BASIC THEORY OF ORDINARY DIFFERENTIAL EQUATIONS
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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
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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
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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 and every t E Z, there exists a rational number r(t, b) E Q such that it - r(t, b)1 < b, (2) Q is an enumerable set; i.e., Q = {r, : j = 1, 2, 3, ... }. Note that the choice of the rational number r(t, 6) is not unique. We prove this lemma in two steps.
Step 1. The set .F contains an infinite sequence { fh : h = 1, 2.... } such that lim ofh(r) exists for every rational number r E Q. h
Proof.
Choose from F a sequence of subsequences F. 1, 2, ... , such that (i) .Fj +1 C .Fj for every j, (ii) t lim o f',,e(rl) = c, exists for every j.
1, 2.... },
j=
To do this, first look at functions in F at t = r1. Using the boundedness of the set {f(r1) : f E .F}, we choose an infinite subsequence F1 = {f 1.t : t = 1, 2,... } from .F so that lim f1,t(r1) = cl exists. Next, look at the sequence F1 at t = r2. 1
+00
Using the boundedness of the set { f l,t(r2) : t = 1, 2.... }, we choose an infinite
subsequence F2 = {j 2 t : t = 1,2.... }from F1 so that lim h,t(r2) = c2 exists. Continuing this way, we can choose subsequences .F,+1 from Y. successively.
Set fh = fh,h (h = 1, 2, 3.... ). Then, ft, E fl for every h >) since fh = fh,h E .Fh C .Fj if h > j. Therefore, lim fh(rj) = c, exists for every j, i.e., lira fh(r) h-+oo h-+oo exists for every rational number r E Q.
-
I. FUNDAMENTAL THEOREMS OF ODES
10
Step 2. The sequence { fh : h = 1, 2.... } of Step 1 converges uniformly on the interval I. Proof For a given positive number f and a rational number r E Q, there exist a positive number b(e) and a positive integer N(r, e) such that
it - rl < b(e),
lfh(t) - fh(r)I N(r, E)
1J(r) - f (t)I < E
whenever
and I > N(r, E),
it - rl < b(e).
Now, observe that
Ifh(t) - ft(t)1 N (r I t, 6 (3 bigg)) 3 I and t > N (r
(t, b (3 )
.3
)
Since the interval I is bounded, we can cover I by a ite number of open intervals 11, I2, ... ,1,(,) in such a way that the length of each interval I1 is not greater than b (11) , where p(E) is a positive integer depending on E. For every t,
/E \
choose a rational number rt(e) E 1t f1 Q. For all t E It, set r t,b1 3) I = rt(E) and set
1
N(e) = maxE N (r,(.E), 3 ) 1
Then, (1.2.1) is satisfied whenever h > N(e) and I > N(E). 0 To construct a solution of the initial-value problem (P) without the Lipschitz condition, approximate the given differential equation by another equation which satisfies the Lipschitz condition. The unique solution of such an approximate problem is called an E-approximate solution. Using the following lemma, an E-approximate solution is found.
Lemma 1-2-4. Suppose that f (t, y) is continuous on a rectangular region
R={(t,y): It - tol
a, Iy-colSERI If (-r, for It - tot < a and all f, (3)
F(t, yJ =
{f(t
on
R,
0
for
It-tot b+1.
The construction of such an f is left to the reader as an exercise. Since f is uniformly continuous on R, properties (1), (2), and (3) of f imply that (4) for every positive number e, there exists a positive number b(e) such that
IF(t,f1)-F(t,f2)i <e
whenever
It - tot < a,
Ig, - f2l < b(e).
Step 2. For every positive number b, there exists a real-valued function p(t, 6) of a real variable t such that (a) p has continuous derivatives of all orders with respect to for -cc < t < +oo,
(b) p(4, b) > 0 for -co < t < +oc,
(c) p(.,b)=0forI{I>b,
(d) f(.5)de = 1. ,o
The construction of such a function p is left to the reader as an exercise. Set 0(g,6) = p(yi, b)p(y2, b) ... p(y,,, b). Then (a') has continuous partial derivatives of all orders with respect to y1, y2,.,.
,y
for ally",
(b') t(#, 6)
0 for all
(c') 0(f, b) = 0 for Iyj > 6, (d')
r +00 ... f
f
+00
'(y, 6)dy1dy2 ... d1M = 1.
Step 3. Set
+- ... 4(t, YJ = 1-00
J+oo
(y - 1, b(E))F(t, ff)d ... dnn,
00
where b(e) is defined in (4) of Step 1. Then, F,(t, y-) satisfies all of the requirements (i), (ii), (iii), and (iv) of Lemma 1-2-4.
Proof Properties (i) and (ii) are evident. Property (iii) follows from the estimate
I
e(t,M1<max{IF(t,10i: all ;t} < iTMaExRIf(T,nl1.
j
00
I. FUNDAMENTAL THEOREMS OF ODES
12
Property (iv) follows from the estimate I A, (t, yi - F'(t, Y) I
=l
j + (y-;),b(E)){F(t,n7)-F(t,)}dg1...dg
r+oo
00
...
_
l
r
(9 - n,b(E)) {F(t,nj - F(t,y-) } dm ... dn.
E tI ... rIl ff
n, b(E))drh ... dnn l = E.
JW- l1 0, q > 0, and (is any real number.
Answer.
Assume that the solution y = 4(x) of the given initial-value problem exists on an interval I = {x : £ < x < xo} for some positive number . Observe that L2(.0')2+ 1.63 +
20-2]
= 2xo(th')2 > 0
on Z. Hence,
(O'(x))2
+ 14(x)-2 < 1(0'(xo))2 + 14(x0)3 + 14(x0)-2 + 14(x)3 2 2 3 2 3
(= Al > 0)
on Z. Therefore, we have (4 (x)) z
< 2M,
4(x)3 < 3M,
4(x)2 >
21Lt
Now, apply Lemma 1-3-1.
The proof of the following result is left to the reader as an exercise.
3. SOME GLOBAL PROPERTIES OF SOLUTIONS
17
Corollary 1-3-4. Assume that f (t, yam) is continuous fort I < t < t2 and all y E R". Assume also that a function fi(t) satisfies the following conditions: (a) and d+' are continuous in a subinterval I of the interval tl < t < t2, (b) (t) = f (t, ¢(t)) in Z. Then, either (i) Qi(t) can be extended to the entire interval t1 < t < t2 as a solution of the
differential equation
j
= f (t, y), or
(ii) limO(t) = oo for some r in the interval ti < r < t2.
t-r
Using Corollary I-3-4, we obtain the following important result concerning a linear nonhomogeneous differential equation dt = A(t)yy + bb(t),
where the entries of the n x n matrix A(t) and the entries of the R"-valued function b(t) are continuous in an open interval I = It : ti < t < t2}. Theorem 1-3-5. Every solution of differential equation (1.3.1) which is defined in a subinterval of the interval I can be extended uniquely to the entire interval I as a solution of (1.3.1).
Proof Suppose that a solution y = d(t) of (1.3.1) exists in a subinterval Z' _ It : r1 < t < r2} of the interval I such that tj < 71 < r2 < t2. Then,
At)I 0, there exists a real-valued function 0(x) such that (1) 0(x) is continuous on an interval 0 < C-x 0 and
if tl1 and tll are small, there exist two positive constants Ko and L such that 1.
om(Z)=co+J f((,0-m-1( ))d(, 0
where the path of integration may be taken to 1be the line segment zaz joining zo to
z in the complex z-plane. Set a = min I a,
M
}. Then,
(i) For every m, the entries of the C"-valued))) function 4m(z) are power series in z - zo which are convergent in the open disk A = {z : iz - zoj < a}, (ii) the sequence { ,,,(z) m = 0,1, ... } satisfies the estimates m !5 II ,c on 0, m = 0, 1, ... , IZ - zojm+l ,4n+l(Z) - om(Z)I (nl + 1),
(iii) the sequence {¢m(z) m = 0, 1,... } converges to ¢(z) = o(z) +
(mk(z)
-
(Z))
k=1
uniformly in 0 as m -+ +oo, (iv) the entries of the C"-valued function b(z) are power series in z - zo which are convergent in 0, (v) the function Q(z) is the unique solution to the initial-value problem
d = f(z,y),
y(zo)
The proof of this theorem is left to the reader as an exercise. The reader must notice that even if a sequence of analytic functions fn(t) of a real variable t converges to a function f (t) uniformly on an interval Z, the limit function f (t) may not be analytic at all. For example, any continuous function can be uniformly approximated by polynomials on a bounded closed interval. In order to derive analyticity of f (t), uniform convergence must be proved on a domain in the complex t-plane.
Observation 1-4-2. The estimates MLm
m+1(Z) - & (Z) 1 :5 (m+ 1)I IZ -
ZOlm+1
in A,
n1 = 0, 1,...
I. FUNDAMENTAL THEOREMS OF ODES
22
+00
imply that there exists a power series Eck(z - zo)k such that k=0 M
m=0,1,2,....
&(z) = E ck(z - z,)' +O(Iz - zol"'+1), k=O
Observation 1-4-3. Since m
&(Z) = $O(Z) + E (&(Z) k=1 00
we obtain (z) - ¢,,,(z) = O(Iz - zol'"+1) and, hence, (z) _ >2Ck(Z - ZO)k. k=0
Observation 1-4-4. Set m
00
Ck(Z - ZO)k + 1: Cm.k(Z - Zo)k.
(Z) =
k=m+1
k=0
Since
cb1 < b in the disk 0, it follows that
a
1Cm,kI
(k = m + 1, m + 2,...).
Note that
4.,k =
1
2?ri
0(z) - ca Izo1=P (z - zo) k+1
for any positive number p < a. This, in turn, implies that ,n
oo
IZ aZOlk
Ck(Z-ZO)kl :5 b k=m+1
k=O
=blzzD a
1 I
1-IZ ZIDI
in the disk A. Example 1-4-5. In the case of the initial-value problem dzy = (1 - z)y2, y(0) = 2, we obtain
O(z) = 2 E(k + 1)zk = k=O
2
(1 - Z)2
EXERCISES I
23
by using the successive approximations Oo(z) = 2
,
01(z) = 2 + 4z - 2z2,
83
02(x) = 2 + 4z + 6x2 -
- 6x4 + 4x5 2x4
¢3(z) = 2 + 4z + 6z2 + 8x3 _ 256z7
+
124z8
63 + 9
2825
82zs
3
68z9
46z10
9
9
9 4z13
22z12
3
23
9
+
56z" 9
2z14
+ 9 - 63
Combining Theorem 1-3-5 and Theorem 1-4-1, we obtain the following theorem:
Theorem I-4-6. If all the entries of an n x n matrix A(t) and C"-valued function b(t) are analytic on an interval Z, then every solution of linear differential equation (1.3.1) is analytic on Z.
EXERCISES I
I-1. Solve the initial-value problem d22 + 2y L = 0,
y(r) = ,lo,
b (r) = 171
The reader must consider various cases concerning the initial data (r, i
I-2 Show that the function f (x y) _ '
1
(3 - (x - 1)2)(9 - (y - 5)2)
Lipschitz condition
If(x,yl)-f(x,y2)I < IY1Y2I if
Ix-1I 5 f,
Iyi-5I < 2,
lye-51 < 2.
1-3. Show that the initial-value problem dy (P)
_
dx -
sm
x3 + 3x + 1
101-y2)
y(5) = 3,
has one and only one solution on the interval Ix - 51 < 7.
