Nonlinear autonomous oscillations: analytical theory

NONLINEAR AUTONOMO.US OSCILLATIONS ANALYTICAL THEORY

MINORU URABE Research Institute for Mathematical Sciences Kyoto University Kyoto, Japan

ACADEMIC PRESS New York

*

London

-

1967

COPYRIGHT

0 1967, BY ACADEMIC PRESS INC.

ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkely Square House, Londen W.1

LIBRARYOF CONGRESS CATALOG CARDNUMBER: 67-16138

PRINTED IN THE UNITED STATES OF AMERICA

Preface This monograph is mainly concerned with the analytical theory of nonlinea1 autonomous oscillations, with the approach based mostly on the author’s work. After some introductory material, in Chapter 5 a moving orthogonal coordinate system along a closed orbit is introduced. In the next four chapters, stability theory and perturbation theory are systematically discussed for general autonomous systems by means of a moving coordinate system. In Chapter 10, the two-dimensional autonomous system is discussed in detail on the basis of results obtained in the preceding chapters. In Chapter 11, a numerical method for determining a periodic solution of the general nonlinear autonomous system is described. To illustrate this, the periodic solutions of the autonomous van der Pol equation for various values of the damping coefficient are computed. Chapter 12, which is based on the work of the author and Sibuya, discusses the center of higher dimension. Chapter 13 discusses a particular inverse problem connected with the period of periodic solutions of the equation

There are, of course, many other topics of importance in the theory of nonlinear autonomous oscillations. These are, however, omitted in the present monograph because they are mainly topological rather than analytical and in order to keep the book from growing inordinately long. The numerical method of Chapter 11 is based on the theory described in the preceding chapters and, at the same time, on the Newton method and step-by-step methods for ordinary differential equations. These two numerical methods are explained in the Appendix in connection with the method of Chapter 11, taking into account recent developments. The Appendix contains some theorems and formulas due to the author which seem to be of use in practical computation. In preparing this monograph, the author endeavored to present a selfcontained and readable account for mathematicians, physicists, and engineers. The author is greatly pleased that this monograph appears in a series vii

viii

Preface

inspired and edited by Richard Bellman. The author wishes to express his hearty thanks to Professor Bellman, who suggested this undertaking and who read through the manuscript making numerous corrections and valuable suggestions. The author would also like to express his appreciation to Academic Press for the excellent work that was done in publication. March I967 Kyoto, Japan

MINORUURABB

Contents PRP..FACE • • • , • • . • • •

• • • • • • • • •

• • • •

• •

• • • • •

vii

I. Analysis of Vecton aod Matrices t .l

1.2 1.3 1.4 1.5

No~;rns of Veccon and Matrices Canonical Forms of Matrioe.s . SequenoeJ and Series of Matrices. Logarithms of Matrices . . . . . Differentiation and Integration of Matrices

3

6 10

ll

2. Basic Theorems Concerning Ordinary Dllreuntial Equations 2.1 2.2

Fundamental Existence Theorems. . . . . . . . . . . . Dependence of the Solution on Initial Conditions and Parameters

17 19

3. Linar Dllrerential Systems 3.1 3.2

4.

Linear Homoge.neous Systems Linear Nonhomogeneous Systems

24 31

Orbits of Autonomous Systems 4.1

(ntroduction . . . . . . . . . . . . .

34

4.2 4.3 4.4

Critical Points of Autonomous Systems . PerioWc So lutions and Closed Orbits . Continuity of Orbits .

37 38 39

S. Mo•lng Orthonormal Systems along a Closed Orbit S.l

5.2 S.3

A Method to Construct a Moving Orthonom1al System along a Closed Orbit. . . . . . . . . . . . . . . . . . . . . . .

42

Equations of Orbits with Respect to the Moving Onhogonal System Multipliers of Solutions of the Normal Variation Equation.

48 Sl

ix

Contents

6. StablUty 6.1 6.2

Definition of Stability. Fundamental Theorems Conoeming StabiHty.

S4 S6

6.3 6.4 6.S

Orbital Stability . . . . . . . . . . Orbital St11bility of Criti
80 82 87

7. Perturbation of Autonomous Systems 7.1 1.2

Fundamental Formulas . . . . . . . . ... . Periodic Solutions of the Perturbed System . . . . . . .

91

7.3

Stability of the Periodic Solution of the Perturbed System .

97

93

8. Perturbation of Fully Oscillatory Systems 8. 1 8.2 8.3 8.4

Univenal Periods . . . . . . . . . . . Preliminary Theorem . . . . . . . Perturbation of a Fully Oscillatory System. An Example . . . . . . . . . . . . . .

102 106 113 115

9. Perturbation of Partially Oscillatory Systems 9.1

The Reduced Form of the Partially Oscillatory S}'lltem

9.2

Perturbation of a Partially OsciiJatory System . . . .

9.3

Stability of the Periodic Solution of the Perturbed System .

9.4

Ae El
. . . . . . .

120 139

IS2 IS9

10. Analysis of Two-Dimensional Autonomous Systems 10.1 10.2 10.3 10.4 10.5

10.6 I 0. 7 10.8 10.9

Fundament11l Formulas . . . . . . . . . . . . Stability of a PeriodJc Solution of the U nperturbed System Perturbation . . . . . . . . . . . . . . . Perturbation of a Fully Oscillatory System. . . . . . . . Fundamental Formulas for Analytic S}'lltems. . . . . . . Stability of a Periodic Solution of the Aealytic Unperturbed System Perturbation of Aealytic S}'lltems . . . . . . . . . Multiplicitie3 of Closed Orbits . . . . . . . . . PerturbaLion of Analytic Fully Oscillatory Systems .

163 166 167 167 179

180 ISS t9S

199

Contents II.

xi

Numerical Computation of Periodic Solutioos 11.1 A Method to Compute a Periodic Solution . 11.2 The Two-Dimensional Case . . . . . . . . . . . . . 11 .3 Periodic Solutions of the Autonomous van dcr Pol Equation.

213

11.4 Remarks . . . . . • . . . . . . . . . . . . • . . . .

225

204

216

12. Ceoter of the Autonomous System 12.1 First Reduclion . . . . . . . . . . . . . . . . . . . . . . . 12.2

The Universal Period of the Orbit . . . . . .

229 234

12. 3 The Canonical Fo rm of the Autonomous System in the Neighborhood

of the Center.

. . . . . . . . . . . . . . . .

239

13. Inverse Problems CoDJHCted with Periods of Oscillations Described by x + g(x) ~ 0 13.1 Preliminaries. . . . . . 13.2 Lemmas. . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . •

244 246

13.3

The Pericxl Function Associated with the Maximum Velocity.

259

13.4

The Period Functions Associated with the AmpUtude and the HalfAmplitud.. . . . . . . . . . . . . . . . . . . • .

270

Appeodix. Tbe Newton Method and Step-by-8tep M ethods for Ordinary Differential Eqnations A.l

The Iterative Method to Solve Equations Numerically

U l

A.2 A.3

The Newton Method . . . . . . . . . . . . . . . A NumcricallUustrati-on of the Newton Method . . . Step-by-Step Methods for Ordinary D ifferential Equations. Convergence or the Run,e.Kuua Method . Convergence of the Multi-Step Methods.

288

A.4

A.S A.6

299 303 309 313

BlliUOGRAPKY . . . . . . . . . - • - . . . . . . . . . . • • . . . . .

323

SUBJECT INDEX • • •

327

• •

• • •

• • • • • .

• • • • • • - • - • • • • .

.

1. Analysis of Vectors and Matrices

In the present chapter, some properties of vectors and matrices necessary for subsequent discussion will be described briefly. For reference, see, e.g., Bellman [l], Halmos [1], etc.

1.1. NORMS OF VECTORS AND MATRICES

I 1,

If x is a vector, we denote its components by x i . By the norm x the nonnegative number corresponding to the vector x such that

we mean

if and only if x = 0,

IxI=o

(Ax(=l

~l.lXl I x + y I 6 Ix I + Iy I

where 1is an arbitrary scalar,

(1.1)

for any vectors x and y.

