically explained in [CL, Chapters 7 and 11]. For the uniform convergence of the Fourier series, it is not necessary to assume that f (x) be twice continuously differentiable. For example, it suffices to assume that f (x) is continuous and f'(x) is piecewise continuous on 1(-7r, 7r), and f (7r) = f(-7r). For those informations, see [Z]. Remark VI-4-11. The results in §§V1-2, VI-3, and VI-4 can be extended to the case when p(x) is continuous on 1(a, b), p(x) > 0 on a < x < b, and p(a)p(b) = 0 under some suitable assumptions and with some suitable boundary conditions. Although we do not go into these cases in this book, the following example illustrates such a case.

Example VI-4-12. Let us consider the boundary-value problem

d f_dy

.

_

(VI.4.5)

and y(l) = 0,

y(x) is bounded in a neighborhood of x = 0,

where u(x) and f (x) are real-valued and continuous on the interval 1(0, 1).

Step 1. Consider the linear homogeneous equation (VI.4.6)

(X2idv)

d

+ u(x)v = 0

on the interval 1(0,1). Since 1 and logs form a fundamental set of solutions of

the differential equation ± (xdx

= 0, a general solution of (VI.4.6) can be

constructed by solving the intergral)equation v(x) = c1 + C2 109X +

10

(logs)10 u(C)V(4)dC, 0

where c1 and c2 are arbitrary constants. In this way, we find two solutions O(x) and

-i(x) of (VI.4.6) such that O(x) and 4'(x) are continuous on 1(0,1), lim O(x) = 1, and lien x¢'(x) = 0, whereas tI'(z) and tY(x) are continuous for 0 < x < 1, r_0+

lim (O(z) - logs) = 0, and urn (xo'(x) - 1) = 0. In general, this step of analy0+

z-+0+

sis is very important. The behavior of solutions at x = 0 determines the nature of eigenvalues of the given problem. Denote also by p(x) the unique soltion of (VI.4.6)

such that p(1)=0andp'(1)=1.

VI. BOUNDARY-VALUE PROBLEMS

168

Step 2. Assuming that 0(l) # 0, set ¢(x AO G(x,

WW

for

¢(F)p(x)

for

0< x < e 0,

then

(VI.5.6)

(C(x)P(x))me-1,

Iym(x> }l < mj

m= 0,1,2....

for (V1.5.4).

Inequality (VI.5.6) is true for m = 0. Assume that (VI.5.6) is true for an m; then,

r+ Isin(((x - r)) I

Iym+i(x,C)J < mi Z

Note that

sin(((x - r))

a

2i(

(1

-

e-2i(z-7)() =

z-r eft(Z-r)

1

e-2it=dz.

Hence,

sln(((x - r)) I < e-q(z-T)(r - x) for r1 > 0. C

Therefore, +00

I ym+1(x,C)I
0,

whereas the function b(() is defined and continuous on -oo < ( < +oo. Furthermore, b(() = O(Ifl-1) as I(I ---. +oo. To simplify the situation, we introduce the following assumption.

6. SCATTERING DATA

173

Assumption VI-6-1. We assume that Iu(x)I
0 and the general solution ci f+ + c2f+ is asymptotically equal to cle`4=+c2e-'4= as x --. +oc. If t = 0, then f+r(x,0) is asymptotically

equal to 1 as x equal to x as x

+oc. Moreover, another solution f+J z

is asymptotically f+ +oc. Therefore, all eigenvalues are A = (ir7)2 < 0. Furthermore,

all eigenvalues of (VI.6.5) are determined by

a(iry) = 0.

(VI.6.6)

This implies that all zeros of a(() for 3(() > 0 are purely imaginary.

Under Assumption VI-6-1, ft(x, () are analytic for 9'(() > - 2. Hence, a(() has only a finite number of zeros for 3(() > 0 (cf. (VI.6.4)). Let S = irlj 0 = 1, 2, ... , N) be the zeros of a(() for (() > 0. Then, f+(x, ir73) are real-valued and f+(x, ins) E L2(-oo, +oo). Set

c) _

r+ao

(j = 1, 2, ... , N),

1

f+(x,n,)2dr

(VI.6.7)

(-oc < { < +oo). Observation VI-6-2. Every eigenvalue of (VI.6.5) is simple, i.e., da(C)

4

;f 0

if

a(() = 0.

Proof.

From a(t;) = 0, it follows that `ia(() d(

i d W[ f+, f-[ 2S d(

- 2 2 W [f+, f-I = 2Zr (u'[f+(, f-] + N'[f+, f- 0 shows that the quantities q, and r({) determine a((). Proof

Set f (() = a(()j

Then,

J

7. REFLECTIONLESS POTENTIALS

175

(i) f (() is analytic for $(() > -2 and (94 0, (ii) ((f(() - 1) is bounded for 3(() >

-2,

(iii) f (() & 0 for Qr(() >- 0 and (54 0. O(log(()) near = 0 and F(() _ Observe that f(() - 1 = O((-'). From this 0((-1) as ( -, 00 on 3(() > 0.

Set also F(() = log(f (()). Then,

observation, it follows without any complication that 1 /'}Oc 2log if (01 F(() - Trif_. -(

for

£(() > 0

(cf. Exercise VI-18). Now, (VI.6.9) follows from If(t) = Ia(()j and loBja(()f2 =

log(Ta(f)22/

Definition V1-6-5. The set {r((), (n,,n2,... ,nN), (cl,c2,... ,CN)} is called the scattering data associated with the potential u(x).

VI-7. Reflectionless potentials is called the reflection coefficient. If this coefficient is zero, the The function potential u(x) becomes a function of simple form. Let us look into this situation.

Observation VI-7-1. If r(() = 0, then b(() = 0. This means that f_(x,t:) = a(t) f+(x, £) = a(t) f+(x, -t) (cf. (§VI-6) ). Therefore, using the relation f-(x, -C)

f+(x,0) =

a(-()

for

3(() !5

0,

we can extend f+ for all ( as a meromorphic function in (. Furthermore, if irlj (y =

1, 2,... , N) are zeros of a(() in 3'(() > 0, then -inj (j = 1, 2,... , N) are simple poles of f+(x, () in ( and = C)f-(x,irli) = -icj f+(x, ins). Ftesidueoff+at -inj = - f-(x,ir,) a9,(x)+ 2E 711+17., 92(x) = 0

(e = 1, 2, ... , N).

Differentiating again, we obtain N

2o[s

E ata (x){gj"(x) + 2r7eg, (x) } + 2ge

(9e(x) + 2gtgt(x)) - u(x)

,=1

N

+ 2E 3=1

gj

ge+g,

9'(x) = 0

(e = 1,2,... N)

or

N

N

Eat,j(x){g,"(x) + 217eg,(x)) - Eaea(x)u(x)9J(x) ,=1

f=1 N

e2nas

aea(x)2%g,'(x) + 2171

+ ,=1

C27jt

ct

)

=

0

(e = 1,2,N). ... ,9e(x)

Thus, we derive N

F'ae,,(x){9,(x) + 217,g,(x)

- u(x)g,(x)}

j=1

N 2171 E atj (x)9, (x) + 217t ct 91(x) ,=1

=0

(t = 1, 2, ... , N),

and (VI.8.3) implies that N

F'at,1(x){9;'(x) + 2s7,g,,(x) - u(x)g,(x)} = 0 ,=1

Therefore, (VI.8.1) follows.

(e= 1,2,... ,N).

VI. BOUNDARY-VALUE PROBLEMS

180

Observation VI-8-2. If we further define f+ by (VI.7.1), then £f+ = (2f+. Proof.

(e-"=f+)11 +

-i

u(e_" f+)

( ,=1

+ 2i(g - ttgJ

g1'

2

N

+i

J=1

-i N

-

g;,

9J

+ 2rligl' - ugJ = 0.

E j=1

(+ tnJ

Observation VI-8-3. Since N

f+(x,it7t) = e-fez 1 J=1

x

g., (x) 711+17.;

en," _ -gt(x) E ct

L2(-oo,+oo),

N numbers -1l,2 (j = 1, 2,... , N) are eigenvalues. N

Observation VI-8-4. Set a

(cf. (VI.6.9) with r

0) and

J=1

N f-(x,() = a(()f+(x, -() =

a(S)e-Cz

1 + iE

g3 (x)

J=1(-ig,

(cf. Observation VI-7-1). Then, £f_ _ (2f_ . Furthermore, since

E gJ (-oo) = 1

(e = 1, 2, ... , N)

J=1171+17J

(cf. (VI.7.2)), we obtain

a(() = 1 - i1 ____

(VI.8.4)

)

j=1(+nJ

In fact, this follows from the fact that both sides of (VI.8.4) are rational functions in ( with the same zeros, the same poles, and the same limits as ( , oo. Observation VI-8-5. The functions f±(.x, () are the Jost solutions of Ly = (2y. Proof.

Note first that lim f+(x. ()e-'t= = 1 (cf. (VI.7.1)). Also, we have

s+oo

lim

gg(-o) = a(()

f+(x,()e-` a, if A =a,

cosh( a --Ax)

if A < a

cos(v1'"T-_ax)

01 (X, A) =

VI. BOUNDARY-VALUE PROBLEMS

190 and

( sin(v1_T_-ax)

A-a 02(x, A) =

x

sinh(x)

if A>a, if A =a,

if A < a.

Therefore,

12 cos(\/-), --at)

AA) =

A) +

(e, A) _

if

A >a,

2

if A=a,

2 cosh( a --At)

if A < a.

Thus, we conclude that in this case, f (A)2 -4 has only one simple zero a (cf. Figure 2).

FicURE 2.

The materials in this section are also found in [CL, Chapter 8j.

EXERCISES VI

VI-1. Assume that u(x) is a real-valued continuous function on the interval 20 = {x : 0 < x < +oo} such that u(x) > rno for x >_ xo for some positive numbers "to and x0. Show that (1) every nontrivial solution of the differential equation (E)

day

&2

- u(x)y = 0

has at most a finite number of zeros on Z0, (2) the differential equation (E) has a nontrivial solution 1)(x) such that

hm q(x) = 0.

Hint. (1) Note that if y(xo) > 0, then y"(xo) > 0. Hence, y(x) > 0 for x > x0 if 1/(xo) > 0.

(2) It is sufficient to find a solution 0(x) such that 0(x) > 0 and 0'(x) < 0 for

x>xo.

EXERCISES VI

191

VI-2. For the eigenvalue-problem (EP)

L2 + u(x)y = Ay,

Y(()) = y(1),

y(0) = y'(1),

where u(x) is real-valued and continuous on the interval 0 < x < 1, (1) construct Green's function, (2) show that (EP) is self-adjoint, (3) show that (EP) has infinitely many eigenvalues.

VI-3. Let A, > A2 >

- > An >

-

be eigenvalues of the boundary-value problem

d2y

+ u(x)y = Ay,

y(a) = 0,

y '(b) = 0,

where u(x) is real-valued and continuous on the interval a < x < b. Show that there exists a positive number K such that 2

n + n2

ir

K

b-a

n

for n = 1, 2, 3, ...

.

VI-4. Assuming that u(x) is real-valued and continuous on the interval 0 < x < +oo and that lim u(x) = +oo, consider the eigenvalue-problem

r+oo

dx2 - u(x)y = Ay,

y(0) cos a - y'(0) sin a = 0,

z

lim

y(x) = 0.

where a is a non-negative constant. Show that (a) there exist infinitely many real eigenvalues AI > A2 > . . . such that lim An = n-.ioo -00, (b) eigenfunctions corresponding to the eigenvalue An have exactly n -1 zeros on the interval 0 < x < +oo. Hint. Let A1(b) > A2(b) > . . . be the eigenvalues of

dx2 - u(x)y = Ay,

y(0) cos a - y'(0) sin a = 0,

y(b) = 0,

where b > 0. Define Am by lim A,(b). See JCL, Problem 1 on p. 2541. 6

+oo

VI-5. Show that if a function O(x) is real-valued, twice continuously differen-

tiable, and ¢"(x) + e-'O(x) = 0 on the interval 10 = {x : 0 < x < +oo} and

if J

J0

O(x)2dx < +oo, then O(x) is identically equal to zero on I

.

VI. BOUNDARY-VALUE PROBLEMS

192

VI-6. Using the notations and definitions of §VI-4, show that +oo

(f,C(f)) = j>n(f,17n)2 n=1 b

=

j

{u(x)f(x)- P(x)f'(x)2}dx + P(b)f(b)f'(b) - P(a)f(a)f'(a),

if f E V(a, b).

VI-7. Assume that u(x) is real-valued and continuous and u(x) < 0 on the interval I(a, b), where a < b. Denote by yh(x, A) the unique solution of the initial-value problem 2 + u(x)y = Ay, y(a) = 0, y(a) = 1, where A is a comlex parameter. Show that (i) O(b, A) is an entire function of A, (ii) ¢(b, A) 54 0 if A is a positive real number, (iii) ¢(b, A) has infinitely many zeros A,, such that 0 > A0 > Al > A2 > and n

- - \ b r- al/ +oo n2

llm

A"

)

\

-

2 .

VI-8. Find the unique solution O(x) of the differential equation

) = y such that

Ix (i)\\\

is analytic at .x = 0 and 0(0) = 1. Also, show that

m(x)

is an entire function of x, (ii) ¢(x) A 0 if x is a positive real number,

(iii) ¢(x) has infinitely many zeros an such that 0 > \o > a1 > A2 > lim 1n = -oc, n-+00 (iv)

nx)dx = 0 if n 96 m.

J0

VI-9. Show that the Legendre polynomials 2"n1

Pn(x)

d" !dx"((x2 - 1)'J

=

(n = 0,1,2,...)

satisfy the following conditions: (i) deg P,, (x) = n (n = 0,1,2,...),

(ii)

J

1

Pn(x)Pm(x)dx = 0 if n 0 m,

(iii) j P n (x)2dx =

1

n (iv)

J

(n = 0,1, 2, ... ),

z

xkPn(x)dx = 0 for k = 0,... , n - 1,

(v) Pn(x) (n > 1) has n simple zeros in the interval Ixl < 1,

.

and

EXERCISES VI

193

(vi) if f (x) is real-valued and continuous on the interval jxj < 1, then

lim f

N-+oo where (f, Pn) =

f

2

N

1

1

f (x) - E (n+ Z) (f, PP)Pn(x))

dx = 0,

n=O

1

f (x)PP(x)dx,

(vii) the series +00 F, (n + 21 (f, PP)Pn(x) converges to f uniformly on the interval n=0

jxj < 1 if f, f', and f" are continuous on the interval jxj < 1. Hint. See Exercise V-13. Also, note that if f (x) is continuous on the interval jxj < 1, then f (x) can be approximated on this interval uniformly by a polynomial in x. To prove (vii), construct the Green function G(x,t) for the boundary-value problem

((I -x2)d/ +aoy=f(W), y(x) is bounded in the neighborhood of x = ±1, 1

1

where ao is not a non-negative integer. Show that f f G(x, )2dx< < +oo. J!

Then, we can use a method similar to that of §VI-4.

1J

1

VI-10. Assume that (1) p(x) and p'(x) are continuous on an interval Zo(a,b) _ {x : a < x < b}, (2) p(x) > 0 on Zo, and (3) u(x, A) is a real-valued and continuous function of (x, A) on the region Zo x ll = {(x, A) : x E Zo, A E It} such that

lim u(x, A) = boo uniformly for x E Zo. Assume also that u(x, A) is strictly A-too decreasing in A E R for each fixed x on I. Denote by O(x, A) the unique solution of the intial-value problem (P(x)L ) + u(x,.\)y = 0

y(a) = 0,

y(a) = 1.

Show that there exists a sequence {pn : n = 0, 1, 2, ... } of real numbers such that (i) pn < 1An-1 (n = 1, 2, ... ), and lim pn = -oo, n .+oo (ii) 4(b, pn) = 0 (n = 0,1, 2, ... ), (iii) ¢(x, .\) 0 on a < x < b for A > po, and q(x, A) (n > 1) has n simple zeros ,-

on a A2 > ... > Am (1 < m < n) be all of the distinct numbers in A. Set (VI1.2.3)

Vj = {j :

is a solution of (VII.2.1) such that A (p) < A-}

for j = 1,2.... Tn. Then, (ii) and (vi) of Lemma VII-1-3 imply that V, is a vector space over C. Set (VII.2.4) 1i = dim Vj (j = 1,2,... ,m). The following lemma states that there exists a particular basis for each space V, which consists of y, solutions whose type numbers are equal to A,. Lemma VII-2-2. For system (VI1.2.1) and y, given by (VII. 2-4), it holds that

(_) yi = n,

(ii) ym < -Ym-1 < ... < y, (ii:) for each j, there exists a basis for V, that consists of y, linearly independent solutions d,,- (v = 1,2,... , y,) of (VIL2.1) such that A (p,,°) = A3. Proof It is easy to derive (i) and (ii) from the definition of V, and the definition of the numbers at, ... , am. To prove (iii), let y,,,, (v = 1, 2, ... . y,) be a basis for Vj. Assume that A, for v = 1,...t, ( for

v=f+1...... ).

It follows from (ii) of Lemma VII-1-3 and definitions of V, and A, that t > 1. Set for

+

for

v = 1.... , t, v = t + l.... , y2.

Then, , , (v = 1, ... , y,) satisfy all the requirements of (iii).

2. LIAPOUNOFF'S TYPE NUMBERS OF A LINEAR SYSTEM

201

Observation VII-2-3. The maximum number of linearly independent solutions of (VII.2.1) having Liapounoff's type number A,, is rye.

Definition VII-2-4. The numbers Al, A2, ... , Am are called Liapounoff's type numbers of system (VIL2.1) at t = +oc. For every j = 1, 2,..., m, the multiplicity of Liapounoff's type number A, is defined by

hj _

(VII.2.5)

for j=1,2,... in- 1, for j = in.

7m

The structure of solutions of (VI1.2.1) according with their type numbers is given in the following result.

Theorem VII-2-5. Let {(1, ¢2r ... , iin} be a fundamental set of n linearly independent solutions of system (VII.2.1). Then, the following four conditions are mutually equivalent:

(1) for every j, the total number of those 4t such that A (y5t) = A, is h, (cf. (VII. 2.5)), (2) for every j, the subset {¢t : A (Qt) < A, } is a basis for V,,

ctt

(3) A

> A) if the constants ct are not all zero,

= max {A (dt)

(4) A

:

ct

0} for every nontrwzal linear combi-

n

nation

FC, it of

e

Wn}

t=1

Proof.

Assume that (1) is satisfied. Then, the total number of those ¢t such that m

A (3t) < A, is Fht = ryr = dims Vj. Hence, (2) is also satisfied. Conversely, t=j assume that (2) is satisfied. Then, the total number of those t such that A (fit) < Aj is equal to dime V) = -,. Hence, (1) is also satisfied (cf. (VII.2.5)).

Assume that (2) is satisfied. Then, if A follows that

ctc

ct4

E V,, and hence

clot =

tt

A, for some ct, it

202

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

for some c e. Since {¢j : t = 1,2,... , n} is linearly independent, all constants cc

must be zero. Thus, (3) is satisfied.

n

Assume that (3) is satisfied. Write a linear combination I:ct4t in the form t=1

m

cj¢t. Then, A

CWI

ci4r 96 0. Hence, (4)

= A), if

j=1 a(ft)=A, a(mt)=a1 follows from (iii) of Lemma VII-1-3.

a(Ot}=a,

Finally, assume that (4) is satisfied. Then, every solution d of (VII.2.1) with A (j) < A, must be a linear combination of the subset {¢t : A ((rt) S Aj }. Hence,

(2) is satisfied. 0 Definition VII-2-6. A fundamental set (01,02, ... , On } of n linearly independent solutions of system (VIL2.1) is said to be normal if one of four conditions (1) - (4) of Theorem VII-2-5 is satisfied.

Since V. C Vm-1 c . C V2 C V1, it is easy to construct a fundamental set of (VII.2.1) that satisfies condition (1) of Theorem VII-2-5. Thus, we obtain the following theorem.

Theorem VII-2-7. If the entries of the matrix A(t) are continuous and bounded on an interval Z = It : to < t < +oo}, system (VIL2.1) has a normal fundamental set of n linearly independent solutions on the interval Z. = Ay" with a constant matrix A, Lia Example VII-2-8. For a system pounoff's type numbers A1, A2, ... , Am at t = +oo and their respective multiplicities

h1, h2, ... , hm are determined in the following way. Let IL I , p2, ... , Pk be the distinct eigenvalues of A and M1, m2, ... , Mk be their

respective multiplicities. Set v, = R(pj) (j = 1, 2,... , k). Let Al > A2 > be the distinct real numbers in the set {v1i v2, ... , vk }. Set h, _

> Am

mr for

t=at dy

j = 1, 2, ... , m. Then, A1, A2, .... Am are Liapounoff's type numbers of = Ay" at t = +oc and h1, h2, ... , hm are their respective multiplicities. The prom of this result is left to the reader as an exercise.

Example VII-2-9. For a system (VII.2.6)

dy dt = A(t)y

with a matrix A(t) whose entries are continuous and periodic of a positive period c..' on the entire real line R, Liapounoff's type numbers )k1, A2, ... , Am at t = +oo and their respective multiplicities h1, h2, ... , hm are determined in the following way:

There exists an n x n matrix P(t) such that (i) the entries of P(t) are continuous and periodic of period w, (ii) P(t) is invertible for all t E R.

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

203

(iii) the transformation y = P(t)z changes system (VII.2.6) to (VII.2.7)

with a constant matrix B. Therefore, (vi) of Lemma VII-1-3 implies that systems (VII.2.6) and (VII.2.7) have the same Liapounoff's type numbers at t = +oo with the same respective multiplicities. Liapounoff's type numbers of system (VII.2.7) can be determined by using Example VII-2-8. Note that if pl, p2,. .. , p" are the multipliers of system (VII.2.6),

then p,

(j = 1, 2,... , n) are the characteristic exponents of system

log[p,]

(VII.2.6), i.e., the eigenvalues of B, if we choose log[p3] in a suitable way. Hence, R(µ1)

(j = 1, 2,... , n).

log[[p2I]

Those numbers are independent of the choice of branches of log[p,].

Example VHI-2-10. For the system

the fundamental set {et

=I J

eu [0] I is normal, but the fundamental set I

[b],

{[] [J} -e is not normal. 2t

,

VII-3. Calculation of Liapunoff's type numbers of solutions The main concern of this section is to show that Liapounoff's type numbers of

a system J = B(t)y" at t = +oc and their respective multiplicities are exactly the = Ay" with a constant matrix A if iimoB(t) = A. same as those of the system t-+0 dt It is known that any constant matrix A is similar to a block-diagonal form

diag[pllm, + W.421,., + Rf2i ... , µkl,,, + Mk],

where µl, ... , Pk are distinct eigenvalues of A whose respective multiplicities are is the mj x mj identity matrix, and M1 is an mj x m, nilpotent m1, ... , mk, matrix (cf. (IV.1.10)). Consider a system of the form (VII.3.1)

10 = A2y1 + EBjt(t)Ut

(j = 1,2,...

t=1

where y2 E C'-,, A. is an n, x n) constant matrix, and B2t(t) is an n, x nt matrix whose entries are continuous on the interval Za = {t : 0 < t < +oo}, under the following assumption.

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

204

Assumption 1. (i) For each j, the matrix A. has the form

Aj = )1In, + E. + N.,

(VII-3.2)

(j = 1,2,... ,m),

where Al is a real number, In, is the nJ x nl identity matrix, E. is an n. x nj constant diagonal matrix whose entries on the main diagonal are all purely imaginary, and Nj is an nj x n, nilpotent matrix, (ii) Am < )1m-1 < ... < 1\2 < Al,

(iii) N,E, = E?NJ (j = 1,2,... ,m), (iv)

t

limp B,t(t) = 0 (j,t = 1,2,... ,m).

The following result is a basic block-diagonalization theorem.

Theorem VII-3-1. Under Assumption 1, there exist a non-negative number and a linear transformation

y , = z ' , + I: T , (t)zt

(VII-3.3)

tj

to

(j = 1,2,... ,m)

with ni x nt matrices T,t(t) such that

(1) for every pair (j, t) such that j

f, the derivative dt tt (t) exists and the

entries of Tat and dj tt are continuous on the intervalI = (t : to < t < +oo)

lim T;t(t) = O (j # e), t-+00 (3) transformation (VII.3.3) changes system (VI1.3.1) to (2)

(VII.3.4) t Lzj

(j=1,2,...,m).

A,+B,,(t)+EBjh(t)Th,(t)I

Proof.

We prove this theorem in eight steps.

Step 1. Differentiating both sides of (VII.3.3), we obtain dzl dt

+

dTit5t

T,tdz"t dt

t#?

df

m

(VII.3.5)

= A) % + E Tj tat t0r

Aj 4' Bjj +

BjhThj

h*

Bjt t + t=1#t i, +

t

Tt V

AjTjt + Bjt +

BjhTht zt,

hit

205

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

where j = 1, 2,... , m. Note that r

in

e=1

n+

2, = F v=1

Vol

1

BjnTnv zv h#v

Rewrite (VII.3.5) in the form

{

-[At+Fj }+ET,t{ Lz' -[At+Fe]zt} t#i

JJJJJJ

_

[AJTJt + Bit +

t#,

1

l

BlhThe - Tlt (At + Ft)

dtt

h#t

Ztf

where j = 1, 2,... , m, and

F t = B» +

B)hTh)

M)

h#j

Define the Tit by the following system of differential equations: (VII.3.6)

T

dtt

(.l r e)

h#t

Then,

-[A,+F,]x",}+Tj dt -[At+Ft]zt}=Q (j=1,2,...,m).

{

This implies that we can derive (VII.3.4) on the interval I if the Tit satisfy (VII.3.6) and condition (2) of Theorem V1l-3-1, and to is sufficiently large.