, m)1).
satisfies the
I. FUNDAMENTAL THEOREMS OF ODES
24
1-4. Suppose that u(x) is continuous and satisfies the integral equation x
u(x) = J sin(u(t))u(t)pdt 0
on the interval 0 < x < 1. Show that u(x) = 0 on this interval if p > 0. What would happen if p < 0?
u -p du = sin(u) dx, u(O) = 0. Thus, the solution u(x) is not identically zero and has a branch point Hint. In case p < 0, the problem is reduced to the initial-value problem
at x = 0. 1-5. Show that if real-valued continuous functions f (x), g(x), and h(x) satisfy the inequalities f
g(x) < h(x) +
f(x) ? 0,
J0*'C
on an interval 0 < x < xo, then g(x) < h(x) +
ono < x
¢(t, t), e)
and (t, ir(t, t), e) E Do
on Z x E.
Assume also that there exists a real-valued function t(e) such that
(1) t(e) E I fore E E, (2) lim t(e) = to, t-w (3) hm 0(t(e), e) = (to) Then:-to Then, lim i/i(t, t) it
to
uniformly on the interval Z. Proof.
We shall prove this theorem in two steps. In Step 1, we prove Theorem 11-1-2 assuming that there exists an open neighborhood U of Co such that (t, tb(t, e)) E 0 on Z x (E n U), where 0 is the open set in the (t, y)-space which was determined by Lemma II-1-1. The existence of such an open neighborhood U of to is proved in Step 2.
Step 1. We can assume without any loss of generality that U is contained in the neighborhood Uo of to which was determined by Lemma 11-1-1. Furthermore, using
condition (3) of Theorem 11-1-2, we choose U in such a way that It (t((),e)I is bounded on U.
II. DEPENDENCE ON DATA
30 Since
(II.1.2)
on Z x £,
tjJ(t, E) = tp(t(E), E) + / t As, j(s, e), E)ds JJJt(E) I
-
we obtain I!(t(E),E)I+MIt-t(E)I onIx(£fU) and Ii(t,E) AT, E)I MI t - rI if (t, c) and (r, E) E Z x (£ n U), where M is the positive constant which was determined by Lemma II-1-1. Therefore, the set F _ E) : E E £ n U} is bounded and equicontinuous on the interval Z.
Let us derive a contradiction from the assumption that ,JJ(t, e) does not converge to fi(t) uniformly on the interval I as e -+ co. This assumption means that max I does not tend to zero as e --+ co; i.e., there exists a positive number p and a sequence (E. : j = 1,2,. .. } such that Ej E£nU lim E; = Eo, and (II.1.3)
m Zx
1
(t,Ej) - v(t)I > p.
Here, use was made of the assumption that the topology of the E-space is separable.
Observe that the sequence { (-, j) : j = 1, 2.... } is a subset of Y. Hence, this sequence is bounded and equicontinuous on Z. Therefore, b y Lemma 1-23 (Arzela-Ascoli), there exists a subsequence v = 1, 2, ... } such that
j., -+ +oo as v - +oo and that tp(t, Ej converges uniformly on I as v -+ +oo. Set l(t) = lim r%(t, on Z. Since (11.1.2) holds for c = Ef for all v, we obtain
J(t) = $(to) + f f(s,rl'(s),EO)ds
on
I.
to
Hence,
f (t, J(t), Eo) on I and 1J(to) = ¢(to). Now, condition (iii) of As-
sumption 1 implies that j(t) = fi(t) identically on the interval Z. This, in turn, implies that lim max I (t, E j ) - $(t) I = 0. This contradicts (1I.1.3). v-+oo LET
Step 2. We shall prove that there exists an open neighborhood U of co such that
UClloand (t,1i(t,E))EAon Ix(EnU). Choose no > 0 so that the rectangular region
Ro(r) = {(t.: It - rl < ao, (y - (r)I %,(t(e), e)
- 4(to)I < M(ao - a) fore E E nU1.
Hence, R((t(e), t%i(t((), e)); a) C Ro(to) C 0i_(t(e), fore E E n U1. Now, using Lemma I-1e)); a) C i for It - t(e)I < a and 1, we can verify that (t, fl(t, e)) E R((t(e), e E EnU1. Also, since lim t(e) = to = rjo, we have It-t(e)I < a for rjo_1 < t < rjo+i
t- !o
and e E E n U1, if the neighborhood U1 of co is sufficiently small.
Using the same argument as in Step 1 for t (t, e) on the interval T,._ j 5 t < T3.+1
and e E E n U1, we obtain lim t(t, e) = fi(t) uniformly on rjo_ 1 < t < 7-,,.+,. In .1
to
particular,
11m1L(T)o_1ie) _ 0(r)o-1)
f
to
and
lim V(T,o+1, e) = d(rjo+1) (-to
Using the same method in the neighborhoods of
and (rio+1,
4(Tjo+i )), we obtain lim e-co1G(t, e) = Q(t) uniformly on rjo-2
-
t< - rjo and r
0 if of 0 for t C013
11-8. Let g(t) be a real-valued and continuous function oft on the interval 0 < t < 1. Also, let A be a real parameter. (1) Show that, if Q(t, A) is a real-valued solution of the boundary-value problem
Ez + (g(t) + A)u = 0,
u(0) = 0, u(1) = 0,
and if 8 &2 (t, A) is continuous on the region A = {(t, A) : 0 < t < 1, a < A < b}, then d(t, A) is identically equal to zero on A, where a and b are real numbers. (2) Does the same conclusion hold if 0(t,,\) is merely continuous on A? 11-9. Let a(x, y) and b(x, y) be two continuously differentiable functions of two variables (x, y) in a domain Do = {(x, y) : IxI < a:, IyI < Q} and let F(x, y, z) be a continuously differentiable function of three variables (x, y, z) in a domain Do = {(x, y, z) : Ixl < a:, IyI < 3, Izl < 7}. Also, let x = f (t, i7), y = 9(t, ij), and z = h(t,17) be the unique solution of the initial-value problem dt
= a(x,y),
x(0) = 0,
L = b(x,y),
= F(x,y,z),
y(0) = n,
z(0) = c(+1). where c(q) is a differentiable function of 11 in the domain 14 = (q: In1 < p). Assume that (t, rl) = (d(x, y), O(x, y)) is the inverse of the relation (x, y) = (f (t,17), g(t, rt)), where we assume that O(z, y) and i,1(x, y) are continuously differentiable with re-
spect to (x, y) in a domain Al = {(x, y)
:
IxI < r, IyI < p}. Set H(x,y) =
h(p(x, y), tp(x, y)). Show that the function H(x, y) satisfies the partial differential equation a(x, y)
+ b(x, y) M = F(x, y, H)
8 y) = c(y). and the initial-condition H(0,
EXERCISES II
39
Hint. Differentiate H(x,y) with respect to (x,y).
11-10. Let F(x, y, z, p, q) be a twice continuously differentiable function (x,y,u,p,q) for (x,y,u,p,q) E 1R5. Also, let x = x(t, s),
z = z(t, s),
y = y(t, s),
p = P(t, s),
of
q = q(t, s)
be the solution of the following system: dx dy
-
OF
_
OF
(x+ y, z, p, q), flq(T, y, z, p, q),
dt dz
OF
OF
dt = pij (x,y,z,P,q) + gaq(x,y,z,P,q),
dp dt
OF
dq dt
OF 8y
OF
8x (T, y, z, P, q) - P 8z (x, y, z, p, q), OF
(x, y, z, P, q) - q az (x, y, z. P, q)
satisfying the initial condition x(O,s) = xo(s),
y(O,s) = yo(s),
p(O,s) = Po(s),
q(O,s) = qo(s),
z(O,s) = zo(s),
where x0 (s), yo(s), zo(s), po(s), and qo(s) are differentiable functions of s on R such
that F(xo(s),yo(s),zo(s),Po(s),go(s))=0,
dzo(s)=Po(s)ds
(s)+go(s)dyo(s)
ds
ds
on R. Show that F(x(t, s), y(t, s), z(t, s), p(t, s), q(t, s)) = 0,
at (t, s) = At, s) 5 (t, s) + q(t, s) flz(t,s) fls
(t, s),
= P(t,s)L(t,s) + q(t,s)ay(t,8) as as
as long as the solution (x, y, z, p, q) exists.
Comment. This is a traditional way of solving the partial differential equation For more details, see (Har2, pp. F(x, y, z, p, q) = 0, where p = 8 and q = 131-143].
H. DEPENDENCE ON DATA
40
II-11. Let H(t, x, y, p, q) be a twice continuously differentiable function of (t, x, y, p, q) in RI. Show that we can solve the partial differential equation Oz
Z 8z\ =0 + H t, x, y,-,
by using the system of ordinary differential equations
8H
dx
dt = dp dt dz
dy 8p'
t=
8H q
,
8H (LL dq _ _ 8H W, 8x ' dt 8H du 8H
= u + p-p + q8q,
dt
OH _ = --6T.
CHAPTER III
NONUNIQUENESS
We consider, in this chapter, an initial-value problem dy
dt = f(t,y1,
(P)
without assuming the uniqueness of solutions. Some examples of nonuniqueness are given in §III-1. Topological properties of a set covered by solution curves of problem (P) are explained in §§III-2 and 111-3. The main result is the Kneser theorem (Theorem III-2-4, cf. [Kn] ). In §1II-4, we explain maximal and minimal solutions
and their continuity with respect to data. In §§III-5 and 111-6, using differential inequalities, we derive a comparison theorem to estimate solutions of (P) and also some sufficient conditions for the uniqueness of solutions of (P). An application of the Kneser theorem to a second-order nonlinear boundary-value problem will be given in Chapter X (cf. §X-1).
111-1. Examples In this section, four examples are given to illustrate the nonuniqueness of solutions of initial-value problems. As already known, problem (P) has the unique solution if f'(t, y) satisfies a Lipschitz condition (cf. Theorem I-1-4). Therefore, in order to create nonuniqueness, f (t, y-) must be chosen so that the Lipschitz condition is not satisfied. Example III-1-1. The initial-value problem dy dt
(III.1.1)
= yt/3
y(to) = 0
has at least three solutions
y(t) = 0
(S.1.1)
(-oo < t < +oc), 3/2
(S.1.2)
[3 t - to)
y(t) _
]I
,
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
0(s, e)
for
s - b(e) < t <s
I
111. NONUNIQUENESS
60
(cf. Figure 19).
FIGURE 19.
Note that
- g(a, j¢(o)j))do + e(t - s) t t'(t,e) - WWI > f [g(a,'p(a,e))
for
t>s
a
and
t)! C j [g(o, v(o, )) - g{O, (o )I)do - e(s - t)
for t
t
Thus, it is concluded that I (t) < t'(t,f) on Z. Letting a proof of Theorem III-5-1. 0
0, we complete the
Example 111-5-2. If an R"-valued function 0(t) satisfies the condition
dj(t)
< C + MI¢(t)t
dt
for
a0 andw(t)>0 on1. (3) g(t, u) is continuous on the set A = {(t, u) : 0 < u < w(t), t E Z}, (4) g(t, u) > 0 on A and g(t, 0) = 0 on Z, (5) the problem
(111.6.1)
du(t) = g(t,u(t)), dt aim u(t) = 0
(t,u(t)) E A
on Z,
(t, y"(t)) E Q
on Z,
has only the trivial solution u = 0 on Z. Let us consider a problem
(111.6.2)
d9(t) dt lim 19(01 = 0, t-.a r(t)
where S1 = {(t, y) : jyj < w(t), t E Z}. The main result of this section is the following theorem.
Theorem 111-6-1. If the function f (t, y-) is continuous on 1 and if I f(t, y-)I :5 g(t, Iyi)
on
Q,
then problem (111.6.2) has only the trivial solution y" = 6 on Z.
III. NONUNIQUENESS
62
Proof Suppose that problem (III.6.2) has a nontrivial solution j(t) on Z. This means
that (a)
,-a
ea+`
(IV.3.8)
=l
In + F, T! Nh
Pj (A).
h=1
Since the general solution of the differential equation (IV.3.9)
is given by (IV.3.7), the following important result is obtained.