Various norms of a vector x can be defined, say, by

I x I = 1 I xi I,

or

I I,

max xi

or

(C I xi 12)1/2.

i

8

i

Corresponding to the norm of vectors, the norm A of an arbitrary matrix

IA I is defined as follows:

I

I

[ A 1 =SUP- Ax 1x1’ where x is an arbitrary vector. From this definition, it readily follows that if and only if A = 0,

IAl=O

I I. A I = I 1 I 1 A 1 + B J5 1-4 1 + p 1,

for any scalar A,

*

(1.3)

IAB16IAl.IBI. In the present monograph, for a vector x with components x i , we shall use two kinds of norms defined by

C 1 xi I

and

(c1 xi 12)1/2. i

i

1

1. Analysis of Vectors and Matrices

2

I 1,

The former norm will be denoted by the notation x and the latter norm will be denoted by the notation x Corresponding to these two kinds of norms of vectors, evidentl! two kinds of norms A and / A are defined for an arbitrary matrix A . For A 1, one can prove

I] I].

/I

I I

I

where aij is the (i,j)-element of A . In fact,

I Ax I = C I C aijxj 5 C (C I aij 1 I xj I) i

j

i

j

consequently,

IA(

~maxCIa,~I. j

i

On the other hand, let j o be the number of the column such that

Ci I aijo

= ma' j

C I aij 1; i

then, for x such that xjo = 1 and xi = 0 f o r j # j,,

I

I

\ A X = C aijol = i

C 1 aijo 1 . 1 i

x

1 7

which implies

Ci I aijoI = inax 2i I a i j I j

6 IA

1.

(1.6)

The two inequalities (1.5) and (1.6) prove (1.4). From (1.4), it is easily seen that when A is an n-by-1 matrix, namely, a column vector, the norm A of the matrix coincides with the norm of the same kind of the vector. The explicit form of 11 A 11 is not given in a simpIe form, but by means of the Schwartz inequality, the following inequality is easily proved :

I I

1.2. Canonical Forms of Matrices

3

1.2. CANONICAL FORMS OF MATRICES

The following fundamental theorem concerning the Jordan canonical form of a matrix is well known. Theorem 1.1. Let ,I1, R2, ..., As be the distinct eigenvalues of the complex n-by-n matrix A, and suppose the elementary divisors ei(l) (i = 1, 2, ..., n) of the matrix 1E - A ( E is a unit matrix) are resolved into factors as follows:

q(1) = (1 - R1)vil ( R

- R2)yi2

Then the matrix A is similar to the matrix

J2

... ...

0

...

0

( R - A,>.(..

Ij, J"

where

and Jik are Vik-by-Vik matrices of the forms

-1, 0 1

0 0

... ...

0'

0

4

1. Analysis of Vectors and Matrices

The elementary divisor ei(A)is by the definition a polynomial of L such that ei(A) = Di(A)/Di-l(A), where D,(A) is the greatest common measure of all i-by-i minors of the matrix AE - A . The form of the matrix A is called the Jordan canonical form. From Theorem 1.1, the following theorem concerning the real canonical form of a real matrix is obtained. Theorem 1.2.

A real n-by-n matrix A is similar to the real matrix

.o

0 .

.o

.. 4

. .

in the realfield. The A i ( i = 1 , 2,

..., I ) are of the forms

where Ai are the real eigenvalues of A . The Bk(k = 1,2,

Bk =

. .

..., m) are of theforms

1.2. Canonieai Forms of Matrices

5

where

are the complex eigenvalues of A .

PROOF. Since the elements of A are all real, the elementary divisors ei(A) of t E - A are the real polynomials. Consequently,when the ei(A) are resolved into factors, the complex eigenvalues of A appear in pairs.Then, inJ, of Theorem 1.1, the matrices of the forms

;I

o . . .

P k * * -

0

1

and

'

o...

. . 1

appear in pairs, where j i k are the complex conjugate numbers of P k . By changing the order of rows and columns, such pairs can be rearranged as follows:

O:

Pk

: o

: o :

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

Pk:

: o

: jik: .................................. 1

0

.. .. . . . ... ... . . . .. .. .. 0 : : 1: .. . . .................................. .. .. .. . . .. . . .. .. .. .. . .. .. .. .. .. . . .. .................................. :1

0

:

:1

0

: .... : :.

:&

1:

iik.

1. Analysis of Vectors and Matrices

6 However,

where U =

['

1 -i

i]. Hence we change the order of rows and columns of r f

of Theorem 1.1 and multiply it by E

T=[

0

y

U

0

from the right-hand side and by T-' from the left-hand side. Then we have the matrix r f of the present theorem. This implies r f of the present theorem is similar to the given A in the complex field. Then the elementary divisors of AE - r f are the same as those of AE - A . But A and A are both matrices in the real field; consequently, we see that A and r f are similar to each other in the real field. Q.E.D.

1.3. SEQUENCES AND SERIES OF MATRICES

For sequences and series of matrices (including vectors), the convergence is defined in terms of norms in the same way as for those of numbers. From this definition, it is evident that the convergence of sequences and series of matrices is the element-wise convergence. Concerning the convergence of a series of square matrices, the following

theorem holds. Theorem 1.3. Let p (>O) be the radius of convergence of the power series

c 00

(z) =

p=o

apzp.

1.3. Sequences and Series of Matrices

7

Then the series of square matrices m

c a$”

f(X)=

p=o

is convergent i f the eigenvalues of the matrix X are all less than p in their ab-

solute values.

PROOF. By Theorem 1.1, one can choose a matrix T so that 2 = T-‘XT is of the Jordan canonical form. Since X = T2T-I and X p = T f P T - ’ ( p = 1, 2, ...), we have only to prove the convergence of the series f ( f ) . However,

where A i are of the forms

Ai =

1‘

-

o . . . o Ai 1

...

0

..

..

Ai

*

o...

1

Ai

and Ai are the eigenvalues of X. Hereafter (1.8) will be abbreviated as

8=

c 0 Ai, i

and 2 is said to be a direct sum of Ai. To prove the convergence off evidently we have only to prove the convergence of f ( A i ) . Let us now rewrite A i as follows: A , = AiE

+ 2,

(g),

(1.10)

I . Analysis of Vectors and Matrices

8

where 0 0 . .

1 0 Z =

. .. .. . .. .. . .

0 1

0 0...1

Since

z2 =

:I-

.o

(1.11)

0

. .o

.

0

0.

0

0

0

1

0

. .

0 .

.o

..

0

... . 1 .. .

0

. . . . . 0.. . 0 1 0 0

1

,

zv=o

. .0

o...o

0

(v is the order of Z ) , we then have

A: = (AiE

+ Z>. = AipE + (f) A:-' 2 + (;) + ... + (v-l) , P

Ap-vil

2'

zv-'

( p = 0,1,2, ...).

Hence we have formally

f (4)

f'(ni1 l!

fll(ni) 2!

0

- 0

. . f (Ail f '(4)* l!

. 0

(Ai) :v - l)! ("- 1)

fl(n,) f (Ail l!

(1.12)

1.3. Sequences and Series of Mutrices

9

I

Now, by assumption, l i 1 < p ; therefore the series

f(4

f'(A,) 7

,

p- (Ai) l)

..'¶

( v - l)!

are all convergent. This implies that the series f ( A i ) is convergent, which proves the theorem. Q.E.D. As is seen from (1.12),

det f ( A i ) = f'(2,). Then, sincef ( X ) = Tf (2)T- formally, we have detf(X) = detf(2) = nf(L,),

(1.13)

i

where f l i f ( L i ) means the product off(&) over all eigenvalues of X. By Theorem 1.3, the series

E

x2 +x -+ -+ l!

* * a

2!

(1.14)

converges for any square matrix X, and the series

x2 x3 x--+ __... 2

3

(1.15)

converges for the square matrix X whose eigenvalues are all less than one in their absolute values. The sums of the series (1.14) and (1.15) will be denoted, respectively, by the symbols ex and log(E X). From (1.13), clearly

+

det ex =

I1e".

(1.16)

i

This implies that the matrix ex is always regular. In addition, we have e x . e - x = E,

because, in the equality

x can be replaced by an arbitrary square matrix X.