Step 2. Let us find a solution T of system (VII.3.6) that satisfies condition (2) of Theorem VII-3-1. To do this, change (VII.3.6) to a system of nonlinear integral equations (VII.3.7)

T,t(t) =

J

exp[(t - s)A,]U,t(s,T(s))exp[-(t - s)A1]ds,

for j 0 e, where the initial points rat are to be specified and

U,t(t,T) = B11(t) + E Bjh(t)Tht - Tjt Btt(t) + E Bth(t)Tht h#t

h961

Using conditions given in Assumption 1, rewrite (V11.3.7) in the form

rt

(VII.3.7')

Tit(t) = J exp 2(a1 -

1

t)(t - s)]

s,s,T(s})ds,

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

206

t and Wjt(r,s,T)

where j (VII.3.8)

[(A,

= exp

- At)r] exp[rEj j exp[rNj[UJt(s, T) exp[-rEtj exp[-rN,].

On the right-hand side of (VII.3.7'), choose the initial points rat in the following way

ra t

(VII.3.9)

1t..,.

( +00

j`

to

if Aj > At

(i.e., j < t),

if A < A

(ie j > t)

Step 3. Let us prove the following lemma. Lemma VII-3-2. Let a be a positive number and let f (t, s) be continuous in (t, s) on the region Z x Z, where Z = {t : to < t < +oo}. Then, (VII.3.10)

t III

ft"

exp[-a(t - s)] f (t, s)dsl
t, we obtain IW,t(r,a,T)I < /C2,3(t) {1 + ITI2} for some positive number 1C2, where ITI = maxIT,t[. Similarly, the estimate t#j I Wjt(r, s, T) I < 1C219(to) { 1 + ITI2) is obtained by choosing a positive number 1C2

sufficiently large, if A., < at, r > 0, and to < s < t.

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

207

Step 5. In a manner similar to Step 4, we can derive the following estimates: IWj1(T,s,T)

- Wj1(r,s,T)I

K3$(t) {1 +ITI + ITI} IT - TI if A, > A1, T < 0, s > t,

1 K3$(to){1+IT{+ITI}IT-T{

if A0, to

(!+o)E(t)

for

t < s < r.

Then,

E(t) + c (1+ 6) E(t)(r - t) < E(t)(1 + (r - t)). This is a contradiction. This, in turn, proves that T

f (s)ds
t.

+430

Since lim E(t) = 0, the integral

t-+o

j+00

e

ft

+

e'('-")f (s)ds exists and

ec(t-e)f(s)ds

condition (2) of Theorem VII-4-3 on an interval I = {t

:

to < t < +oo} for a

large to.

Step 2. Now, construct Q(t) by using equation (VII.4.9) and condition (2) of Theorem VII-4-3. To do this, let 4?(t. s) be the unique solution of the initial-value problem

dY = A(t)Y, dt

Y(s) = In.

Then, (VII.4.9) is equivalent to the following linear integral equation: (VII.4.10)

Q(t) = J

1

(t, s)R(s) f 1 + Q(s)145(t, s)'1ds,

where 4?(t, s) = diag(Fl (t, s), F2(t,1s), ... , Fn(t, s)],

= exp [ I A,(r)d; {

F, (t, s)

(j = 1,2,. .. ,n).

JJJ

Step 3. Letting q,k(t) and rik(t) be the entries on the j-th row and the k-th column of Q(t) and R(t), respectively, write the integral equation (VII.4.10) in the form

(VII.4.10')

ggk(t) =

J

exp

[ft

Aik(r)drr)k(s) + F 11

A=1

ds,

215

4. A DIAGONALIZATION THEOREM

where j, k = 1, 2,... , n and the .1jk(t) are defined by (VII.4.5). Note that lexp

t

t ( J(Ajk(r)dr} I

I

= exp I J Djk(T)dr]
0.

Step 4. Define successive approximations by

gojk(t) = 0,

gPJk(t) = j

t

l

exp LJ

rk(S) +

t Ajk(r)dTJ

h=1

rja(S)gp-I;hk(s) ds,

where p = 1, 2, .... Then, we obtain I gpuk(t) I < eK (1 + nC]J

r(s)ds on the

to

interval I={t:to rJh(S)hk(s)J ds

exp

h=1

\.k(T)dr] {r)k(S) +

+ s

n h-1

for any a such that to t(e). This complete the proof of Theorem IV-4-3.

such that Iq,k(t)(
0 on the interval Zo, (3) the derivative p'(t) of p(t) is absolutely integrable on Io,

Then, Theorem VII-4-3 applies to system (VII.4.14) and yields the following theorem (cf. IHN2]).

Theorem VII-4-9. If a function p(t) satisfies the conditions (1), (2), and (3) given above, every solution of equation (IV.4.12) and its derivative are bounded on the interval Zo.

The proof of this theorem is left to the reader as an exercise. Note that condition (3) implies the boundedness of p(t) on the interval 1 . If p(t) is not absolutely integrable, set

z = [I2 + q(t)E]t,

(VII.4.15)

where I2 is the 2 x 2 identity matrix, q(t) is a unknown complex-valued function, and E is a constant 2 x 2 unknown matrix. Then, the transformation (VII.4.15) changes system (VI I.4.14) to (VII.4.16)

dt = [Iz + q(t)E]-1 { [ip(t)h12A0

- 4p(t) A11 [12 + q(t)E] J

dtt) E} u,

where

Ao =

Al =

Anticipating that (i) q(t)I and Ip'(t)I are of the same size, (ii) Jq'(t) I and Ip"(t)I are of the same size, choose q(t) and E so that two off-diagonal entries of the matrix on the right-hand side of (VII.4.16) become as small as Ip'(t)I2 + [p"(t)I. In fact, choosing

p'(t) , q(t) = 8-ip(t)312

E = [01 -11 0J

219

S. SYSTEMS WITH ALMOST CONSTANT COEFFICIENTS

we obtain

(12 + q(t)E]'

= 1 + I9(t)2 (12

-

q(t)E(

and q(t)E]-1 { [1(t)1/2

(12 +

Ao - 4p(t) A11 (12 + q(t)E1 jip(t)1/2A0 _ p'(t) EAo - p'(t)

1

1 + q(t) 2

8p(t)

4p(t)

dt) E}

Al

+g(t}AoE+E(p,p',p") - q(t) dtt)12} I 1 -r q(t) 2

I

lp(t)1/2Ao- p,(t)12+E(p,p,p')-q(t)dq(t)12 dt

4p(t)

where

EAo = -AoE =

E2 = -12,

0 11

,

Il 0

and E(p, p', p") is the sum of a finite number of terms of the form

a (P (t)

12

+ $ p"(t)

p(t)h/2

with some rational numbers or and /3 and some positive integers h. Applying Theorem VII-4-3 to system (VII.4.16), we can prove the following theorem.

Theorem VII-410. Suppose that (1) a function p(t) is continuous on the interval Zo = {t : 0 < t < +oo}, (2) there exists a positive number c such that p(t) > c > 0 on the interval ID, (3) +00

(Ip(t)12 + Ip'(t)I } dt < +oo. 0

(4)

limop'(t) = 0.

Then, every solution of equation (VII.4.12) is bounded on the interval 7o.

The proof of this theorem is left to the reader. Condition (3) of Theorem VII-4-10 does not imply the boundedness of p(t) on the interval Io. For example, Theorem VII-4-10 applies to equation (VII.4.12) with p(t) = log(2 + t).

VII-5. Systems with asymptotically constant coefficients In this section, we apply Theorem VII-4-3 to a system of the form (VII-5.1)

d9

where A is a constant n x n matrix and V(t) is an n x n matrix whose entries and their derivatives are continuous in t on the interval Zo = It : 0 < t < +oo} under the following assumption.

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

220

Assumption 4. The matrix A has n mutually distinct eigenvalues p 1, u2, ... , An, and the matrix V(t) satisfies the conditions

lim V(t) = 0 t

+00

and

+ Cc

+

[V'(t)[dt < +oo.

o

Let A1(t), 1\2 (t) .... , an(t) be the eigenvalues of the matrix A+V(t). Then, these are continuous on the interval I. Furthermore, it can be assumed that

lim ),(t) = µj t-+,

(j = 1,2,... ,n).

Choose to > 0 so that A) (t),.\2(t), ... , an(t) are mutually distinct on the interval

I={t:to 0, i.e., f+00 0

JR(t)jdt < +oo.

Observation VII-5-5. Let us look into the case when the matrix A has multiple eigenvalues. To do this, consider system (VII.3.1) under Assumption 1 given in §VII-3. By virtue of Theorem VII-3-1, system (VII.3.1) is changed to system (VII.3.4) by transformation (VII.3.3). Furthermore, (VII.3.4) is changed to (VII.5.3)

dt

[ N 3 + R 3 (t)) u 3

(j = 1, 2, ... m),

where

R3(t) = e-'E, B33(t)+E B3h(t)Ttj(t)

etE,

(j=1,2,...,m)

h #)

by the transformation (.) = 1, 2, ... , m). z3 = explt(A3I., + E, )]u3 Therefore, we look into the following two cases.

5. SYSTEMS WITH ALMOST CONSTANT COEFFICIENTS

223

Case VII-5-5-1. If N is an n x n nilpotent matrix such that N' = 0 and R(t) is an n x n matrix whose entries are continuous on the interval 0 < t < +oo and satisfy the condition (VII.5.4)

+

jt2(T_1)IR(t)Idt < +oo,

we can construct a fundamental matrix solution 45(t) of the system dy dt

(VII.5.5)

such that lim

t -.+00

= (N + R(t))y'

e-' N4(t) = I, , where I is the n x n identity matrix. In fact, the

transformation y" = etNii changes system (VII.5.5) to

dii

= e-tNR(t)e1NU. Since

leftNI < Kjtjr-1 for some positive number K, the integral equation X(t) = In

j

e-R(r)erNX(r)dT can be solved easily.

Case VII-5-5-2. Let N be an n x n nilpotent matrix such that Nr = 0 and let R(t) be an n x n matrix whose entries are continuous on the interval 0 < t < +oo and satisfy the condition +00

tr-1IR(t)jdt < +oo.

(VII.5.6) 10

Exercise V-4 shows that this case is different from Case VII-5-5-1. As a matter of fact, in this case, we can construct a fundamental matrix solution 'P(t) of system (VII.5.5) such that (VII.5.7)

lira t-(k-t) (41(t)

t-+00

-

etN) c" = 0',

whenever

Nkc_ = 0,

where k < r. We prove this result in three steps. Step 1. First of all, if an n x n matrix T(t) satisfies condition (VII-5.7), then We N'5 c' = 0 if 41(t)c = 0 for large t. In fact, noice first that lim e tr if f!

etNt: Also, note that lim t-+oo tktkl

NPe = 0 for p = F + 1, ... , r - 1. = 0 if a(t)e = 0 and Nkcc = 0. Therefore, N'lE = 0 if k = r. Hence, Nr-2c' = 0. In this way, we obtain c" = 0. Thus, we showed that if the matrix %P(t) satisfies the differential equation (N + R(t))* and condition (VII.5.7), then %P(t) is a fundamental matrix solution of (VII.5.5).

Step 2. To prove the existence of such a matrix W(t), we estimate integrals of ske(t-")NR(s) with respect to s, where 0 < k < r - 1. Observe that ske(t_e)N =

r1

Sk(t - S)hNh h1

h=0

Let us look at the quantity sk(t - s)h. Note that 0 5 k 5 r - 1 and 0 5 h < r -

1.

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

224

Case 1. k + h < r - 1. In this case, since sk(t - s)h = sk+h C t _ 11, define

-

s

Sk(th'

Jt

8)h N°R(s)ds = / t

8k(th! S)h NhR(s)ds.

Case 2. k + h > r. In this case, look at sk(t - 8)h = D-1)p

(hp)

Sk+pth-p

p_Q

Subcase 2(i). k + p < r - 1. In this case,

sk+pth-p = Sr-I

\s

r-1-u=k+p,u+v=h-p. Since

1 t", where

v = h - p - p = (h-p) + (k+p) - (r-1) = (k+h) - (r-1) = k - (r-1-h), we obtain 0 < v < k. Now, in this case, define h

t

Irt

Sk+pth-PNhR(S)dS

j

= J+ao p) sk+Pth-PNhR(s)ds.

Subcase 2(ii). k + p > r. In this case,

Sk+pth-p = Sr-1(tlµt",

r-l+p=k+p, v -p=h -p.

where

Since

v = h - p + p = (h-p) + (k+p) - (r-1) = (k+h) - (r-1) = k - (r-i-h), we obtain 0 < v < k. Also, note that h-p'

P

A(oo) are ei_)Iof p ' where the a, are the eigenvalues of A(oo). Now, apply ,

P

Corollary VII-3-7.

VII-6. For each of two matrices A(t) given below, find a unitary matrix U(t) analytic on (-oo, oo) for each matrix such that U-' (t)A(t)U(t) is diagonal or uppertriangular. 1 t 0 r (a) A(t) = Ot 0J, (b) A(t) = t 1 + 2t 0 11

i

L

0

0

1+t2

Hint. See [GH]. VII-7. Show that if a function p(t) is continuous on the interval 0 < t < +oo and lim t-Pp(t) = I for some positive integer p, the differential equation d2 t +p(t)y = +oo 0 has two linearly independent solutions rlf(t) such that t

77.k(t) =

P(t)-114(1 +o(1))exp [±ij t

)ds(, JJ

77, (t)

= P(t)114(+1 +o(1))exp

[f

iJ t

d4]

,

to

as t +oo, where to is a sufficiently large positive number and o(1) denotes a quantity which tends to zero as t - +oo. Hint. Use an argument similar to Example VII-4-8. m-1

VII-8. Let Q(x) = x' + E ahxh and P(x, e) _ ,Tn+l + Q(x), where in is a h=o

positive integer, x is an independent variable, and

am-1) are complex

VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

232

parameters. Set A(x, f) =

{(0) 11.

Also, let Ro, po, and ao be arbitrary but

fixed positive numbers. Suppose that 0 < po < 2. Show that the system d:V

= A(x,f)y,

y = 17121

has two solutions y, (x, f) (j = 1, 2) satisfying the following conditions: (a) the entries of y, (x, f) are holomorphic with respect to (x, f, ao,... , am_ 1) in the domain

D={(x,f,ao,...,am_,): rEC, 0 O, co > O, ro > O, k(t) > O and bounded fort > 0, lim k(t) = 0,

t+x

and A(ro) = {(t, yj : 0 < t < +oo, Iy1 < ro). The following theorem is the main result in this section. Theorem VIII-2-1. If the real part of every eigenvalue of A is negative, the tnvial solution g = 0 of (VIII. 2.1) is asymptotically stable as t -i +oo. Proof.

We prove this theorem in four steps. Step 1. Let A, (j = 1, 2,... , n) be the eigenvalues of A. Then, RIA,I (j = 1, 2, ... ,

n) are Liapounoff's type numbers of the system d= Ay (cf. Example VII-2-8). t Therefore, choosing a positive number p so that 0 < p < -9RIA,I f o r j = 1, 2, ... , n, we obtain IeXpIAtll < Ke-"t on the interval To = {t : 0 < t < +oc} for some positive constant K. Step 2. Fixing a non-negative number T, change system (VIII.2.1) to the integral equation

g(t) =

e(t-T)Ag(T)

+ fTte(t-')Ag(s,g(s))ds.

If Ig(t)I < b for some positive number b on an interval T < t < T1, then ly(t)l

T

to the form e"(t-T)Ig(t)I 0,

K61 T.

Step 4. If Jy(t)J < b < 1 for 0 < t < T, there exists a positive number is such that

Id

t)

I < n19(t)j, for 0 < t < T. This implies that Jy-(t)J < Jy-(0)JeK' as long

as Jy(t)I < 6 < I for 0 < t < T. Therefore, Jy(T)J is small if Jy(0)j is small. Thus, it was proven that (VIII.2.3) holds for t > T if Iy(0)J is small. This completes the proof of Theorem VIII-2-1. O

Example VIII-2-2. For the following system of differential equations d = A#+ where

y=

[y Y

j

A=

r -0.4

-2

-

(Y2

1

.2

( 1 + y2) I 11 J

the trivial solution y = 0 is asymptotically stable as t -' +oc. In f act, the characteristic polynomial of the matrix A is PA(a) _ (A + 0.3)2 + 1.99. Therefore, the real part of two eigenvalues are negative. Remark VIII-2-3. The same conclusion as Theorem VIII-2-1 can be proven, even if (VIII.2.2) is replaced by (VIII.2.4)

19(t, y-)J < (k(t) + h(t)Jyj")Jy7

for

(t, y) E A(ro),

where p is a positive number, and two functions h(t) and k(t) are continuous for

t > 0 such that k(t) > 0 for t > 0, lim k(t) = 0, h(t) > 0 fort > 0, and Liapounoff's type number of h(t) at t = +oo is not positive (cf. [CL, Theorem 1.3 on pp. 318-319]). Remark VIII-2-4. The converse of Theorem VIII-2-1 is not true, as clearly shown in Example VIII-1-5. In that example, the eigenvalues of the matrix A are -1 and 0, but the trivial solution is asymptotically stable as t -+ +oo. Also, in that example, solutions starting in a neighborhood of 0 do not tend to 0 exponentially as t -+ +oo.

Remark VIII-2-5. Even if the matrix A is diagonalizable and its eigenvalues are all purely imaginary, the trivial solution is not necessarily stable as t -+ +oo. In such a case, we must frequently go through tedious analysis to decide if the trivial solution is stable as t -+ +oo (cf. the case when g'(a) > 0 in Example VIII-1-6). We shall return to such cases in IR2 later in §§VIII-6 and VIII-10 .

243

3. STABLE MANIFOLDS

Remark VIII-2-6. If the real part of an eigenvalue of A is positive, then the trivial solution is not stable as it is claimed in the following theorem.

Theorem VIII-2-7. Assume that (i) A is a constant n x n matrix, (ii) the entries of y(t, yam) are continuous and satisfy the estimate

for (t,y-) E o(ro) _ {(t,y-) : 0 5 t < +oo, Iy-I

0, e(t, y-)

> 0 for (t, y) E

and

dy

= Ay + §(t, y-) is not stable as t -. +oo. dt A proof of this theorem is given in (CL, Theorem 1.2, pp. 317-3181. We shall prove this theorem for a particular case in the next section (cf. (vi) of Remark VIII-3-2). Then, the trivial solution of the system

An example of instability covered by this theorem is the case when g'(a) < 0 of Example VIII-1-6. The converse of Theorem VIII-2-7 is not true. In fact, Figure 3 shows that the trivial solution of the system I

= -yI,

dY2

= (yi + yi)112

is not stable as t - +oo. Note that, in this case, eigenvalues of the matrix A are -1 and 0. Y2

Yt

FIGURE 3.

VIII-3. Stable manifolds A stable manifold of the trivial solution is a set of points such that solutions starting from them approach the trivial solution as t --. +oo. In order to illustrate such a manifold, consider a system of the form (VIII-3-1)

dx

= AIi +

LY = Alb + 92(40,

where Y E iR", 11 E R n, entries of R"-valued function gl and RI-valued function g2 are continuous in (1, y-) for max(IiI, Iyj) 5 po and satisfy the Lipschitz condition 19't(_,y)

- 9,(f,nil 5 L(P)max(Ii-.1, Iv-ffl)

(1=1,2)

VIII. STABILITY

244

for max(jil, Iy1) < po and max(11 i, Ii1) < po, where po is a positive number and u oL(p) = 0. Furthermore, assume that gf(0, 0) = 0 (j = 1, 2). Two matrices AI and A2 are respectively constant n x n and m x m matrices satisfying the following condition: le(t-9)A' I < Je(t-s)A21 < K2e-02(t-8)

for

Kie-o1(t-a)

(VIII.3.2)

t

for

t > s, t < s,

where K, and of (j = 1,2) are positive constants. Condition (VIII.3.2) implies that the real parts of eigenvalues of AI are not greater than -01, whereas the real parts of eigenvalues of A2 are not less than -02. Assume that

a1 > 02.

(VIII.3.3)

Let us change (VIII.3.1) to the following system of integral equations: I

x(t, c) = e1AI C + J e(1-a)A' gi (Y(s, c-), y(s, c))ds, 0

(VIII.3.4)

At, c ) =

Jt to0 e(t

8)A2 2(x(s, c), y(s, c))ds.

The main result in this section is the following theorem.

Theorem VIII-3-1. Fix a positive number c so that a1 > 02 + E. Then, there exists another positive number p(() such that if an arbitrary constant vector c' in R' satisfies the condition KI I c1 < be constructed so that

1(0,61=6 and

(VIII.3.5)

2 , a solution (i(t, c), y"(t, cam)) of (VIII.3.1) can

max(Ii(t, c)I, I y(t, c)I) < p(e)e-(Ol -t)t

for 0 < t < +oo. Furthermore, this solution (i(t, cl, #(t, c)) is uniquely determined by condition (VIII.3.5). Proof Observe that if

max(Ix(t,c)I,Iy(t,c)I)
Ph, and yam' = yl'

yP,,-.

h=1

The following theorem is a basic result concerning formal simplifications of system (VIII.4.1).

Theorem VIII-4-2. Under Assumption VIII-4-1, there exists a formal power series

(VIII.4.3)

P'(u) = Pou" +

EPP,, IpI>2

in a vector u E C" with entries {ul,... ,u"} such that (i) Po is an invertible constant n x n matrix and PP, E C", (ii) the formal transformation y' = P(u") reduces system (VIII 4. 1) to (VIII.4.4)

j = Boii + g"(u)

4. ANALYTIC STRUCTURE OF STABLE MANIFOLDS

247

with a constant n x n matrix Bo and the formal power series g(ui) _ E upgp Ipi>2

with coeficients gp in Cn such that (iia) the matrix B0 is lower triangular with the diagonal entries Al.... , An,

and the entry bo(j, k) on the j-th row and k-th column of B0 is zero whenever A, j4 Ak,

(iib) for p with IpI > 2, the j-th entry gp, of the vector gp is zero whenever n

A1

(VIII.4.5)

1: PhAh h=1

Proof Observe that if y" = P(d), then d9 =

Po +

dt

1: [ L

p

p

Pp P22 Pp

Pp

un Pp

...

W>2

Boi + F 4Lpgp Ipl>2

and

Ay + f(y) = A Pod + F u-'pPp + E P(u)p fp. IpI>2

IPI>_2

Furthermore, [Plup,5, P2upP-p

Po + Ipl>2

poop P_p

Bo u'

...

u2

ul

Un

J

n

= PoBou" +

phAh lpl>2

u"PPp + 1rif(ur+1,...,u,,)=(0,....0).

Observation VIII-4-4. Under the same assumption on the Aj as in Observation VIII-4-3, set (ur+1, ... , un) = (0,... , 0). Then, the system of differential equations has the form on (u1,... dud

dt

=

A3u3

+

QJ,huh

(VIII.4.10)

+

9(P,.

.P04ul1

... u p ,

( i=1 , -...- - , r)-

A,=p,a,+ +p.a.,lai>2

r

r

Observe that since A. -

ph is sufficiently large, the right-hand

ph Ah # 0 if h=1

h=l

members of (VIII.4.10) are polynomials in (ul,... ,ur).

249

4. ANALYTIC STRUCTURE OF STABLE MANIFOLDS

Observation VIII-4-5. Assume that R[Ah+1J PhAh h=1

h=1

r

for some positive number a if E ph is large. In this case, a iajorant series for P h=1 can be constructed.

The construction of such a majorant series is illustrated for a simple case of a system dy = Ay + f (y), dt

where A is a nonzero complex number and AM is a convergent power series in y given by (VIII.4.2). According to Theorem VIII-4-2, in this case there exists a dd = Ad, where formal transfomation ff = d + Q(u) such that dt

This implies that E AIpJi1PQg, = AQ(u) + f (u" + IeI>2

IpI?2

Set f (ii + Q(u)) _

VIII. STABILITY

250

ui"A,. Then, lal>2

Ap

a= AOPI-1) for all p (IpJ > 2). Set 1

' (y1= > IfP11F IpI>2

1

Then, F(y-) is a majorant series for ft y-). Determine a power series V (u-) =

uupii

IpI>2

by the equation v =

ICI 0(u"

iBa. Then,

+ v'). Set F(u + V(U)) _ Ipl>2

vp =

B

Ii

for all p (JpJ > 2).

It can be shown easily that Y(ul) is a convergent majorant series of Q(ur). This proves the convergence of Q(ii). Putting P(u") and the general solution of (VIII.4.10) together, we obtain a particular solution y" P(#(t, cl) of (VIII.4.1), where

= (t,cj = (e'\It7Pi(t,cl,... ,cr),... ,e*\1 'Wr(t,cl,... ,cr),0,... ,0).

This particular solution is depending on r arbitrary constants c = (cl,... , c,.). Furthermore, this solution represents the stable manifold of the trivial solution of

(VIII.4.1)if W(Aj)>0for j#1,...,r. Remark VIII-4.7. In the case when y, A, and AM are real, but A has some eigenvalues which are not real, then P(yl must be constructed carefully so that the

particular solution P(i(t, c)) is also real-valued. For example, if A =

a b

b

a

,

the eigenvalues of A are a ± ib. If IV = ['] is changed by ul = 1/i + iy2i and

=

- iy, system (VIII.4.1) becomes dul OFF

(VIII.4.11)

due dt

= (a + ib)ul +

9P,,p2u 'uZ Pl+P2>2

_ (a - ib)u2 +

,

where gP,,- is the complex conjugate of g,,,p,. If a 54 0, using Theorem VIII-4-2, simplify (VIII.4.11) to (VIII.4.12)

= (a - ib)v2

dtl = (a + ib)vl, ddt

S. TWO-DIMENSIONAL LINEAR SYSTEMS

251

by the transformation PP,.P2'UP11t?221

VI + p, +p2>_2

(VIII.4.13)

7P,.p2V2P1VP13.