Theorem IV-3-2. (i) If R(,\,) < 0 for j = 1, 2, ... , k, then every solution of (IV. 3.9) tends to 06 as t -' +00, (ii) if R(Aj) > 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 0. Hence, we define a as the order of the singularity of (V.7.7) at x = 0. System (V.7.2) can be reduced to an equation (V.7.7) (cf. §XIII-5). The following theorem due to J. Moser [Mo) concerns the order of a given singularity-
Theorem V-7-9. Let A(x) = x-a E x'A be an n x n matriz, where p is an =o
integer, A are n x n constant matrices, A0 y6 0, and the power series is convergent. Set
Im(A) = max (P_ 1 + n,0) , µ(A) =
minlm(T-1AT-T-1dx)J
136
V. SINGULARITIES OF THE FIRST KIND
Here, r = rank(Ao) and min is taken over all n x n invertible matrices T of the
form T(x) = x-9 E x°T,,, where q is an integer, T are constant n x n matrices, and the power series is convergent. Note that det T(x) 0 0, but det To may be 0. Assume that m(A) > 1. Then, we have m(A) > p(A) if and only if the polynomial
P(A) = vanishes identically in A, where In is the n x n identity matrix. The main idea is to find out p(A) in a finite number of steps. Using the quantity
p(A), we can calculate the order of the singularity at x = 0. The criterion for m(A) > p(A) is given in terms of a finite number of conditions on the coefficients of A(x). There is a computer program to work with those conditions. In particular, in this way, we can decide, in a finite number of steps, whether a given singularity at x = 0 is regular. Notice that IP(x)I can be estimated at z = 0 for the matrix P(x) of Theorem V-72, using similar estimates for fundamental matrix solutions of (V.7.2) and (V.7.4).
Therefore, an analytic criterion that a singularity of second kind at x = 0 be a regular singularity is that in any sectorial domain V with the vertex at x = 0, every solution fi(x) satisfies an estimate
c K[xj'
(x E V)
for some positive number K and a real number m. These two numbers may depend on 0 and V. In fact, Theorem V-7-2 and its remark imply that a fundamental matrix
solution of (V.7.6) is P(x)xA0. The matrix P(x) has, at worst, a pole at x = 0 if and only if !P(x)I < Kjxjr in a neighborhood of x = 0 for some positive number K and a real number p (cf. [CL, §2 of Chapter 4, pp. 111-1411). Another criterion which depends only on a finite number of coefficients of power series expansion of the matrix A(x) of (V.7.2) was given in a very concrete form by W. Jurkat and D. A. Lutz [JL] (see also 15117, Chapter V, pp. 115-1411). Now, let us look into the problem of convergence of the formal solution which was constructed in Theorem V-1-3. Let n
P = Eah(x)Jh h=0
be a differential operator with coefficients ah(x) in C{x). Assume that n > 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<x 0 (j = 1, 2) for x > 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),
VI. BOUNDARY-VALUE PROBLEMS
170
Step 2. Proof of (i) and (ii): To prove (i) and (ii), it suffices to prove that if
( 21x1 + 2
e(x) -
2
for
x < 0,
for
x > 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
M for some positive constant M, then f+(x, () = e'(= for x > M, while f- (x, () = e-K= for x < -M.
VI-6. Scattering data For real (, 1+ (x, () = f+ (x, -(), where f denotes the conjugate complex of f . Both f+ and f+ are solutions of Gy = (2y and
I f+(x, f) - e
ZI
< C(x)P(x)
dl (x, + i(e-+ dx
1 + ICI
r+
C() +
r))u(r)e(rdrl
(x) 16
for -oo < x < +oo and -oo < f < +oo. Let W(f,gj denotes the Vlwronskian of {f,g}. Then,
W If+, f+l - I f+(x, ()
f+, (x, ()
-2i(
(-oo
0 and (VI.6.4)
a(() = 1 + 0((-1)
as
(-' 00
on
`3(() > 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- -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-`<x = 1 -
(cf. (VI-7.1) and (VI.8.4)).
i=1
This implies that
lim f+(x,-()e"x = a(()a(-() = 1. lim f-(x,()e'S= = a(() x---oo t--oo
O
9. DIFF. EQS. SATISFIED BY REFLECTIONLESS POTENTIALS
Observation VI-8-6. Note that f_ (x, ) = a(() f+(x, -{) =
181
f+(x, ) for ( =
0. Hence, r(C) = 0. Thus, we conclude that u(x) is
C real. This means that reflectionless.
Remark VI-8-7. For the general r((), the potential u(x) can be constructed by solving the integral equation of Gel'fand-Levitan: +00 +
K(x, r) + F(x + r) +
F(x + r + s)K(x, s)ds = 0,
0
where
F(x) = 1R J
+
2cL` cle-2n,=
oo
J=1
Find K(x, r) by this equation. Then, the potential is given by u(x)
8x
(x,0).
If we set r({) = 0 in this integral equation, we can derive (VI.7.2) and (VI.7.3). Details are left to the reader as exercises (cf. (Ge1LJ ).
V1-9. Differential equations satisfied by reflectionless potentials Suppose that u(x) is a reflectionless potential whose associated scattering data are given by 0, (r11, r12, ... flN ), (cl, c2,... , cN) },where 0 < ril < ]2 < < rily. It was proven that one of the Jost solutions is given by N
f+(T,() = e`t=
1 - j_1i (+
Furthermore,
9j (T)
N f+(x, -() = e`= 1 -
g1 (r)
-( + trj7
is also a solution of
dX2
- (u(x) - A)y = 0, where J = (2.
Therefore, N
(VI.9.1)
P(x, A) = 11 ! 1+11- f+(T,C)f+(x, -() A 3=1
)
satisfies the differential equation 3
(E)
2 dz3
+ 2(u(x) - A) dP +
ddx
)P = 0.
VI. BOUNDARY-VALUE PROBLEMS
182
Let
+oo
P = E an
(S)
(PO 3
0)
n=0
be a formal solution in powers of A-1 of differential equation (E). Then,
0 = P [-- + 2(u(x) - A)P + u'(x)P P2
,
+ (4)2 + (u(x)
- A)P2 J
and, hence,
A(P2)' = I - PZ it + (4)2 + u(x)P2 }, . Therefore, (i) the coefficients pn can be determined successively by
po = ca, 2C Pn+l =
where co is a constant, n
n
n
t=o
t=o
c=o
uE pip.-t 2 E PtFri-1 + 4E Ptpn-t +
nPtPn+1-t r
1=1
(n = 0,1,2,...), (ii) the coefficients pn are polynomials in u and its derivatives with constant coefficients,
(iii) the formal power series (S) is uniquely determined by the condition P = 1 for u = 0, +00 G"( u) ( Go = 1), then the ()lv i f we denote the unique q P of (iii) by G u, A ) = An Y
(
n=0
general formal solution of (S) is given by P(u, A) = P(0, A)G(u, A).
Example VI-9-1. A straight forward calculation yields
Gl =
u Z'
G2 =
3u2 - u" 8
G3 =
10u3 - lOuu" - 5(u')2 + u(4) 32
Observe that P(x, A) of (VIA 1) is a polynomial in the potential u(x) satisfies a differential equation (VI.9.2)
of degree N. Therefore,
GN+1(u) + a1GN(u) + a2GN-1(u) + ... + QNG1(u) = 0
for some suitable constants a,, a2, ... , 0N. More precisely speaking, it is shown above that P(u, A) = P(0, A)G(u, A). Compute the coefficients of A-(N+1) on both sides of this identity. In fact, N
N
u(x) = 2
g}(x)
4 1`
e2,7ixgj(x)2 171
C.,
i=1
1=1
183
10. PERIODIC POTENTIALS
implies that u = 0 if and only if gl = 0 (1 = 1, 2, that P(0, A) = 11
C1 +
-
,
N). This, in turn, implies
A
and
Hence, ao = 1 and
EN
1 +
3=1
A=C2.
where
G(u, A) = f+(x, C)f+(x, N
+
2
= 1 =1
Example VI-9-2. The function u(x) = -2q2 sech2(17(x /+ p)) is a reflectionless
1c)
(cf. Example VIpotential corresponding to the data {r7, c}, where p = 2 In 7-5). In this case, G2(u)+172G1(u) = 0. Also, G3(u)+(n1+7)2 2(u)+rI14Gi(u) _ 0 in the case when N = 2 and {r(C) = 0, (i71, n2), (cl, c2)} are the scattering data. In particular, G3 (U) + 13G2(u) +36G1(u) = 0 if i1 = 2 and '12 = 3.
Remark VI-9-3. The materials of §§VI-5-VI-9 are also found in (TDJ.
VI-10. Periodic potentials In this section, we consider the differential equation
2 + (A - u(x))y = 0
(VI.10.1)
under the assumption that (I) u(x) is continuous for -co < x < +oo, (II) u(x) is periodic of period 1, i.e., u(x + e) = u(x) for -oo < x < +oo. The period a is a positive number and A is a real parameter. Denote by 01(x, A) and 02(x, A) the two linearly independent solutions of (VI.10.1) such that (VI.10.2)
1(o
01(0,A) = 1,
A) = 0,
02 (0, A) = 0,
01(e, A)
02(t, A)
d2(0,A) = 1.
Set .D(A)
=
O1 (t"\)
(e A)
The two eigenvalues Z1(A) and Z2 (A) of the matrix periodic system (VI.10.3)
d dx
are the multipliers of the
[Y2J = [u(--O-A 0, [y21
VI. BOUNDARY-VALUE PROBLEMS
184
(cf. Definition IV-4-5).
It is easy to show that det [1(A)] = 1 for -oo < A
2, two multipliers are real and distinct, i.e.. ZI(A) > 1, -1 2, if f(A) < -2,
(ii) if I f (A) I < 2, two multipliers are complex and distinct, and I Zt (A)
I
=1
and
I Z2(A) 1
= 1,
(iii) if f (A) = 2, then Z1(A) = Z2(A) = 1.
(iv) if f(A) _ -2, then Z1(A) = Z2(A) = -1. Observation VI-10-2. If f (A) = 2, let
ICCl2
be an eigenvector of 4;(A) associated J
with the eigenvalue 1. This means that cl and c2 are two real numbers not both zero and that Cl d ' (t, A) + c2 d-i (t, A) = C2 -
c141(t, A) + C202(1, A) = c1,
Set 6(x,,\) = cl o1(x, A) + c202(x, A). Then, d(x, A) is a nontrivial solution of (VI.10.1) such that .0(t, A) = 0(0,A)
and
LO (t, A) =
(0, A).
This implies that O(x, A) is a nontrivial periodic solution of (VI.10.1) of period t.
Observation VI-10-3. If f (A) = -2, it can be shown that equation (VI.10.1) has a nontrivial solution O(x, A) such that
0(t, A) = -0(0,A)
and
(t, A) = -
(O, A)
10. PERIODIC POTENTIALS
185
This implies that ¢(x, A) satisfies the condition
¢(x + e, A) = -&, A)
for - oo < x < +oo. Observation VI-10-4. Cconsider the eigenvalue problem (VI.10.5)
2 + (A - tL(x) y = 0,
y(o) = 0,
y(e) = 0.
Applying Theorem VI-3-11 (with -A instead of A) to problem (VI.10.5), it can be shown that (VI.10.5) has infinitely many eigenvalues (VI.10.6) µI < i2 < which are determined by the equation (VI.10.7) t2(£, A) = 0. The eigenfunction 02 (x, has exactly n -1 zeros on the open interval 0 < 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<x 0. Since e-;`(=-() sinh(µ(x - )) is bounded as p - +oo, it can be shown that e-v(x-E)+G(x,
, A) =
e-al=-E)
such{µ x- f ))
+ O(
on the interval 0 < x - < t as A -e -oo. Thus, we derive lim
t'(4+f,.,A) = 1. (sinh(e))
Step 2. There exists an entire function R(A) of A such that
sin(Pf) R( A ) =
J
a
sinh(CX)
for
A > 0,
for
A 0, ( 0, (ii) f (() is analytic for `£( > 0, (iii) ((f (() - 1) is bounded for !a( > 0,
(iv) f (() # 0 for !'( > 0, (0 0. Denote by SR the semicircle {(: 1(I = R,0 < arg(< r) which is oriented counterclockwise, where R is a positive number. Show that 1
1
(a) (b)
2ri IR
F(() = tai
F(z) dz =
2Ri
j
F'(()
f F(() 0
d{ + 7 f- g
where fl() is the complex conjugate of F().