Equality (1.17) evidently implies

(1.17)

10

1 . Analysis of Vectors and Matrices 1.4. LOGARITHMS OF MATRICES

As is shown in the preceding section, the matrix A = ex is always regular. Conversely, we shall now prove the following theorem.

Theorem 1.4. For any regular square matrix A , there is a matrix X such that ex

=

A.

(1.19)

The matrix X satisfying (1.19) will be called the logarithm of the matrix A, and it will be denoted by X = log A .

PROOF.We have only to prove the theorem for the case where A is of the Jordan canonical form. For, if T is chosen so that A = T-'AT is of the Jordan canonical form and such A has a logarithm 2,then

A = TAT-' = T exp[%]T-', and this implies that A has a logarithm TgT-'. Since the matrix of the Jordan canonical form is a direct sum of the matrices of the form (l.lO), we have only to prove the theorem for the case where A is of the form A = 1E

+ Z.

(1.20)

Here Z is the matrix of the form (1.1 l), and 1# 0 since A is regular by assumption. Now consider the power series x(z) = log(1

= log1

=

log1

+ z)

+ log +

[:

-z

( + 1) 1

-

- - 1z

2

212

+ - 213

3A3

-

...I.

For this series, evidently the equality ex(') = 1 + z

holds formally. Then, in this equality, z can be replaced by the matrix Z. However, for the v-by-v matrix Z of the form (l.ll), Z' = 0; consequently, for the polynomial

1.4. Logarithms of Matrices

+

x(Z) = l o g I . E

[; z -

1 2 -2 2A2

11

1 + -z

3

313

(1.21) we have ex(')

This proves the theorem.

=

JE

+ z = A.

Q.E.D.

As is seen from (1.21), the logarithm of a real regular matrix is not necessarily real. This is natural, since even the logarithm of a real number is not necessarily real. However, as for a real number, we can prove the following theorem.

Theorem 1.5. The square of any real regular square matrix always has a real logarithm.

PROOF. Let A be an arbitrary real regular square matrix. Then, by Theorem 1.2, we can take a real regular matrix T so that T-'AT has the form A of Theorem 1.2. For such A, evidently

o...o

AI2

0 0

o . . * o

A,' .

.

.A,?

(1.22)

0

. BIZ

.

BZ2 -0

.

-

Now

+ 2)' = Ai2E + 21,Z + Z 2 ,

Ai2 = (Ail?

. . Bm2.

I . Analysis of Vectors and Matrices

12

where 2 is the matrix of the form (1.11). Using the techniques employed in the proof of Theorem 1.4, let us consider the power series Vi(Z) = log(liz = logli2 =

logl?

+ 242 + 2)

.>1 + [(i +2 ;(1 +

+ log

[ + 6,+ 1

z

-z

1

z2)

-z

;2

-

1 z2),

+

...I.

These are real power series, since li are nonvanishing real numbers. For qi(z), evidently we have

eqi(') = li2+ 21iz + '2 formally ; hence, replacing z by 2, we have = li2E

+ 2AiZ + 2'

= Ai2.

(1.23)

However, the powers of Z higher than a certain degree are all zeros; consequently, the real power series ~ ~ (turn 2 )to the real polynomials of 2. Then the equality (1.23) implies that Ai2 have the logarithms Vi= ~ ~ ( 2 ) . The analogous techniques can be applied to Bk2.In fact,

(1.24)

where

1.4. Logarithms of Matrices

However, since

ak

+ J-I

/?k

z 0,

consequently, there is a complex number P k exp ( p k

+

13

m

g

k

)

=

+

such that

J T O k

@k + f i p k .

Then

consequently, as is shown in the proof of Theorem 1.2, @k

-/?k

a]

[pk

Pk

=

[

'-'

+ J-Ok

0

0

Pk

-

J-Ok

0 = exp

U-' Pk

-

JTgk

(1.26)

where U =

[

1

i]

-i

. Equality (1.26) expresses that

[

Pk

log Mk =

-Ok a

I , Analysis of Vectors and Matrices

14

This implies

J.

Pk

logM,2

=

2

-Gk

[ok

(1.27)

Now, from (1.25), det Mk # 0; consequently, from (1.24), we have

B; = Mk2(E + 2ML

*

z + M;’.

z’),

regarding k f k as scalars. Then, we see that analogously to A,’, the matrices Bk2 also have the real logarithms Wk

=

$k(z)= log Mk2 + [ ( 2 M L 1 . Z

ML2’Z2)

Needless to say, in the right-hand side, the powers of Z higher than a certain degree are all zero as in q i ( Z ) of (1.23). From the results obtained, A i 2 = exp V,

and

consequently, if we put 0..

V’

.o

. .

. . v ,

Lo . then, from (1.22), we have

T - ~ A ~= Tex,

1.5. Diflerentiation and Integration of Matrices

15

which implies

This proves the theorem, since T and X are both real.

Q.E.D.

1.5. DIFFERENTIATION AND INTEGRATION OF MATRZCES Let X be an n-by-m matrix whose elements x i j are functions of a real variable t . Then the matrix X may be regarded as a function of t, and this will be denoted by X = X ( t ) . For such matrix functions (including vector functions), the convergence, continuity, differentiation, and integration are defined in terms of norms in the same way as for ordinary functions. From this definition, it is then evident that these operations are element-wise. For differentiation, the usual rules for sums and products are also valid for matrix functions. However, since matrices are not commutative in multiplication, care must be taken to keep the order of multiplication in differentiation of products. Thus, for X = X ( t ) , we have d(X2) - d X _dt dt

X . x + x . d-, dt

but in general we do not have

Concerning the derivatives of X - ' ( t ) and etA, we have the following theorem.

Theorem 1.6. (1.29) d - eiA = AefA= etAA dt

for any square matrix A .

(1.30)

PROOF. Equality (1.29) readily follows from differentiation of the identity X ( f ) .X--'(t) = E.

1 . Analysis of Vectors and Matrices

16

To prove (1.30), consider the (i,j)-element xij(t) of the matrix ezA.Then, since ezA = E + -t A + -t2A 2 + -t3A 3 *-., (1.31) l! 2! 3!

+

we have Xij(t) =

d,,

t +Uij l!

(1)

t2

(2)

f -aij

2!

t3 + 3! -@ +

- * a ,

(1.32)

where Jij is the Kronecker delta, and u$) are the (i, j)-elements of AP( p = 1, 2, . .). Since (1.3 1) is convergent for any finite value of t , the series (1.32) are all convergent for any value of t. Then

.

which implies d t tZ ezA = A + - A z + - A 3 +

dt

This proves (1.30). Q.E.D.

1:

2:

...,

2. Basic Theorems Concerning Ordinary Differential Equations

Some basic theorems concerning ordinary differential equations, necessary for subsequent discussion, will be stated in this chapter. The proof for most of the theorems will be omitted, however, since these could be found in almost all current textbooks (see, e.g., Coddington and Levinson [I]). The quantities appearing in the sequel are all reals unless otherwise mentioned.