V2 +

P,+p2>2

Now, system (VIII.4.12), in turn, is changed back to - = At by wI = and w2 =

VI - V2

2i

,

V 2 V2

where (w1, w2) are the entries of the vector tu. Observe that U l + u2

w1 + E

2

p,+p2>2

uI - u2

U'2 +

2i

g2.P,,pzu'P1 u" p, +p3 22

where ql,p,,p2 and g2,p,,p2 are real numbers. Similar arguments can be used in general cases to construct real-valued solutions. (For complexification, see, for example, [HirS, pp. 64-65].) For classical works related with the materials in this section as well as more general problems, see, for example, [Du).

VIII-5. Two-dimensional linear systems with constant coefficients Throughout the rest of this chapter, we shall study the behavior of solutions of nonlinear systems in 1R2. The R2-plane is called the phase plane and a solution curve projected to the phase plane is called an orbit of the system of equations. A diagram that shows the orbits in the phase plane is called a phase portrait of the orbits of the system of equations. As a preparation, in this section, we summarize the basic facts concerning linear systems with constant coefficients in R2. Consider a linear system dy =

(VIII.5.1)

dt

Ay,

where

E R2 and A is a real, constant, and invertible 2 x 2 matrix. Set p = trace(A) and q = det(A), where q ¢ 0. Then, the characteristic equation of the matrix A is 1\2 - pA + q = 0. Hence, two eigenvalues of A are given by

AI =

2

+

4

q

and

A2 = 2 -

4

- q.

It is known that p = AI +A2,

q = AIA2,

and

Al - A2 =2

P2 - q. 4

VIII. STABILITY

252

Also, let t and ij be two eigenvectors of A associated with the eigenvalues Al and A2, respectively, i.e., At = All;, l 0 0, and Au = A2ij, ij 0 0. Observe that, if y(t) is a solution of (VIII.5.1), then cy(t +r) is also a solution of (VIII.5.1) for any constants c and T. This fact is useful in order to find orbits of equation (VIII.5.1) in the phase plane.

Case 1. Assume that two eigenvalues Al and A2 are real and distinct. In this case, two eigenvectors { and ij are linearly independent and the general solution of differential equation (VIII.5.1) is given by 1!(t) = cl eAI t

+ c2ea'tti = e)lt (cl +

c2e(1\2-,,,)t11= ea2t[cie(AI -a2)t (+ c2i17,

where cl and c2 are arbitrary constants and -oo < t < +oo. 2

la: In the case when Al > A2 > 0 (i.e., p > 0, q > 0 and 4 > q), the phase portrait of orbits of (VIII.5.1) is shown by Figure 4. The arrow indicates the direction in +oo. Note that as which t increases. The trivial solution 0 is unstable as t and -oo , the solutions y(t) tends to 0 in one of the four directions of t -ij. The point (0, 0) is called an unstable improper node. 2

lb: In the case when 0 > Al > A2 (i.e., p < 0, q > 0 and 4 > q), the phase portrait of orbits of (VIII.5.1) is shown by Figure 5. The trivial solution 0 is stable as t --i +oo. The point (0, 0) is called a stable improper node. 4

FIGURE 5.

FIGURE 4.

1c: In the case when Al > 0 > A2 (i.e., q < 0), the phase portrait of orbits of (VIII.5.1) is shown by Figure 6. The trivial solution 0 is unstable as Itl +oo. Note that as t -+ -oo (or +oo), only two orbits of (VIII.5.1) tend to 0. The point (0, 0) is called a saddle point. 2

Case 2. Assume that two eigenvalues Al and A2 are equal. Then, q = 4 and

Al=A2=29& 0. 2a: Assume furhter that A is diagonalizable; i.e., A = 212i where 12 is the 2 x 2 identity matrix. Then, every nonzero vector c is an eigenvector of A, and the general

solution of (VIII.5.1) is given by y"(t) = exp

[t]e.

In this case, the phase portrait

of orbits of (VIII.5.1) is shown by Figures 7-1 and 7-2. As t -+ +oo, the trivial solution 6 is unstable (respectively stable) if p > 0 (respectively p < 0). Note that,

5. TWO-DIMENSIONAL LINEAR SYSTEMS

253

for every direction n, there exists an orbit which tends to 6 in the direction n" as t tends to -oo (respectively +oo). The point (0, 0) is called an unstable (respectively stable) proper node if p > 0 (respectively p < 0).

p0 FIGURE 6.

FIGURE 7-1. 2b: Assume that A is not diagonalizable; i.e., A = p

FIGURE 7-2.

212 + N, where 12 is the 2 x 2

identity matrix and Nris a 2 x 2 nilpotent matrix. Note that N 0 and N2 = O. Hence, exp[tA] = exp 12t] {I2 + tN}. Observe also that a nonzero vector c' is an

eigenvector of A if and only if NcE = 0. Since N(Nc) = 0, the vector NcE is either 6 or an eigenvector of A. Hence, NE = ct(-) , where is the eigenvector of A which was given at the beginning of this section and a(c) is a real-valued linear homogeneous function of F. Observe also that at(c) = 0 if and only if c' is a constant multiple of the eigenvector . The general solution of (VIII.5.1) is given

by y(t) = exp

[t] {c + ta(c7}, where c" is an arbitrary constant vector. In this

case, the phase portrait of orbits of (VIII.5.1) is shown by Figures 8-1 and 8-2. The trivial solution 6 is unstable (respectively stable) as t - +oo if p > 0 (respectively p < 0). The point (0, 0) is called an unstable (respectively stable) improper node if p > 0 (respectively p < 0).

p>0

p

and

4 Al =a+ib,

A2=a-ib,

a=

2,

4

Note that b > 0. Set A = a12 + B. Then, B2 = -b212, since two eigenvalues of B

VIII. STABILITY

254

are ib and -ib. Therefore, exp[tB] = (cos(bt))I2 +

Smbbt)

B.

3a: Assume that a = 0 (i.e., p = 0). Then, the general solution of (VIII.5.1) is given by '(t) = exp[tB]c = (cos(bt))c"+

slnbbt)

Bc, which is periodic of period 2b

in t. The phase portrait of orbits of (VIII.5.1) is shown by Figures 9-1 and 9-2. The trivial solution 0 is stable as It 4 +oo. The point (0, 0) is called a center. It is important to notice that every orbit y(t) is invariant by the operator . In fact, r,

B j(t)

T

=

cos(bt) b

b

sin bt +

Bc-(sin(bt))c = (cos (bt + ))e+

b

2 Be= 17 (t +

In other words, dy(t) = by (t + Note that "" is 1 of the period dt 2b) 2b 4

j 2b

)

2r

.

FIGURE 9-1.

b

FIGURE 9-2.

3b: Assume that a 0 0 (i.e., p 0 0). Then, the general solution of (VIII.5.1) is given by y(t) = exp[at]il(t), where u(t) is the general solution of dt = Bu. The solution 0 is unstable (respectively stable) as t - +oo if p > 0 (respectively p < 0). The orbits, as shown by Figures 10-1 and 10-2, go around the point (0, 0) infinitely many times as y(t) -- 0. The point (0, 0) is called an unstable (respectively stable) spiral point if p > 0 (respectively p < 0).

(0.0)

p>0 FIGURE 10-1.

p 2. Therefore, there exists an R 2-valued function P(g) whose entries are convergent power series in a vector u" E R2 with real coefficients such that LP (0) 8u

_

12, and the transformation y = P(u) reduces system (VIII.6.1) to dt = Au (cf. Remark VIII.4.7). This, in turn, implies that the point (0, 0) is also a stable spiral point of (VIII.6.1) and that the general solution of (VIII.6.1) is y = P(eAC1, where c" is an arbitrary constant vector in 1R2.

Case 4. If the point (0, 0) is a saddle point of the linear system

= Ay, the

eigenvalues At and A2 of A are real and At < 0 < A2. Construct two nontrivial and real-valued convergent power series fi(x) and rG(x) in a variable x so that ¢(eA'tcl) (t > 0) and ,G(eA2tc2) (t < 0) are solutions of (VIII.6.1), where cl and c2 are arbitrary constants (cf. Exercise V-7). The solution y' = *(eAt tct) represents the stable manifold of the trivial solution of (VIII.6.1), while the solution y" =1 (eA2tc2) represents the unstable manifold of the trivial solution of (VIII.6.1). The point (0, 0)

6. ANALYTIC SYSTEMS IN R2

257

is a saddle point of (VIII.6.1). In the next section, we shall explain the behavior of solutions in a neighborhood of a saddle point in a more general case. dy Case 5. If the point (0, 0) is a center of the linear system - = Ay, both eigenvalues at

of A are purely imaginary. Assume that they are ±i and A = 1i

y" = 2 L

01

Set

J.

[t1]. Then, the given system (VIII.6.1) is changed

v, where v =

1

[0

J

2

to

du dt

(VIII.6.2)

-

Ig(vl,v2)l 9(27)

012, v1)

where +00

g(tvl, v2) = ivl +

9P'9111

012,V0 = -iv2 +

2,

p+9=2

p.9v2v1 p+9=2

Here, a denotes the complex conjugate of a complex number a. Let us apply Theorem VIII-4-2 to system (VIII.6.2).

Observation VIII-6-1. First, setting Al = i and A2 = -i, look at Al = p1A1 + p2A2 and A2 = Q1A1 +g2A2, where p1, p2, q1, and q2 are non-negative integers such that p1 + P2 > 2 and ql + q2 > 2. Then, p1 - 1 = p2 and q2 - 1 = q1. This implies

that system (VIII.6.2) can be changed to du1

(VIII . 6 . 3)

dt

=

i W ( ulu2 ) u1,

d = -iC due

0 ( u1U2 ) U2,

+oo

where w(z) = 1 + E Wmxm, by a formal transformation m=1

v = f(17) =

(T)

ul + h(ul,u2) u2 + h(u2,u1)

Here,

+0

+00

h(ul,u2) _

hp.qupU2,

h(u2,u1) _

_ hP,9u'lU2

p+9=2

p+q=2

In particular, h(ul, u2) can be construced so that (VIII.6.4)

the quantities hp+1,p are real for all positive integers p.

Observation VIII-6-2. We can show that if one of the Wm is not real, then y = 0 is a spiral point (cf. Exercises VIII-14). Hence, let us look into the case when the +00

Wm are all real. Furthermore, if a formal power series a(t) = 1 + real coefficients am is chosen in a suitable way, the transformation (VIII.6.5)

u1 = a(B11j2)131,

u2 = W10002

with m=1

VIII. STABILITY

258

changes (VIII.6.3) to another system d,31

= in(AiQ2)Qi,

2 _ -0#10002,

where f2(() is a polynomial in C with real coefficients which has one of the following two forms: 1, Case I Q(o 1 + coS"'°, Case II

-

where co is a nonzero real number and tno is a positive integer.

Observation VIII-6-3. Let us look into Case I. Assume that system (VIII.6.3) is

(VIII.6.3.1)

due

dul

dt

= iu1, dt

=

-iu2.

Note that transformation (VIII.6.5) does not change (VIII.6.3.1). Using a transformation of form (VIII.6.5), change h(ul, u2) so that (VIII.6.6)

1,P = 0

for all positive integers p.

Since (u1,u2) = (ce`t,de-'t) is a solution of system (VIII.6.3.1) with a complex arbitrary constant c, the formal series

v = f(-) =

(FS-1)

cell + h(eett,

cue-,t)

ce-u + h(ce-tt cett)

is a formal solution of system (VIII.6.2) which depends on two real arbitrary constants. Let

u(t, , ) _

(S)

Vt + H(t, fe-'t

be the solution of system (VIII.6.2) satisfying the initial condition

t'(0, f, 6 = III].

(C)

Note that +«0

H(t, , ) _

HP,q(t)ef°, D+9=2

H(tto = E

P+v=2

are power series in { and which are convergent uniformly on any fixed bounded interval on the real t line.

6. ANALYTIC SYSTEMS IN R2

259

Using (VIII.6.6), we obtain

I 0

Zx

r 2* h(Ce-'t, cett)e'Ldt

h(ceic, ee ")e-'tdt = 0,

= 0.

0

0

Fix c and c by the equations

c = £ +T7r-

/

2x

H(s, t:, )e-"ds and e = { + 2 J2w H(s, {, {)e'sds. 0

0

Then,

= c + -(c, c)

t; = e + ?(c, c),

and

where =(c, e) and c(e, c) are convergent power series in (c, e). Now, we can prove that two formal solutions +h(ce",Ce-'t)

ce`t

v'(t,c+F(c,e),e+ =2(c' i!))

and ee-tt +

h(&-'t, cent )

j

of system (VIII.6.2) are identical. The first of these two is convergent; hence, the second is also convergent. This finishes Case I.

Observation VIII-6-4. Let us look into Case II. Assume that system (VIII.6.3) has the form (VIII.6.3.2)

dtl = iul(l +

dt2 = -iu2(1 +

co(u1u2)'0),

co(uiu2)"'°)

Note that we have (VIII.6.4). Since (ul,u2) = (ce't(1+co(ce)'"°) ee-u(1+c0(ct)'"0)) is a solution of system (VIII.6.3.2) with a complex arbitrary constant c, the formal series ce't(I+co(ce)'"0) + h(ceit(l+co(ct)"'O),

(FS-2)

ee-'t(1+c0(ct)_0))

v = f (u') = h(ce-tt(1+co(ce)'"° ),

ee-'t(1+c0(ce)'"0) +

celt(1+co(ct)"O) )

is a formal solution of system (VIII.6.2) which depends on two real arbitrary constants. Again, as in Observation VIII-6-3, let (S) be the solution of system (VIII.6.2) satisfying the initial condition (C). Set (VIII.6.7)

t; = ?(c, cJ = c + h(c, c)

and

= =(c, c) = e + h(e, c).

Then,

I

ce1t(1+co(ct)"°)

+

h(ceit(1+co(ce)'"°),ee-'t(l+co(ct)"'0)

ee-tt(1+c0(ct)'"0) + h(ee-it(t+co(ct)'"O),ceut(l+co(et)"O))

VIII. STABILITY

260

as formal power series in (c, e). This series has a formal period

T(cc) =

27r

1 + co(ce)m0

with respect to t, i.e., E(c, c)

(VIII.6.8)

v(T(cc), =[c, e], E[c, c]) E(e, C)

Solving (VIII.6.7) with respect to (c, e), we obtain two power series in

c=

and

e = e(

,

£).

t) = T(c(., t)e(, £)). Then, (VIII.6.8) becomes

Set

(VIII.6.9)

Using (VIII.6.9) as equations for P({,£), it can be shown that the formal power series is convergent. This implies that 1/7+o

C(S,S)C(CS)

[(CO)

1/

J

is convergent. (Here, some details are left to the reader as an exercise.) On the other hand,

T(ct)

{{ese

0 T(cz)

H(s,e-.(1+c0(c4),"O)ds

+ 1Cesa(1+co(ct)'"°)

L

h(ceps(1+Co(cr)m0),ce-1e(1+co(Ce)m°))J

+ +00

x e-fs(1+G)(cz)m°)ds = c 1 + Ehp+l,p(cz)P P=1

and `T(cC) J/

{e

0

+ H(s, , ) }

+oo e'8(1+co(cc)"° )ds

1 + t he+1,P(cz)p P=1

since the quantity hP+1,P are real (cf. (VIII.6.4)). This proves that

C(Cr/

) is conver-

C(l, 7)

gent. Hence, c(l;, ) and 4) are convergent. Thus, the proof of the convergence of A g) in Case II is completed. Thus, it was proved that if all of the coefficients w,,, on the right-hand side of system (VIII.6.3) are real, the point (0,0) is a center of system (VIII.6.1). The general solution of (VIII.6.1) can be constructed by using various transformations of (VII.6.1) which bring the system to either (VIII.6.3.1) or (VIII.6.3.2). Periods of solutions in t are independent of each solution in Case I, but depend on each solution in Case H.

7. PERTURBATIONS OF AN IMPROPER NODE AND A SADDLE PT. 261

Remark VIII-6-5. The argument given above does not apply to the case when the right-hand side of (VIII.6.1) is of C(O°) but not analytic. A counterexample is given by _

2

[h(1)

dt

where h(r2) = e-1jr2 sin I

r

h(r)

r2 = Y1 + y2+

y'

I. Using the polar coordinates (r,O), this system is

changed to dt = rh(r2)

and

and periodic solutions are given by r2 =

1 MR

d with any integers nt. This is an

example of centers in the sense of Bendixson (cf. [Ben2, p. 26] and §VIII-10). For more information concerning analytic systems in JR2, see [Huk5].

VIII-7. Perturbations of an improper node and a saddle point In the previous section, the structure of solutions of an analytic system in JR2 was showen by means of the method with power series. Hereafter, in this chapter, we consider a system dy _

(VIII.7.1)

dt

Ay + §(y-),

under the assumption that (i) A is a real constant 2 x 2 matrix, (ii) the entries of the R2-valued function §(y) are continuous in g E JR2 near 0' and satisfies the condition lim 9(y)

(VIII.7.2)

y-o Iy1

= 0.

In this section, it is also assumed that the initial-value problem (VIII.7.3)

df = Ay + §(y-),

g(0) = g

has one and only one solution as long as g belongs to a sufficiently small neighborhood of 0.

Remark VIII-7-1. In the case, when the entries of an 1R2-valued function f (Z-) are continuously differentiable with respect to the entries z1 and z2 of i in a neigh-

borhood of i = 0 and f (0) = 0, write the differential equation dt = f(E) in form (VIII.7.1) by setting A

=

of (o) i9

aft 0z' Lof2 8x1

(o)

eft 19x2

(o) and

aft az2

( )

g(y)=f(y)-Ay.

VIII. STABILITY

262

Using thepolar coordinates (r, 0) in the g-plane, write the vector g in the form

y=r

[c?}. Then, system (VIII.7.1) is written in terms of (r, B) as follows: d=r[a, i cos2(B) + a22 sin2(9) + (a12 + a21) sin(g) cos(9)] + 91(-1 COs(0) + 92() sin(g),

(VIII.7.4)

r dte

=r[-a12 sin2(0) + a21 COS2(0) + (a22 - all) sin(e) cos(0)] - 91(y-) sin(g) + 92 M cos(e),

all a12 J and g"(y) = [iW)l Here, use was made of the formula

where A =

92 M

22

a21

dr =

dy1

dt

dt

r de

cos (9) + dye sin(6),

dt

= -dy1 sin(g) + dye cos(9). dt

dt

dt

In this section, we consider the case when the two eigenvalues Ai and A2 of the

matrix A are real, distinct, and at least one of them is negative, i.e., A2 > Al and Ai < 0. Throughout this section, assume that all = Al, a22 = A2, and a12 = a21 = 0. Then, in terms of the polar coordinates (r, 0), system (VIII.7.1) can be writen in the form (VIII.7.5)

dt = d9

1

r [Ai(cos(8))2 + A2(sin (e))2 +

(e) + g2(g)sin (e))]

= (A2 - Ai )sin (0) cos (0) + r

(e) + 92(y-)cos (e))

Observation VIII-7-2. The point (0,0) is not a proper node of (VIII.7.1). In fact, if (r(t), 9(t)) is a solution of (VIII.7.5) such that r(t) 0 and e(t) --+ w as

t - +oo or -oo, then

d8 --* (A2 - AI) sin w cos w as t -+ +oo or -oo. Hence,

sin w cos w = 0. This implies that w = -7r, -

7r,

0, or

2

Observation VIII-7-3. For any given positive number e > 0, there exists another positive number p(e) > 0 such that

de dt

>0

for

- 7r + e < 0 < - 2 - e,

0

for

-2

for

e AI, find a real number wo such that 0 < wo < 2 and tanwo =

-1. Then, for any given positive number 2

E > 0, there exists another positive number p(E) > 0 such that

>0 dr dt

0 0, lim r(t) = 0, and lim 9(t) = 0. (cf. Figure 20). t-+oo t-+oo (I +E)Yj

.V

(I +e)Yi

FIGURE 19.

FIGURE 20.

Observation VIII-7-10. If we assume some condition on smoothness of §(y-), there is only one orbit of (VIII.7. 1) that tends to the point (0, 0) in the direction

9 = 0 as t - +oc. Theorem VIII-7-11. Assume that all = A1, a22 = A2, and a12 = a2I = 0, and that

(1) A2 > \1 and Al < 0, (2) g(j) is continuous in a neighborhood of

(3)y-.o lim " =

0,

lye

(4)

a9(y

exists and is continuous in a neighborhood of 6.

2

Then, there exists a unique orbit y(t) _ [01(1)1 such that 02(t)

01(t) > 0

for

urn 01(t) = 0,

i t-.+oo

t > 0,

lim 02(t) = 0,

b2(t)

= 0.

lim t--+co 01(t)

Proof.

The existence of such orbit is seen in previous observations. To show the uniqueness of such orbit, rewrite (VIII.7.1) in the form I

(VIII.7.8)

dye = 1\2Y2 + 92(Y) _ dy1 AIyI+91(91

All yz + a y192(y - (Aty1)291(Y) 1+ 1 91()J)

\Y1

AIYI

Next, introduce a new unknown u by y2 = y1u 1or u =

y2 yI.

Then, (VIII.7.8)

becomes JI2u

1

( VIII.7.9 )

y1du + u dy1

= (L2) u+

1y192(YIU) - \I-91(y1u) 1

1 +

Iy1gI(y1u) '\

VIII. STABILITY

266

where u =

[:1].

Observe that (a) 161 = 1 if Jul < 1, (b) -L (d) = U

g(0, y2)

-0 Y2

= d, and (c)

- 1 < 0. i

Write system (VIII.7.9) in the form

l

- 1 I u + G(yi.u).

Q 1

G(yi, u) =

-

A2u

92(yiu) - a

91(yl U

1

1 + -9i(y1iZ) aiyi

ac

It is easy to show that lim 5:u (y1, u) = 0 uniformly on Jul < 1. Hence, there exists

v --o

a non-negative valued function K(yj) such that

!G(yi, u) - G(y1, v)! < K(yi )lu - vl whenever Jul < 1, !v! < 1 and !y1! is sufficiently small. Furthermore, slim K(yt)

=0. Let u = 01(yi) and u = ip2(yi) be two solutions of (VIII.7.9) such that (1) ti i (yi) and .'2(y1) are defined for 0 < yi < r) for some sufficiently small

positive number 'I, (2)

lim i'1(yi) = 0 and

vl

o+

lim i2(yl) = 0.

v,

Set '(yi) = 01 (y1) - s (yi ). By virtue of uniqueness, assume without any loss of generality that y(yi) > 0 for 0 < yi < ri. Then, for a sufficiently small positive r?, d y1 -)tt'(yi) for 0 < y1 < i , where 7 = 2 1 I < 0. This implies that J

d [y1 "V,(yi)j < 0 for 0 < yi < rj. Hence, 0 < y1 lim yi 'W(yi) = 0 for Y:- 0 dyi 0 < y1 < 17. Therefore, ;1(yi) = 0 for 0 < y1 < r). The materials of this section are also found in (CL, §§5 and 6 of Chapter 151.

VIII-8. Perturbations of a proper node In this section, we consider a system of the form (VIII.8.1)

dt

= All +

where y" E R2,

under the assumptions that (i) A is a negative number, (ii) the entries of the R2-valued function g(y} are continuous in a neighborhood of 0,

(iii) 1§(y -)l < K!y"l1+i in a neighborhood of 6, where K is a nor-negative number and v is a positive number. The main result is the following theorem.

267

8. PERTURBATIONS OF A PROPER NODE

Theorem VIII-8-1. Under the assumption given above, the point (0, 0) is a stable proper node of system (VIII.8.1) as t --, +oo. Before we prove this theorem, we illustrate the general situation by an example.

Example VIII-8-2. Look at the system

dt = -y + 9(yj,

(VIII.8.2)

9(y _ (yi + y2)

[

y

where y E R2 with the entries yl and y2 It is known that the point (0,0) is a stable proper node of the linear system dt = -y. To find the general solution of (VIII.8.2), change (VIII.8.2) to (VIII.8.3)

by the transformation y' = e-'Z. Next, introduce a new independent variable s by s = 2e-2t. Then, system (VIII.8.3) becomes (VIII.8.4)

ds

-g(z)

Observe that

If a solution y(t) of system (VIII.8.2) satisfies an initial condition

9(o) = n =

(VIII.8.5)

171 ?12

'

the corresponding solution z(s) = et y'(t) of system (VIII.8.4) is the solution of the l initial-value problem ds = r(7)2 f zz2

z" C 2

r(i) =

(VIII.8.6)

/

= rj, where

(n,)2 + (m)2.

Hence,

zl(s) = r(ir)sin(r(i)2s + 6(rl)),

z2(s) = r(i)7)cos(r(1-7)2s + 0(r1)),

where

z, (0) = r(rl sin

{ Z2(0) = r(n7) cos

2

111 = r(i) sin(

in = r( n

oos

(

0(rl 2

VIII. STABILITY

268

(cf. Figure 21). The general solution of (VIII.8.2) is 2

yI(t) = e-tr(7f)sin (L(7 a-2t y2(0 = e-tr(771 cos

(!2e_2t

where y(0) and r(fl) are given by (VIII.8.5) and (VIII.8.6), respectively. Every orbit of (VIII.8.2) goes around the point (0, 0) only a finite number of times. Figure 22 shows that the point (0, 0) is a stable proper node as t -+ +oo.

FIGURE 22.

FIGURE 21.

Remark VIII-8-3. If condition (iii) of the assumption given above is relaxed, the point (0, 0) may become a stable spiral point (cf. Exercise VIII-10). Proof of Theorem VIII-8-1.

We prove Theorem VIII-8-1 in two steps. To start with, using the polar coordinates, write system (VIII.8.1) in the form

dr (VIII.8.7)

J

dt

= Ar + gI (y) cos(9) + g2(y) sin(g),

dO

r t = -g1(y) sin(g) + g2(y COs(9) (VIII.7.4)). By virtue of the assumption given above, there exists a positive number ro such that (cf.