(`1( > 0,1(1 < R),
(`'( 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
At, r < 0, and s > 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<s A,,
K21300){I +C2}
if A, < '\1
2
I TP .iat(t)
I
1
$pl?1
y Ip1?1
B, (t, e,
6. AN APPLICATION OF THE FLOQUET THEOREM
229
Then, equation (VII.6.12) is equivalent to (VII.6.13)
8
p = B0(E)P1,p - P1,pBo(E) + Q1,p(t, E) -
where
E
(VII.6.14) Q1,p(t,E') =
(B1,ptP1,p2 - Pi,p2Hl,ps) + B1,p.
P1+p2=p, 0p1IJ{r21>_1)
Hence,
{c(
Pi,p(t,e-) =
+ 1 exp[-sBo(0
(VII.6.15)
x 1 Q1,p(s, ) - H1,$,(e)) exP[sBo(E))ds} exp[-t B0(Z)1,
where C(e) and H1,p(e) are n x n matrices to be determined by the condition that P1,p(t,e) is periodic in t of period w, i.e., exp[wBoQ-)1C(E) - C(i) exp[wBo(E)1
(VII.6.16)
- exp[wBo(E)1
_ - exp[wBo(E)1
J0
exp[-sBo(e)1H1,p(i) exp[sBo(i)1ds
exp[-sBo(e)1Ql.p(s, e) exp[sBo(t1ds. 0
It is not difficult to see that condition (VII.6.16) determines the matrices C(ep) and Hl,p(ej. Then, the matrix P1,p(t, E) is determined by (VII.6.15). The convergence of power series P1 and H1 can be shown by using suitable majorant series. 0 In the same way as the proof of Theorem VII-6-3, the following theorem can be proven.
Theorem VII-6-6. The transformation
it = {In + Pi (t,h(t),h'(t)))v"
(VII.6.17)
changes differential equation (VII.6.10) to
dv = f H(i(t)) + H1(h(t), h'(t)) dt l (VII.6.18)
- [In
+ P1(t, h(t), hh'(t)),'1
+ h (t) a_ (t, h{t), h'(t))J j v
I hj(t) aP1(t, h(t), h'(t})
l ,
230
VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS
where P1(t, E, µ) and H1(E, µ) are those two matrices given in Lemma VII-6-4.
Observe that under assumption (VII.6.3), we obtain
In + P1(t,h(t),h'(t))j_1
f0+00
8Ej
(VII.6.19)
(t, h(t), h'(t))J dt < +oo.
+ hJ (t) a.
Observe also that the matrix H(h(t))+H1(h(t), h'(t)) does not contain any periodic
terms. Furthermore, H(6) + H1(6, 6) = H(0). It is clear that the derivative of H(h(t)) is not necessarily absolutely integrable over to < t < +oo. Therefore, in order to apply the argument of MI-5, the matrix H(h(t)) must be examined more closely in each application. Details are left to the reader for further observation.
EXERCISES VII
VII-l. Find the Liapounoff's type number of each of the following four functions
f (t) at t = +00: (1) exp [t2sin
(1L1)
]'
sint(dt](3)
(2) exp Lfo
inin(exp[3t],exp[5t]sin67rt]),
(4) the solution of the initial-value problem y"-y'-6y = eat, y(O) = 1, y'(0) = 4.
VII-2. Find a normal fundamental set of four linearly independent solutions of the system dy = Ay" on the interval 0 < t < +oo, where
A=
252
498
4134
698
-234
-465
-3885
-656
15
30
252
42
-10
-20
-166
-25
Hint. See Example IV-1-18.
VII-3. Let 4i(x) be a fundamental matrix solution of the system
=
log(l+x)
e=
expx2
x3
1+ 1+ x
sin x
exp(e')
cosx
arctanx
on the interval 0 < x < +oo. Find Liapunoff's type number of det 4i(x) at x = +oo.
EXERCISES VII
231
VII-4. Assuming that the entries of an n x n matrix A(t) are convergent power series in t-1, calculate lim p log I (t) for a nontrivial solution ¢(t) of the differential equation t
= A(t)9 at t = +oo.
Hint. Set t = e".
VII-5. Let
= xP-'A(x)i be a system of linear differential equations such that y E C" is an unknown quantity, p is a positive integer, the entries of an n x n matrix A(x) are convergent power series in x-1, and lim A(x) is not nilpotent. Show that dY
there exists a solution g(x) of this system such that lim r-.+oo number for some real number 6.
ln(l y-(Teie)j)
rp
is a positive
Hint. Set x = re!e for a fixed 0 and t = rp. Then, the given system becomes d#
e'
e+'0 PO = -A(re's)y'. Note that lim e'-A(re'B) _ -A(oo). The eigenvalues
r-.+oo P
P
ei>'
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<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
0
for
0 < s < 1,
where rig (s) (j = 1, 2) are the entries of the vector r"(s) (cf. Figure 9).
-
__11
(0.0)
FIGURE 7.
FIGURE 8.
n2=0
FIGURE 9.
For 0 0. Show 0
that (x(t), x'(t)) is an orbitally asymptotically stable orbit as t phase plane.
+00 in the (x, x')
Hint. Use Exercise IX-4-
IX-6. Show that there exists a nontrivial periodic orbit of the system dI7
=f (y),
y=
Ly2J,
W) Lf2h (Y-)
where (1) the entries of the 1R2-valued function f (y-) is continuously differentiable on the entire y-plane,
301
EXERCISES IX
and f(y)
(2) f(6) (3)
ifil96
of
(O) = i, eft (0) = 8,
ay1
OW
liM
,
2Z-2 (0)
+oo(ft(Y1,Y2) + yr) and
(4) Ivll+l
(p)
= -2 and
ay1
= 1, +oo(f2(yt,N2) +
1Y11+liM
yz) exist.
IX-7. Given that Y1 - y2
y2
find the total number E of elliptic sectorial regions, the total number H of hyperbolic sectorial regions, and the total number of parabolic sectorial regions in the = f (y, E) in the neighborhood of the isolated stationary point 0 of the system following two cases: (i) E = 2 and (ii) E _
dt
2. Also, find I y(0) for e 96 0.
IX-8. Let us consider a system
dt = f (y),
(S3)
where the entries of the l 3-valued functionf is continuously differentiable on the entire y-space R3. Assume that (S3) has a periodic orbit pit,ip) of period 1 such that f (r'(t, o)) 0 0. Assume also that the first variation of system (S3) with respect to the solution p(t,' o), i.e., d9
Of (r(t,io))u,
dt
has three multipliers p1 = 1, p2i and p3 satisfying the condition: 1P21 < 1 and jp3j > 1, respectively. Construct the general orbits pit, r) of (53) such that distance(At,17),
C(s'7o)) tends to0ast-+too. IX-9. Show that the differential equation d-t2 + (x + 3)(x + 2) does not have nontrivial periodic solutions.
drt
+ x(x + 1) = 0
Hint. Two stationary points are a node (0,0) and a saddle (-1,0). Furthermore, x2
setting f(x1,x2) =
, we obtain divf(x1,x2) _
-(xt + 3)(x1 + 2)x2 - xt(xt + 1) J -(x1 + 3)(x1 + 2) < 0 if x1 > -1. Also, use index in §IX-5. IX-10. For the system I
= f(x,y) = x(1 - x2 - y2) - 3y,
= g(x,y) = y(1 - x2 - y2) + 3x,
dt
(1) find and classify all critical points, (2) find dt
= f (x, y)
+ 9(x, y)
.-
5
IX. AUTONOMOUS SYSTEMS
302
for the function V (x, y) = x2 + y2, dV = 01, (3) find the set S = dt (4) examine if S is an invariant set, (5) find the phase-portrait of orbits.
{(x):
IX-11. For the system dx
dt = dt = dz
dt =
f(x,y,x) = - x(1 - x2 _
y2)2
+ 3xz + y,
9(x,y,z) = - y(1 - x2 - y2)2 + 3yz - x,
h(x,y,z) = - z -
3(x2 + y2),
(1) find all critical points and determine if they are asymptotically stable, (2) find 8V + h(x, y, x) 8V dV = f (x, y, x) 8x 8V + g(x, y, z) 8y 8z dt for the function V(x, y, z) = x2 + y21+ z2, dV (3)
find thesetS=((x,y,z):
=0},
find the maximal invariant set M in S, find the phase portrait of orbits. IX-12. Find the a phase portrait of orbits of the system (4) (5)
dt = y,
dt = y + (1 - x2)x2(4 - x2).
IX-13. Let f (x, y) and g(x, y) be real-valued, continuous, and continuously differentiable functions of two real variables (x, y) in an open, connected, and sim-
ply connected set D in the (x, y)-plane such that 8f (x, y) + Lf (x, y) j4 0 for all dxt
(x, y) E D. Show that the system = f (x, y), d = g(x, y) does not have any nontrivial periodic orbit that is contained entirely in D. IX-14. Let p(z) be a polynomial in a complex variable z and deg p(z) > 0. Set x = 3t(z] and y = Q [z]. Verify the following statements. (i) In the neighborhood of each of stationary points of system (A)
d = R[p(z)],
dt =
`3`[6 (x)],
there are no hyperbolic sectors. Also, system (A) does not have any isolated nontrivial periodic orbit. (ii) In the neighborhood of each of stationary points of system (B)
= 3R[p(z)], L = -`3'[p(x)],
dt there are no elliptic sectors. Also, system (B) does not have any nontrivial periodic orbits.
303
EXERCISES IX
IX-15. Find the phase portrait of orbits of the system
d = x2 - y2 -3x+2,
d _ -2xy + 3y.
Hint. See Exercise IX-14 with p(z) = (z - 1)(z - 2). dx
dy
IX-16. Find explicitly a two-dimensional system dt = f (x, y), d t= 9(x, y) so that it has exactly five stationary points and all of them are centers. IX-17. Consider a system dy dt = f (y1,
(S4)
where the entries of the R'-valued function f are analytic with respect toy in a domain Do C R'. Assume that (i) system (S2) has a periodic orbit pit, ijo) of period 1 which is contained in the domain Do, (ii) AP-Tt,no)) 0 (iii) for any open subset V of Do which contains the periodic orbits p(t, i"p), there
exists an open subset U of V which also contains p'(t, o) such that for any point i in U, the orbits p'(t, i of (S4) is contained in V and periodic in t. Show that if U is sufficiently small, for any point i in U, we can fix a positive period is bounded and analytic with respect to i in any simply of p1t, y) so that connected bounded open subset of U.
Hint. Apply the following observation. Observation. Let Do be a connected, simply connected, open, and bounded set in 1Rk and let T2 (j = 1, 2, ...) be analytic mappings of Do to l . Suppose that, for any pointy E Do, there exists a j such that T3 [yl = y, where j may depends on Y. Then, there exists a jo such that Tjo [yj = y for all y E Do. Proof.
Set E j _ {y E D o : T j [ y 1 = Y - ) .
j = 1, 2, ... .
Then, (1) E. is closed in Do, +00
(2) Do = U E,, 3=1
(3) Do is of the second category in the sense of Baire.