2.1. FUNDAMENTAL EXISTENCE THEOREMS The most basic theorem is the following well-known fundamental existence theorem.

Theorem 2.1. Consider a system of differential equations dx/dt = X ( t , x ) ,

(2.1)

where x and X ( t , x ) are the vectors of the same dimension. Suppose X ( t , continuous in the closed region D : ( t - ~ l s a , I ~ - ( l s b

X)

is

(a,b>O)

and that it satisfies a Lipschitz condition

I X(t, x') - X ( t , x") I =< L I x' - x" I

( L > 0)

(2.2)

for every (t, x ' ) and (t, x") belonging to D. Then, there is one and only one solution x = 4 ( t ) of (2.1) passing through a point (7, t ) on the interval

I

t

- z I 5 a'

= min(a, b / ~ ) ,

I

where M = max 1 X ( t , x) and b / M = co when M = 0. 17

(2.3)

2. Basic Theorems on Diferential Equations

18

In this theorem, the derivatives of the solution at the end points of the interval (2.3) should be understood to be the suitable one-sided derivatives. This convention will be used hereafter without further mention. In the Lipschitz condition (2.2), the constant L is called the Lipschitz constant. When the function X ( t , x) is defined in the domain R of the tx-space, the function X ( t , x) is said to satisfy a Lipschitz condition locally in R, if, for any point (to,xo) of R, there is a neighborhood of (to, xo) contained in R in which X ( t , x) satisfies a Lipschitz condition. From Theorem 2.1, it is then seen that Eq. (2.1) has one and only one solution x = 4(t) passing through any point of the domain R if X ( t , x) is continuous and satisfies a Lipschitz condition locally in a. However, such a solution exists in general on a broader interval than the one specified in Theorem 2.1, since by Theorem 2.1 the solution can be continued beyond the interval specified, as long as the trajectory x = 4 ( t ) (the curve represented by the equation x = q5(t)) lies in the domain R. In connection with Theorem 2.1, it is well known that X ( t , x ) satisfies a Lipschitz condition locally in the domain 0 when it is continuously differentiable with respect to x in R. The system (2.1) has a solution passing through a point (z, c) if X ( t , x ) is merely continuous in the region R. In such a case, however, the solution passing through a point (7, <) is not unique in general. A Lipschitz condition i s indeed a sufficient condition for the uniqueness of such a solution, but it is not a necessary condition. The general theorems connected with the existence and uniqueness of the solution are omitted in the present monograph, however, since Theorem 2.1 is enough for our study of nonlinear oscillations. The following theorem is a fundamental existence theorem for a system of linear differential equations. Theorem 2.2. Consider a system of linear differential equations dx/dt

= A(t)x

+ b(t),

where x and b(t) are vectors of the same dimension, and A ( t ) is a matrix. r f A(t) and b(t) are continuous on the interval I , then any solution of (2.4) exists on I .

2.2. Dependence of the Solution on Initial Conditions and Parameters 19 2.2. DEPENDENCE OF THE SOLUTION ON INITIAL CONDITIONS AND PARAMETERS

The continuous dependence of the solution on initial conditions and parameters is guaranteed by the following theorem.

Theorem 2.3. Consider a system of diferential equations dxldt = X ( t , X, a),

(2.5)

where x andX(t, x,a ) are the vectors of the same dimension, and a is aparameter vector. Suppose X ( t , x, a ) E C [R x IT],* where R is a domain of the tx-space and ll is a domain of the a-space; X ( t , x, a ) satisfies a Lipschitz condition

I X ( t , x’, a) - X ( t , x”, a) 1 s L 1 x‘ - x” 1

(2.6)

uniformly in at for every (t, x‘) and (t, x “ ) belonging to R. Let x = #(t, 5, a) be the solution of (2.5) for an arbitrary a E ll such that #(T, <, a ) = 5 for an arbitrary (7,<) E R, z beingfixed. r f t h e solution x = #(t, to,ao) exists on the closedfinite interval I and the trajectory x = #(t, to, ao) lies in R.for t E I , then the solution x = 4(t, 4, a ) also exists on the interval I, provided 5 - to and a - a. are sujiciently small, and 4(t, 5, a ) converges to #(t, to,ao) uniformly on the interval Z as 5 + toand a ao.

I

1

I

I

--+

This theorem means that the solution x = #(t, t, a ) is continuous not only in t but also in 5 and a. The topological study of nonlinear oscillations is largely based on this theorem. The following theorem guarantees further the differentiability of the solution x = #(t, 5, a) with respect to 5 and a under some additional conditions.

Theorem 2.4. In system (2.5), suppose X ( t , x, a) E C:,,[R x Ill$( p 2 l), where i2 is a domain of the tx-space and Il is a domain of the a-space. Let x =

n.

* t

This means X ( t , x , a) is continuous in the domain R x This means (2.6) holds for every U E I I and, moreover, the Lipschitz constant L is independent of a. $ This means X ( t , x , a) is p times continuously differentiable with respect to x and tl in the domain R x

n.

2. Basic Theorems on Differential Equations

20

4(t, 5, a) be the solution of (2.5) for an arbitrary a E ll such that #(t, t, a) = < for an arbitrary (2, t) E R, z being-fixed. If the solution x = $(t, 5, a ) exists on the closedjnite interval I, and the trajectory x = 4(t, 5, a) lies in R for t E I, then 4(t, t, a) E C:,.* for every t E I. In system (2.5), let us suppose X ( t , x, a) E Ci,,[R x n]. Then, by Theorem 2.4, +(t, <,a ) E C:,.. Now, by the definition o f 4(t, 5, a),

identically in t , 5, a. Hence, differentiating both sides of (2.7) with respect to ti and ak, we have

and

where Xx(t, x , a) is a Jacobian matrix of X(t, x, a) with respect to x and

The right-hand sides of (2.8) and (2.9) are, however, continuous with respect to t, 5, and a. Hence, in the left-hand sides of (2.8) and (2.9), we can invert the order of differentiation, and thus we see that a4(r, 5, a)/agi satisfy the linear homogeneous differential system dy/dt and a$(t,

5,

= X,[r,

#(t, 5, a), ~

IY,

(2.10)

satisfy the linear differential system d.ldt = X,[t, 4(t, 5, a), alv

+ X,,[t,

4 @ 9

<,

4, .I.

(2.11)

The linear homogeneous differential system (2.10) is called the first variation equation o f (2.5) with respect to the solution x = 4(t, t, a). From the definition o f 4(t, 5, a), 4(T,

*

This means $(t,

5, a) = <

t, a) is p times continuously differentiable with respect to ( anda.

2.2. Dependence of the Solution on Initial Conditions and Parameters 21

identically in

5 and a. Therefore,

where +<(t, 4, a) is a Jacobian matrix of +(t,

r, a) with respect to 5, and

The equalities (2.12) give the initial conditions by which 84(t, 5, .)/at, and 6@(t,5, a)/da,, respectively, are determined as the solutions of the linear differential systems (2.10) and (2.11). In the study of nonlinear oscillations, various perturbation techniques are frequently used, but they are mostly based on Theorem 2.4. Now we shall proceed to the case where X(t, x , a) in (2.5) is analytic with respect to x and a. A functionf(x) of a single variable x is said to be analytic with respect to x at x = a if f ( x ) can be expanded into power series (2.13) andf(x) is said to be analytic with respect to x in the interval Z if f ( x ) is analytic with respect to x at any point of I. Let p (> 0) be the radius of convergence of the power series (2.13); then it is convergent for complex x such that x - a c p. Consequently, for complex x , by virtue of the power series (2.13), we have a complex function f ( x ) that is regular in the circle Ix - a 1 c p . In what follows, such a complex function will be called the complex extension of a real analytic function. It is evident that the complex extension takes the same real value as the original function for real value of the independent variable. Whenf(x) is analytic in the interval Z, the complex extension of f ( x ) is evidently defined in the domain D containing the real interval Z, and it is regular in D . The situations are the same even when a function is dependent on several independent variables. Making use of the above fact, one can easily prove the following fundamental theorem concerning real analytic differential systems.

I

1

Theorem 2.5.

In the real system (2.5), suppose the complex extension of X ( t , x , a) with respect to x and a is continuous and regular with respect to x and a in the region

2. Basic Theorems on Differential Equations

22

D:It-zlIa, where z,

to,and a.

Ix-5,,1<6,

]a-ao]<6

(a,b,6>0),

are real;

1 X ( t , x, a) I 5 M

( M > 0)

in D ;

the Jacobian matrix X,(t, x , a) of X(t, x , a) with respect to x isboundenin D. Then, for any real

5 and a such that

1 5 - to 1 < b/2

and

1 c1 - a. 1 < 6,

a unique real solution x = 4(t, 5, a ) of (2.5) such that the interval

I t - z I 5 a'

=

(2.14)

4(z, 5, a) = 5 exists on

min (a, b/2M),

(2.15)

and the function +(t, 5, a) is analytic with respect to 5 and a in the domain (2.14).

PROOF. As in the case of real systems, we consider the successive approximations ( n = 0, 1, 2, { X n ( t , 5, a)} a*.),

defined recursively by xo(t, 5, a) = xn+l(t,5, a) =

5,

s.'