(VIII.8.8)

2Ar < Ar + gi (y-)cos(9) + g2(y) sin(g)
0, the variable t can be regarded as a function of r, i.e., t = t(r), where t(p) = 0 and lim t(r) = +oo. Use r as the independent variable. Then, (VIII.8.7) becomes ( V1II . 8 . 10 )

d9

Tr

F (r, 0 ) _

-gi (y-)sin(g) + g2(Y)cas(e) r[Ar + g1(y) cos(0) + g2(y) sin(0)J

Let 0 = 4D(r) = 0(t(r)) (0 < r < ro) be the solution of (VIII.8.10) satisfying the initial condition (b(p) = w. Then,

fi(r) = w +

JP

(0 < r < ro).

r F(s, I(s))ds

Estimate (VIII.8.8) implies that 2Ar2

I - gt (y sin(0) + 92(Y) cos(0) I

0 < r < ro.

for

to

J

F(s, $(s))ds =

P

r(t)

ar2

(I g1(y)I + Ig2(y1I)

Now, by virtue of condition (iii), the improper integral lim r r-.0 JP F(s, 4(s))ds exists. Observe that 0(t) = 4(r(t)) = w +

F(s, (s))ds and that P

F(r, 0)
0, is decreasing,

and tends to 0 as t -4 +oo. Set 0(t) = $(r(t)). Then, it is easily shown that (r(t), B(t)) is a solution of (VIII.8.7) such that limo 0(t) = 1i mt(r) = c. t

+0

0

The materials of this section are also found in (CL, §3 of Chapter 151.

VIII-9. Perturbation of a spiral point In this section, we show that (0,0) is a stable spiral point of (VIII.7.1) as t -, +oo

if (0, 0) is a stable spiral point of the linear system the following theorem.

= Ay". The main result is

Theorem VIII-9-1. If (i) two eigenvalues of A are not real and their real parts are negative, (ii) the entries of the L2-valued function ff(y-) is continuous in a neighborhood of the point (0,0),

(iii) lira MI = 6,

g-alyl

then the point (0, 0) is a stable spiral point of (VIII. 7.1) as t

+oo.

Proof.

Assume that (VIII.9.1)

all = a22 = a < 0

and

a12 = -a21 = b 0 0.

10. PERTURBATIONS OF A CENTER

271

Then, from system (VIII.7.4), it follows that dr

ar + 91(yjcos(9) + 92(ylsin(g),

dt

rt

= -br - 91(y-) sin(g) + g2(Y)cos(0). 1at

Therefore, a positive number ro can be fouind so that r(t) < r(0) exp

J for t > 0

if 0 < r(0) < ro. This, in turn, implies that lim r(t) = 0 and that dt < t -+oo

if

2 b > 0, while de > - 2 if b < 0, whenever 0 < r(0) < ro. Therefore, lim00(t) = -oo if b > 0 and lim 0(t) = +oo if b < 0, whenever 0 < r(0) < ro. Thus, we conclude t too that (0, 0) is a stable spiral point of system (VIII.7.1) as t -+ +oo. 0

The materials of this section are also found in fCL, §3 of Chapter 151-

VIII-10. Perturbation of a center We still consider a system (VIII.7.1) under the assumption that the entries of the RI-valued function §(y) is continuous in a neighborhood of 0 and satisfies condition (VIII.7.2). In this section, we show that if the 2 x 2 matrix A has two purely

imaginary eigenvalues Al = ib and A2 = -ib, where b is a nonzero real number,

then the point (0,0) is either a center or a spiral point of

(VIAssume

without any loss of generality that the matrix A has the form

[°b

(VIII.7.4) becomes

and system

b J

_ (0) , dt = 91Mcos (0) + 92(y-)sin

dr

(VIII . 10 . 1)

dB dt

= -b + r 1

(B) +

92(y')cos (B)).

Observe that system (VI I I.10.1) can be written in the form (VIII.10.2)

dr = d8

91(y')cos (0) + 92(y-)sin (0)

-b + r (-91(yjsin (0) + g2(y-')cos (0))

If a positive number p is sufficiently small, the solution r(0, p) of (VIII.10.2) satisfying the initial condition r(0, p) = p exists for 0 < 0 < 27r.

Case 1. If there exists a sequence (pm : rn = 1, 2, ...) of positive numbers such that lim p,,, m-.+00

=0

and r(2xr, Pm) = Pm

(m. = 1, 2, ... ),

then the point (0,0) is a center of (VIII.7.1) in the sense of Bendixson (Bent, p. 261 (cf. Figure 23).

VIII. STABILITY

272

Case 2. Assume that b < 0. If there exists a positive number po such that

r(2ir, p) > p

(respectively < p)

whenever 0 < p < po, then the point (0, 0) is an unstable (respectively stable) spiral point as t - +oo (cf. Figures 24-1 and 24-2).

FIGURE 24-1.

FIGURE 23.

FIGURE 24-2.

Example VIII-10-1. The point (0,0) is a center of the system d

[ yuj

dt [y2J

since

dr d8

+

yi +y2 I

yy1J

= 0.

Example VIII-10-2. The point (0, 0) is an unstable spiral point of the system d

dt

= [ Y2

l

Y2

l

-Yi J

+ ` y 2I + Y22 yI

Y2

as t - -oc, since dr = r2, which has the solution r = 1 r(0) = dt ro - t r

ro. (Note: t < ro.)

+oo when t

1

ro

and

Example VIII-10-3. For the system

dt

= - y,

d y = x + x2 - xy +

cry2,

the point (0, 0) is (a) a stable spiral point if a < -1, (b) a center if a = -1, and (c) an unstable spiral point if a > -1. Proof.

Use the polar coordinates (r, 0) for (x, y) to write the given system in the form

(1+rcos8(cos20-cos0sin8+asin28))

= r2sin8(cos28-cos0sin8+asin20).

273

10. PERTURBATIONS OF A CENTER

Setting

too

r(9, c) _ E r,,, (9)c"', where r(0, c) = c, m=1

and comparing r(2r, c) and c for sufficiently small positive c, it can be shown that

r1(t) = 1,

r3(2r) =

r2(2r) = 0,

(1 + Or 4

Thus, r(27r, c) < c if a < -1. Therefore, (a) follows. Similarly, r(2r, c) > c if a > -1. Therefore, (c) follows. x2 - xy - y2 = (x - +2 ) (x - (1-2 ) . Set In the case when a

W = - (1 X - (1 +

Then. a =

,/)y

2

and v = x - (1 du

Therefore, changing x and y by u =

(1

)y, the given system is changed to

2

dv

Wv(1 + u),

= ---u(1 + v).

dt

Hence, on any solution curve, the function W2(v

- ln(1 + v)) + (u - ln(1 + v)) =

u2

t2W2v2

+

is independent of t. This implies that in the neighborhood of (0, 0), orbits are closed

curves. This shows that (0, 0) is a center. 0

Example VIII-10-4. The point (0,0) is a center of the system dy dt

= bAy' +

9(y),

y= y'J y2

if (1) b is a nonzero real number, (2) A = 101

"

1J, (3) the entries of the R2-

valued function g"(yj is continuous and continuously differentiable in a neighborhood

of 0, (4) lim

g-6 1Y1

= 6, and (5) there exists a function M(y) such that M is positive

valued, continuous, and continuously differentiable in a neighborhood of 0 and that 8(Mf1)(y) + 8(M 2)(y..) = 0 in a neighborhood of 0, where f, (y-) and f2 (y-) are

the entries of the vector MY + §(y).

Proof The point (0, 0) is either a center or a spiral point. Look at the system dt

=bAy + 9(y)

and

7t- = M(y-

VIII. STABILITY

274

Using s as the independent variable, change the given system to ds = M(y)(bAy+ g"(y)). Upon applying Theorem VIII-1-7, we conclude that (0, 0) is not asymptotically stable as t - ±oo. Hence, (0, 0) is a center. The materials of this section are also found in [CL, §4 of Chapter 15] and [Sai, §21 of Chapter 3, pp. 89-1001.

EXERCISES VIII

VIII-I. For each of the following five matrices A, find a phase portrait of orbits of the system

d:i

= AY.

I 14

1]'

[ 11

[ 15 11]'

31 '

5

1

[1

5

VIII-2. Find all nontrivial solutions (x, y) = (d(t), P (t)). if any, of the system

-xy2, _ -x4y(1 + y) dt = satisfying the condition lim (th(t),tb(t)) = (0,0). c-.+00 VIII-3. Let f (x, y) and g(x, y) be continuously differentiable functions of (x, y) such that (a) f (0, 0) = 0 and g(0, 0) = 0,

(b) (f(x,y),g(x,y)) # (0,0) if (x,y) 0 (0,0), (c) f and g are homogeneous of degree m in (x, y), i.e., f (rx, ry) = rm f (x, y) and g(rx, ry) = rmg(x, y). Let (r, 8) be polar coordinates of (x, y), i.e., x = r cos 8 and y = r sin 0. Set F(8) = f (cos 0, sin 8) and G(O) = g(cos 8, sin 8).

(I) Show that

dr dt

(S)

= rm(F(8) cos 8 + G(8) sin 8),

d8

= rm-1(-F(8)sin8 + G(O)cosO).

dt

(II) Using system (S), discuss the stability property of the trivial solution of each of the following three systems: (i)

4ii

= x2

-

y2

= x3 (x2 + y2) - 2x(x2 + y2)2,

(iii)

d

= x4 -

6x2y2 + y4,

dt

= 2xy; = -y3(x2 + y2);

dy = 4x3y - 4xy3.

EXERCISES VIII

275

VIII-4. Let J be the 2n x 2n matrix defined by (IV.5.2) and let H be a real constant 2n x 2n symmetric matrix. Show that the trivial solution of the Hamiltonian system

a = JHy is not asymptotically stable as t

+oo.

V11I-5. Show that the trivial solution of the system dy'

d = Ay" + 9(t, y-) +oo if the following conditions are satisfied: is asymptotically stable as t (i) A is a real constant n x it matrix, (ii) the real part of every eigenvalue of A is negative, (iii) the entries of the IRI-valued function g(t, y-) are continuous in the region

0(ro) = To x D(ro) = {(t, yl : 0 < t < +oo, Iyj < ro) for some ro, (iv) g(t, yj satisfies the estimate 19(t, y)

1

eo(t, y')Iyj

_
- 0. t+0

Hence, if r(0) is small, r(t) is bounded by r(0) and This implies that if 1#(0)I is small, g(t) is bounded

and lim y(t) = 0'. Look at

t-+o

y(t) = eAtetN {(o) +

-sN9(y($))d$]

VIII. STABILITY

276

If we set y(t) = e' e'Nu, we obtain

r e-aae-aN9(g(s))ds.

d(t) = y-(0) +

0

Choose e > 0 so that 1 - e >

Then, condition (4) implies that

1

+ 6(+

+00

= 9(0) +

e-aae_a'

0

exists. Now,

(1) if d(+oo) = 0, then d(t) = 0 identically, since, in this case,

d(t) =

f

e-X$e-ajV9(y(s))ds;

+a

0 0, then t line-a`y"(t) = i"(+00);

(2) if t7(+oo) =

02

(3) if iZ(+oc) = RZJ with 2 # 0, then limo ( e _) L

Hence, the point (0, 0) is a stable improper node of the given system.

VIII-7. Determine whether the point (0,0) is a center or a spiral point of the system

dt

= -y,

dt = 2x + r3 - x2(2 - x)y.

VIII-8. Show that the point (0, 0) is the center of the system

_ -x + xy2 - yg.

+ xy3 - y7,

dt = y

Hint. This system does not change even if (t, y) is replaced by (-t, -y).

VIII-9. For the system dty = 27x + 5y(x2 + y2),

3y + 5x(x2 + y2), dt =

find an approximation for the orbit(s) approaching (0,0) as t - +oo. VIII-IO. Show that the point (0, 0) is a stable spiral point of the system

as t-++oc.

d

[y,]

dt

y2

[yj

+

1

- yIn yi + y2 --

[

y2

yi

277

EXERCISES VIII

Hint. If we set yl = r cos 0 and y2 = r sin 0, the given system becomes

dr _

dO

-r'

dt

1

dt = In r'

VIII-11. Suppose that a solution ¢(t) of a system dy _

dt

AJ + §(y-)

satisfies the condition

lim t-r 4(t) = 0 for a positive number r. Show that if (1) A is an n x n constant matrix, (2) the n-dimensional vector W(if) is continuous in a neighborhood of 0, (3) lien 9(Y) = 0, V-6 !y1

then ¢(t) = 6 for all values oft. (Note that the uniqueness of solutions of initial-value problems is not assumed.) VIII-12. Assume that (i) AI.... .. 1n are complex numbers that are in the interior of a half-plane in the complex A-plane whose boundary contains A = 0,

(ii) there are no relations \, = plat + P2 1\2 + + pnan for j = 1, ... , n and non-negative integers p,,... , pn such that pi + - - + pn >_ 2,

(iii) f(y' =

yam' ft, is a convergent power series in ff E Un with coefficients jp1>2

fpEC'. Show that there exists a convergent power series ((u) _

ul'Qp with coefficients Inl>2

_

dg

Qp E Cn such that the transformation if = u" + Q(ur) changes the system - _ A#+ f (y-) to

j

dt

= Au, where A = diag[.1t, A2,4, ... , A ].

VIII-13. Show that the trivial solution of the system dx

- -st(x +X22) + x2ezl+ 2,

is aymptotically stable as t

dr

dt

-22(22 + 22) _ Xlez'+ 2

+oo. Sketch a phase portrait of the orbits of this

system.

VIII-14. In Observation VIII-6-2, it is stated that if one of the then y" is a spiral point. Verify this statement.

VIII-15. Discuss the stability of the trivial solution of the system

j1 = x21 as t - +00.

722 = xi(I - x1)

is not real,

VIII. STABILITY

278

VIII-16. Find the general solution of the system

= 223 + 2j + X122 + 22,

= 3x4 + explicitly.

+ X122 t 2122 + 21X3 + x223,

CHAPTER IX

AUTONOMOUS SYSTEMS

In this chapter, we explain the behavior of solutions of an autonomous system dg = f (y). We look at solution curves in the y-space rather than the (t, y-)-space. dt Such curves are called orbits of the given system. In general, an orbit does not tend to a limit point as t -+ +oo. However, a bounded orbit accumulates to a set as t -4 +oo. Such a set is called a limit-invariant set. In §IX-1, we explain the basic properties of limit-invariant sets. In §IX-2, using the Liapounoff functions, we explain how to locate limit-invariant sets. The main tool is a theorem due to J. LaSalle and S. Lefschetz [LaL, Chapter 2, §13, pp. 56-71] (cf. Theorem IX-2-1). The topic of §IX-4 is the Poincare-Bendixson theorem which characterizes limitsets in the plane (cf. Theorem IX-4-1). In §IX-3, orbital stability and orbitally asymptotic stability are explained. In §IX-5, we explain how to use the indices of the Jordan curves to the study of autonomous systems in the plane. Most of topics discussed in this chapter are also in [CL, Chapters 13 and 16], [Har2, Chapter 7], and [SC, Chapter 4, pp. 159-171].

IX-1. Limit-invariant sets In this chapter, we explain behavior of solutions of a system of differential equations of the form

dg

(IX.1.1)

fW

where y" E Ill;" and the entries of the RI-valued function f (y) are continuous in the entire y-space II8". We also assume that every initial-value problem (IX.1.2)

dy

dt

= Ay),

y(0) = if

has a unique solution y = p(t, y7). System (IX.1.1) is called an autonomous sys-

tem since the right-hand side f (y) does not depend on the independent variable t.

Observe that p(t + r, r) is also a solution of (IX.1.1) for every real number T. Furthermore, At + r, rl) = p(r, il) at t = 0. Hence, uniqueness of the solution of initial-value problem (IX.1.2) implies that p(t +r, rt) = p(t, p(-r, rt)) whenever both sides are defined. For each il, let T(rl) be the maximal t-interval on which the solution p(t, r) is defined. Set C(rt) = {y" = p(t, rl) : t E Z(rl)}. The curve C(rl) is called the orbit passing through the point il. Two orbits C(iji) and C(rj2) do not intersect unless they are identical as a curve. In fact, (IX.1.3)

C(ijl) = C(i@

if and only if 7)2 E C(il, ). 279

IX. AUTONOMOUS SYSTEMS

280

If f (n) = 0, the point ij is called a stationary point. If , is a stationary point, the Generalizing property (IX.1.3) of consists of a point, i.e., C(17) = orbit orbits, we introduce the concept of invariant sets which play a central role in the study of autonomous system (IX.1.1).

Definition IX-1-1. A set M C R is said to be invariant if i E M implies C(n) C M. For example, every orbit is an invariant set.

Remark IX-1-2. If M1 and M2 are invariant sets, then M1 UM2 and M1 f1M2 are also invariant. For a given set f2 C 1R", let Ma (A E A, an index set) be all invariant subsets of S2, then U M>, is the largest invariant subset of Q. AEA

Hereafter, assume that every solution p(t, nJ) is defined for t > 0, i.e., (t : t > 0) C Z(n). In general, lim Pit, nom) may not exist. However, if p(t, n) is bounded for t > 0, the orbit C(171 accumulates to a set as t -+ +oo. This set is very important in the study of behavior of C(n) as t -+ +oo.

Definition IX-1-3. Let

C+(i7-7)

denote the set of all

y"

E

II8"

such that

lim p(tk, n ) = y f o r some increasing sequence {tk : k = 1, 2, ... } of real numbers

k-++oo

lim tk = +oo. The set C+(n) is called the limit-invariant set for the

such that

k-+oo initial point r7.

The basic properties of

are given in the following theorem.

Theorem IX-1-4. If p(t, n) is bounded fort > 0, then C+(n) is nonempty, bounded, closed, connected, and invariant. Proof.

(1) C+ (1-71 is nonempty: In fact, let {sk : k = 1,2,... } be an increasing sequence of real numbers such that lim sk = +oo. Since p(t, n) is bounded for t > 0, k-.+oo

there exists a subsequence {tk : k = 1, 2.... } such that lim tk = +oo and that

lim p(tk,

k-.+oo

exists. This limit belongs to f_+ (17).

(2) The boundedness of C+(n7) follows from the boundedness of p(t,i7) immediately. (3)

is closed: To prove this, suppose that

lim yk = y for 9k E L+(1-7).

k-+oo it follows that ilk = It must be shown that y" E L+(771. Since g k E lira p(tk,t, r 7 ) for some {tk,t : I = 1, 2, ... } such that lim tk,l = +00. Choose tk t-+oo t-.+oo

so that lim tk,t,, = +oo and 19k - P(tk,t,,,'Th !5 ! Then, since ly - P(tk,t4, n)I k-.+oo k

+ ly - ykl, we obtain lim oP(tk,l,,, n) = y E C+(1_7) k

(4) C+(777) is connected: Otherwise, there must be two nonempty, bounded, and closed sets Sl and S2 in R" such that

281

2. LIAPOUNOFF'S DIRECT METHOD

(1) S1 n S2 = 0, (2) S1 U S2

L+(rf)

yl E S1, y2 E S2}. Note that d > 1. Then, So is not empty, bounded, g: distance(y, SI) =

Set d = distance(S{{l, S2) = min{lyl - y21

:

0.

Set also So =

and closed. Furthermore, So n C+(i) =20. Choose two points y"I E SI and y ' 2 E S 2 and t w o sequences {tk : k = 1, 2, ... } and {sk : k = 1,2,...) of

lim tk = +oo, lim sk = +oo, real numbers so that tk < Sk (k = 1, 2, ... ), k-+oo k-+oo r), SO < y"1, and k lim G p1sk, r7) = 92. Assume that limp P(tk, k

Then, there exists a Tk for each k such that 2 and distance(p(sk, r), SI) > tk < Tk < sk and p-(Tk, rl) E So. Choose a subsequence (at : e = 1,2.... ) of {rk : k = 1, 2, ... } so that lim at = +oo and lim p(at, r) = # exist. Then, 2.

Y E So n C+ (Y-)) = 0. This is a contradiction. (5) C+ (1-7) is invariant: It must be shown that if ff E C+ (r), then p1t, y) E L+(1-7) f o r all t E 11(y). In fact, there exists a sequence {tk : k = 1, 2, ... } of real numbers such that Iii o tk = +oo and k lim o P-14, r) = 17. From (IX.1.1) and the continuity k

of P(t,y) of y", it follows that for each fixed t. k

li

+co

tk, n) =

k

UM P(t, Pptk, r1)) = Pit, y) E L+(7).

El

The materials of this section are also found in [CL, Chapter 16, §1, pp. 389-3911 and [Har2, Chapter VII, §1, pp. 144-1461.

IX-2. Liapounoff's direct method In order to find the behavior of pit, r)) as t +oo, it is important to locate L+(rl). As a matter of fact, if p(t, r)) is bounded for t _> 0 and if a set M contains ,C+ (1-7), then p(t. r)) tends to M as t -+ +oo, i.e., lim inf(1p(t, r)) - yj : y E M) = 0. t +a: Otherwise, there must be a positive number co and a sequence {tk . k = 1, 2.... } of real numbers such that lim tk = +00, limo P140 7) exists, and ipjtk, >)) -y"1 eo k k for all y E Jul. Hence, lim -1) V M. This is a contradiction. Keeping this k-.too fact in mind, let us prove the following theorem (cf. [LaL, Chapter 2, §13, pp. 56-71]).

Theorem IX-2-1. Let V(y) be a real-valued, continuous, and continuously differentiable function for Jyl < ro, where 0 < ro < +oo. Set

fDt = J#: V (y-) < e}, St = {y" E DI : V-(y) J(y) = 0}. Mt = the largest invariant set in St, yl fl( F'

a`,

where Vi(y) ' f(y) _ ayJ fJ(y), y J-1

, and f( y) _ yn

.

fn (y)

Suppose

IX. AUTONOMOUS SYSTEMS

282

that there exist a mat number t and a positive number r such that 0 < r < ro, Dt C {y : Iy1 < r}, and Vg(y-) f (y) < 0 on De. Then, L+ (1-7) C Me for all n E Vt. Proof.

Set u(t) = V(p(t,rl")). Then,

dot)

= V9(r(t,r7))'

dtp(t,i) =

du(t) < 0 as long as p(t, t) E Vt. This implies that u(t) < V(1-7) < e dt for r) E Vt as long as p(t, n-) E Vt. Hence, p(t, rl E Dt for t > 0 if ij E Vt. Consequently, Ip(t, rlI < r for t > 0. It is known that C+ (171 is nonempty, bounded, C Vt. closed, connected, and invariant (cf. Theorem IX-1-4). Furthermore, d t) < 0 if Let po be the minimum of V(rl') for Iy7 < r. Then, po < u(t) and it E 1)t. Hence, lim u(t) = uo exists and no > po. This implies that V(y) = uo Therefore,

for all y E C+(t)). Since C+(71 is invariant, V(p(t, y)) = uo for all y E L+(171 and

t > 0. Therefore, VV(p(t, y)) &I t, y)) = 0 for all y E C+ (q) and t > 0. Setting C St if i E Vt. Hence, t = 0, we obtain Vc(y) f (y) = 0 if y E C+(rl, i.e., C+() C Mt for if E Vt. The following theorem is useful in many situations and it can be proved in a way similar to the proof of Theorem IX-2-1.

Theorem IX-2-2. Let V (y) be a real-valued and continuously differentiable function for ally E lR' such that VV(y) f (y) < 0 for all y" E R". Assume also that the

orbit p(t, rte) is bounded fort > 0. Set S = 1g: Vi(y) f(y) = 0) and let M be the largest invariant set in S. Then, L+ (7-1) C M.

The proof of this theorem is left to the reader as an exercise. In order to use Theorem IX-2-2, the boundedness of p(t, advance. To do this, the following theorem is useful.

must be shown in

Theorem IX-2-3. If V (y) is a real-valued and continuously differentiable function for all y" E lR^ such that VV(y) f(y) < 0 for Iyi > ro, where ro is a positive number, and that lim V (y) = +oo, then all solutions p(t, r)7) are bounded for t > 0. 1Q1+00

Proof.

It suffices to consider the case when p(to,rl > ro for some to > 0. There are two possibilities: (1) IP-(t,r1)I > ro fort > to,

(2) IP(t,n')I > ro for to < t < t, and Ip1ti,17)1 = ro for some tl > to. Case (1). In this case, dt V (p( t, r)1) = Vj(p(t, rl) f (p(t, 4-7)) < 0 fort > 0. Therefore, V (r(t,

V (r(to, t))) for t > to. Hence, p(t, tt") is bounded for t > 0.