Hence, for some jo, the set Ego contains a nonempty open subset (cf. Baire's Theorem). Since Tea is analytic, we obtain Tao [ 7 = y" for all y" E Do. O For Baire's Theorem, see, for example, [Bar, pp. 91-921.
CHAPTER X
THE SECOND-ORDER DIFFERENTIAL EQUATION dt2 + h(x)
+ g(x) = 0
In this chapter, we explain the basic results concerning the behavior of solutions of a system 112
[-h(yi)112 - 9(111)
as t -- +oo. In §X-2, using results given in §IX-2, we show the boundedness of solutions and apply these results to the van der Pol equation (E)
x + E(x2 - 1)
d +x=0
(cf. Example X-2-5). The boundedness of solutions and the instability of the unique
stationary point imply that the van der Pol equation has a nontrivial periodic solution. This is a consequence of the Poincar&$endixson Theorem (cf. Theorem IX-4-1). In §X-3, we prove the uniqueness of periodic orbits in such a way that it can be applied to equation (E). In §X-4, we show that the absolute value of one of the two multipliers of the unique periodic solution of (E) is less than 1. The argument in §X-4 gives another proof of the uniqueness of periodic orbit of (E). In §X-5, we explain how to approximate the unique periodic solution of (E) in the case when a is positive and small. This is a typical problem of regular perturbations. In §X-6, we explain how to locate the unique periodic solution of (E) geometrically as e - +oo. In §X-8, we explain how to find an approximation of the periodic solution of (E) analytically as a +oc. This is a typical problem of singular perturbations. Concerning singular perturbations, we also explain a basic result due to M. Nagumo [Na6] in §X-7. In §X-1, we look at a boundary-value problem
y = F (t, y, d i )
y(a) = o,
,
y(b) = 8.
Using the Kneser Theorems (cf. Theorems 1II-2-4 and III-2-5), we show the existence of solutions for this problem in the case when F(t, y, u) is bounded on the entire (y, u)-space. Also, we explain a basic theorem due to M. Nagumo [Na4] (cf. Theorem X-1-3) which we can use in more general situations including singular perturbation problems (cf. [How]). For more singular perturbation problems, see, for example, [Levi2], [LeL], (FL], [HabL], [Si5], [How], [Wasl], and [O'M].
304
1. TWO-POINT BOUNDARY-VALUE PROBLEMS
305
X-1. Two-point boundary-value problems In this section, first as an application of Theorems 111-2-4 and 111-2-5 (cf. [Kn]), we prove the following theorem concerning a boundary-value problem (X.1.1)
d2
= F t, y, dt)
y(a) = a,
,
y(b) = Q
Theorem X-1-1. If the function F(t, yt, y2) is continuous and bounded on a region 11 = {(t, Y1, y2) : a < t < b, lyt I < +oo, Iy2I < +oo}, then problem (X.1.1) has a solution (or solutions). Proof.
For any positive number K, the set Aa = {(a, or, y2) : Iy21 < K} is a compact and connected subset of ft We shall show that A0 satisfies Assumptions 1 and 2 of §111-2 for every positive number K. In fact, writing the second-order equation (X.1.1) as a system (X.1.2)
dyt
dt
_
- Y2,
dye
dt
= F(t, yt, y2),
we derive y1(t) = y1(a) + J y2(s)ds, a
y2(t) = y2(a) + JF(s,yi(s),y2(s))ds. ` Hence, if (a, y1(a), Y2 (a)) E A0, we obtain
Jy2(t)I < K + M(b - a), ly1(t)I < lal + [K + M(b - a)](b - a), where I F(t, yi, Y2)1 < Al on Q. Therefore, Ao satisfies Assumptions 1 and 2 of §111-2. Thus, Theorem 111-2-5 implies that SS is also compact and connected for every c on the interval Te = {t : a < t < b}. We shall prove that if K > 0 is sufficiently large, the set Sb contains two points (771,772) and (t;t, (2) such that (X.1.3)
n, < 0 < (1.
In fact, by using the Taylor series at t = a, write yt (b) in the form
yt(b) = a + y2(a)(b - a) +
2 dt
where c is a certain point in the interval 10. Since
(c) (b - a)2, I ddt2
(c)
M, the quantity
yI(b)I can be made as large as we wish by choosing Iy2(a)I sufficiently large. Thus, there are two points (nt, n2) and ((1, (2) in Sb such that (X.1.3) is satisfied. Since the set Sb is compact and connected, there must be a point (Q, () in the set Sb. This implies the existence of a solution of problem (X.1.1). 0
X. THE SECOND-ORDER DIFFERENTIAL EQUATION
306
Example X-1-2. Theorem X-1-1 applies to the following two problems: (X.1.4)
y(a) = a,
dt2 + sin y = 0,
y(b)
and
y(b) = Q
0, y(a) = a, + p-+-, = However, Theorem X-1-1 does not apply to
(X.1.5)
dt2
(X.1.6)
y(b) = Q.
y(a) = a,
d 22 + y = 0,
For more general cases, the following theorem due to M. Nagumo (Na4] is useful.
Theorem X-1-3. Assume that (i) a real-valued function f (t, x, y) and its derivatives az and
09 f
are continuous
in a region V = {(t, x, y) : (t, x) E A, -oo < y < +oo}, where 0 is a bounded and closed set in the (t, x) -space; (ii) in the region D, the function f satisfies the condition
1f(t,x,y)1 : 0(IyI),
(I)
where 0(u) is a positive-valued function on the interval 0 < u < +oo such that
+°° udu
(II)
+00;
(iii) two real-valued functions wi (t) and w2(t) are twice continuously differentiable
on an interval a < t < b and satisfy the conditions for wi(t) < w2(t) a < t < b, Ao
= {(t x)-a'Y; (2) two curves x = x(t, c) and x = wl (t) meet for some ton the interval a < t < b if c < -ry. Proof.
For part (1), by virtue of Lemma X-1-4, x'(r,c) > a < r < t. The proof of part (2) is similar.
La
if (r,x(t,c)) E Ao for
Proof of Theorem X-1-3.
Now, let us complete the proof of Theorem X-1-3. The main point is that when two curves x = x(t,c) and x = w2(t) or two curves x = x(t,c) and x = w, (t) meet, they cut through each other. So look at Figure 1.
(b, B) (a. A)
t=a
r=b FIGURE 1.
Example X-1-6. Theorem X-1-3 applies to the boundary-value problem (X.1.10)
d2x dt2
_
x(0) = A,
x(1) = B
Ax'
if A is a positive number. In fact, assume that 0(u) is a suitable positive constant. If
wi(t) = sinh(ft) -a and w2(t) = sinh(ft)+$ with two positive numbers a and ,3 such that -a < A < j3 and sinh(/) - a < B < sinh(VrA_) +,Q, all requirements of Theorem X-1-3 are satisfied.
If A is negative, Theorem X-1-3 does not apply to problem (X.1.10). Details are left to the reader as an exercise.
2. APPLICATIONS OF THE LIAPOUNOFF FUNCTIONS
309
X-2. Applications of the Liapounoff functions In this section, using the results of §IX-2, we explain the behavior of orbits of a system
as t
[yj
d
(X.2.1)
Y2
__
y[-h(yi)y2
dt
- 9(yi)
l)y2- 9(y1) and f (yl = [_h(yi
+oo. Set y = I yl
JLet us assume that
J
h(x), g(x), and dd(x) are continuous with respect to x on the entire real line R. Also, we denote by p(t,r) the solution of (X.2.1) satisfying the initial condition y(0) = n Fi rst set V (y) = + G(yl ), where G(x) = fox g(s)ds. Then, 2Y2
09
= [9(yi ), Y21,
Set also, S = { g : l
8y'
-
f (y-)
= -h(yi)
y22.
R Y) = 0 y. Then, U E S if and only if either h(yl) = 0 or JJJ
Y2 = 0.
Observation X-2-1. Denote by M the set of all stationary points of system (X.2.1), i.e., M = {9: g(yl) = 0, y2 = 01. Then, M is the largest invariant set in S if the following three conditions are satisfied:
(1) h(x) > 0 for -oo < x < +oc, (2) h(x) has only isolated zeros on the entire real line IR, (3) g(x) has only isolated zeros on the entire real line R.
The proof of this result is left to the reader as an exercise (cf. Figure 2, where 0 and 0). By using Theorem IX-2-2, we conclude that lim p(t, y) = 17 E M if conditions t-+00 (1), (2), and (3) are satisfied and if the solution p(t,g) is bounded for t _> 0. Note that G+(rt) is a connected subset of M. In Observation X-2-1, the boundedness of the solution p(t, i) for t _> 0 was assumed. In the following three observations, we explore the boundedness of all solutions of (X.2.1). Set 0,
G(x) =
o
g(s)ds
and
H(x) =
Jo
f
X
h(s)ds.
o
Observation X-2-2. Every solution p(t, q) of (X.2.1) is bounded for t > 0 if (i) h(x) > 0 for -oc < x < +oo and (ii) lim G(x) = +oo. 1x1
+00
This is a simple consequence of Theorem IX-2-3. In fact,
lim V (Y-) = +oo. 191-.+00
310
X. THE SECOND-ORDER DIFFERENTIAL EQUATION
Observation X-2-3. Every solution pit, of (X.2.1) is bounded for t > 0, if (i) h(x) > 0 for -oo < x < +oc, (ii) urn IH(x)l = +oo, and (iii) xg(x) > 0 I=t-+o0
for -oc < x < +oo. Proof Change system (X.2.1) to (X.2.2)
d
[zl]
Z2 - H(z1)
dt
22
-g(z1)
by the transformation
Y2 = z2 - H(zi).
Y1 = Z1,
Denote by qqt,() the solution of (X.2.2) such that z(0) _ . Set V, (z-) = zz2 + G(z1). Then, L9 V,
ay -
[g(zl), z21,
8z"
,P = -g(z1)H(zl)
Note that g(x)H(x) > 0 for -oo < x < +oo and that dt V1(ggt,C))
Hence, setting q"(t, S) =
- z(g-(t,C)) F(glt,S)) S 0 z1(t,
for
t > 0.
= c > 0, we obtain
and V,
2[z2{t,S)]2
0,
since G(x) > 0 for -oo < x < +oo. Therefore, Figure 3 clearly shows that q1 t, C) is bounded for t > 0. This implies that all solutions of (X.2.2) are also bounded for
t>0.
{s2, 0)
z2=0
Y2=0 (43,0)
(41.0)
di-1
FtcuRE 2.
z1=0
F1cuRE 3.
2. APPLICATIONS OF THE LIAPOUNOFF FUNCTIONS
311
Observation X-2-4. Every solution p(t, >)1 of (X.2.2) is bounded for t > 0 if (i)
=
lim H(x) = +oo, (ii) = --Cc lim H(x) = -oo, (iii) g(x) > a for x > as > 0, +00
and (iv) g(x) < -a for x < -ao < 0, where a and ao are some positive numbers. Proof.
In Observation X-2-3, the Liapounof function
G(zi) = 2[y2 + H(yl)]2 + G(yi)
Vi(z) =
was used. Now, let us modify Vl to a form
V2(yl = 2[y2 + H(yi) - k(yi)J2 + G(yl) Then, OV2
09
and
_
[[y2
- dyyt )1 + 9(yi ),
+ H(yi) - k(yi )J {h(yi)
(Y ')
az
dy,
1
Y2 + H(1h) - k(yi) I
J {y22
J
+ [H(yi) - k(yi)J y2}
- 9(yi) [H(yi) - k(yl)J Using (i) and (ii), three positive numbers M, a, and c can be chosen so that
fa [H(x) - cJ > 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<x ±V(i(x,a2))
for
0<x V(E(ro(a2),a2)) - V(E(O,a2)) > 0
for goa for ro(a) a on this t-interval. Regarding zi (t, a) as a function of z2(t, a) for z2(r1(a), a) < z2 < z2(ro(a), a), we obtain
(II) 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 r2 and t > 0, where r is a positive number r, r>
(v) G(x) = J g(t)dC --' +oo as jx[
+oo,
0
(vi) 1 je(r)Idr is bounded for 0 < t < +oo. 0`
Show that every solution (x(t), y(t)) of the system dy = dx _ -f(x,y,t)Y dt -
Y
'
df
is well defined and bounded for 0 < t < +oo.