5 + X [ s , xn(s,5, a), a1 ds

(H = 0,

1,2, ...).

Here, however, we suppose that 5 and a are the arbitrary complex numbers satisfying (2.14). As in the case of real systems, it is then easily proved that the sequence (Xn(t,

<,a)>

(n = 0, 1,2, ..)

is indeed constructed on the interval (2.15) so that

1 xn(t, 5, a) - t o I


(n = 0, 1,2, ...)

on the interval (2.15). Further, from the boundedness of X,(t, x, a), it is easily proved that the sequence {xn(t,5, a)> (a = 0, 1, 2,

*

- .)

is convergent uniformly with respect to t, 5, and a. As is easily seen, this implies the unique existence of the desired solution x = 4 ( t , 5, a) and the analyticity of the function +(t, 5, a) with respect to 5 and a. Q.E.D.

2.2. Dependence of the Solution on Initial Conditions and Parameters 23 Applying Theorem 2.5 step by step, one can easily prove the following theorem. Theorem 2.6.

In the real system (2.5), suppose the complex extension of X ( t , x, a ) with respect to x and a is continuous and regular with respect to x and a in the domain R x ll, where R is a domain of the tx-space and ll is a domain of the or-space. Let x = $(t, c, a ) be the real solution of (2.5) for an arbitrary real a E ll such that $(z, (, or) = 5 for an arbitrary real (z, 5) E SZ, T beingfixed. Zfthe solution x = $(t, 5, a ) exists on the closedfinite interval I, and the trajectory x = 4 (t, 5, a ) lies in Q for t E I, then the function # ( t , 5, u)isanalytic with respect to 5 and a for every t E I. If, in the system (2.5), X ( t , x, a) is analytic not only with respect to x and a but also with respect to t, then, by considering the complex extension of X ( t , x , a) with respect to t, x, and a, we have a theorem similar to Theorem 2.5; hence we have the following theorem corresponding to Theorem 2.6. Theorem 2.7. In the real system (2.5), suppose the complex extension of X ( t , x , a) with respect to t, x, and a is regular in the domain R x ll, where R is a domain of the complex tx-space, and ll is a domain of the complex a-space. Let x = #(t, (, a) be the real solution of (2.5) for an arbitrary real a E I T such rhat $(T, 5, a ) = ( for an arbitrary real (z, 5 ) E R, z being fixed. Zf the solution x = $(t, a) exists on the open finite interval I, and the trajectory x = $(t, 5, a) lies in R for t E I, then the function $(t, 5, a ) is analytic with respect to t, (, and a for every t E I.

c,

When the perturbation technique is applied to the system of the form (2.5) satisfying the conditions of Theorem 2.6 or 2.7, one can obtain the results in more detail using Theorem 2.6 or 2.7 than from Theorem 2.4.

3. Linear Diflerential Systems

The present chapter describes some basic properties of linear differential systems necessary for subsequent discussion. 3.1. LINEAR HOMOGENEOUS SYSTEMS

Consider a linear homogeneous system dx/dt = A(t)x,

(34

where A(?) is continuous on the interval I. Let n be the order of the matrix A(t) and x = +"'(t) ( j = 1,2, .., n) be n solutions of (3.1). By Theorem 2.2, the solutions x = +")(t) ( j = 1,2, . ., n) all exist on the interval I. Let +,(I) be the ith component of the vector + ( j ) ( t ) and @(t)be the matrix whose (i,j)elements are +ij(t). Then one can prove the following theorem.

.

.

Theorem 3.1. det @ ( t ) vanishes identically on I, or else det @(t)never vanishes on I.

PROOF.Put det @(t)= A(t) and A ( t ) = (aij(t));then we readily obtain d ! ! = kaii(f) - A(t). dt i=i

This means A(t) is a solution of the equation dyldt = a(t)y,

(3.3)

where n

a(t) =

Catj(t). i= 1

If the solution y = y(t) of (3.3) vanishes at some t = T E I, then y(z) = 0. However, y = 0 is evidently a solution of (3.3). Therefore, by the uniqueness of the solution, the solution y = y(t) must coincide with the solution y = 0; 24

3.1. Linear Homogeneous Systems

25

in other words, y ( t ) must be zero throughout I. This proves the theorem. Q.E.D. From the definition of @(t),

d@(t)/dt= A(t)@(t),

(3.4)

which shows X = @(t) is a solution of the matrix differential equation

dXldt = A(t)X.

(3.5)

This equation is called the associated equation of (3.1). The solution X = @(t) of the associated equation (3.5) is called the fundamental matrix of the original equation (3.1) if det @(t) # 0 on I. The fundamental matrix of a linear homogeneous system always exists, because there are always solutions x = & ( j ) ( t ) ( j = 1, 2, .. ., n) such that & i j ( ~ ) = dij for any t = t E I. The following theorem gives a general form of an arbitrary solution of the linear homogeneous system in terms of a fundamental matrix.

Theorem 3.2. Let @ ( t ) be any fundamental matrix of (3.1). Then any solution x = &(t) of (3.1) is expressed as

444 = @,(t)c,

(34

where c is a constant vector. Similarly, any solution X = Y ( t ) of the associated equation (3.5) is expressed as Y(t) = @(t)C,

(3.7)

where C is a constant matrix.

PROOF. Let x = 4(t) be an arbitrary solution of (3.1). Since det @ ( t ) # 0 on Z,@)-'(t) exists on Z, and hence we may put @-1(t)f$(t)= u(t). For u(t), we have then

= -@-'(t)

9 ! dt

*

a+(t)&(t)

+ @-'(t)-

__

dt

(3.8)

26

3. Linear Differential Systems

(see Theorem 1.6). Now, since x solutions of (3.1) and (3.5),

=

d'(t) - - - A(t)4(t) dt

$(t) and X

=

@(t) are, respectively, the

d sA(t)@(t). dt

and

=

Hence, from (3.8), we have du(t)/dt = -@-'(t)A(t)+(t)

+ @-'(t)A(t)$(t)

= 0.

This implies @-'(t)$(t) = c, that is, $ ( t ) in a similar way. Q.E.D.

=

@(t)c.Equality(3.7) is derived

The following theorem is evident from Theorem 1.6. Theorem 3.3. One of the fundamental matrices of

linear homogeneous system

c1

d.x/dt = A X

(3.9)

with constant coejicients is given by @(t) = etA.

By this theorem, any solution x

=

$(t) of (3.9) is given by

x = +(t) = et Ac.

Consequently, if the solution x

(3.10)

(3.11)

$(t) satisfies the initial condition

= $(TI

=

5,

(3.12)

we have erAc = <,

which implies c

=

e-'*[.

Hence, substituting this into (3.1 l), we have x

=

$(t)

=

e(z-r)Ac,

(3.13)

which is the form of the solution satisfying the initial condition (3.12). Since the fundamental matrix of the form (3.10) will be used often in what

27

3.1. Linear Homogeneous Systems

follows, it should be useful to investigate the form of (3.10) in greater detail. Let A be a Jordan canonical form of A , and put A = T-'AT. Then @(t) = exp[tTAT-']

=

Texp[t2.f]T-'.

(3.14)

Since 2.f is a direct sum of the matrices of the form I E + 2 [see (1.10) and (l.ll)], exp[rA] is a direct sum of the matrices of the form exp t(AE + Z ) . However, by (1.12),

exp t(AE

1

0

-

t I!

1

tZ

t I!

0 .

. .o'

+ Z) = etA -

2!

1

0 tv- 1

f-2

______

. . . -t

.1

consequently, exp[d] is a direct sum of the matrices of the above form. Hence we see that the column vectors of the matrix T exp[tA] consist of the sets of the vectors (e"p(t),

e'*p'(t), . .., et'pp('-')(t))

(' = d/dt),

(3.15)

where p(t) is a vector whose components are the polynomials of t with the degree v - I at most. In (3.15), evidently 1is an eigenvalue of the matrix A , and v is an order of the matrix 1E + Z . Now, as is seen from (3.14), the column vectors of Q(r) are the linear combinations of those of T exp[tA]; therefore, the column vectors of @(t) are all of the forms

C e'liPi(t), i

where Izi are the eigenvalues of A , and P,(t) are the vectors whose components are the polynomials o f t . For the periodic linear homogeneous system, one can prove the following theorem.