Case (2). In this case, it can be shown that V(p(t, 71)) 0

(IX.3.16)

and

1t1 < 60, where ko and bo are suitable positive numbers. First fix three positive numbers ko, ro, and 6o so that Ico > 2, kobo < ro, and K (r) < 2 for 0 < r < ro. Then, from (IX.3.5), (1X.3.10), (IX.3.11), (IX.3.14), and a (IX.3.15), it follows that + t

(I)

< K(ko[

t9i (s,+u(s,&ds

JL0

K(a

itt

e-O°ds

If1)

and

jexE(t -

exp (tBJ ( +

e-2otlrj + S

t f eo"ds 0

Se

of

I1+

Next, set lIJt ({) = sup

K(

(eot 1

(1 +

J+9i(s,t,ii(s,())ds

< K(ko

< ko1 1e-OL

(t, 4)1 : t > 0) if the entries of an R"-valued func-

tion t(i(t, £) are continuous for (IX.3.16) and Then,

I

)

-

l}} < kolle-ot for (1X.3.16).

ft 9i(s,tG2(s,&ds l +oo

at

< 2IItLI -

1211(&-ot

287

3. ORBITAL STABILITY and

f exp[(t-s)B]92(s,))ds t

f+ exp((t-s)B](s,(IV)

- 2[[i1 -


im(t,

m --+oc

uniformly for (1X.3.16) and the limit %i(t, { is a solution of integral equations (1X.3.14) satisfying condition (IX.3.15) for (IX.3.16). Note that i(O,t;) 0

where a(t) = f gl (s, z/i(s, })ds. This implies that 00

Step 5. Set

00,6 = Then, (IX.3.18)

Plt,vio)

is a solution of system (IX.3.1) such that

At, f) - pjt, i o)[ 5 Ko{f [e-°`

for

(IX.3.16),

where Ko is a positive constant. Define an (n - 1)-dimensional manifold M by

M = (g=4(0'6: 01io)) +

This yields (IX.3.20) df11

Wdue

and

dT

42(r) +

y' dt

2(T) f(y(t))dr dt

-

(

2(t)'

((t)

due _ dT 93(r)

+ u1

42(T) 1 dr l u1 dQ2(T)\

dr

dge(r)

+ (f(p1r,TIo))

d9d(T)) uz]

dT

((t)

d93(r)1

u1 - 93(t)

dg3(r))

= [f(PIT'770)) f(PIT,, )) + (fcvlr,no)) (IX.3.21)

+ u2 dQ3(r)

d-,

dT J u2, dr

u1

x [f(Ar, *lo}).N(t))} -', f(

where a" b denotes the usual dot product and we assumed that

q"1(r} = 1 (j =

2, 3).

Note that

AM)) = RAT, 7lo))+ul

89

(PIT,r, ))42(r)+

u2

f(YlT,r/o))93(T)+O(ItII +Iu2I) 09

Hence, from(IX.3.20) and (IX.3.21), we derive dt 77.

= 1 + 00U11 + Iu21)

IX. AUTONOMOUS SYSTEMS

290

and (IX.3.22) dul

dr

42(r) '

- (o) d7-

93(T) '

ul + [i(T).

l

.

_d_r)

W

Jut -

(0)

(FFIr, 1I0))42(r) 1

d9drr)) -

ul +

(r))1 u2 + O(lul l + 1U21),

[i(r).

(i(t). ddrr)) ul - (t(t). d (r)

U2

r, 90))93(r) 11 U2

J U2 + O(lulI + 1U21)

Let Q(t) be the 3 x 3 matrix whose column vectors are (at, qo)), fi(t), and ,fi(t), i.e., Q(t) = [ f (p1t, rlo)) 6(t) q"3(t)! . The transformation w = Q(t)v changes the linear system dtr, (S2)

_

L(PIt,rl *6

to

dv'

=

0

/31(t)

32(t)

0

atl(t) a12(t)

0

a21(t)

16,

a22(t)

Using these notations, write (IX.3.22) in the form dul d7-

(IX.3.23) 1

= all(r)ul + a12(r)u2 + 91(T,ul,u2),

dug

dr

= a21(r)u1 + a22(r)u2 + 92(T,u1,u2),

where

K(r)(lul-V11+1u2-V21)

191(r,u1,u2) -

(i = 1,2)

with K(r) > 0 and limK(r) = 0 for lull + lull < r and Ivll + 1v21 < r. Observe r-O that the two multipliers of the linear system

j

du'

= [all(t) a12(t) a21(t)

a22(t)

U

are also multipliers of system (S2). Therefore, using system (IX.3.23), Theorem IX-3-4 can be proven. In general, we obtain more precise information concerning the behavior of solutions in a neighborhood of a periodic solution in this way. The materials of this section are also found in 1CL, Chapter 13, §2, pp. 321-3271.

4. THE POINCARE-BENDIXSON THEOREM

291

IX-4. The Poincare-Bendixson theorem In this section, we explain the structure of C+(i-) on the plane. Consider an R2-valued function f (y) of y E R2 such that the entries of f (y7) are continuously differentiable on the entire g-plane R2. Denote again by pit, ill the unique solution of the initial-value problem dg = f (y-), y(0) = i. The main result of this section is the following theorem due to H. Poincare [Poll] and 1. Bendixson [Ben2].

Theorem IX-4-1. Suppose that the solution pit, qo) is bounded for t > 0 and that C+(ilo) contains only a finite number of stationary points. Then, there are the following three possibilities: (i) C+(i-Io) is a periodic orbit,

(ii) C+(i o) consists of a stationary point, (iii) C+(i)) consists of a finite number of stationary points and a set of orbits each of which tends to one of these stationary points as I ti tends to +oo. To prove this theorem, we need some preparation.

Definition IX-4-2. A finite closed segment t of a straight line in R2 is called a transversal with respect to f if f (y1 0 0 at every point on t and if the vector f (y) is not parallel to t at every point on t (cf. Figure 3). Observation IX-4-3. For every transversal t and every point i , the set tnC+( contains at most one point (cf. Figures 4-1 and 4-2). I

PU,

FIGURE 3.

FIGURE 4-1.

FIGURE 4-2.

The following lemma is the main part of the proof of Theorem IX-4-1.

Lemma IX-4-4. If pit, qo) is bounded fort > 0 and if there exists a point i7 E C+(7) such that C+ (11) contains a nonstationary point, then C+(i ) is a periodic orbit.

Proof.

We prove this lemma in three steps. Since C+(qo) is invariant and rj1 E C+(7 ), it follows that p1t,i7l) E C+(jo) for all t. Furthermore, C+(ijt) C C+(io) since C+(ilo) is closed.

Step 1. Let rj be a nonstationary point on C+(t7 ). Also, let t be a transversal with respect to f that passes through I. Then, t n C+(10) = {77} since rj E G+(ijl) C C+(no).

292

IX. AUTONOMOUS SYSTEMS

Step 2. Since tj E C+(r)l ), there exists a sequence {tk : k = 1,2.... } of real lira tk = +oo and p1tk, ill) E t (k = 1, 2, ... ). Note that numbers such that m+oo p(tk, ill) E C+(ilo). Therefore, p(tk, m) = #(k = 1,2.... ). This implies that there exist two distinct real numbers rl and T2 such that it = p(rl, ill) = (r2, ill) and hence p(t, rl = p(t, p(ri , ili )) = p1t, p(r2i ill )). This, in turn, implies that p(t + T1, ill) = p(t + r2i ill) for t > 0. Therefore, the orbit C(ill) is periodic in t of period Irl - r21. Furthermore, C(ill) C C+(rjo). Note that there is no stationary point on C(qj ).

Step 3. Since C+(ilo) is connected, it follows that distance(C(ijl), C+(ilo)-C(ni)) = 0. Hence, if C(ijl) # L+ (10), there exists a sequence { k E C+(i o) : k = 1, 2.... } such that tk C(rj,) (k = 1, 2, ...) and lim £k = { E C(ill ). Assume that k-.+oo

there exists a transversal t such that E e and lk E t (k = 1, 2.... ). Note that l; E C(ill) C C+(ilo). Then, k = t c C(ill) (k = 1, 2, ... ). This is a contradiction. Thus, it is concluded that C+(rlo) = C(iji). Now, we complete the proof of Theorem IX-4-1 as follows. Proof of Theorem IX-4-1If C+ (, ) does not contain any stationary points, then (i) follows (cf. Lemma IX-4-4). If C+(no) consists of stationary points only, we obtain (ii), since C+(ilo) is C+(7-) for connected. If f-+ (Q contains stationary and nonstationary points, then any point it E C+(ilo) does not contain nonstationary points (cf. Lemma IX-4-4). This is true also for t < 0. Hence, (iii) follows.

Observation IX-4-5. In cases (i) and (iii), the set 1R2 - C+(ilo) is not connected. Furthermore, if an orbit C(i)) is contained in C+(i o), two sides of the curve p1t, tl") belong to two different connected components of R2 - C+(t ). In fact, if we consider a simple Jordan curve C which intersects with the orbit C(tl at n transversally, then the curve p(t,,o) intersects with C in a neighborhood of two distinct points on C infinitely many times (cf. Figure 5).

Theorem IX-4-6. If C(ijo) n C+(ilo) 0 0, then C+(ijo) = C(rjo) and either no is a stationary point or the orbit C(ilo) is periodic. Proof In this case, C(ilo) C C+(ilo). If C+ (8o) contains nonstationary points, the orbit C(i o) consists of nonstationary points. Choose a transversal a at ilo. Then, it can be shown that C(ilo) is periodic in a way similar to the proof of Lemma IX-4-4, since r1o E C+(tlo).

Observation IX-4-7. If C(ilo) n L+(no) = 0, then lim distance(plt,ilo) and t-+oo

,C+(Q) = 0. It follows that if C+(ilo) consists of a stationary point then lim p(t, irjo) If C+(ijo) is a periodic orbit, then C+(rjo) is called a limit t+oo cycle (cf. Figures 6-1 and 6-2).

293

5. INDICES OF JORDAN CURVES

FIGURE 6-1.

FIGURE 5.

FIGURE 6-2.

The materials of this section are also found in [CL, Chapter 16, §§1 and 2, pp. 391-3981 and (Har2, Chapter VII, §§4 and 5, pp. 151-1581. In [Hart[, the PoincareBendixson Theorem was proved without the uniqueness of solutions of initial-value problems.

IX-5. Indices of Jordan curves In this section, we explain applications of index of a Jordan curve in the plane to the study of solutions of an autonomous system in 1R2. Consider again an R'-valued function f (y-) of y E 7,k2 whose entries are continuously differentiable on the entire y-plane R. Denote also by pit, J) the unique solution of the initial-value problem

dg dt

= f (y1, yi(0)

To begin with, let us introduce the concept of indices of Jordan curves. Let C be a Jordan curve y = il(s) (0 < s < 1) with the counterclockwise orientation (cf.

Figure 7). Assume that &(s)) # 0 for 0 < s < 1. Set u(s) = f ('l(s))

If(s))1

(0
M

x > a > ao,

for

-a [H(x) + c] > M

for

x < -a < -ao.

Also choose a function k(x) so that for x > 2a, for x!5 -2a,

{c (1)

(II}

k(x) =

Jk(x)J < c

-C dk(x)

and

>0

dx

for

- on < x - +oo

and dk(x)

dx

>

m > 0

for

JxJ < a

for some positive number m (cf. Figure 4). k

k=c

x

k=-c x=-2a

x=-a

x=O

x=a

FIGURE 4.

x=2a

,

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

312

If a positive number b is chosen sufficiently large,

OV2 .f

0

for

or 1y21 > b.

lyil ? 2a,

In fact,

-g(yl) [H(bi) - k(yl))

(A)

- a IH(yl) - cI < -M < 0

for yl > a,

a IH(y1) + cj < -M < 0

for yl < -a

and

Y2 + IH(yj) - k(yi )11/2 ? 1

(B)

for

1yi 15 2a, 1y21 > b

if b > 0 is sufficiently large. Therefore, (i)

av2

' f = -g(yi) IH(yi) - k(yi)1 < 0

(u)

f
0 is sufficiently large. Since lim G(x) = +oo, Theorem IX-2-3 implies that every solution of (X.2.2) 1XI

+00

is bounded.

Example X-2-5. For the van der Pol equation 2 + e(x2 - 1) dt + x = 0,

where a is a positive number, h(x) = e(x2 - 1) and g(r) = x. Hence,

G(x) = 2x2

and

H(x) = e (3x3 - x)

Therefore, conditions (i), (ii), (iii), and (iv) of Observation X-2-4 are satisfied. This implies that every solution of the van der Pol equation (

X.2.3 )

d

yl

dt

y2

y2

-E(Y - 1)y2 - yi,

is bounded for t > 0. System (X.2.3) has only one stationary point 6. It is easy to see that 0 is an unstable stationary point as t -+ +oo. Therefore, using the Poincaare-Bendixson Theorem (cf. Theorem IX-4-1), we conclude that there exists at least one limit cycle. In §X-3, it will be shown that system (X.2.3) has exactly one periodic solution.

3. EXISTENCE AND UNIQUENESS OF PERIODIC ORBITS

313

Example X-2-6. For a given positive number a, the system [yi

(X.2.4)

d

_

y2

[

Y2 J

-aye - sin (yl )

satisfies three conditions (1), (2), and (3) of Observation X-2-1. But, (X.2.4) does not satisfy conditions of Observations X-2-2, X-2-3, and X-2-4. Therefore, in order to prove the boundedness of solutions of (X.2.4), we must use some other methods.

[ij.

In fact, using the Liapounoff function V (y) = -yZ - cos (yl ), we obtain

ev. f =

WY=

2Since V (y-)

-aye < 0, where f (y =

t

a

lim

+oo

exists for every solution p-(t, ii) of (X.2.4). Now, observe that (1) We must have a < 1. Otherwise we would have y2(t,,)2 > 2(a+cos(yi(t,rte))

for t > 0. This implies that dt V (pl t, r7-')) < -2a(o - 1) < 0. This contradicts (X.2.5).

(2) If -1 < a < 1, the solution p(t, 7) must stay in one of connected components of the set {y" : V(y) < a + e < 1} for large positive t. Those connected components are bounded sets. (3) In case a = 1, we can show the boundedness of p(t, t) by investigating the behavior of solutions of (X.2.4) on the boundary of the set {g: V(y-) < 1).

X-3. Existence and uniqueness of periodic orbits In this section, we prove the following theorem (cf. (CL, p. 402, Problem 51).

Theorem X-3-1. Assume that (i) two real-valued functions h(x) and g(x), and X < +00,

dg

(x) are continuous for -oo
0 for x > 0, (iv) h(0) < 0,

(v) H(x) =

f h(s)ds has only one positive zero at x = a, 0

(vi) h(x) > 0 for x > a,

(vii) H(x) tends to +oo as x -- +oo. Then, the system (X.3.1)

d [Yyj dt

[-h(yi)y22- 9(yi)

has exactly one nontrivial periodic orbit and all the other orbits (except for the stationary point 0) tend asymptotically to this periodic orbit as t +oo. Proof.

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

314

Change system (X.3.1) to

zi)

d

(X.3.2)

dt Lz2J

Lz2-

by the transformation Y2J

Lz2

-zH(zi),

Setting

V(z) =

+ G(z1),

2

where

G(x) =

J0

z g(s)ds

and

z=

[z2] look at the way in which the function V(z) changes along an orbit of (X.3.2). For example, dt V

(zI = 9(zl)[z2 - H(zi)] - z29(z1) = -g(z1)H(z1)

along an orbit of (X.3.2). Hence, dz2

g(zl)

z2 - H(zl)'

dz2

dV

9(z1)H(z1)

dV

dz1

z2 - H(z1)'

dz2

dzi

z2 - H(zi)

dzi

9(z1)

_

H(zi)

along an orbit of (X.3.2).

z1(t' a)be the orbit of (X.3.2) such that z(0, a) Observation 1. Let z(t, a) = IZ24,a)]

[0].

Then, V (i(0, a)) = 2 a2. There exists exactly one positive number ao such h

that [

z1(ao, ao) =

0]

and

z2 (t, ao) >0 for 0:5t < oo

for some positive number co, where a is the unique positive zero of H(x) given in condition (v) (cf. Figure 5).

Observation 2. Since

0 when z2 = H(z1), there exist two positive numbers

r(a) and Q(a)such that

z(r(a), a)

0

-Q(a)

and

0 < zi (t, a) :5a for 0< t < r(a)

if 0 < a < ao. Also, since H(x) < 0 for 0 < x < a, we obtain dtV(z'(t,a)) = -g(zi(t,a))H(zi(t,a)) > 0

for 0 < t < r(a)

3. EXISTENCE AND UNIQUENESS OF PERIODIC ORBITS

315

except for a = ao and t = oo. Therefore,

V(z"(r(a),a)) - V(z(0,a)) > 0

(A)

for

0 < a < a0 (cf. Figure 6).

(0.ao)

zI = 0

FIGURE 6.

FIGURE 5.

Observation 3. If ao < a, there exists a positive number r0(a) such that J z1(ro(a), a) = a,

0 < z1(t, a) < a for 0 < t < ro(a), for 0 < t < ro(a)

10 < z2(t,a) - H(zl(t,a))

(cf. Figure 7). In particular ro(ao) = Co (cf. Observation 1). If the variable t is restricted to the interval 0 < t < 7-0 (a), the quantity z2(t, a) can be regarded as a function of z1(t,a), i.e.,

z2(t,a) = Z(zl(t,a),a), where Z(x, a) is a continuous function of (x, a) for 0 < x < a and a > a0, and continuously differentiable for 0 < x < a and a > ao except for x = a and a = a0. Furthermore, Z(x, al) < Z(x, a2) for 0:5 x:5 a if a0 < a1 < 02 (cf. Figure 7).

Set 2(x, a) = I Z(z, a) J . Then, d ,

dx

V(Z(x,a))

-

g(x)H(x)

Z(x,a) - H(x) >

for

0

0 V(zi(ri(ai),ai))-V(zlro(ai),ai)) > V(z(ri(a2),a2))-V(z(ro(a2),a2)) for ao < al < a2 in a way similar to Observation 3 (cf. Figure 8).

Observation 5. If ao < a, there exists a positive number r(a) such that r(a) > 71(a), z1(r(a), a) = 0, and 0 < zi (t, a) < a for r1(a) < t < r(a) (cf. Figure 9). Note that z2(t, a) < H(zi (t, a)) < 0 for r2 (a) < t < r(a). Again, regarding Z2 (t, a) as a function of zi (t, a) in the same way as in Observation 3, we can derive

(III) V(z(r(ai),al))-V(z-'(ri(ai),ai)) > V(i(r(a2),a2))-V4flri(a2),a2)) > 0 if a0 < aI < a2 (cf. Figure 9).

Observation 6. Thus, by adding (I), (II), and (III), we obtain (B)

V(z(r(aI),at)) - V(z(0,a1)) > V(zr(a2),a2)) - V(z0,a2)) > 0

if a0 < al < a2. This implies that the function G(a) defined by

g(a) = V((r(a),a)) - V(z(0,0 = Zz2(r(a),a)2 - 2a2 is strictly decreasing for a > ao as a -+ +oo. Also, 9(a) > 0 for 0 < a 5 a0 (cf. (A)).

Observation 7. Since dV

lim jz21-+oo dzi z1lim00

a

=0

uniformly

= +oo uniformly

for 0 < z1 < a, for

- oo < z2 < +oo,

3. EXISTENCE AND UNIQUENESS OF PERIODIC ORBITS

317

it follows that lim (V(i(ro(a),a)) - V(z(0,0J = 0, a+oo

lim

(V(z(rl(a),a)) - V(z(ro(a),a))J = -oo,

lim

JV(z(r(a),a)) - V(i(ri(a),a))J = 0.

a+oo a-.+00

Therefore, lim Cg(a) = -oo.

(C)

a-'+00

Thus, we conclude that g has exactly one positive zero a+, i.e.,

(X.3.3)

9(a)

Zz2(r(a), a)2

-

2a2

> 0,

0 0 is sufficiently small, the slope dw, of any orbit is small at a point (wl, w2) far away from the curve C : w2 = 3 - w!. This implies that every orbit moves toward the curve C almost horizontally (cf. Figure 12). Observation X-6-2. From Observation X-6-1 a rough picture of orbits of (X.6.2) is obtained (cf. Figure 13).

FIGURE 12.

FIGURE 13.

Actually, defining the curve Co by Figure 14, we prove the following theorem.

Theorem X-6-3. For a sufficiently small positive number J3, the unique periodic orbit of (X.6.2) is located in an open set V(/3) such that the closure of V(f3) contains the curve Co and shrinks to CD as 0 -+ 0+. Proof of Theorem X-6-8. In eight steps, we construct an open set V(13) so that (1) the closure of V(O) contains the curve Co, (2) the closure of V(J3) shrinks to the curve Co as O 0+,

(3) if an orbit of (X.6.2) enters in the open set V(!3) at r = ro, then the orbit stays in V(/3) for r > ro. Step 1. Fixing a number a(3) > 2, we use the line segment

C1(a) = j wi, a(3)3 - a((3) I

:

0 < wi 0)

AL2 + f (x,y,LY) = 0

satisfying the conditions y(O) = Y(0) and ly'(0) - Y'(0)l l< p. Then,

ly(x) - Y(x)I < {h, + on the interval 0 < x
0)

to the systern

du

(X.7.1)

dx

dv

= v,

F(X, u, v, a),

where

F (x, Y(x), v + d

)

I + F(x, u, v,.)l < (e + AM) + Kf ul,

as long as (x, u, v) is in the regi/on

l

Do = {(x, u, v) : 0 < x < t, Jul Lv

())

< Lv

for

v > 0,

for

v < 0,

in Do. Hence, in Do, (X.7.2))

F(x, u, v, A) < -Lv + Klul + (e + AM) F(x, u, v, A) > -Lv - Klul - (E + AM)

for

v > 0,

for

v

0.

329

8. A SINGULAR PERTURBATION PROBLEM

Step 2. Suppose that two functions wl (x, .A) and w2(x, A) satisfy the following conditions: (X.7.3)

0 < w2(x,A) < pe-' + b(x), 10 < wl(x,.A) < a(x), w2(0,.1) > p, wl(0,A) > 0, wi (x, A) > W&, A),

Aw2(x, A) > -Lw2(x, A) + Kwl (x, A) + (E + AM)

on the interval 0 < x < e. Then, as long as (x, u, v) is in the region

Dl = {(x,u,v): 0 -aw2(x, I\).

Look at the right-hand side of (X.7.1) on the boundary of V. Then, it can be easily seen that if a solution of (X.7.1) starts from Dl, it will stay in Dl on the interval 0 < x < e. Step 3. Show that two functions

Jwi(x,A) =

J0

w2(x A) =

+ 62(1+1),

A)l

x

pe-Lx/A + Alex:/L

ApA e + AM + K62(e + 1) where bl = L2 + , satisfy the requirements (X.7.3) if f, A, L and a positive constant 62 are sufficiently small. Observe that two roots of AX2 +

LX -K=0 are -L -(o and (o = L

+O(A).

Step 4. Note that

wl (x, A) = (L (1 - e-Lx/A) +

bK I (eKx/L - 1) + 82(x + 1).

/

To complete the proof, look at Jul < wl (x, A) as 62 -4 0.

The inequality 2( a, )as 62 I VI v< wr dy dx

Note that

0, yields the following estimate of

dY(x) < pe-Lx/A +

fE

dx

+

L

lim a-Lx/A

0

=0

L

if

(h + M1 ] ex'/L. L

x > 0.

J

d

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

330

X-8. A singular perturbation problem In this section, we look at behavior of solutions of the van der Pol equation (X.4.1) as a --+ +oo more closely. Setting t = Er and A = c2, let us change (X.4.1) to d2x 2 J1dr2 + (x -

1)dx

dr

+x=

0.

Set A = 0. Then, (X.8.1) becomes

(x2 - 1) d + x = 0.

(X.8.2)

Solving (X.8.2) with an initial value x(O) = xo < -1, we obtain x = 0(r), where 2 )2 - In I0(r) I = -r + 2 - In(-xo).

Observe that d0(r)

0(r) >0 0(7)2-1

dr

if

d(r) < -1.

2

Note also that setting ro = 2 - In(-xo) -- 2 > 0, we obtain ,(ro) = -1 and 45(7-)2

- 1 > 0 for 0:5 r < ro. The graph of 0(r) is given in Figure 22.

FIGURE 22.

(i) Behavior for 0 < r < 7-o -bo, where do > 0: Let us denote by x(r, A) the solution to the initial-value problem (X.8.3)

d2z

(x - 1) dx x = 0, dr- + 2

a dr2 +

x(O, A) = xo,

40,A)

= 7-7,

where the prime is d7- and q is a fixed constant. Using Theorem X-7-1 (Na6J, we derive

I

Ix(r, A) - O(T)I 5 AK, Ix'(r, A) - 4'(r)l 5 I7-7 -

0'(0)le-P'1x

+ AK,

331

8. A SINGULAR PERTURBATION PROBLEM

for 0 < T < To - 60, where K and a are suitable positive constants.

(11) Behavior for lx+1l< o: First note that lim p0'(T) _ +oo. Set ¢'(ro-2bo) _

1Pi > 0. Then, there exists T1(A) for sufficiently small A > 0 such that { O ro. Hence, if an orbit (x(t), y(t)) satisfies conditions that x(t)2 + y(t)2 > ro for to < t < ti, we obtain

V(x(t),y(t))

for

to 0, the inequality

fT

f (x(t))dt > 0 implies that this orbit is orbitally asymptotically stable o

(cf. Exercise IX-5). (3) Set

a(t) = V(x(t), x'(t)) = Then,

da(t) dt

(

x'(t))2 2

f

+

_ -f(x(t))(x'(t))2

x(t)

.

Therefore, we obtain

dA J f(x( t) )dt = - Jr (x '

(I)

)

2

Now, investigate the quantity (x'(t))2 as a function of A along the periodic orbit (x(t), x'(t)). For a fixed value of A, compare (x'(t))2 for different values of x(t), using the assumptions on f and g. Step 1. First the following remarks are very important. (a) If we set y = x', the given differential equation is reduced to the system

dt = -f (x)y -

dt = y,

(S)

A careful observation shows that the index of the critical point (0, 0) is 1, while the index of the critical point (-2,0) is -1. There are only two critical points of (S). (c) The periodic orbit and any line {x = a constant) intersect each other at most twice.

(d) The critical point (-2,0) should not be contained in the domain bounded by the periodic orbit. To show this, use the fact that the index of any periodic orbit is 1. These facts imply that the periodic orbit should be confined in the half-plane x > -2. 2

Step 2. A level curve of V (x, y) = 2 +

r:

J0

is a closed curve if the curve is

totally confined in the strip jxj < 1. In particular, (1, 0) and (-1, 0) are on the same

level curve V (x, y) = / 0

1

Io

_ 1 g(C)d{. Since A(t) is increasing as long as

jx(t)I < 1, the periodic orbit cannot intersect the level curve V(x,y) = / 0

,

Therefore, the periodic orbit must intersect the lines x = 1 (as well as the line x = -1) twice. Denote these four points by (1,s1(A)), (1,-rt(B)), (-1,-q(C)), and (-1, 77(D)), where r)(A), n(B), q(C), and q(D) are positive numbers.