- g(x) + e(t)
335
EXERCISES X
Hint. Set V (x, y) = 2 + G(x). Then, d = - f (x, y, t)yz + e(t)y. There exists a positive number ro > r such that V(x, y) > 4 for x2 + y2 > 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 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, A > 0, and k > 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
2
IIY1Ia,A = max{IIy,II5,A :1 < j < n},
(iii) fo(0) = 0, (iv) (3(0) is invertible. Show that there exists a unique power series E C[[x]]n satisfying system (S) and the condition ¢(0) = 0. Furthermore, E (C[[x]],)n. Hint. Write system (S) in a form y" = go + x9+1 1P dx Banach fixed-point theorem in terms of the norm II IIs.A-
+ E y'gp and use the ipl>2
XI-8. Let D(r) = {z : IzI < r}, S(r, p) = {z : 0 < IzI < r, I arg zI < p}, and a power +w
series f (z, e) = E fn(e)zn is convergent uniformly in a domain D(ro) x S(r, p), n=o
where ro, r, and p are positive numbers. Assume that the coefficients fn(e) of the series f are holomorphic in S(r, p) and admit asymptotic expansions in powers of e as e -' 0 in S(r, p). Suppose also that limo fo(e) = 0 and limo f, (c) 0 0. Show C1-
that there exist a positive number rt and a function O(e) such that (1) O(e) is holomorphic in S(rt, p) and admits an asymptotic expansion in powers of a as e -+ 0 in S(rt, p), (2) l ¢(e) = 0, and (3) f (c(e), e) = 0 identically in S(rt, p).
XI. ASYMPTOTIC EXPANSIONS
368
X1-9. Consider a formal power series
#,
_
F(x, y,
(P)
y
fa,,n2,
ip,l+IpI>o yi
where
, fn:,as E
%' _ yn
(p21,
z1
C[[x]]
"
, pi = (pll,... ,pln), and Pz =
xm
... , p2",) with non-negative integers P1k and p2k. Denote by Fg(x, y z) the
Jacobian matrix of ! with respect to W. Assume that power series (P) satisfies the conditions (1) f;(0,6,6) = 0, (2) Fy(0, 0, 0') is invertible, (3) fn, a, E E(s, A)" for some s > 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, <argx --o
as e - 0 in sector (XII.1.5) with coefficients holomorphic in the domain (XII.1.10)
1x(< 60-
Let A(x, e) be the n x n matrix whose (j, k)-entry is a.,&, e), respectively (i.e., A(x, e) = (ajk(x, e)). Then, A(x, c) admits an asymptotic expansion (XII.1.11)
A(x, e) ^_- E e"A,,(x) =o
as a -+ 0 in sector (XII.1.5), where the entries of coefficient matrices (ajk,,(x)) are holomorphic in domain (XII.1.10). The following second assumption is technical and we do not lose any generality with it.
Assumption II. The matrix Ao(0) has the following S-N decomposition Ao(0) = diag
[µh,02,... ,p1 + N,
where Al, µ2f ... , An are eigenvalues of Ao(0) and .,V is a lower-triangular nilpotent matrix.
Note that [Ni can be made as small as we wish (cf. Lemma VII-3-3). The following third assumption plays a key roll.
Assumption III. The matrix Ao(0) is invertible, i.e., t<J#0
(j = 1, 2,. .. , n).
In §§XII-2 and XII-3, we shall prove the following theorem.
XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER
374
Theorem XII-1-2. Under Assumptions I, II, and III, system (X11.1.1) has a solution (XII.1.12)
(j = 1,2,... ,n)
vJ = pj (x, c)
such that (r) pi(x,e) are holomorphie in a domain jxl < b,
(XII.1.13)
0 < lei < p,
l arg el < a,
where b, p, and a are suitable positive constants such that 0 < 6 < bo, 0
cos(ka) > 0
for
f E S(r,a) if
ka < 2.
Let us change l/ in (XII.5.1) by
Ii = exp(-ce-'16.
(XII.5.10)
Then, (XII.5.1) is reduced to (XII.5.11)
el" = exp[m-kJf(x,eXP[-ce-k]u,e).
Set
f (x+ FJ, e) = f (x, 0, e) + A(x, e)il + E yPfp(x, e). JpI>2
Then, exp[cE-k1f(x, eXp[-cE-k]u, e)
= eXPfce-kJf (x, 0, e) + A(x, e)U +
explc(1 - lpl)E-k]ul fp(x, e)' ip1>2
Using a method similar to the proof of Theorem XII-1-2, we construct a bounded solution u = tlr(x, e) of (XII.5.11). Therefore, system (XII.5.1) has a solution of
the form y" _ ¢(x,e) = expl-ce-k]y,(x,e). This completes the proof of Lemma XII.5.1.
The main result of this section is the following theorem, which was originally conjectured by J: P. Ramis.
5. GEVREY ASYMPTOTIC SOLUTIONS IN A PARAMETER
387
Theorem XII-5-2. Assume that (i) f (x, y', e) is holomorphic in (x, y, e) on a domain 0(6o) x f2(po) x S(ro, ao), where bo, po, ro, and ao are positive numbers, (ii) f (x, y, e) admits an asymptotic expansion F(x, y, e) of Cevmy orders as a --+ 0 in S(ro, ao) uniformly in (x, y-) E A(bo) x f2(po), where s is a non-negative number,
(iii) the matrix Ao(x) given by (XIL5.4) is invertible on 0(bo), (iv) it holds that
on 0(bo).
lim f (x, 0, e) = 0
(XII.5.12) Then,
(1) (XIL5.1) has a unique formal solution
Plx, e) _
(X1.5.13)
c-
eep'e(x)
with coefficients p"t(x) which are holomorphic on A(bo), (2) there exist three positive numbers 6, r, and a such that (XII. 5.1) has an actual solution ¢(x, e) which is holomorphic in (x, e) E A(5) x S(r, a) and that ¢(x, e) admits the formal solution P(x, e) as its asymptotic expansion of Gevrey order max {
1, s } as e - 0 in S(r, a) uniformly in x E L1(b). a
J11
Proof.
If a positive number & is sufficiently small, for every real number 6, there exists
a C"-valued function fe(x,y,e) such that (a) fe(x, y', e) is holomorphic in (x, y", e) on a domain A(60) x f2(po) x Se(ro, &), where (XIL5.14)
Se(ro,&) = {e: jarge - 61 < &, 0 < fej < ro},
(b) fe(x, y, e) admits an asymptotic expansion F(x, y e) of Gevrey order s as e - 0 in Se(ro, &) uniformly in (x, y) E 0(bo) x f2(po), where s is the nonnegative number given in Theorem XII.5.2. Such a function fe(x, y, e) exists if & is sufficiently small (cf. Corollary XI-2-5). In particular, set (XII.5.15)
fo(x, b, e) = f (x, v, e)
Let y = & (x, e) be a solution of the system (XII.5.16)
e°
= fe(x, y, e)
such that &(x, e) is holomorphic and bounded in (x, e) E i (b1) x Seri, al) for suitable positive numbers 61, r1, and al. Using Theorem XII-1-2, it can be shown that such a solution of (XII.5.16) exists.
388
XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER
Suppose that So,(ri,al) nSo,(ri,al) 0 0. Set
/i(x, E) = 4 (x, E) - 4 (x, E)
(XII.5.17)
for (x, E) E A(5j) X {So,(r1,a1)nSo.(rl,a,)}. Then, obi satisfies the following linear system
E°L = B(x, E)/+ 6(x, E),
(XII.5.18) where (XII.5.19)
B(x,E)W(x,E) =
fet(x,iez(x,E),E),
6(x,0 =
Note that (XII.5.20)
B(x, c) _
JlOfJ (x,tiet(x,E) + (1 -
and that, if s > 0, then jb(x,E)I -1. In this theorem of G. D. Birkhoff, the entries of the matrix B(t) are polynomials in t. However, even though we can choose P(t) so that x = 0 is, at worst, a regular singular point of (Eo), the degree of B(t) with respect to t may be very large. In order that the
degree of B(t) with respect to t is at most k + 1 so that x = 0 is a singularity of the first kind of system (Eo), we must impose a certain condition on A(t) (cf. Exercises XII-8, XII-9, and XII-10). For interesting discussions on this matter, see, for example, [u2], [JLP], [Ball], and [Ba12j. A complete proof of Theorem XII-6-1 is found, for example, in [Si17, Chapter 3]. In this section, we prove a result similar to Theorem XII-6-1 for a system of linear differential equations f '!L
(XII.6.1)
= A(x,e)y,
dx
under the assumption that a is a positive integer, y" E C", and A(x, e) is an n x n matrix whose entries are holomorphic with respect to complex variables (x, e) in a domain (XII.6.2)
X E Do,
[e[ < 60,
where Do is a domain in the x-plane containing x = 0, and 6o is a positive constant. Let 00
A(x,e) = E ekAk(x)
(XII.6.3)
k=0
be the expansion of A(x, e) in powers of e, where the entries of coefficients Ak(x) are 00
holomorphic in Do. We assume that the series >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<wj+(r+1)w< (2q+)1r_v}
,w
q is an integer, N is a sufficiently large positive constant, and -y is a sufficiently
small positive constant. For each j, there exists at least one integer q, such that the real half-line defined by x > 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 in its interior, their common part D(N, ry) is given by the inequalities (XIII.2.16)
l xI > 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 r,
the set of indices J is divided into three groups:
-2r+ry' <wj -(r+1)t r - y'. These inequalities imply that 2 r - 2 ry' > 0 and
-(r + 1)t < -(r + 1)t - 27' + 17r < (r + 1)t' + 27' - 2r < (r + 1)t'. In D(N), let
1 = Nr+1 exp i{ (r + 1)t' + ifl
12'Y'
- Zr}J
,
C2Nr+lexp i -(r+1)t-+2r} 1,Y
,
413
2. BASIC ESTIMATES
For every t E D(N), the paths of integration in t(N) are defined in the following manner: (a) For j E G1, from (XIII.2.37) and (XIII.2.38.1), it follows that -27r + 27'
sin
(j E G2)-
1
(-') for E D(Nh) and j E G1 U G2 .
Moreover, (XIII.2.34) implies
arg( - o) -
2(r + 1)(e+ e') +
2ry1
and (XIII.2.28) and (XIII.2.29) imply (r + 1)(t + t') < n - ry'. Hence,
I arg( - {o)
(XIII.2.44)
-
ir
- 27
Observe that idol = Nn+' implies 1812 = M2 + t2 + 2cMt, where M = Nh+1 and (cf. Figure 4).
c= -cos(rr-0) with 0=
Let b = sin ('). Then,
0 0. Hence, for r > 0, (XIII.2.47) implies
that Y(r) < M . On the other hand, Y(r) satisfies the differential equation dY = {
M+h
if M >
D2 + r2 } h(r + 1) .
1i2 1b
< (r + 1)
1
M2
Y -1. Since
r D2+72
lit, Ib
2(r + 1)
Here, a use was made of the inequality M -< D. Since Y(0)
Ro, Iu-1 < do, lµl < p o, IeI < Eo, I arg EI < po}, where R0, 60, µo, co, and po are some positive constants, (b) f admits a uniform asymptotic expansion in Do s/ r/x, a/,
e)
ti
+0 Ehfh(x, . p)
as
e - 0,
h=0
where coefficients fh (x, y, i) are single-valued, bounded, and holomorphic in the domain
Ao = {(x, 9, #) : IxI > Ro,
I vi < 6o,
Iµl < po},
3. PROOF OF THEOREM XIII-1-2
419
(c) if f (x, y, µ', e) = fo (x, #,c) + A(x, µ, e)y + O(I y12), then
fo(oo,µ',e) = 0
and
detA(oo,0,0) 0 0.