28

3. Linear Diferential Systems

Theorem 3.4.

Consider a linear homogeneous system dxjdt

=

A(t)x,

(3.16)

where A(t) is continuous and periodic of period 0 > 0. Let @(t)be an arbitrary fundamental matrix of (3.16). Then there is a regular matrix C such that @(t

+ 0 ) = @(t)C.

(3.17)

c = eoB

(3.18)

If we put and (3.19)

@(t)e-'B = P(t), then P(t) is periodic of period

0

and, by the transformation x = P(t)Y,

(3.20)

the given system (3.16) is transformed to the following system of constant eflcients:

(3.21)

dy/dt = By.

PROOF. By assumption, *d consequently, replacing t by

t

+ 0,we have

d@(t dt

+

because A(t

= A(t)@(t);

dt

O)

= A(t)@(t

+ 0)= A(t). B y Theorem 3.2, @(t

CO-

+ 0)

we then have

+ 0)= @(t)C.

Since @(t)is a fundamental matrix of (3.16), det @(t + w) # 0, which implies det C # 0. Thus we have (3.17) for a regular matrix C.

3.1. Linear Homogeneous Systems

29

Since C is a regular matrix, there is a matrix B satisfying (3.18), by Theorem 1.4. Then, from (3.17) and (3.18), we have @(t

+ o)= @(t)ewB,

and hence,

+ w ) e - w B= ( ~ ( t ) .

@(t Then

@(t + o ) e - w B e - f B= Q(t)e-'B,

which implies

+ w) = P ( t ) ,

P(t

since

e-wBe-tB

=

e-~B-ZB

(3.22)

- e-(tew)B.Equality (3.22) expresses that P ( t )

is periodic of period o. Lastly, substitute (3.20) into (3.16); then we have

)*d

y

+ P(t) dY - = A(t)P(t)y.

dt

dt

However, from (3.19),

= A(t)@(t)e-rB- cD(t)eetBB = A(t)P(t) - P(t)B

(see Theorem 1.6). Hence, [A(t)P(t) - P(t)B]y

+ P(t)

2 = A(t)P(t)y; dt

that is, P(t) dY - = P(t)By. dt

This implies (3.21), since det P(t) # 0 as is seen from (3.19). This completes the proof. Q.E.D.

The eigenvalues of the matrix C satisfying (3.17) are called the multipliers ofsolutions of (3.16). If we take another fundamental matrix 'Dl(t) of (3.16), then, by the above theorem, a regular matrix C1 satisfying @,(t

+

0)=

'Dl(t)C,

(3.23)

30

3. Linear Differential Systems

is again determined. Since m1(t)can be written as Q l ( t ) = @(t)T

(3.24)

by Theorem 3.2, from (3.23), we have @(t

+ w)T = @(t)TCl.

This implies @(t

+

0) =

@(t)TCIT-',

(3.25)

since det T # 0 as is readily seen from (3.24). Comparing (3.25) with (3.17), we then have

c

=

TC,T-',

or

c1= T - ~ C T . This shows the eigenvalues of C 1 coincide with those of C . In other words, the multipliers of solutions of (3.16) are independent of choice of a fundamental matrix of (3.16). This means the multipliers of solutions of aperiodic linear homogeneous system are uniquely determined by the system itself. The eigenvalues of the matrix B defined by (3.18) are called the characteristic exponents of (3.16). As is readily seen from (3.18), between the multipliers Ai of solutions and the characteristic exponents pi,there hold the relations

A i = eoPi,

or

pi = (l/o)* log li

(3.26)

(see the proof of Theorem 1.3). From the above relations, it is evident that the characteristic exponents are also independent of the choice of a fundamental matrix. As is remarked after the proof of Theorem 1.4, the matrix B defined by (3.18) is not always real even if the given system (3.16) is real. However, for the real system (3.16), by Theorem 1.5, we can always find a real matrix B' such that C 2 = exp[2oB'].

(3.27)

For such a matrix B', the matrix P l ( t ) = @(t)exp[ - tB']

(3.28)

is evidently periodic of period 20. Now, as in the case of P(t) defined by (3.19), by the transformation x = Pl(t)Y,

(3.29)

3.2. Linear Nonhomogeneous Systems

31

the given system (3.16) is transformed to the system dyldt = B’y.

(3.30)

Thus we have the following theorem.

Theorem 3.5. Consider a real linear homogeneous system (3.16), where A ( t ) is continuous and periodic of period o > 0. Let @ ( t ) be an arbitrary real fundamental matrix of (3.16) and C be a matrix satisfying (3.17). If we define a real matrix B’ by (3.27) and define Pl(t) by (3.28), then P l ( t ) is periodic of period 2 0 and, by the transformation (3.29), the given real system (3.16) is transformed to the real system (3.30) with constant coefficients.

Let pi’ be the eigenvalues of the matrix B ; then, from (3.27),

Ai2 = exp[2mpi]. Comparing these with (3.26), we thus have

Ai2 = exp[2wpi] = exp[2wpi], which implies %(pi’)

=

%(pi)

(% expresses the real part);

that is, the eigenvalues of B and B’ are mutually equal in their real parts. This will be of use when we discuss the stability of the solutions (e.g., see Corollary 1 to Theorem 6.1).

3.2. LINEAR NONHOMOGENEOUS SYSTEMS Consider the following linear nonhomogeneous system dx/dt = A(t)x

+ b(t),

(3.31)

where A(t) and b(t) are continuous on the interval I. By Theorem 2.2, any solution of (3.31) exists on I. In the present section, the form of a solution of this type will be sought. Consider the following homogeneous system corresponding to (3.31): dYW = A(t)Y,

(3.32)

3. Linear Diferential Systems

32

and let @(t)be an arbitrary fundamental matrix of (3.32). Since det @(t)# 0 on I, for any solution x = d(t) of (3.31), let us put Then

(3.33) consequently we have d*U(t)

+ @(t) du(t) dt = A(t)@(t)u(t)+ b(t).

dt

(3.34)

Since @(t)is a fundamental matrix of (3.32), d@(t) - - - A(t)@(t); dt consequently, from (3.34), we have

that is, = W'(t)b(t).

dt By integration, we then have U(t) =

s:

@-'(s)b(s)ds +

C,

(3.35)

where z is an arbitrary value belonging to I, and c is a constant. Substituting (3.35) into (3.33), we thus have

+ @(t)

$(t) = @(t)c

(3.36)

This proves the following theorem.

Theorem 3.6. Consider a linear nonhomogeneous system (3.31), where A(t) and b(t) are continuous on the interval I. Let @(t)be an arbitrary fundamental matrix of a corresponding homogeneous system (3.32); then any solution of (3.31) is given by (3.36), where z is an arbitrary value belonging to I, and c is an arbitrary constant.

3.2. Linear Nonhomogeneous Systems

33

By Theorem 3.2, any solution of the homogeneous system (3.32) is of the form (3.6). Then, by comparing (3.33) with (3.6), it can be seen that the former equality is derived from the latter one by replacing the arbitrary constant c by the function u(f).In this sense, the above method to obtain the solution of a nonhomogeneous system by the substitution (3.33) is called the method of variation of constants. By means of Theorem 3.6, the solution x = 4(t) of (3.31) such that 4(z) = 5 for any given 5 is obtained easily as follows: x

=

4(t) = Y(t)( + Y(t)

jrt

Y-'(s)b(s)ds,

(3.37)

where Y ( f )is a fundamental matrix of (3.32) such that Y(z) = E.