337

EXERCISES X Step 3. Set

= V(1, rl(A)), AB = V (1,

AA

-n(B)),

Ac = V(-1, -n(C)), AD = V (-1, rl(D))

Then, Set

AA > AD-

AC > AD,

AC > AB,

AA > AB,

Z(A) = { A : AA > A > max(AB, AD) }, Z(B) = { A : AB < A < min(AA, )'C) }, Z(C) = { A : Ac > A > max(AB, AD) }, Z(D) = { A: AD < A < min(AA, Ac) }.

Note that the interval lo = JA: min(AA, Ac) > A > max(AB, AD)} is contained in Z(A) n Z(B) n Z(C) n I(D). Now,

(a) on the arc between (1, rl(A)) and (1, -rl(B)) of the periodic orbit, regard y as a function of A and denote it by yi (A), where AB < A 5 AA, (Q) on the arc between (1, -rl(B)) and (-1, -r1(C)) of the periodic orbit, regard y as a function of A and denote it by y2 (A), where AB < A < AC, (ry) on the are between (-1, -rl(C)) and (-1, rl(D)) of the periodic orbit, regard y as a function of A and denote it by yi (A), where AD < A < AC, (6) on the arc between (-1, rl(D)) and (1, rl(A)) of the periodic orbit, regard y as a function of A and denote it by yz (A), where AD :5 A < AA. Then, it can be shown that yi (A)2 < y2 (A)2 yi (A)2 < y2 (A)2

on on

I(A),

yi (A)2 < y2 (A)2

on

I(C),

yi (A)2 < y2 (A)2

on

Z(D).

Z(B),

Step 4. Now, fixing a AO E Z(A) n Z(B) n 1(C) n T(D), evaluate the integral (I) as follows: rT

J

{JA8 f(s(t))dt = -

rc

dA

w yi (A)2 +

J

dA

y2 (A)2

+'Ac

where dA

J\A rac

dA

_

ra°

AC

dA

Joe y2 (A)2 = Jaa y2 (A)2 I"

dA

ra°

+

(A)2,

dA y2(A)2,

Jib rAc

dA

dA

yi (A)2 = - JaD yi (A)2 - ao yi ao ys (A)2

aD y2 (A) 2

Jao

IA a

dA

yi

lao

as yi (A)2

AA

yl (A)2 +

r"

dA

yi (A)2

dA

rAD

y2

(A)2.

(A)2,

dA

1

y2 (A)2 T ,

X. THE SECOND-ORDER DIFFERENTIAL EQUATION

338

Thus, we conclude that

T f (x(t))dt > 0.

J

This implies that every periodic orbit is orbitally asymptotically stable. Hence, there exists at most one periodic orbit.

X-8. For a nonzero real number c and a real-valued, continuous, and periodic

function f (t) of period T which is defined on -oo < t < +00, find a unique solution 0(t, E) of the differential equation

E

dt = -y + f (t)

which is periodic of period T. Also, find the uniform limit of 0(t, c) on the interval

-oo 0. f 0+

Hint. Look at E dt = c - y2. Then, c = Q + a2 > a2. Hence, y2 increases to c very quickly. Then, use Theorem X-7-1 [Na6], or find the solution explicitly (cf. Exercise I-1).

X-11. Find, if any, solution(s) y(t, c) of the boundary-value problem

4

dt2 + 2ydty

= 0,

y(0) = A, y(1) = B,

in the following six cases: (1) 0 < B < A, (2) B = A, (3) B > IAA, (4) B = -A > 0, (5) CBI < -A, and (6) B < 0 < A, assuming that e is a positive parameter. Also, find lim y(t, e) for 0 < t < 1 in each of the six cases. e

0+

EXERCISES X

339

Hint. Use explicit solutions together with the Nagumo Theorems (Theorems X1-3 and X-7-1) on boundary-value and singular perturbation problems. See also Exercises X-10 and X-12, and [How].

X-12. Let f (x, y, t, e) be a real-valued funcion of four real variables (x, y, t, e). Assume that (i) 0(t) is a real-valued, continuous, twice-continuously differentiable function

on the interval Zo = {t

:

0 < t < 1) and satisfies the conditions 0 =

f (¢(t), 4'(t), t, 0) and 4(1) = B on Zo, where B is a given real number, (ii) the function f (x, y, t, e) and its partial derivatives with respect to (x, y) are continuous in (x, y, t, e) on a region R = {(x, y, t, e) : Ix - ¢(t)I < rl, IyI < +00, t E Zo, 0 < E < r2}, where r1 and r2 are positive nmbers, (iii) If (4(t), 4'(t), t, e) I < Ke for t E Zo, where K is a positive number,

(iv) there exists a positive number µ such that Of (x, y, t, e) < -p on R, (v) there is a positive-valued and continuous function '(s) defined on the interval J+00

+oo and that If (x, y, t, e)I < +'(IyI) on

0 < s < +oo such that R,

(vi) A is a given real number. Then, there exists a positive number co such that for each positive f not greater than co, there exists a solution x(t, E) of the boundary-value problem d2x

f d#2 = f (X,

d , t, E) ,

x(0, e) = A, x(1, c) = B

such that Ix(t,E) - di(t)I < IA - h(0)led=te-ft/` +C2e on Zo, x'(t, e) - 4'(t)

C3e e

+ co for 0 < c5e 0. This is an example of M=0

an asymptotic representation of an actual solution by means of a formal solution. In this chapter, we explain the asymptotic expansions of functions in the sense of Poincare and in the sense of the Gevrey asymptotics. In the Poincare asymptotics, flat functions are characterized by the condition lim E f) = 0 for all positive intexm gers m, whereas in the Gevrey asymptotic, flat functions are characterized by the condition If (x) I exp(cIxI-k) < M as x -i 0, where c, k, and M are some positive numbers. Generally speaking, the Poincare asymptotics is too general for the study of ordinary differential equations. A motivation of the Gevrey asymptotics is also given by the Maillet Theorem (cf. Theorem V-1-5). In §XI-1, we summarize the basic properties of asymptotic expansions of functions in the sense of Poincare. The Gevrey asymptotics is explained in §§XI-2-XI-5. For more information concerning the Poincare asymptotics, see, for example, [Wasl]. The Gevrey asymptotics was originally introduced in [Wat] and further developed in [Nell. To understand the materials concerning the Gevrey asymptotics of this chapter, [Ram 1], (Ram 21, [Ram3], [Si17, Appendices], [Si18], and (Si19] are helpful.

XI-1. Asymptotic expansions in the sense of Poincare In this section, we explain the asymptotic expansions in the sense of Poincare. Let x = a be a point on the extended complex x-plane. Consider a formal power series 00

(XI.1.1)

P(x) = E c,(x - a)m. M=0 342

1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCAR$

343

Let D be a sector in the x-plane with vertex at x = a and Do be a neighborhood of x = a in D. Assume that f (x) is defined and continuous in Do. Definition XI-1-1. The formal series (XI.1.1) is said to bean asymptotic (series) expansion of f (x) as x -k a in V if for every non-negative integer N, there exists a constant KN such that N

(XI.1.2)

f(x)->2cm(x-a)m

< KN Lx - aIN+r,

N = 0, 1, 2,- ..

M-0

for all x in Do. Such an asymptotic relation is denoted by f(x)

(XI.1.3)

p(x)

as x

a in D.

This definition of an asymptotic expansion of a function was originally given by H. Poincare [Poi2J. Before we explain some basic properties of asymptotic expansions, it is worthwhile to make the following remarks.

Remark XI-1-2. The vertex x = a can be x = oo. In that case, the asymptotic on

C nx-"'.

series is in the form m=0

Remark XI-1-3. Assume that f (x, t) is a function defined and continuous in (x, t) for x in D and t a parameter in a domain H in the t-plane. A formal power series N

F c,,,(t)(x-a)'", where c,(t) is a function oft, is said to bean asymptotic (series) M=0

expansion of f (x, t) as x 0 in V if for every non-negative integer N, there exists a function KN(t), independent of x, such that N

f(x, t) -

Cm(t)(x - a)m cm(x - a)'+Ei(x)(x-a)N+i M=0

N

and g(x) _ > 'ym(x - a)m + E2(x)(x - a)N+1 Then, there exists two constants m=0

KN and LN such that (XI.1.10)

IE1(x)I

(XI.1.20)

(n = 1, 2, 3, ...) as x --+ a in D.

M=O

Assume that {f(x)In = 1,2,3.... } converges uniformly to a function f (x) in a subsector Do of D with vertex x = a. Then, lira Cnm = Cm

00

f (x) ^- E cm(x - a)'

as x -+ a in Do.

M=0

Proof The assumption implies that for each non-negative integer N, there exists a positive constant KN, independent of n, such that N

(XI.1.23)

a)'1

I

(n = 1,2,3,... )

:5

M=0

for x in a neighborhood of a in D. Furthermore, for each pair of positive integers (j, k), there exists a positive constant bjk such that (XI.1.24)

If,(x) - fk(x)I S bjk

for x in Do, where (XI.1.25)

bjk-+0

as j, k -+ oo.

Put N

(XI.1.26)

pjN(x) =E cjm(x - a)n' M=0

(j = 1, 2, 3, ... ).

1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCARE

349

Then, (XI.1.27)

(j = 1,2,3,...).

Ifj(x) -P)N(x)I 5 KNIx-alN+1

Thus, (XI.1.28)

IP) N(x)-PkN(x)I 5 5)k+2KNIx-alN+l

(j,k=1,2,3,...).

In particular, (XI.1.29)

Ic)o - ckol < b)k+2Kolx-al

(j,k= 1,2,3,...)

for N = 0 and x in Do. Therefore, (XI.1.25) and (XI.1.29) imply that lim c)o = CO exists.

Assume that lim c),,, = c,,, exist for m < N. Then, from (XI.1.28), it follows 1

that

00

N-1

E (C),,, - Ckm)(x - a)m + (CjN - CkN)(x - a)N M=0 < S)k + 2KNIx - alN+1

(j,k = 1,2,3,...)

for x in Do. Thus, IC)N - CkNI Ix - aIN < e)k + 61k + 2KNIx - alN+1 where e)k

k=1 ,2 , 3 . .

. .

0 as j, k -. oo. Hence, Ic)N -CkNl :5-

aIN +2KNIx-al

11

Iz

(j,k= 1,2,3,...).

Therefore, lim c)N = cN. Consequently, lim c),, = c,,, for all m. ) --00 l-00 Furthermore, since (XI.1.27) holds independently of j, we obtain N

f(x) - > cm(x - a)m < KNlx - alN+1 M=0

for x in Do. Thus, (XI.1.22) holds.

The following basic theorem is due to E. Borel and J. F. Ritt.

Theorem XI-1-13 ((Bor( and [Ril). For a given formal power series in (x - a) (XI.1.30)

P(x) = > c,,,(x - a)m m=0

and a sector with vertex x = a

(XI.1.31)

D={xIO 0 are suitable numbers independent of 1. Set

(X1.2.4) +00

Then, there ex,sts a formal power series p = E a ..xm E C[[x)]. such that ¢t E M=0

A. (r, at, bt) and J(41) = p for each 1. There are various situations in which Gevrey asymptotic expansions arise. To illustrate such a situation, let ¢(x) be a convergent power series in x with coefficients

in C. For two positive numbers r and k, set k

x

rr

J0

0(t)e-('1x)}tk- 1 dt.

This integral is called an incomplete Leroy transform ofd of order k. +oo

Theorem XI-2-4. For every ¢ =

cx"j E C{x}, it holds that m=0 R

if

4,k(0) E Al/k P,-2k'2k

355

2. GEVREY ASYMPTOTICS and

+00

r 1 1 + k Cm2'",

J(tr,k(4)) _ M=0

where k and p are any positive numbers and r is any positive number smaller than the radius of convergence of 0. Proof.

The following proof is suggested by B. L. J. Braaksma. For an arbitrary small positive number e, let {Sl (e), S2(e), ... , SN(c)} be a good covering at x = 0, where

St(E) _ {x : I argx - dtI < 2k - e, 0 < jxj < r} ir, and set

with real numbers dt such that -ir < d 1 < . - - < dN

J OtW _

(t = 1, 2, ... , N).

I. 4t(x) = tr,k(0t)(xe-d`) In particular, choose dt = 0 for some t. Then, {Sj(e),S2(e),... ,SN(e)) and {01, 02, . . , ON} satisfy conditions (1), (2), and (3) of Theorem XI-2-3. In fact, (1) and (2) are evident. To prove (3), note that -

k Qe(x) =

zk

/ re'd'

J

r

0(a)e-(o/s) o

'da,

0

where the path of integration is the line segment connecting 0 to re{d'. Therefore, of (x) - bt+1(x) =

k X

j

0(a)e-(ol=)'ak-1d,

t

where the path ryt of integration is the circular arc connecting re'd'+1 to re'd'. Statement (3) follows, since

e-(°/=)`

/

exp - t fx

cos[k(arg a - arg x)J

and kj arga-argxj < 2 -ke for a E ?Y and x E St(e)f1St+1(E) Since a is arbitrary, it follows that

61 E Allk I P, de - r,-, dt +

2k

(t = 1, 2, ... , N).

Furthermore, J(01) can be computed easily (cf. Exercise XI-13). +00

Observe that a power series p = 1: a,nxm E C[[xjj belongs to C[[xJ), if and only m=0

+ao

if d(x) =

I'(l

sm)xm corollary of Theorem XI-2-4.

+

belongs to C{x}. Therefore, we obtain an important

XI. ASYMPTOTICS EXPANSIONS

356

Corollary XI-2-5. For any p E G[[x]], and any real number d, there exists a function ¢(x) E A. d - 2 , d + ) such that J(O) = p. (r,

2

This corollary corresponds to the Borel-Ritt Theorem (cf. Theorem XI-1-13) of

the Poincar6 asymptotics. Also, this corollary implies that the map J , d + 2) -+ c[[x]], is onto. A8 (r, d - s7r 2 Theorem XI-2-3 is a corollary of the following lemma.

Lemma XI-2-6. Assume that a covering {St : e = 1, 2,... , N} at x = 0 is good and that N functions 61(x), 62(x), ... , 6v (x) satisfy the following conditions: (i) 6t is holomorphic on St n St+1, 7exp[-Alxl-k] on St n St+1, where -y > 0, A > 0, and k > 0 are (ii) I6t(x)I suitable numbers independent of e. Define s by (XI. 2-4). Then, there exist N functions ), (x), t.b2(x), ... ,'IPN(x) and a +oo

formal power series p = >

E C[[x]], such that

m=o

(a) 4,t E A. (r, at, bt) and J(4't) = p, where St = {x : 0 < I x[ < r, at < arg(x) < bt} (e = 1, 2, ... , N), (b) 6t(x) _ t(x) - i/'e+l(x) onSt n St+1. Let us prove Theorem XI-2-3 by using Lemma XI-2-6. Proof. Set

5t(x) = 4t(x) - 41+1(x)

(e = 1, 2, ... , N).

Then, there exist N functions 4111(x), 02(z), ... , ON (x) satisfying conditions (a) and (b) of Lemma XI-2-6. In particular, (b) implies that

Oe(x) - 4,t+1(x) = t t(x) - 4Gt+1(x)

(e = 1, 2,... , N)

on St n St+1. This, in turn, implies that

01(x) - 4't(x) = 4,1+1(x) - 4Gt+1(x)

(e = 1, 2,... , N)

on St n St+1. Define a function 0 by

4,(x) = 4t(x) - 4Vt(x)

on St

(e = 1, 2, ... , N).

Then, 0 is holomorphic and bounded for 0 < IxI < r. Therefore, 0 is represented by a convergent power series. Since .01 = -01 + 0, Theorem XI-2-3 follows immedi-

ately. 0 We shall prove Lemma XI-2-6 in §XI-5. Because the Gevrey asymptotics of functions containing parameters will be used later, we state the following two definitions.

3. FLAT FUNCTIONS IN THE GEVREY ASYMPTOTICS

357 00

Definition XI-2-7. Lets be a positive number. A formal power series

am(u-')em

m=0

is said to be of Gevrey order s uniformly on a domain V in the u-space if there exist two non-negative numbers Co and C, such that (XI.2.5)

Iam(i )I < Co(m!)sCr

for u E D and rn = 0, 1, 2, .... Set V = D(60, a0, Qo) = {e : 0 < Ie] < 5o, ao < arg e < (30} and W = D(b, a, Q).

Definition XI-2-8. Let s be a positive number. A function f (u e) is said to admit 00

an asymptotic expansion

on D if

am(u")em of Gevrey order s as c

0 in V uniformly

m=0

(i) > am(i )em is of Gevrey order s uniformly on D, m=0

(ii) for each W such that a0 < a < 0 < X30 and 0 < b < 60, there exist two non-negative numbers KW and Lµ such that N

(X1.2.6)

f (u, e) - > am('tL)cm
0 and A > 0, and (4) the power series i[ fn, a, Ij, Aye' is a ia,I+?P2I?0

convergent power series in y and F. Show that there exists a unique power series 92 in (x, z) such that (i) drr,, E C[[x]]", (ii) b(0,0) = 0, and (iii) ¢(x, z) lack>o

F(x, (x, z), z) = 0. Also, show that (a, E E(s, B)" for some B > 0 and the power series E II( a' i" is a convergent power series in z. IP3i>o

Hint. This is an implicit function theorem. Write f in the form f = fo + fiy + F'gv' z"P2 fn,,r,, , where fo and fn,,n, are in E(s, A)", $ E M"(E(s, A)), and >' is the sum over all (pi i p) such that either jell = 0 and [g-,j = I or Jial j + jp2j > 2. Here, M.(R) denotes the set of all n x it matrices with entries in a ring R. Assumption (1) implies that fo(0) = 0. Assumption (2) implies that -t(0) is invertible. Therefore, I ' exists in M"(C[[x]],), and hence II4 'jI, A < +oo for some A > 0, where III-' IIB,A = sup{jj4V' f Ij, A : f E E(s, A)n, Ij f [j,.A = 11. Since IIfIIe,B 5 11A .,A for f E (C[[x]],)" if B > A, assume without any loss of generality that A = A. Then, t' IF = 90 + y" + zP2§p., p,, where g"o E E(s, A)", 9a,,a, E E(s, A)", 90(0) = 0, and 'II9n,,a2I1a,A9" i is a convergent power series ul in y" and F. Consider the equation 0 = u"+ y"+ F,'#P- z g"n, ,n, , u" = . . Then, un

there exists a unique power series 5(x, u",

_

lip ' z'na"n,

such that

fa,l+lpiI>I

dn,,a, E E(s, A)" and 0 = u'+ a'+

identically. The unique power series satisfying conditions (i), (ii), and (iii) is given by (S)

(x, z_) = 5(x, 90, zfl

Now, consider the equation &'$121 Is,A

(R) y'=u".+z-V Gn,,p,

where

Ga,,i7 =

ER". 9P% ,P2 Il .w

EXERCISES XI

369

Equation (R) has a unique solution y = E 9P'z-r'pp,,p, such that Qp,,p, E 1p1I+1P21>>-1

]R" , the entries of pP,,p, are non-negative, and the series is a convergent power series in u and z1 . Use this series as a majorant to show that defined by (S) satisfies conditions (iv) and (v).

XI-10. Assume that a covering (S,,$2,... , SN} at x = 0 is good and that N functions 01(x), 02(x),... , ON (x) satisfy the conditions: (1) 0t(x) is holomorphic in St,

(2) of(x)^_-Oasx-.0 in St, (3) 10e(x) - Ot+1(x)l < -yexp[-AIxI-kJ on St n St+1i where -y > 0, A > 0 and k > 0 are suitable numbers independent of e. Show that there exists a positive number H such that

[4t(x)I < Hexp[-AIxI-kJ

in

St.

Hint. See [Si15J.

XI-11. Assume that a covering {S1, S2, ... , SN } at x = 0 is good and that N functions 01(x), 02(x),... , y` N (x) satisfy the conditions (1) 4,(x) is holomorphic on St, (2) 4c(x) is bounded on St, (3) we have IOt(x) - 01+1(x)l < Knlxl'

(n = 1,2,...)

on St n St+1,

where K are positive numbers. Show that there exists a formal power series p = > a,,,xm E d[[xJJ such that for m=0

each t, we obtain ¢t E A(S1) and J(¢1) = p, where the notations A(S) and J are defined in Exercise XI-1.

XI-12. Assume that a covering {S1, S2.... , SN } at z = 0 is good and that n x n matrices 4i1(x),t2(x),... ,4N(x) satisfy the following conditions: (1) the entries

of tt(x) belong to A(S1 n St+1) and (2) J(tt) = I,. Show that there exists a formal power series Q = I,, + E xmQm having constants n x n matrices Qm as m=1

coefficients, and n x n matrices P1(x), P2(x), ... , PN(x) such that (i) for each e, the entries of P1(x) and P1(x)-1 belong to A(S1), (ii) J(P1) = Q (t = 1, 2,... , N),

(iii) It(x) = PI(x)-1Pt+1(x)(e= 1,2,... ,N), where the notation A(S) and J are defined in Exercise XI-1. Also, show that if the entries of (t(x) belong to A.(St), respectively, then the entries of P,(x) and P, 1(x) also belong to A,(Se), respectively. Hint. See [Si17, Theorem 6.4.1 on p. 150, its proof on pp. 152-161, and §A.2.4 on pp. 207-2081.

XI. ASYMPTOTIC EXPANSIONS

370

XI-13. Prove the formula 00

M=0

which is given in Therorem XI-2-4.

Hint. Assume that j argx1 < 2k - b, where b is a small positive number. Then, k

(r'/) v"'/ke °do

k rT tme-(t/x)ktk-Idt = xm J0

0

r (1 +

m)

T

xm - xm

Jr°O

J (t/z)k

Om/ke-`do.

Hence, N

r (i + m ) Cmxm - E cmxm

1,,k(6) _

k

M=0

/r

m-0

+00

+ k J xk

0

Om/ke °d0 (t/k)k

cmtm

e-(t/x)ktk-ldt.

m=N+1

XI-14. Let a be a positive number larger than 1. Also, let

SJ={x: a, 26o (Ak(x)I is convergent uniformly k=0

in Do. The main result of this section is the following theorem, which was originally proved in [Hsl].

Theorem XII-6-2 ([Hsl]). For each non-negative integer m, there is an n x n matrix P(x, e) satisfying the following conditions: (i) the entries of P(x, e) are holomorphic in (x, e) in a domain (XII.6.4)

x E D1i

jeJ < 60,

where V1 is a subdomain of Do containing x = 0, (ii) P(x, 0) = In for X E V1 and P(0, e) = In for je[ < 6o, (iii) the system (XII.6.1) is reduced to a system of the form (XII.6.5)

e°

dii

m

o-1

k=0

k--O

_{kA(X) E + em+1 r` c'Bk(: W

)

u

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

392

by the transformation P(x, e)u,

(XII.6.6)

where Bk(x) (k = 1, 2, ... , or - 1) are n x n matrices whose entries are holomorphic for x E D1.

Remark XII-6-3. In case when a = 1, (XII.6.5) becomes dil ed

En

CkAk(x)+em+IB0(x)I!

.

k=0

In particular, if a = I and m = 0, (XII.6.5) has the form e

dil

= {Ao(x) + eB0(x)} u.

It should be noticed that in Theorem XII-6-2, if we put m = 0, then the right-hand side of (XII.6.5) is a polynomial in a of degree at most a without any restrictions on A(x, e). We prove Theorem XII-6-2 by a direct method based on the theory of ordinary differential equations in a Banach space. (See also [Si8j.) Proof of Theorem XII-6-2.

The proof is given in three steps.

Step 1. Put (XII.6.7)

a

m+o

k=0

k--0

P(x, e) = I + Em+1 [-` ekPk(x) and B(x, e) = E EkBk(x),

where

(XII.6.8)

Bk(x) _

(k = 0,1, ... , m),

J A&. (x)

Bk-m-1(x)

(k = m+ 1,m+2,... ,m+a).

From (XII.6.1), (XII.6.5), and (XII.6.6), it follows that the matrices P(x,e) and B(x, e) must satisfy the equation (XII.6.9)

e

QdP

= A(.x, e)P - PB(x, e).

From (XII.6.3), (XII.6.7), (XII.6.8), and (XII.6.9), we obtain k

(XII.6.10)

Am+1+k(x) - Bm+1+k(x)+ E{Ak-h(x)Ph(x) - Ph(x)Bk-h(x)} h=O

(k=0,1,...,a-1)

6. ANALYTIC SIMPLIFICATION IN A PARAMETER

393

and

dPk(x) dx

o+k

Ad+k-h(x)Ph(x)

=A m +1+ o +k(x) + h=o

(XII.6.11)

o+k (k = 0,1,2,...),

E Ph(x)Bo+k-h(x)

h=k-m where

Ph(x) = 0

(XIL6.12)

if

h < 0.