Let )q, A2, ... , )1n be n eigenvalues of A(oo, 0', 0) and set wf = arg A, (j _ 1, 2, ... , n). Note that the w, are not unique. Set also S(pl, p2) _ {x : Pi < argx < p2}. Let a and r be two non-negative integers. Assume that the sector S(pl, p2) is contained in the sector 32
- min{wj} + apo < (r+ 1)argx
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
xlf2, with Hi being n x n constant matrices. As in t=o
§V-4, the operator C can be represented by the matrix H1 f12
01
J,, +A
and J_ _ [Oq]
where H =
I...
Here, 0, is the nr x oo zero matrix and I,,,, is the oc x oo identity matrix. Let
Qo=So+No and A=S+N be the S-N decomposition of 0o and that of A in the sense of §V-3. Also, let Po, P1, P2, ... , Pm, ... be n x n constants matrices and let AO be an n x n diagonal
matrix such that det Po 0 0 and that
(1)
SoPo = PoAo,
Ao =
A 1
0
0
0
0
0
A2
0
0
0
0
0
A3
0
0
0
0
0
0
An
and (2)
SP=PAo,
where P1
P=
P2 P0
P.
Note that At are eigenvalues of So. Hence, A, are eigenvalues of 00. Show that
(a) the matrix S represents a multiplication operation: y -+ o(x)y for an n x n matrix o(x) = So + O(x) whose entries are formal power series in x,
XIII. SINGULARITIES OF THE SECOND KIND
452
(b) the matrix N represents a differential operator Lo[yM = x4+1 fy + v(x)y for an n x n matrix v(x) = No +O(x) whose entries are formal power series in x, (c) C[f, = £o[yl + o(x)y and Go[o(x)yj = a(x)Go[yl, +00
(d) if we set P(x) _ F xmP,,,, then P(x)-1o(x)P(x) = A0, m=0
(e) if we set /C[uZ = P(x)-'C[P(x)uZ and Ko[u] = P(x)-1Go[P(x)ul, then K[u"j = Ko[ul + Aou,
(f) if we set Ko[i ] = xq+1
du
Vii
+ vo(x)u,
vo(x) =
.
vn1
then
vjk(x) = 0 if .\.7 # Ak (cf. [HKS]).
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INDEX
Asymptotic solution, 374 Asymptotic stability, 236, 300 a sufficient condition for, 241 Atiyah, M. F., 109 Autonomous system, 279
,A.(r,a,b), 353 Abel's formula, 80 Abramowitz, M., 141, 449, 450 Adjoint equation, 79 Airy function, 449 Algebraic differential equation, 109, 353 Analytic differential equation, 20 Analytic simplification of a system, at a singular point, 421 with a parameter, 391 Approximate solutions, c-, 10 Approximations, successive, 3, 20, 21, 378, 418 Arzelh, C., 9
Balser, W., 371, 391, 401, 435, 443 Baire's Theorem, 303 Banach space, 25, 26, 103, 366, 394 differential equation in, 25, 394 Bartle, R. G., 303 Basis of the image space, 73 Bellman, R., 71, 144, 148, 197 Bendixson, 1., 27, 279, 291, 304 Bendixson center, 261 Bessel function, 125 Bessel inequality, 162 Birkhoff, G. D., 372, 390 Block-diagonalization theorem, 190, 204, 380, 422, 423
Arzelb-Ascoli Lemma, 9, 13, 30 Ascoli, G., 9
Associated homogeneous equation, 97, 99 Asymptotics, Gevrey, 353 Poincar6, 342
Asymptotic behavior of solutions, 197 of a linear second-order equation, 226 of linear systems, 209, 213, 219 Asymptotic (series) expansion, 343, 353 definition, 343 differentiation, 347 integration, 347 inverse of, 346 of a holomorphic function, 345 of Ei(z), 352 of Gevrey order s, 353 of Log(o(z)), 352 of uniformly convergent sequences, 348 product, 344 sum, 344 Taylor's series as, 345 uniform with respect to a parameter,
Block solution, 107
Blowup of solutions, local, 17 Borel, E., 349, 356 Borel-Ritt Theorem, 349, 356 Boundary-value problem, 144 of the second-order equation, 305 Sturm-Liouville, 148 Bounded set of functions, 9 Bourbaki, N., 69 Braaksma, B. L. J., 359, 355, 443 Branch point, 44
C, field of complex numbers, 20 C", set of all n-column vectors in C, 20 C[xJ, set of all polynomials, 109 C{x}, set of all convergent power series, 109 C[[xJJ, set of all formal power series, 109 C{x}", set of all convergent power series with coefficients in C", 110 C[[xfJ", set of all formal power series with coefficients in C", 110 C[[xJJ set of all power series of Gevrey
343, 357, 361
of Gevrey order s, 357 uniqueness, 343 Asymptotic reduction of a linear system with singularity, 405 of a singularly perturbed linear system, 381
order s, 353 462
463
INDEX C-algebra, 70
Cayley-Hamilton Theorem, 71, 220 Canonical transformation of a Hamiltonian system, 92, 94 Cauchy, A. L., 1 Cauchy-Euler differential equation, 140 Caratheodory, C., 15 Center, 255, 257, 261, 272 perturbation of, 271 Cesari, L., 197 Characteristic exponents of an equation, 89, 203 Characteristic polynomial of a matrix,
in S-N decomposition, 74, 119, 122 Diagonalization theorem, Levinson's, 213 Differentiability with respect to initial values, 35 Differential operator C, 120, 151, 169, 424, 437, 443 S-N decomposition, 122 normal form, 121 Dirac delta function, 8 Distribution, of L. Schwartz, 8 of eigenvalues, 159 Dulac, H., 235, 251 Dwork, B., 8
71, 72
of a function of a matrix f(A), 84, 85 partial fraction decomposition of the inverse of, 72, 76 Chevalley, Jordan-, decomposition, 69 Chiba, K., 197 Classification of singularity of a homogeneous linear system, 132 Coddington, E. A.,14, 28, 36, 58, 65, 69, 108, 137, 138, 144, 148, 153, 162, 191, 197, 225, 233, 246, 266 274, 281, 290, 293, 298, 313
Commutative algebra, 109 Commutative differential algebra, 109, 362
Commutative differential module, 110 Comparison theorem, 58, 144 Complexification, 252 Contact point, exterior (interior), 295 Conti, R., 15, 235, 279
Continuity of a solution, 17, 29 with respect to the initial point and initial condition, 29, 31 with respect to a parameter, 31 Convergence of formal solution, 113, 118 Coppel, W. A., 144, 148, 197 Covering, good, 354 Cullen, C. G., 71 Cyclic vector, 424, 426
Date, E., 144 Deligne, P., 403, 424 Denseness of diagonalizable matrices, 70 Dependence on data, 28, Devinatz, A., 197 Diagonalizable matrix, 70
E(s, A), 366 E(s, A)", 367 Eastham, M. S. P., 197, 217 Eigenfunction, 162 Eigenvalues of a matrix, 70, 71 Eigenvalue-problem, 153, 154, 156 of a boundary-value problem, 153, 186 Eigenvectors of a matrix, 70 Elliptic sectorial domain, 296, 302 Equicontinuity, 9 equicontinuous set, 47, 55, 63
Euler, Cauchy-, differential equation, 140 Existence, 1, 12 of solution of an initial-value problem, 1, 3, 12, 21
with Lipschitz condition, 3 without Lipschitz condition, 12 of solution of a boundary-value problem, 148 of S-N decomposition of a matrix, 74 Existence theorem for a nonlinear system at a singular point, 405 Existence theorem for a singularly perturbed nonlinear system, 420 cxp[Al, exponential of a matrix A, 80 exp[tAl with S-N decomposition, 82 Extension of a solution, 16 of a nonhomogeneous equation, 17 Finite-zone, potentials, 189 First variation of an equation, 284 Flat of Gevrey order s, 342, 367, 385 Flatto, L., 341 Floquet, G., Theorem, 69, 80, 225 Formal power series, 109, 343, 349
INDEX
464
Formal power series (cont.) as an asymptotic series, 350 of Gevrey order s, 353, 355
with a parameter as an asymptotic series, 351 Formal solution, 109, 342 convergence, 115, 118 Taylor's series as, 345 uniqueness, 115 Fourier series, 165, 166, 167 Fuchs, L., 137
Fundamental matrix solution, 78, 105, 125, 122 of a system with constant coefficient, 81
with periodic coefficient, 87, 88
of an equation at the singularity of the first kind, 125 of an equation at the singularity of the second kind, 136, 432, 433 Fundamental theorem of existence and uniqueness, 3, 12 without Lipschitz condition, 8 Fundamental set of linearly independent solutions, 78 normal, 202 Gel'fand, 1. M., 181
Gel'fand-Levitan integral equation, 181 General solution, 7, 105 Gerard, R., 108, 112, 125 Gevrey asymptotics, 353 Gevrey order 8, 356, 367 asymptotic expansion of, 353 flat of, 342, 367, 385 formal power series, 354 uniformly on a domain, 357, 361 Gevrey property of asymptotic solutions at an irregular singularity, 441
solutions in a parameter, 385 Gingold, H., 104, 197, 217, 231 Global properties of solutions, 15 GL(n, C), general linear group, 69 Good covering, 354 Green's function, 150, 154 Gronwall, T. If., Lemma, 3, 398
Haber, S., 304
Hamiltonian system with periodic coefficients, 90
Harr inequality, 27 Harr's uniqueness theorem, 27 Harris, W. A., Jr., 142, 143, 197, 433, 447 Hartman, P., 14, 27, 28, 39, 58, 69, 144, 148, 153, 162, 197, 246, 281, 293, 298
Heaviside function, 8 Hermit polynomials, 450 Higher order scalar equations, 98 Hille, E., 103 Hirsch, M. W., 69, 251
Holomorphic function asymptotic to a formal series, 349, 356 Homogeneous systems, 78 Homomorphism of differential algebras, 359, 362, 365 Howes, F. W., 304, 339
Hsieh, P. F., 78, 104, 108, 130, 197, 217, 231, 372, 424, 436, 452 Hukuhara, M, 25, 26, 50, 108, 131, 197, 212, 218, 261, 403, 428, 447 Hukuhara-Nagumo condition, 218 for bounded solution of a second order equation, 212 for bounded solution of a system, 218 Hukuhara-Turrittin theorem, 428 Humphreys, J. E., 69 Hyperbolic sectorial domain, 296, 302 Hypergeometric series, 138 Improper node, 252, 254 perturbation of, 261 stable, 252, 253, 256 unstable, 252, 253 Independent solutions, 78 Index of isolated stationary point, 293 Index of a Jordan curve, 293 Indicial polynomial, 112 Infinite-dimensional matrix, 118 Infinite-dimensional vector, 120, 393 Infinite product of analytic function, 196 Initial-value problem, complex variable, 21 partial derivative of a solution as a solution of, 33 Initial-value problem, real variable, 1, 28, 32, 39
INDEX
nonhomogeneous equation, 96 nonlinear, 1, 16, 32, 39 second order equation, 324, 333 Inner product, 151 Instability, 243 Instability region, 189 Integral inequality, 377, 418 Invariant set, 280, 299 Irregular singular point, 134, 441 Iwano, M., 447 Iwasaki, 1., 138
J, map of A,(r,a,b) to C[[x]] 353 one-to-one, 359 onto, 356 Jacobson, N., 69 Jordan-Chevalley decomposition, 69 Jordan canonical form, 421, 426 Jordan curve, index of, 293 Jost solution, 168, 180, 194 Jurkat, W., 136, 391, 401, 402, 435