The fundamental matrix Y(t) satisfying the above initial condition is, however, readily obtained from an arbitrary fundamental matrix @(t) of (3.32) by

Y ( f )= @(t)@.-1(z), because @(t)@-'(z) is again a fundamentaI matrix of (3.32), by Theorem 3.2. When the matrix A(t) in (3.31) is a constant matrix, writing A ( f )as A , by Theorem 3.3, we have ~ ( t =)

Hence, by (3.37), for the solution x = +(t) of (3.31) such that 4(z) = 5, we have rt

(3.38)

4. Orbits of Autonomous Systems

4.1. INTRODUCTION

The ordinary differential equation that does not explicitly contain the independent variable, say t, is caIled an autonomous system. The equations f

+ g(x) = 0

= d/dt)

(a

and

x - A(1 - x z p

+x =0

(van der Pol equation; see, e.g., McLachlan [l], pp. 41-46, 148-151) are familiar examples of autonomous systems. As is well known, these equations can be rewritten in the form of the first-order systems as follows: 5 = y,

3

=

-g(x)

5 = y,

3

=

-x

and

+ A(l

- x')y.

Therefore, in general, it is convenient to write an autonomous system in the form of the first-order system as follows: dx/dt

= X(X),

(4.1)

where x and X(x) are the real vectors of the same dimension. In connection with the autonomous system (4.1), the x-space is called the phase space, and the curve represented by the solution x = q(t) in the phase space is called the orbit of (4.1). The curve represented by the same solution x = q(t) in the tx-space is called the trajectory of (4.1). It is then evident that the orbit is a projection of the trajectory upon the phase space. If x = q(t) is a solution of the autonomous system (4.1), then

('

q'(t) = X[q(t)]

=

Wt);

consequently, for any z, (d/dt)q(t

+ z)

= q'(t

+ T) = X[q(t + z)], 34

35

4.1. Introduction

+

which implies that x = q ( t T ) is also a solution of the original system (4.1) for any T . This means geometrically that an autonomous system always admits of a family of parallel trajectories whose projection is a single orbit. Because of this property, the orbits are used more often than the trajectories in studying autonomous systems. In the sequel, we suppose that the function X ( x ) satisfies a Lipschitz condition locally in the domain D of the phase space. Then evidently, in the txspace, X ( x ) is continuous and satisfies a Lipschitz condition locally in the cylindrical domain D x L, where L is a real line. Hence, by Theorem 2.1, for any T E L and 5 E D, a solution x = q ( t ) of (4.1) such that P(T) = 5 exists in the neighborhood of t = T , and such a solution is unique. From this, one can easily prove the following theorem.

Theorem 4.1.

Through any point P(5) of the domain D, there exists one and only one orbit of (4.1).

PROOF. The existence of an orbit of (4.1) passing through the point P(5) is evident from the existence of a solution x = p(t) of (4.1) such that p(z) = 5 for any T E L . Suppose there are two orbits x = q l ( t ) and x = q z ( t ) of (4.1) passing through the point P(5). Then, for some values T~ and T ~ , Cpl(T1)

=

'pz(Tz)

=

5-

+

Since x = ql(t)is a solution of (4.1), x = p l ( t - zz zl)is also a solution of (4.1). However, this solution takes the value C ~ ~ ( =T 5~ for ) t = zz. Then this solution must coincide with the solution x = p 2 ( t ) because of the uniqueness of the solution of (4.1); that is, we must have q z ( t ) = pl(t - T~ + T~). This implies the two orbits x = q l ( t ) and x = q z ( t ) coincide with each zl) other, since the orbits represented by x = q l ( t ) and x = pl(t - T~ are the same as the curves in the phase space. This proves the uniqueness of the orbit passing through the point P(5). Q.E.D.

+

Theorem 4.1 does not hold for orbits of non-autonomous systems. Namely, for nonautonomous systems, one can define the phase space and the orbit in the same way as for autonomous systems ;but, for non-autonomous systems, the orbit passing through a point of the phase space is not unique in general, even if the solution satisfying a given initial condition might be unique. In

4. Orbits of Autonomous Systems

36

fact, the trajectories in the tx-space passing through the points lying on a line parallel to the t-axis have in general different projections upon the phase space, and this implies the non-uniqueness of the orbits passing through a point of the phase space. When the point Q(q) lies on the orbit x = q ( t ) passing through the point P ( t ) , we shall say that the point Q is reached from the point P in the time z $5 = cp(0) and q = cp(z). This phraseology is based on the following theorems. Theorem 4.2.

If the point Q is reached from the point P in the time z, then the point P is reached from the point Q in the time - z. PROOF. Suppose P(<)and Q(q) are on the orbit x = cp(t) of (4.1), and 5 = cp(0) and r = cp(z). Since x = p ( t ) is a solution of (4.1), x = q ( t + 7 ) is also a solution of (4.1). However, this solution takes the value q ( z ) = q for t = 0, and the value cp(0) = 5 for t = -7. T h i s means the point P(5) is reached from the point Q(q) in the time -z. Q.E.D. Theorem 4.3. If the points P,((,)and P,(c,) are on the orbit x = q ( t ) of(4.1), and 5, = cp(zl) and 5, = cp(z,), then the point P, is reached from the point P, in the time z, -

21.

+

z,) is also a PROOF. Since x = q ( t ) is a solution of (4,1), x = q ( t solution of (4.1). This solution takes the value cp(zl) = tl for t = 0, and the value q(z,) = 5, for t = z, - 7,. This implies the point P, is reached from Q.E.D. the point P, in the time z, - z,.

Theorem 4.4.

If the point P,(c,) is reached from the point Po(t0)in the time zl,and the point P,((,) is reached from the point P,((,)in the time z, then the point P,(5,) is reached from the point Po(to)in the time z1 7,.

+

PROOF. By assumption, there are two orbits x such that cpl(0) =

50,

'~1(71) = 51

=

cpl(t) and x = cp&)

4.2. Critical Points of Autonomous Systems

37

and

cpm =

(Pz(T2)

51,

=

52.

+

Since x = cpl(t) is a solution of (4.1), x = pl(t z1)is also a solution of (4.1). However, this solution takes the value 'pl(zl) = tl for t = 0; consequently, by the uniqueness of the solution of (4.1), cpdt

+ 71) = cpdt).

Then we have 52

=

cpZ(TZ)

=

cpl(T2

+ 71).

Since cp,(O) = to,the above equality means that the point P2(t2)is reached from the point Po(&,) in the time x1 z2. Q.E.D.

+

4.2. CRITICAL POlNTS OF AUTONOMOUS SYSTEMS

The point P ( { ) of the domain D is called a critical point of (4.1) if X ( < ) = 0.

The point of D that is not a critical point is called an ordinary point of (4.1). If P(5) is a critical point of (4. I), then x = 5 is evidently a solution of (4.1). The trajectory corresponding to this solution is a straight line parallel to the t-axis, and the orbit corresponding to this solution is a single point P(@. In the sequel, only the orbits that are not criticalpoints will be called orbits. Let P ( ( ) be an arbitrary critical point of (4.1). Then the point P(5) can never be reached along any orbit in a finite time; for, if P(<)is reached from some other point in a finite time T along the orbit x = cp(f), then (P(T) = <,and this implies the existence of two different trajectories x = 5 and x = cp(t) passing through the same point (7, 5 ) of the tx-space. From this the following theorem readily follows.

Theorem 4.5.

If the point

Q[cp(t)]

approaches a criticalpoint P(<) of(4.1) along the orbit -+ co or - co.

x = cp(t) of (4. l), then it must be that t

+

-

PROOF. Suppose the theorem does not hold. Then there is a sequence (7,) such that neither T,,+ + 00 nor T, + - co and cp(z,) -+ { as n 00. Since neither T, + +00 nor T,,+ -00 as n + a,there is a subsequence

4. Orbits of Autonomous Systems

38

of {z,} that converges to a certain finite number z. Then, for large nk, the point z[, (~(z,,,)] is close to the point (2, 9). Then, by Theorem 2.3, the solution x = $ ( t ) of (4.1) such that $(z) = cp(z,) exists on the interval 1 for sufficiently large 1. Since J t -z

{z,}

1

x = q ( t ) = $(t

- ,z,

+ z)

as is readily seen, this implies that the solution x o f t containing z. Then we see that cp(z) =

=

cp(t) exists on an interval

r,

(44

since znk+ z and cp(z,) + 5 as nk -, co. The equality (4.2) says that the critical point P ( t ) is reached in a finite time along the orbit x = cp(t). This is a contradiction, since the critical point can never be reached in a finite time along any orbit. This proves the theorem. Q.E.D.