It should be noted that the formal power series P and B that satisfy the equation (XII.6.9) are not convergent in general. In order to construct P as a convergent power series in e, we must choose a suitable B. To do this, first solve equation (XII.6.10) for B,,,+1+k(x) to derive

Bm+1+k(x) = Am+I+k(x)+Hm+1+k(x;P0,P1,... Pk)

(XII.6.13)

(k=0,1, ..,Q-1),

where H, are defined by (XII.6.14)

H , = 0,

(k = 0,1, ... , m), k

k

Hm+I+k(x; Po, PI, ... , Pk) = E{Ak-h(x)Ph - PhAk-h(x)) h=0

- E PhHk-h, h=0

(k = 0,1,...,a - 1). Denote by P an infinite-dimensional vector {Pk : k = 0, 1, 2,... }. Then, by substituting (XII.6.13) into (XII.6.11), we obtain dPk(x)

(XII.6.15)

dx

= fk(x; P)

(k = 0,1, 2, ... ),

where o+k

o+k

fk(x; P) = Am+1+o+k(x) + E Ao+k-h(x)Ph - 1: PhA,,+k-h(x) h=0

(XII.6.16)

h=k-m

a+k

-

E PhHo+k-h(x;P)

(k=0,1,2,...).

h=k-m

Denote by -F(x; P) the infinite-dimensional vector { fk(x; P) : k = 0,1, 2,... }. Then, equation (XII.6.15) can be written in the form (XII.6.17)

dP = F(x; P). dx

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

394

Solve this differential equation in a suitable Banach space with the initial condition P(0) = 0. Actually, we solve the integral equation

P(x) = f0F(;P())de.

(XII.6.18)

This is equivalent to the system (XII.6.19)

Pk(x) =

ffk(;P())de

(k = 0,1, 2, ... ).

If P is determined, then the matrices Bk are determined by (XII.6.13) and (XI1.6.14).

Step 2. We still assume that A(x, c) is holomorphic in domain (XII.6.2). Denote by B the set of all infinite-dimensional vectors P = {Pk : k = 0, 1, 2.... } such that (i) Pk are n x n matrices of complex entries, 00

(ii) j:boilPkll < oo, where IlPkll is the sum of the absolute values of entries k=0 of Pk.

For each P, define a norm 11P11 by 00

(XII.6.20)

IIPII = Ebo IlPkll k=0

Then, we can regard B as a Banach space over the field of complex numbers. Now, we can establish the following lemma.

Lemma XII-6-4. Let F(x; P) be the infinite-dimensional vector whose entries fk(x,P) are given by (XII.6.16). Then, for each positive numbers R, there exist two positive numbers G(R) and K(R) such that (XII.6.21)

for IIPII 2 EkA,,,+1+o+k(x) + k=0

a-1

- k=0 >

l

\ k=0 EkAk(x)/

(e'Pk) k=0

k

- Ek=OEk hE Ak-h Ph 0-1

1

M+V

C*

k=o Ekpk/

k

ek E Ph[Ak-h(x) + Hk-h(x;P)1 h=O

Hence, Lemma XII-6-4 follows immediately.

l }.

r

k o Ek [Ak(x) +

Ik(x; PA)

395

EXERCISES XII

Step 3. We construct the matrix P(x, e), solving the integral equation (XII.6.18) by the method of successive approximations similar to that given in Chapter I. By virtue of Lemma XII-6-4, we can construct a solution P(x) in a subdomain Dl of Do containing x = 0 in its interior. Since (XII.6.18) is equivalent to differential equation (XII.6.15) with the initial condition P(0) = 0, the solution P(x) gives the desired P(x, e). The matrix B(x, c) is given by (XII.6.13) and (XII.6.14). 0

EXERCISES XII 00

XII-1. Find a formal power series solution y = F, e'd,,,(x) of the system of m=0

differential equations CO

LY = Ay" + > Embm(x), m=0

where y E C", A is an invertible constant n x n matrix, and 5,n (x) and bm(x) are C"-valued functions whose entries are holomorphic in x in a neighborhood of x = 0. XII-2. Using Theorem MI-4-1, diagonalize the system dy = e dx

0 11-X

1+x1 ex

y'

where

[1.

XII-3. Using Theorem XII-4.1, find two linearly independent formal solutions of each of the following two differential equations which do not involve any fractional powers of e. E2d2

(1)

Y

2 + y = eq(x)y,

(2) a2 + y = eq(x)y,

where q(x) is holomorphic in x for small jxj.

Hint. If we set '62 = e, differential equation (2) has two linearly independent solutions e4=/00(x, 0) and a-1/190(x, -/3). The two solutions

= Q reiz/°4(x, Q) -

-Q),

do not involve any fractional powers of e.

XII-4. Let x be a complex independent variable, y" E C", z" E C, e be a complex parameter, A(x, y, z, e) be an n x n matrix whose entries are holomorphic with respect to (x, y, z, e) in a domain Do = {(x, y, X, e) : jxj < ro, jyi < Pi, Izl < p2, 0 < jej < ao, j arg ej < flo}, f (x, y, z, e) be a C-valued function whose entries are holomorphic with respect to (x, y, 1, e) in Do, and #(x, ,F, e) be a C"-valued function whose entries are holomorphic with respect to (x, z, e) in the domain Uo = ((x, zl, e) : jxj < ro, jzj < p2,0 < jej < ao, j argej < /3o}. Assume that the entries of the matrix

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

396

A(x, if, z, c), the functions f (x, y, z, e), and §(x, .F, e) admit asymptotic expansions as f - 0 in the sector So = {E : 0 < IEI < ao, I arg EI < /30} uniformly in (x, y, z) in the domain Ao = {(x, y, z) : IxI < ro, Iv'I < pl, I1 < p2} or (x, z-) in Vo = {(x, z) : I xI < ro, Izl < p2}. The coefficients of those expansions are holomorphic with respect to (x, y", I) in Do or (x, z-) in V0. Assume also that det A(0, 0, 0, 0) 0 0. Show that the system f dy = A(x, y, z, E)y + E9(x, Z, f),

= Ef (x, g, Z, f)

has one and only one solution (y, z) = (¢(x, f),

e)) satisfying the following

conditions:

(a) the entries of functions O(x, E) and '(x, f) are holomorphic with respect to (x, e) in a domain {(x, f) I xI < r, 0 < lei < a, I arg EI < /3} for some positive numbers r, a, and 0 such that r < ro, a < ao and /3 < /io, (b) the entries of functions (x, c) and i/ (x, c) admit asymptotic expansions in :

powers of f as c - 0 in the sector if : 0 < lei < a, I argel < (3} uniformly in x in the disk (x : Ixi < r), where coefficients of these expansions are holomorphic with respect to x in the disk {x: Ixi < r}, (c) 0(0,c) = O for f E {E : 0 < IEI < a, I arg EI < ,0}.

Hint. Use a method similar to that of §§X11-2 and XII-3.

XII-5. Find the following limits:

j

f (t) sine I

r / c-0 0

(1) lim

at

1

f (t) sin 1

\E

dt and (2)

lim

I dt, where a is a nonzero real number and f (t) is continuous

and continuously differentiable on the interval 0 < t < 1.

XII-6. Discuss the behavior of real-valued solutions of the system

Edg = rEE

-1 +Ex1 y as e -40,

y"_ [y2J

where

XII-7. Discuss the behavior of real-valued solutions of the following two differential

equations as f

O+: (1) E2

d3 y

+ 2 + xty = O,

(2) E3

1

(1

x)y = O.

XII-8. Assume that (i) the entries of a C"-valued function Ax, y", e) are holomorphic with respect to (x, y", e) in a domain A1(Eo) X II(po) X O2(ro), where b0, po, and ro are positive

numbers and

0l(6o)={xE(C:IxI R' > 0 and e E S, (b) r(x, e) = 0 as a - 0 in S uniformly for jx[ > R', (c) for a sufficiently large M' > 0, the transformation v = [I +x-(h1'+1)r(x,e)jw changes (s") to

= xk

dw

M

N

` x-mam(f) + x -(N+1) E ? bm(e) w, Lm=o M-

whereb,,,(e)-0ase-0 inSform>k+1. XII-13. Let A(t) be a 2 x 2 matrix whose entries are holomorphic in a disk It it[ < po} such that the matrix (M)

P

()' {kA(!)P(1)

-P

:

(-X1)1

is not triangular for any 2 x 2 matrix P(t) such that the entries of P(t) and P(t)-1 are holomorphic in the disk D = It : jtj < po}. Show that there exists such a 2 x 2 ) matrix P(x) for which matrix (M) has the form xk B Gl with a 2 x 2 matrix B(t) wh ose entries are polynomials in t and whose degree in t is at most k + 1.

Hint. See [JLP]. This result can be generalized to the case when A(t) is a 3 x 3 matrix (cf. [Ball) and [Bal2J).

XII-14. Let P(t) be a 2 x 2 matrix such that the entries of P(t) and holomorphic in a disk V = it : jtj < po} and that the transformation y" = P

P(t)-1

are

(!)

u"

X

changes the system dil dx

[ l +x-1 I.

0

x-3

1 1-x'1Jy'

y =

[ i12,

to

dil

ds

Blx/u,

u=

u1 112

with a 2x 2 matrix B(t) whose entries are polynomials in t. Show that the degree of the polynomial B(t) is not less than 2. Also, show that there exists P(t) such that the degree of the polynomial B(t) is equal to 2.

Hint. To prove that there exists P(t) such that the degree of the polynomial B(t) -17+ at is equal to 2, apply Theorem V-5-1 to the system t d'r = 1 +tbt ] v with t .ii L suitable constants a, Q, y, and b. To show that the degree of the polynomial B(t) is not less than 2, assuming that the degree of the polynomial B(t) is less than 2, derive a contradiction from the following fact:

402

XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

The transformation y = 0

0

dy

rl+x'1

dx

l x-3

dil dx =

[lx-I

u changes the system 0

1

xtJ',

to

1-x-11

u,

u = IuaJ.

XII-15. Let A(t) be a 2 x 2 triangular matrix whose entries are holomorphic in a disk it : Its < po}. Show that there exists a 2 x 2 matrix Q(x) such that (i) the entries of Q(x) and Q(x)-I are holomorphic for jxi > o and meromorphic at x = oo, (ii) the transformation y" = Q(x)t changes the system dx = xkA

\ x/

y to

_

xkB 1 f u" with a 2 x 2 matrix B(t) whose entries are polynomials in t and whose degree in t is at most k + 1.

Hint. See IJLP]. This result can be generalized to the case when A(t) is a 3 x 3 matrix (cf. [Ball] and [Bal2j).

CHAPTER XIII

SINGULARITIES OF THE SECOND KIND

In this chapter, we explain the structure of asymptotic solutions of a system of differential equations at a singular point of the second kind. In §§XIII-1, XIII-2, and XIII-3, a basic existence theorem of asymptotic solutions in the sense of Poincar6 is proved in detail. In §XII-4, this result is used to prove a block-diagonalization

theorem of a linear system. The materials in §§XIII-I-XIII-4 are also found in [Si7J. The main topic of §XIII-5 is the equivalence between a system of linear differential equations and an n-th-order linear differential equation. The equivalence is based on the existence of a cyclic vector for a linear differential operator. The existence of cyclic vectors was originally proved in [De1J. In §XIII-6, we explain a basic theorem concerning the structure of solutions of a linear system at a singular point of the second kind. This theorem was proved independently in [Huk4] and (Tull. In §XIII-7, the Newton polygon of a linear differential operator is defined. This polygon is useful when we calculate formal solutions of an n-tb-order linear differential equation (cf. [Stj). In §XIII-8, we explain asymptotic solutions in the

Gevrey asymptotics. To understand materials in §XIII-8, the expository paper [Ram3J is very helpful. In §§XIII-I-XIII-4, the singularity is at x = oo, but from §XIII-5 through §XIII-8, the singularity is at x = 0. Any singularity at x = oo is changed to a singularity of the same kind at = 0 by the transformation x = I

XIII-1. An existence theorem In §§XIII-1, XUI-2, and XIII-3, we consider a system of differential equations (XIII.1.1)

= x'f,(x,L'i,v2,... ,un)

(.7 = 1,2,... n),

where r is a non-negative integer and ff (x, v1, t,2,... , vn) are holomorphic with respect to complex variables (x, v1, v2, .... vn) in a domain (XIII.1.2)

JxJ>No,

JargxI No,

J argxj < cto,

where coefficients fjk(v-) are holomorphic in the domain (XIII.1.6)

jvl < bo.

Furthermore. we assume that

fio(d) =

(XIII.1.7)

0

(j = 1, 2,... , n).

Observation XIII-1-1. Under Assumption I, fo(x), ajh(x), and fl,(x) admit asymptotic expansions oe

00

f70kx-k,

f}0(x)

fjp(x) ti

1: fjpkx-k, k=0

k=1

ajhkx-

a,h(x) ti00E k=0

(j, h = 1, 2,... , n)

as x oo in sector (XIII.1.5) with coefficients in C. Let A(x) be then x it matrix whose (j, k)-entry is ajk(x), respectively (i.e., A(x) = (a,h(x)). Then, A(x) admits an asymptotic expansion 00

(XIII.1.10)

A(x) '=- E x-kAk k=0

as x - oo in sector (XIII.1.5), where Ak = (ajhk). The following assumption is technical and we do not lose any generality with it.

Assumption II. The matrix A0 has the following S-N decomposition: (XIII.1.11)

Ao = diag [µ1,µ2, ... 44n] + N,

where µ1,µ,2, ... , µ are eigenvalues of Aa and N is a lower-triangular nilpotent matrix.

Note that WI can be made as small as we wish (cf. Lemma VII-3-3). The following assumption plays a key roll.

405

1. AN EXISTENCE THEOREM

Assumption III. The matrix A0 is invertible, i.e., p,54 0

(XIII.1.12)

(j=1,2,...,n).

Set

w = arg x,

(XIII.1.13)

1 ,2 , .n ) .

w , = arg p3

and denote by D, (N,'y,q) the domain defined by D3 (N, y, q)

(XIII.1.14)

l

x:

Ixi > N, jwj < ao,

(2q_)r+Y N is contained in the interior of V., (N, y, qj). Set n

(XIIL1.15)

7)(N,7) = n V, (N,y,qj). =1

In §§XIII-2 and XIII-3, we shall prove the following theorem.

Theorem XIII-1-2. If N-1 and y are sufficiently small positive numbers, then under Assumptions 1, II, and III, system (XIII. 1.1) has a solution (XHI.1.16)

U = 1,2,... , m),

vj = pj (x)

such that (i) p,(x) are holomorphic in D(N,-y), (ii) p, (x) admit asymptotic expansions 00

(XIII.1.17)

Pi (z)

>2 P,kx-k

(j = 1, 2, ... , m)

k=1

as x - oo in 7)(N,7), where p,k E C. To illustrate Theorem XIII-1-2, we prove the following corollary.

Corollary XIII-1-3. Let A(x) be an n x n matrix whose entries are holomorphic and bounded in a domain Ao = {x : lxi > Ro} and let f (x) be a C'-valued function whose entries are holomorphic and bounded in the domain Do. Also, let A1,A2, ... An be eigenvalues of A(oo). Assume that A(oo) is invertible. Assume also that none of the quantities Aje1ka (j = 1, 2,... , n) are real and negative for a real number 9 and a positive integer k. Then, there exist a positive number e and a solution y"= J(x) of the system d-.yi =

xk-1A(x)y+x-1 f(x) such that the entries of

XIII. SINGULARITIES OF THE SECOND KIND

406

O(x) are holomorphic and admit asymptotic expansions in powers of x-1 as x -+ 00 in the sector S = {x : IxI > Ro, I argx - 01 < Zk + e}. Proof.

The main claim of this corollary is that the asymptotic expansion of the solution is valid in a sector I argx-8 < 2k +e whose opening is greater than k So, we look at the sector D(N, y) of Theorem XIII-1-2. In the given case, ao = +oo, r = k -1, and µ, = Aj (j = 1,... , n). The assumptions given in the corollary imply that

w, + k6 -A (2p + 1)7r,

where wj = arg Aj (j = 1, ... , n),

for any integer p. Therefore, for each 3, there exists an integer q j such that

either - it < w, - 2qjr, + k8 < 0

or

0 < w, - 2gjir + k8 < rr.

Therefore, either

- 3r.

< wj - 2q, 7r + k8 - 2 < w) - 2q,rr + kO +

2

0 and a sufficiently small e > 0 if we use xe-i8 as the independent variable instead of x.

XIII-2. Basic estimates In order to prove Theorem XIII-1-2, let us change system (XIII-1-1) to a system of integral equations.

Observation XIII-2-1. Expansion (XIII.I.17) of the solution pj (x) 00

(XIII.2.1)

Epjkx_k

uJ =

(j = 1,2,... n)

k=1

must be a formal solution of system (XIII.1.1). The existence of such a formal solution (XIII.2.1) of system (XIII.1.1) follows immediately from Assumptions I and III. The proof of this fact is left to the reader as an exercise.

2. BASIC ESTIMATES

407

Observation XIII-2-2. For each j = 1,2,... , n, using Theorem XI-1-14, let us construct a function z3(x) such that (i) z j (x) is holomorphic in a sector (XIII.2.2)

I argxl < ao,

In l > No,

where No is a positive number not smaller than No,

(ii) zj (x) and

(XIII.2.3)

dz) admit asymptotic expansions z,(x)

L.f,kx-k

dz)

and

k=1

E(-k)P,kx-k-1

k=1

as x - oo in sector (XIII.2.2), respectively. Consider the change of variables

v, = u, +z,(x)

(XIII.2.4)

(j = 1,2,... ,n).

Denote (zl, z2, ... , z,a) and (u1, U2,. , isfies the system of differential equations -

du,

(XIII.2.5)

-

= x'g,(x,U-)

by i and u, respectively. Then, u' sat-

(.7 = 1,2,... ,n),

where dx Set

m (XIII.2.6)

g,(x, u") = go(x) + >2 b,k(x)uk +00E b,p(x)iI k=1

(j = 1, 2,... , n).

jp1>2

In particular, (XIII.2.7)

92o(x) = f, (x, a) - x-' d

dxx)

0

(j = 1, 2,... , n)

and

b3k(x)-ajk(x)=O(IxI-')

(j,k=1,2,...,n)

as x -+ oo in sector (XIII.2.2). Thus,

()III.2.8)

b,k(x) = a,k(oc) +o(IxI-1)

as x -' oo in sector (XIII.2.2).

(j, k = 1,2,... , n)

408

XIII. SINGULARITIES OF THE SECOND KIND

Observation XIII-2-3. Set

(j = 1,2,... ,n).

g1(x,u") = u, uj +R1(x,u')

(XIII.2.9)

Then, by virtue of Assumption III, (XIII.2.7), and (XIII.2.8), for sufficiently small positive numbers No 1 and 5, there exists a positive constant c, independent of j, such that for any positive integer h, the estimates (XIII.2.10)

+bhlxl-(n+1)

lR3(x,u)I ( clui

(j = 1,2,... n)

and (XIII.2.11)

IR,(x,u) - R, (x,1U') I < clu"- u l

(j = 1,2,... ,n)

hold whenever (x, u") and (x, u"') are in the domain (XIII.2.12)

lxl > No,

luil N,

-t < argx < e',

409

2. BASIC ESTIMATES

where t and f are positive constants. It is noteworthy that if x E D(N, y), then x satisfies the inequalities (XIII.2.17)

argxI N, x E S(p1, p2), 1141 < ul, 0 < IEI < e1,1 arg el < p0},

p1 and E1 are sufficiently small,

(ii) the solution ¢(x, µ, e) admits an uniform asymptotic expansion +oo

F, x-k&(µ, e)

as

x -. oo

k=1

in So, where coefficients &(µ,e) are bounded, holomorphic, and admit uniform asymptotic expansions in powers of a as c -+ 0 in the domain {(µ, e) Iµ1 < {31, 0 < lei < el, I argel < po}, (iii) the solution 1'(x, N, e) admits an uniform asymptotic expansion: +oo

(x, , e) '

ehWh(x, µ)

as

f -4 0

h=0

in So, where coefficients >Gh(x,#) are bounded, holomorphic, and admit uni-

form asymptotic expansions in powers of x'1 as x - oo in the domain {(x,µ') : Ixf > N, x E S(p1,p2), Jill < p1}. FLrther more, there exist functions tc(x, µ, e) for t = 0,1, ... such that (a) these functions ¢'c satisfy conditions (i) and (is) given above, (b)

h=0

A complete proof of Theorem XIII-3-4 is found in [Si7J and (Si101.

420

XIII. SINGULARITIES OF THE SECOND KIND

Remark XIII-3-5. In the proof of Theorem XHI-1-2, we used the assumption that the matrix Ao on the right-hand side of (XIII.1.10) is invertible (cf. Assumption III of §XIII-1). Without such an assumption, we can prove the following theorem. Theorem XIII-3-6. Let F(x, yo, yr, ... , yn) be a nonzero polynomial in yo, yl, 00

,

an,x-°`

yn whose coefficients are convergent power series in x'1, and let p(x) = M--0

E C[(x-1)) be a formal solution/ of the differential equation (XIII.3.3)

F(x,y,Ly,..., fn)=0. \arg

Then, for any given direction x = 0, there exist two positive numbers 6 and a and a function 0(x) such that (i) ¢(x) is holomorphic in the sector S = {x E C:0< [xj No,

f argx[ < ao,

where No and ao are positive numbers. Assume that the matrix A(x) admits an asymptotic expansion in the sense of Poincare 00

(X1II.4.3)

A(x) E x'"A &'=o

as x - oo in sector (XIII.4.2), where coefficients A are constant n x n matrices. Suppose also that Ao = A(oo) has a distinct eigenvalues A1, A2, ... , J1t with mul-

tiplicities n1, n2, ... , nt, respectively (nl + n2 +

+ nt = n). Without loss of

generality, assume that Ao is in a block-diagonal form: ()UII.4.4)

AO = diag [Al, A2, ... , At) ,

4. A BLOCK-DIAGONALIZATION THEOREM

421

where Aj are of x n, matrices in the form A,

(XIII.4.5)

(j=1,2,...,1).

=A,In, +Nj

Here, A , is a lower-triangular nilpotent n, x n, matrix. The main result of this section is the following theorem (cf. (Si7J).

Theorem XIII-4-1. Under the assumption (XIIL4.3) and (XIII.4.4), there exists an n x n matrix P(x) whose entries are holomorphic in a sector (XIII.4.6)

I argxj < al,

IxI > N1,

where Ni 1 and a1 are sufficiently small positive numbers, such that (i) P(x) admits an asymptotic expansion 00

(XIII.4.7)

P(x)

Ex-"p-

(Pp = In),

"=0

as x -' oo in sector (X111.4.6), where coefficients P" are constant n x n matrices,

(ii) the transformation y = P(x)zi

(XIII.4.8) reduces system (X111.4.1) to

di

(XIII.4.9)

,

= x B(x)i,

where B(x) is in a block-diagonal form (XIII.4.10)

B(x) = diag [BI (x), B2(x),... , Bt(x)J

.

Hen, B,(x) an n, x n. matrices and admit asymptotic expanswns 00

(XIII.4.11)

B, (x) ^-- E xBj" "=o

as x

oo in (XIII.4.6), where coefficients B,, are constant n) x n) matrices.

Proof.

From (XIII.4.1), (XIII.4.8), and (XIII.4.9), we derive the equation (XIII.4.12)

dP

= xr(A(x)P - PB(x)]

that determines the matrices P(x) and B(x). Set (XIII.4.13)

A(x) = Ao + E(x),

B(x) = A0 + F(x),

P(x) = In + Q(x).

XIII. SINGULARITIES OF THE SECOND KIND

422

Then, E(x) = O(x-1), F(x) = O(x'1), and Q(x) = O(x-1). Furthermore, (XIII.4.12) becomes dQ = x''[AoQ - QAo + E - F + EQ - QF]. dx

(XIII.4.14)

Write each of three matrices E(x), F(x), and Q(x) in a block-matrix form according to that of Ao in (XIII.4.4), i.e., (XIII.4.15)

F(x) = diag]Fl,F2,... ,Ft], Ell E11 E12 ... ... E2t E21 E22

Q11

E(x) _

Q12

...

Qlt

Q(x) = Ell

E2

Eu

.

Qtt where EJk and QJk are nJ X nk matrices and F, are nJ x nJ matrices. Set

QJJ = 0

(XIII.4.16)

Q11

Q2t

(j = 1,2,... ,t).

From (XIII.4.4), (XIII.4.14), (XIII.4.15), and (XIII.4.16), it follows that

(j=1,2,...,t)

F,=EJJ+FEJhQh,

(XIII.4.17)

h#J and

(XIII.4.18)

dQjk

= x' [43Qik - QJkAk + EJk +

EJhQhk - QJkAk]

(j j4 k).

h*k

Substituting (XIII.4.17) into (XIrII.4.18), a system of nonlinear differential equations dQJk dx

(XIII.4.19)

= x' LAJQJk - QJkAk +

EJhQhk h*k

{

- QJk (Ekk +

EkhQhk) + EJkJ

(j # k)

h*k

is obtained. Since it is assumed that .11i ... , Al are distinct eigenvalues of Ao and that A0 is in the block-diagonal form (XIII.4.5), upon applying Theorem XIII-1-2 to (XIII.4.19) we can construct a desired holomorphic solution Qjk(x) of (XIII.4.19) 00

which admit an asymptotic expansion Qjk(x) > x "Qjk" (j, k = 1, 2,... , 3; j "=1

k), where Qjk" are constant nj by nk matrices. Defining Fj by (XIII.4.17) and then B(x) by (XIII.4.13), the proof of Theorem XIII-4-1 is completed. 0

Theorem XIII-4-1 concerns the behavior of solutions of system (XIII.4.1) near x = oo. Since it is useful to give a similar result concerning behavior of solutions near x = 0, we consider, hereafter in this section, a system of differential equations (XIII.4.20)

xa+1 dY = A(x)y,

423

4. A BLOCK-DIAGONALIZATION THEOREM

where d is a positive integer and the entries of n x n matrix A(x) are holomorphic in a neighborhood of x = 0. Also, assume that A(0) is in a block-diagonal form

()III.4.21)

+N2,...