K, the field of fractions of C[[x[], 424 K[u'J, differential operator, 123 Kaplan, J., 197 Kato, J., 44, 66 Kato, T., 103 Kimura, II., 138 Kimura, T., 197 Kneser, H., theorem, 41, 47, 305 Kohno, M., 78, 108, 130, 452
Komatsu, H., 8
C, differential operator, 120, 151, 169, 424, 437, 443 normal form, 121 calculation of, 130 C+, limit-invariant set, 280 Laplace transform, inverse, 103 LaSalle, J., 279, 281 Lebesgue-integrable function, 15 Lee, E. B., 68, 106 Lefschetz, S., 279, 281 Legendre, equation, 141 polynomials, 141, 192 Leroy transform, incomplete, 354 Lettenmeyer, F., 142 Levelt, A. H. M., 108, 125, 435 Levin, J. J., 340
465
Levinson, N, 14, 28, 36. 65, 69, 108, 137, 138, 144, 153, 162, 191, 197, 225, 233, 246, 266, 274, 281, 290, 293, 298, 304, 340, 341
Levinson's diagonalization theorem, 21:3 Levitan, B. M., 181 Levitan, Gel'fand-, integral equation, 181 Liapounoff, A., 197 Liapounoff function, 239, 309, 311 Liapounoff's direct method, 281 Liapounoff's type number, 198 calculation of, 203 multiplicities, 202 of a function, 198
of a system at t = oc, 201, 204 of a solution, 199 properties, 198 Lie algebra, 90 Limit cycle, 292 Limit invariant set, 280 Lin, C.-H., 370 Lindelbf, E., 1, 28, 359 Lipschitz condition, 3, 43, 64
constant, 3 sufficient condition for, 5 existence without, 8 Local blowup of solutions, 17 Logarithm of a matrix, 86 log[ 1 + ltf ], 86, 88 log[8(w)[, 88 log[8(w)2], 88, 94 log[s], 86, 88 log[S2], 88 Lutz, D. A., 136, 197, 391, 401, 402, 435
M,,(C), set of all n x n matrices in C, 69 MacDonald, 1. C., 109 Magnus, W., 196 Mahler, K., 112 Maillet, E., 112, 353, 441 Malgrange, B., 438 Manifold, 45 differential equation on, 45 stable, 243 unstable, 246 Markus, L., 68, 69, 96, 106 Matrix, 69 functions of a matrix f (A), 84
466
INDEX
Matrix (cont.) diagonalizable, 70
in S-N decomposition, 74 infinite-dimensional, 118 nilpotent, 71 in S-N decomposition, 74 norm of a matrix A, 11A11, 69 semisimple, 70
symplectic, 93, 95 upper-triangular, 70 Maximal interval, 16 Maximal solution, 52, 54, 56, 66 Minimal solution, 52, 54, 66 Milnor, J. W., 298 Moser, J., 135 Mullen, F. E., 232 Multiplicity of an eigenvalue of a matrix, 72, 74 Multiplicity of Liapounoff's type number, 201 of a system of equations with constant coefficients, 202 of a system of equations with periodic coefficients, 202 Multipliers, of an autonomous system, 284 of a system with periodic coefficients, 89, 183, 203 of periodic orbits, 318, 319
N(1), Newton polygon of G, 437, 439 Nagumo, M., 27, 50, 197, 212, 218, 304, 306, 327, 330, 333, 334, 342 Nagumo condition for uniqueness of solution, 64, 65 Nevanlinna, F., 342 Nevanlinna, R., 359 Newton polygon, 437, 439 Nikiforov, A. F., 141 Nilpotent matrix, 71 in S-N decomposition, 74 Node, 253, 254 improper, 253, 255 proper, 253, 256 Nonhomogeneous equation, 17 Nonhomogeneous initial-value problem, 96
Nonlinear equation, 18, 372, 384
Nonlinear initial-value problem, 17, 21 28, 32, 41
Nonstationary point, 291, 292 Nonuniqueness of solution of an initial-value problem, 41 Norm, (in C[[x]], 366
in C[[x]l", 367
in CI[x]j 366 of a continuous function, 162 in E(s, A)", 367 of a matrix A, 1PAIl, 69
of a vector, 1, 345 of an infinite-dimensional vector, 394 Normal form of a differential operator, 121
Normal fundamental set, 202 Null space of a homomorphism, 366
Olver, F. W. J., 138, 141, 449, 450 O'Malley, R. E., 304 Orbit, 251, 279 periodic, 304, 318, 319
of van der Pol equation, 313, 318 Orbitally asymptotical stability, 283 Orbital stability, 283 Orthogonal sequence, 166 Osgood, W. F., 63 Osgood condition for uniqueness of solution, 63, 65
Palka, B. P., 196 Parabolic sectorial domain, 296 Parseval inequality, 166 Partial differential equation, 39 Peano, G., 1, 28 Periodic coefficients, system with, 87 Periodic orbit, 291, 297, 318, 319 of van der Pol equation, 313, 318 Periodic potentials, 183 Periodic solution, 184, 288
of van der Pol equation, 313, 318 Perron, 0., 57, 212, 443 Perturbation, of initial-value problem, 43 of a center, 271 of a proper node, 266 of a saddle point, 263 of a spiral point, 270 of an improper node, 263 Peyerimhoff, A., 391, 401, 402
INDEX
Pfaffian system, 448 Phase plane, 251 Phase portrait of orbits, 251, 302 Phillips, R. S., 103 Phragmen-Lindelof theorem, 359, 370 Picard, E., 1 Poincarc, H., 279, 291, 304 Poincarc asymptotics, 342 Poincarc-Bendixson Theorem, 291, 293 Poincarc's criterion, 300 Popken, J., 112 Potentials, 175, 178 finite-zone, 189, reflectionless, 175, 177, 178, 181 periodic, 183
Projection, P,(A), 72 properties, 73 Proper node, 253, 254 perturbation of, 266 stable, 253, 255, 267 unstable, 253 R, real line, I R", set of all n-column vectors in R, 1 Rabenstein, A. L., 36, 70, 101 Ramis, J.-P., 342, 360, 363, 371, 386, 420, 443
Reflection coefficient, 175 Reflectionless potentials, 177, 178, 181 construction, 178
Regular singular point, 131, 133, 134, 394 Ritt, J. F., 349, 356
S,,, set of all diagonalizable matrices, 70 S-N decomposition, 74, 75 existence, 74
of a differential operator, 121
of a function of a matrix f(A), 84 of a matrix for a periodic equation, 88 of infinite order, 118, 120 of a real matrix, 75 uniqueness, 75 Saddle point, 252, 255, 256 perturbation of, 261 Sansone, G., 15, 235, 279
Saito, T., 274 Sato, Y., 336
Scalar equations, higher order, 98 Scattering data, 172, 175
467
Schwartz, L., 8 Second-order equation, boundary-value problem, 304, 308 Sectorial region, 296 elliptic, 296, 302 hyperbolic, 296, 302 parabolic, 296 Self- adjoi nt ness, 151, 155
Semisimple matrix, 70 Shearing transformation, 209 Shimomura, S., 138 Sibuya, Y., 44, 66, 69, 78, 96, 108, 112, 130, 131, 136, 142, 143, 197, 217, 227, 232, 304, 342, 360, 363, 365, 369, 371, 372, 381, 382, 390, 391, 392, 399, 419, 420, 424, 436, 438, 447, 448, 452
Singular solution, 42 Singular perturbation of van der Pol equations, 330 Singularity of a linear homogeneous system, 132, 134 of the first kind, 113, 132, 136 of the second kind, 132, 134, 136, 403 n-th-order linear equation, 436 regular, 131, 133, 134, 394 irregular, of order k, 134, 441 Smale, S., 69, 251 Solution, asymptotic, 374 asymptotically stable, 236, 241 independent, 78 fundamental matrix, 78, 105, 125, 222 periodic, 184, 288 of van der Pol equation, 313, 318 singular, 42 trivial, 236, 243 uniqueness, 1, 3, 21 Solution curves, 42, 49 Spiral point, 254, 255
perturbation of, 270 stable, 254, 256, 270, 272 unstable, 254, 272 Sperber, S., 112 Stability, 235 Stability region, 189 Stationary point, 280, 291, 312 isolated, 294 Stable manifolds, 243 analytic structure, 246
INDEX
468
of solution of an initial-value problem, 1, 3, 21, 79 Osgood condition, 63, 65 sufficient conditions for, 61, 63 Unstable manifold, 246 Upper-triangular matrix, 70 Urabe, M., 288
Stegun, 1. A., 141, 449, 450 Sternberg, W., 403, 438 Stirling formula, 358, 364 Sturm-Liouville problem, 148 Subalgebra, 109 Successive approximations, 3, 20, 21, 378, 418
Sufficient condition, for asymptotic stability, 241 for uniqueness, 61, 63 for unstability, 243 Summability of a formal power series, 371, 442, 443 Symplectic group Sp(2n, R), 92, 93 matrix of order 2n, 92, 93 107
Uvarov, V. Q., 141
Tahara, H., 112 Tanaka, S., 144 Taylor's series, 345 Transform, incomplete Leroy, 354 Transformation, shearing, 209 Transversal, to orbit, 291, 292 with respect to a vector, 291 Trivial solution, 236, 243 Turrittin, H. L., 391, 403, 428 Turrittin, Hukuhara-, theorem, 428 Two dimensional system with constant
Vector, infinite-dimensional, 120, 393 norm of, 393 Vector space of solutions, 78
V image of P, (A), 73 direct sum for C", 73 van der Pol equation, 312 for a large parameter, 322 for a small parameter, 319 singular perturbation of, 330 Variation of parameters, formula of, 101
Wasow, W., 108, 304, 342, 372, 382, 449 Watson, G. N., 141, 342, 359
Weinberg, L., 142, 447 Weyl, H., 69 Whittaker, E. T., 141 Winkler, S., 196 Wintner, A., 197 Wronskian, 145, 149, 168, 172
coefficients, 251
Unbounded operators, 103 Uniformly asymptotic series, 343, 357,
Xie, F., 197, 217
x"C{x}", 110
361
x"C((Fx)l", 110
Uniqueness, Lipschitz condition, 1, 3, 64 Nagumo condition, 64, 65 of S-N decomposition of a matrix, 75 of asymptotic expansion, 343 of formal solution, 115 of solution of a boundary-value problem, 150
Yoshida, M., 138 Zeros, of eigenfunctions, 185 of solutions, 144 Zygmund, A., 167
468
Universitext
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Moise: Introductory Problems Course in Analysis and Topology Morris: Introduction to Game Theory Polster: A Geometrical Picture Book Porter/Woods: Extensions and Absolutes of Hausdorff Spaces Ramsay/Rlchtmyer: Introduction to Hyperbolic Geometry Reisel: Elementary Theory of Metric Spaces Rickart: Natural Function Algebras Rotman: Galois Theory Rubel/Colliander: Entire and Meromorphic Functions Sagan: Space-Filling Curses Samelson: Notes on Lie Algebras
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Universitext The authors' aim is to provide the reader with the very basic knowledge necessary to begin research on differential equations with professional ability. The selection of topics should provide the reader with methods and results that are applicable in a variety of different fields. The text is suitable for a one-year graduate course, as well as a reference book for research mathematicians. The book is divided into four parts. The first covers fundamental existence, uniqueness, and smoothness with respect to data, as well as nonuniqueness. The second part describes the basic results concerning linear differential equations, and the third deals with nonlinear equations. In the last part, the authors write about the basic results concerning power series solutions.
Each chapter begins with a brief discussion of its contents and history. The book has 114 illustrations and 206 exercises. Hints and comments for many problems are given.
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