A(0) = diag[A11,,,

+JVt],

where a1, ... , .1t are distinct eigenvalues of A(0) with multiplicities n1, n2, ... , nt, + nt = n), and, for each j, .W is a lower-triangular and respectively (n1 + n2 +

nilpotent nj x nj matrix. Comparing the present situation with that of Theorem XIII-4-1, we notice the following two differences:

(a) singularity is at x = 0 in the present situation, while singularity is at x = 00 in Theorem XIII-4-1,

(b) The power series expansion of A(x) is convergent in the present situation, while A(x) in Theorem XIII-4-1 admits only an asymptotic expansion in a sector containing the direction arg x = 0. We can change any singularity at x = 0 to a singularity at x = oo by changing 1. Also, any direction argx = 8 can be changed the independent variable x by x to the direction arg x = 0 by rotating the independent variable x. Furthermore, the asymptotic expansion P of Po(x) and the expansion b of Be are formal power series satisfying the equation xd+i dP = AP - PB. This implies that two matrices P and b are independent of P. Hence, using Corollary XIII-1-3, the following result is obtained.

Theorem XIII-4-2. Let A(x) be an n x n matrix whose entries are holomorphic in a neighborhood of x = 0. Also, let d be a positive integer. Assume that the matrix A(0) is in block-diagonal form (XIII.4.21), where al, ... , at are distinct eigenvalues of A(0) with multiplicities n1, n2, ... , nt, respectively (n1 + n2 +, + nt = n), and for each j, JVj is a lower-triangular and nilpotent n, x nj matrix. Fix a real number 0 so that (a, - Xk)e-utO V EP for j 96 k. Then, there exists two positive numbers be

and ee and an n x n matrix Pe(x) such that (a) the entries of Po(x) are holomorphic and admit asymptotic expansions in powers of x as x - 0 in the sector Se = Ix: 0 < IxI < be, I arg x - 01 < + to }, 00

(b) if >2 xmP,,, is the asymptotic expansion of Pe(x), then this expansion is m=0

independent of 8 and Po = I,,,

(c) the transformation (XIII.4.22)

PB(x)u" changes system (XIIL4.20) to a system xd+1

= Be(x)u,

where the matrix Be(x) is in a block-diagonal form

Be(x) = diag (B1e(x), B2e(x), ... , Bte(x)] For each,, Bje is an n, xn2 matrix which admits also an asymptotic expansion (XIII.4.23)

in x as x--i0 in So. The main claims of this theorem are

XIII. SINGULARITIES OF THE SECOND KIND

424

(i) the asymptotic expansion of Pe(x) is independent of 0, (ii) the opening of the sector So is larger than k.

Proof of Theorem XIII-4-2 is left to the reader as an exercises.

Remark XIII-4-3. Using Theorem XIII-3-4, we can generalize Theorem XIII4-1 to the system e°dx = xr'A(x, µ, e)y", where r and o are non-negative integers, y" E C^, and A(x, µ', e) is an n x n matrix with the entries that are bounded and holomorphic with respect to the variable (x, µ, e) E C x C'° x C in a domain Do = {(x, µ, e) : 1x1 > N, Iµ1 < µo, 0 < 1e1 < eo, 0 < 1 arg e1 < po}. Also, assume 00

that A admits a uniform asymptotic expansion A(x, µ, e) - > ehAh(x, µ) in Do h=0

as a -+ 0, where coefficients Ah (X, µ) are bounded and holomorphic in the domain {(x,µ) : 1x1 > N, 1µ1 < µo}. A complete proof of this result is found in [Si7] and [Hs2].

XIII-5. Cyclic vectors (A lemma of P. Deligne) In the study of singularities, a single n-th-order differential equation is, in many

cases, easier to treat than a system of differential equations. In this section, we explain equivalence between a system of linear differential equations and a single n-th-order linear differential equation. Let us denote by 1C the field of fractions of the ring C[(x]] of formal power series in x, i.e., K=

I

p E C[Ix]J, 9 E C[Ix]j, 9 34 0} 9:

Also, denote by V the set of all row vectors (cl(x), c2(x), ... , c,,(x)), where the entries are in the field K. The set V is an n-dimensional vector space over the field 1C.

Define a linear differential operator C : V - V by G[vl = by + 7l(x) (v' E V ), where b = x and S2(x) is an n x n matrix whose entries are in the field 1C. We

d

first prove the following lemma.

Lemma XIII-5-1 (P. Deligne [Del]). There exists an element iio E V such that {v"o, Gv"o, G2v"o,

... , G"-lvo} is a basis for V as a vector space over IC.

Proof.

For each nonzero element v of V, denote by µ(v'' the largest integer t such that {ii, Cii, C2v, ... ,.CeV) is linearly independent over K. In two steps, we shall derive

a contradiction from the assumption that max{µ(v') : v' E V} < n - 1.

5. CYCLIC VECTORS

425

Step 1. First, we introduce a criterion for linear dependence of a set of elements of V. Consider a set {i11, ... vm } C V, where m is a positive integer not greater than

n = dim, V. Let 6j = (c31,c12,... ,Cin) (j = 1,2,... ,m). Set .7 = {(jl,... ,jm)

1 < it < j2 < . < jm < n}, and introduce a linear order .7 -' { 1, 2,... , (M-)) in the set J. Let us now define a map

(.) Vm={(v1,V2,...,Vm): 61 EV (j=l,...,m)} -+ Id.) (VI, ... , Vm) --+ V1 A v2 A ... A vm, where

1,AV2A ... A vm

f

cI31

det C-i 1

Cj32

...

clim

:

:

Cmj2

C-3-

(jl,... ,jm) E .7

:

A v,,,:

It is easy to verify the following properties of V1 A v"2 A

1m_1 A Um

Vk_1 A

=

(1)

Vk_1 A

+

Vm_1 A v"m

v'k_1 A

Vm_1 A Vm,

V1 A ... A Vk_1 A (a Vk) A vk+I A ... A Vm_1 A Vm (2)

=

6k_1 A irk

A vm),

for all aEIC,

VIA.-.A irk A ...A...AVjA ... A Vm = -(11A...A iY A

(3)

A Vm),

(4) a sety{ {6j, 62,... , Vm } E V is linearly dependent if and only if v'1 AV2 A 6 in K('^).

A 17m =

Step 2. Fix an element Vo of V such that µ(6o) = max{p(V) : v" E V} < n-1. Since p(VO) < n-1, another element w of V can be chosen so that {VO, Cv"o, CZi3o, ... , CnOv',

w"} is linearly independent, where no = max{µ(v") : v E V}. Set v' = vo+Ax"'O E V,

where A E C and m is an integer. Then, Cwt = Cw"o + C'(Ax-ti) = Crvo + Axm(C + m)tw. Since {V, Cv", ... , Cn0+1V} is linearly dependent, it follows that v A DU A A CnOv' A C"O+I V = 6 for all A E C and all integers in. Note that v A,06 A A C"OiYA C"°+16 is a polynomial in A. Since this polynomial is identically zero, each coefficients must be zero. For example, the constant term of this polynomial is i6o A CVO A ... A 00+I V"o. This is zero since {iio, Ciio..... G"0+Iuo} is linearly dependent. Compute the coefficient of the linear term in A of the polynomial. Then, w A C60 A ... A Cna+1v'o + v'Y A ... A 00 Vo A (C +

m)n0,+1ti

no

+ 1:60A...ACi-IVoA (C+m)lw A CJ+'A A...ACnb+Ii = 0 J=1

XIII. SINGULARITIES OF THE SECOND KIND

426

identically for all integers m. The left-hand side of this identity is a polynomial in in of degree no + I. Hence, each coefficient of this polynomial must be zero. In particular, computing the coefficient of mn0+1, we obtain vOAGvOA

. A GnbtloAw' =

0. This is a contradiction, since {vo, Gvo, ... , L"Ovo, w} is linearly independent. This completes the proof of Lemma XIII-5-1. Definition XIII-5-2. An element vo E V is called a cyclic vector of G if {6o, Lu0, L2v'o, ... , Ln-1%) is a basis for V as a vector space over X.

Observation XI II-5-3. Let uo be a cyclic vector of Land let P(x) be the n x n maUo

trix whose row vectors are {vo,G'o,... ,L' 'io}, i.e., P(x) =

Gvo

. Then,

Gn-lvo

nv"o o

L21 yo

=

and, hence, setting A(x) = L[P(x)]P(x)-1 = bP(x)P(x)-l +

P(x)f2(x)P(x)-1, we obtain 0 0

1

0

0

1

0 0

..

...

0 0

A 0

ap

0 0 0 ... 1 al a2 a3 ... an-1

with the entries a, E C. Thus, we proved the following theorem, which is the main result of this section.

Theorem XIII-5-4. The system of differential equations y1

(XIII.5.1)

by' = fl(x)y", where g _ Y.

becomes

bu" = A(x)u,

(XIII.5.2)

if y is changed by u = P(x)y". System (XIll.5.2) is equivalent to the n-th-order differential equation n-1

(XHI.5.3)

bnq -

arbrq = 0, where q = yl. 1=o

5. CYCLIC VECTORS

427

Example XIII-5-5. (1) Let us consider the system y1

by" = 0,

(a)

y=

where

yn

The transformation

u = diag[l, X'... , xn-11y

(T)

changes system (a) to

bu = diag[0,1,... ,n - 1]u.

(E)

Further, the transformation 1

2 22

--

(r)

U

2n-1

changes (E) to the form 0

1

0

0

0

1

0 0

0

0 w,

0

0

ao

a1

0 a2

0 a3

1

...

an-1

where ao, a,,.. . , an-1 are integers. Hence, the transformation

w" =

1

1

1

...

1

0

1

2

..

n- 1

0

1

22

0

1

2n-1

...

(n - 1)2

... (n -

,x°-1

y

1)n-1

changes (a) to (E'). This implies that vo = (1, x, this case. (II) Next, consider the system (b)

diag [1, x,...

x2'. ..

xn-1) is a cyclic vector in

by = Ay,

where A is a constant diagonal n x n matrix. Choose a transformation similar to (T) of (a) to change system (b) to (E")

oiZ = A'u

XIII. SINGULARITIES OF THE SECOND KIND

428

so that A' is a diagonal matrix with n distinct diagonal entries. Then, a transformation similar to (T') can be found so that (E") is changed to (E') with suitable constants ao, al, ... , an_ I .

XIII-6. The Hukuhara-Turrittin theorem In this section, we explain a theorem due to M. Hukuhara and H. L. TLrrittin that clearly shows the structure of solutions of a system of linear differential equations of the form (XIII.6.1)

where the entries of the n x n matrix A are in K (cf. §XIII-5). In order to state this theorem, we must introduce a field extension f- of K. To define L, we first set

+a E a.nxm1" : a.., E C and M E Z M=M

I

,

+a where 7L is the set of all integers. For any element a = F a,nxn'/° of K,,, we M=M +00

define x

ji by x da = F M=M

\ v) amx'nl '. Then, K, is a differential field. The field

+oo

L is given by L = U K which is also a differential field containing K as a subfield. L=1

Furthermore, L is algebraically closed. The Hukuhara-T rrittin theorem is given as follows.

Theorem XIII-6-1 ({Huk4j and [Tu1j). (XIII.6.2)

There exists a transformation

y' = UI

such that

(i) the entries of the matrix U are in L and det U ;j-1 0, (ii) transformation (XIII.6.2) changes system (X111.6.1) to (XIII.6.3)

xd

= Bz',

where B is an n x n matrix in the Jordan canonical form (XIII.6.4)

B=diag[Bl,B2,...,Bpj, Bj=diag[BJi,Bj2i...,B,,n,], Bjk =AI In,,. + Jn,,,.

429

6. THE HUKUHARA-TURRITTIN THEOREM

Here, I,,,, is the njk x njk identity matrix, J.,,, is an n 1k x n3k nilpotent matrix of the form

(XIII.6.5)

0 0

1

0

0

1

0

0

0

0

0

0

Jn",

and the Aj are polynomials in xfor some positive integer s, i.e., d,

(XIII.6.6)

Aj = > A

x-'18

where

all E C

(j = 1, 2, ... , p)

r=O

and (XII1.6.7) A,d,

0

if dJ > 0

A., - A, are not integers if i 0 j.

and

Proof.

Without loss of generality, assume that the matrix A of system (XIII.6.1) has the form 0 0

0 0

1

0

0

1

0 0

...

0

0

...

1

a1

0 a2

0

00

a3

"'

an-1

A

where ak E 1C (k = 0, 1, ... , n - 1) (cf. Theorem XIII-5-4). Set E= A Then,

an- 1

n In.

trace [E] = 0.

(XIIi.6.8)

Consider the system

X E = E.

(XIII.6.9)

Case 1. If there exists an n x n matrix S with the entries in X such that det S 0 0 dii = A(x)u", where the and the transformation w = Sii changes (XIII.6.9) to x entries of A are in C[[x]], then there exists another n x n matrix S with the entries in X such that det S 96 0 and the transformation w = Suu (XIII.6.10) dig

= Aoii, where the entries of the matrix A0 are in C. changes (XIII.6.9) to x F irthermore, any two distinct eigenvalues of A0 do not differ by an integer (cf. Theorem V-5-4). Hence, in this case, system (XIII.6.1) is changed by transformation (XIII.6.10) to dil

xaj = [an 1 In+Ao]u.

This proved Theorem XIII-6-1 in this case.

XIII. SINGULARITIES OF THE SECOND KIND

430

Case 2. Assume that there is no n x n matrix S with entries in X such that det.S 0 and the transformation w = Su' changes (XM.6.9) to xdu = A(x)u, where the entries of A are in C([x]]. Since 1

0

0

0

1

0

0

- ln an_1 E = 0

0

0

ap

al

a2

- ln an_1

1

an-2

a3

an-1 --

-an-1,

a cyclic vector can be found for system (XIII.6.9) by using the matrix W defined by

win

w11

with writ

Wnn

[wll ... Win] = (10 ... 01, [w,,1

... wJn] = V1-1[1 0

01,

cn]) = x[c1 - - cn]E. The matrix W is lower[c1 triangular and the diagonal entries are {1, ... ,1}, i.e., where V([c1

1

(XIII.6.11)

0 1

W=

0 0

0 0

1

If (XIII.6.9) is changed by the transformation v" = Wtv", then (XIU.6.12)

X!LV

_

x

+ WEJ W-Y.

It follows from (XIII.6.8) and (XIII.6.11) that ( XIII . 6 . 13 )

t racel x

+ WE]W- l =

0.

Also,

x 11

0 0

1

0

0

1

0

0

0

0 0

0 0

+ WE] W-1 = V[ W]W-1

W i31 /32

0

l

22

An-1

A

6. THE HUKUHARA-TURRITTIN THEOREM

431

where 1k E )C (k = 0, 1,... , n - 1). In particular, from (XIII.6.13), it follows that On-1 = 0. Under our assumption, not all Qt are in C[[x]]. Set 9 = (t: at V C[[x]]} +00

and set Also, /3t = x-u" E

Qtmxm

(t E 3), where, for each t, the quantity µt is a

m=0 +oo

positive integer, 1: /3tmxm E C[[xJ] and #to 54 0. Set m=0

k=max(n µt t

:IE91. J

Then, ut < k(n - t) for every t E J and µt = k(n - t) for some t E J. This implies that

Of =

(I)

L. m>-k(n-t)

Qt,mxm

(t=0,1,... n- 1)

and

Qt = x-k(n-t) (Ct + xqt)

(11)

for some t such that k(n - t) is a positive integer, ct is a nonzero number in C, qt E C[[x]], and Qt,m E C. We may assume without any loss of generality that k = h for some positive integers h and q. 9

Let us change system (XIII.6.12) by the transformation

v" = diag [1, x-k, ... , x-(n-1)kI u". Then, (XIII.6.14) where

(XIII.6.15)

0

1

0

0

0

0

1

0

F=

+ kxkdiag [0,1, ... , n 0

0

0

0

70

71

72

73

1]

and

7t = xk(n-t)o,

=:

m>-k(n-t)

In particular, 7n-1 = 0.

$,,mxm+k(n-t)

E C[ [x'1911

(0 < t < n - 1).

XIII. SINGULARITIES OF THE SECOND KIND

432

+oo

Setting F = E xm/9Fm, where the entries of F,,, are in C, we obtain m=o 0

1

0

0

C1

C2

C3

01

0

Fo= CO

...

0

Cn_2

where the constants co, cl ... , cn_2 are not all zero. This implies that the matrix F0 must have at least two distinct eigenvalues. Hence, there exists an n x n matrix T such that

(1) T =

XM19Tm, where the entries of the matrices Tm are in C and To is m=o

invertible,

(2) the transformation

y = Ti

(XIII.6.16)

xd

changes system (XIII.6.14) to a system

a block-diagonal form G = [

0

0

,

= x-kGxi with a matrix G in

where Gl and G2 are respectively

G2 J

n1 x nl and n2 x n2 matrices with entries in C[[x1/91and that nl + n2 = n and n., > 0 (j = 1, 2) (cf. §XIII-5). Therefore, the proof of Theorem XIII-6-1 can be completed recursively on n. 0 Observation XIII-6-2. In order to find a fundamental matrix solution of (XIII.6.1), let us construct a fundamental matrix solution of (XIII.6.3) in the following way:

Step 1. For each (j, k), set 4'1k = xA,0 eXp[AJ (x)] exp[(log x)Jf, ], where

-t/s

if

dj = 0,

if

d1>0.

Step 2. For each j, set

Dj = diag ['j1, where J. = diag [J,,,,, J,,72 , ... J,mJ ] .

xAJ0 exp[A,(x)] exp[(logx)Jj],

6. THE HUKUHARA-TURRITTIN THEOREM

433

Step 3. Set A

= ding [Al(x)In A2(x)I,,,, ... ,

C = diag [,\101., + J1, 1\20I., + J2, ...

,

Then, (XIII.6.17)

4) = diag [451, 4i2, ... , 4ip] = xC exp[A]

is a fundamental matrix solution of (XIII.6.3), where xC = diag [xA1OxJ, xl\2OxJ2'

... 'X aPOxJP]

,

xJ, = exp[(logx)JjJ.

The matrix (XIII.6.18)

U4i = Uxc exp[A]

is a formal fundamental matrix solution of system (XIII.6.1), where U is the matrix of transformation (XIII.6.2) of Theorem XIII-6-1. The two matrices exp[A] and xC commute.

Observation XIII-6-3. Theorem XIII-6-1 is given totally in terms of formal power series. However, even if the matrix A(x) of system (XIII.6.1) is given analytically, the entries of U of transformation (XIII.6.2) are, in general, formal power series in xl1", since the entries of the matrix T(x) of transformation (XIII.6.16) are formal power series in general. Transformation (XIII.6.16) changes system (XIII.6.14) to a block-diagonal form. Therefore, in a situation to which Theorem XIII-4-1 applies, transformation (XIII.6.2) can be justified analytically. The following theorem gives such a result.

Theorem XIII-6-4. Assume that the entries of an n x it matrix A(x) are holomorphic in a sector So = {x E C 0 < lxJ < ro, I argxi < ao} and admit asymptotic expansions in powers of x as x 0 in So, where ro and ao are positive numbers. Assume also that d is a positive integer and y" E C'. Let S be a subsector of So whose opening is sufficiently small. Then, Theorem XIII-6-1 applies to the system (XIII.6.19)

xd+1dy = A(x)f dx

with transformation (XIII.6.2) such that the entries of the matrix U of (XIII.6.2) are holomorphic in S and each of them is in a form x,00(x) where p is a rational number and O(x) admits an asymptotic expansion in powers of xl"° as x 0 in S, where s is a positive integer.

Observation XIII-6-5. In the case when the entries of the matrix A(x) on the right-hand side of (XIII.6.19) are in C[[x]l and A(0) has n distinct eigenvalues, the matrix A(x) also has n distinct eigenvalues A1(x), A2(x), ... , which are in C((xJJ. Furthermore, the corresponding eigenvectors p"1(x), p"2(x), ... ,15n(x) can be constructed in such a way that their entries are in Chill and that p""1(0),7"2(0), ... ,

XIII. SINGULARITIES OF THE SECOND KIND

434

p",,(0) are n eigenvectors of A(O). Denote by P(x) the n x n matrix whose column

vectors are p1(x), p2(x), ... , p,+(x). Then, detP(O) 36 0 and P(x)-lA(x)P(x) = diag[A1(x),A2(x),... ,An(x)]. This implies that the transformation y = P(x)ii changes system (XIII.6.19) to (XIII.6.20)

xd+1 dx

{diagiAi(x)A2(x).... , An(x)]

- xd+1P(x)-1 d ( )

It is easy to construct another n x n matrix Q(x) so that (a) the entries of Q(x) are in C[(x]], (b) Q(0) = 4,, and (c) the transformation i = Q(x)v changes system (XIII.6.20) to (XIII.6.21)

xd+1 dv

dx

= diag [ft1(x), µ2(x), ... , µn(x)] v,

where µ1(x), µ2(x), ... , i (x) are polynomials in x of degree at most d such that A, (x) = it) (x) + O(xd+1) ( j = 1, 2, ... , n). Therefore, in this case, the entries of the matrix U of transformation (XIII.6.2) are in 1C.

Observation XIII-6-6. Assume that the entries of A(x) of (XIII.6.19) are in C([x]]. Assume also that A(O) is invertible. Then, upon applying Theorem XIII-6-1

to system (XIII.6.19), we obtain following theorem.

d1 s

= d for all j. Using this fact, we can prove the

Theorem XIII-6-7. Let Q;i1(x) and Qi2(x) be two solutions of a system (XIII.6.22)

xd+1 dy = A(x)yf + x f (x),

dx

where d is a positive integer, the entries of the n x n matrix A(x) and the C"-valued function 1 *(x) are holomorphic in a neighborhood of x = 0, and A(O) is invertible. Assume that for each j = 1, 2, the solution ¢,(x) admits an asymptotic expansion

in powers of x as x - 0 in a sector S3 = {x E C : lxl < ro, aj < arg x < b3 }, where ro is a positive number, while aJ and bi are neat numbers. Suppose also that S1 n S2 0. Then, there exist positive numbers K and A and a closed subsector S = {x : lxl < R,a < argx < b} of Sl nS2 such that K exp[-Alxl -d] in S. Proof.

Since the matrix A(O) is invertible, the asymptotic expansions of1(x) and ¢2(x) are identical. Set 1 (x) = ¢1(x) -$2(x). Then, the C"-valued function >G(x) satisfies system (XIII.6.19) in S, n$2 and ii(x) ^_- 6 as x 0 in S1nS2. By virtue of Theorem XIII-6-4, a constant vector 66 E C" can be found so that r%i(x) = U4D(x)6, where lb(x) is given by (XIII.6.17). Now, using Observation XIII-6-6, we can complete the proof of Theorem XIII-6-7.

435

6. THE HUKUHARA-TURRITTIN THEOREM

Observation XIII-6-8. The matrix A = diag [AI In,, A2In3, ... , API,y] on the right-hand side of (XIII.6.18) is unique in the following sense. Assume that another formal fundamental matrix solution Uxcexp[A] of system (XIII.6.1) is constructed with three matrices U, C, and A similar to U, C, and A. Since the matrices Uxc exp[A] and UxO exp[A] are two formal fundamental matrices of sys-

tem (XHI.6.1), there exists a constant n x n matrix r E GL(n, C) such that Uxc exp[A] = U5C exp[A]I' (cf. Remark IV-2-7(1)). Hence, exp[A]T exp[-A] = x-CU-IUxC. Using the fact that r is invertible, it can be easily shown that A = A if the diagonal entries of A are arranged suitably. For more information concerning the uniqueness of the Jordan form (XIII.6.3) and transformation (XIII.6.2), see, for example, [BJL], [Ju], and [Leve].

Observation XIII-6-9. The quantities A.,(x) are polynomials in x1l'. Set w = 2a[!] and x 1/a = wx 1/a Then, i = x. Therefore, if z 1/e in Ur Cexp[A] is exp replaced by zI/', then another formal fundamental matrix of (XIII.6.1) is obtained. This implies that the two sets {Aj (i) : j = 1, 2,... , p} and (A.,(x) : j = 1, 2,... , p) are identical by virtue of Observation XIII-6-8.

Observation XIII-6-10. A power series p(x) in x1/' can be written in a form a-1

Ax) = Exh1'gh(x), where ql(x) E C[[x]] (j = 0, 1, ... , s - 1). Using this fact and h=0

Observation XIII-6-9, we can derive the following result from Theorem XIII-6-1.

Theorem XIII-6-11. There exist an integer q and an n x n matnx T(x) whose entries are in C[[x]] such that (a) det T (x) 96 0 as a formal power series in x, dil

(b) the transformation y" = T(x)t changes system (XII1.6.1) to x = E(x)iZ with an it x it matrix E(x) such that entries of x9E(x) are polynomials in x. The main issue here is to construct, starting from Theorem XIII-6-1, a formal transformation whose matrix does not involve any fractional powers of x in such a way that the given system is reduced to another system with a matrix as simple as possible. A proof of Theorem XIII-6-11 is found in [BJL]. Changing the independent variable x by x-1, we can apply Theorem XIII-6-1 to singularities at x = oo. The following example illustrates such a case.

Example XIII-6-12. A system of the form P(x) where P(x) = xm+

y = lyd,

0]b,

ahxin is a positive odd integer, and the ah are complex h=1

numbers, has a formal fundamental matrix solution of the form x

F(x)

1

0

0 f1 x-1/2 ] l I

1

-111

e

0

0 eE(t,a) I

XIII. SINGULARITIES OF THE SECOND KIND

436

where 1/2

m

+

E

k=1

+00

+ E bk(a) xk

ak xk

k=1

2

E(x, a) = (m 2) x(m+2)/2 + +

i

1( -1)h

54hh!r(h +)

[

is a formal solution of the differential equaton function.

d2-xZ

XIII-13. Show that the differential equation

xexp [_x3/2} 3

-xy = 0, where r is the Gamma-

d2-xZ

- xy= 0 has a unique solution

O(x) such that (1) b(x) is entire in x and (2) ¢(x)exp [3x3/21 admits the formal series p(x) exp [x3I2J as its asymptotic exansion as x -oc in the sector I arg xj