4.3. PERIODIC SOLUTIONS AND CLOSED ORBITS In the theory of autonomous oscillations, the principal subjects are the periodic solutions of the autonomous system. Let x = cp(t) be a nonconstant periodic solution of the autonomous system (4.1). Then, as is readily seen, the function cp(t) has the least positive period o,and any other period of cp(t) is an integral multiple of o.In the sequel, such a least positive period o will be called the primitive period of cp(t>* In connection with periodic solutions, we have the following theorem. Theorem 4.6. Let x = cp(t) be any solution of (4.1) such that cp(zl) = cp(z2) for z1 # then the function q ( t ) is periodic in t with the period z2 - zl.

+

7,;

PROOF. Since x = cp(t) is a solution of (4.1), x = cp(t z2 - zl)is also a solution of (4.1). However, this solution takes the value cp(z2) = cp(zl) for t = zl.Consequently, by the uniqueness of the solution of (4.1), we have

+ z2 - 71) = cp(t>. This shows the function q ( t ) has the period z2 - zl. cp(t

Theorem 4.7 follows from Theorem 4.6.

Q.E.D.

4.4. Continuity of Orbits

39

Theorem 4.7. The orbit C represented by the solution x = q ( t ) is simply closed if and only i f x = q ( t ) is a nonconstant periodic solution.

PROOF. Suppose x = q(t) is a nonconstant periodic solution of (4.1) of the primitive period w . Then the orbit C represented by x = q ( t ) is evidently a curve represented by x = q(t) for 0 5 t S o.Now, if q ( r , ) = q ( z 2 ) for some z1 and z2 such that 0 z1 < z2 5 w, then z2 - z1 must be a period of q(t), by Theorem 4.6; consequently, we must have z2 - z1 = w. Hence, z1 = 0 and z2 = o.This implies that C is simply closed. Conversely, suppose the orbit C represented by the solution x = q ( t ) is simply closed. Then there are two different values z1 and z2 of t such that C is represented by the equation x = q ( t ) for z1 6 t 5 z2 and P(71)

= 40(4.

(4.3)

By Theorem 4.6, the equality (4.3) implies that q ( t ) is periodic in t of the period z2 - zl.Since C is simply closed, q ( t ) cannot be a constant. Hence we see that x = q ( t )is a nonconstant periodic solution of (4.1). This completes the proof. Q.E.D. In what follows, the orbit that is simply closed will be called the closed orbit. Then, by Theorem 4.7, the problem of seeking a nonconstant periodic solution of the autonomous system is reduced to the geometrical problem of seeking a closed orbit of the autonomous system in the phase space. In the sequel, for brevity, the period of the periodic solution represented by the closed orbit will be called simply the period of the closed orbit. 4.4. CONTINUITY OF ORBITS

Consider the autonomous system dx/dt = X ( X ,a),

(4.4)

where a is a parameter vector. Let us suppose that X ( x , a) E C[D x IT], where D is a domain of the phase space and ll is a domain of the @-space; X ( x , a) satisfies locally a Lipschitz condition with respect to x uniformly in a. Then, for orbits of (4.4) lying close to each other, we have the following theorem.

40

4. Orbits of Autonomous Systems

Theorem 4.8.

Let P(5) and Q(q) be the points on the orbit C of (4.4) corresponding to = uo E ll and S be the smooth hypersurface crossing C at the point Q. Then any orbit C' of (4.4)corresponding to arbitrary a close to a. crosses the hypersurface S in the neighborhood of the point Q if C' passes through the point close to P.

a

PROOF.Let x = 4(r, 5, a) be the solution of (4.4)corresponding to arbitrary a E ll such that 4(0, (, a) = for arbitrary 5 ED. Then the orbit C is represented by the equation x = 4(ty 5, ao), and one may suppose that q = 4(z, 5, ao) for some T. Let f ( x ) = 0 be the equation of hypersurface S; then, by assumption,f ( x ) is continuously differentiable with respect to x, and

c axi

*

Xi(& ao) # 0

(4.5)

i

for x = q, where xi and X i ( x , a) are, respectively, the components of the vectors f ( x ) and X ( X , a). Since an orbit C' passing through point close to P(5) is represented by the equation x = 4(t, 5, a), the intersection of C' with S is a point of the orbit C' corresponding to t .= z', which is a root of the equation

fC4(t, r, 41 = 0.

(4.6)

By assumption, Eq. (4.6) is evidently satisfied by t = z when [ = 5 and a = ao. On the other hand, since x = 4(t, a) is a solution of (4.4), we have

c,

=

af [4(t,.5, a)] 1axi i

*

Xi

[4(4 C, a), ~

IY

(4.7)

where #Ji(t,[, a) are the components of the vector 4(t, 5, a). The quantity (4.7),however, does not vanish for t = z, 5 = 5, and,^ = a,, by (4.5). Thus, by the theorem on implicit functions, we see that Eq. (4.6) has indeed a unique solution t = t(c, a ) such that t(c, a ) -P z, as 5 +. 5 and a -,%. In addition, by (4.9,we have

4.4. Continuity of Orbits

41

(t,

t

for (C, u) close to uo), since cb[t(T, a), 5, a] + cb(7, 5, ao) as t; and a + uo. These imply that orbit C' crosses hypersurface S in a point close to Q, provided a is sufficiently close to a. and point is sufficiently close to +

c

point P ( t ) . This proves the theorem. Q.E.D.

In the proof, the quantity t(C, a) is the time required to reach hypersurface S from point c along orbit C'. Therefore, the fact that t ( [ , a) + z as 5 + 5 and a + a. means that the time required to reach hypersurface S from point ( is continuously dependent on point c and parameter a.

5. Moving Orthonormal Systems along a Closed Orbit

The search for closed orbits is a problem in the geometry of the phase space. Therefore, in the mathematical study of autonomous oscillations, it is convenient to use a moving coordinate system along an orbit, as is frequently done in differential geometry. In the present chapter, a simple method for the construction of a moving orthonormal system along a closed orbit will be introduced; then the equations of orbits with respect to the moving orthonormal system will be derived and some of their properties will be described. The discussion of the present chapter is based mostly on the author’s paper c41.

5.1. A METHOD TO CONSTRUCT A MOVING ORTHONORMAL SYSTEM ALONG A CLOSED ORBIT A local moving coordinate system can be easily constructed. Consequently, by linking each local coordinate system step by step, one can easily construct a moving coordinate system along a whole orbit. However, the coordinate system constructed in such a way does not necessarily return to the first one after rounding the orbit when the orbit is closed. This is not convenient for research in the area of closed orbits. For our purpose, it is desirable to construct a moving coordinate system so that the system may be uniquely associated with points in the neighborhood of a closed orbit. In what follows, it will be shown that this is possible for the autonomous system

dx/dt

= X(X),

(5.1)

where X ( x ) satisfies a Lipschitz condition locally. Suppose X ( x ) is defined in the domain D of the phase space and the system 42

43

5.1. A Method to Construct a Moving Orthonormal System

(5.1) has a nonconstant periodic solution x = q ( t ) lying in D.* Then we have the following lemma due to Diliberto and Hufford [ 11. Lemma 5.1.

If the dimension n of the phase space is greater than 2, then there is a unit vector e l such that x [ q ( t ) ] never coincides with - e l , where X ( 4 = X(4lll X ( 4 11-

(5.2)

PROOF. Let C be a closed orbit represented by the periodic solution x = p ( t ) ; then C does not contain any critical point (see 4.2). Therefore X [ q ( t ) ] never vanishes for any value of t, and consequently there are positive constants M and m such that

2 I[ X [ q ( t ) l 1 2 m > 0.

(5.3)

From this, it is evident that W [ q ( t ) ] is defined for every value of t . Now C is covered by a finite number of the neighborhoods in which a Lipschitz condition is satisfied. Let (Uk}be the set of these neighborhoods. Then one can prove that there is a positive number 6, such that points x’and x” on C lie in one of (uk}whenever (1 x‘ - x” (1 < 6,. Otherwise, there will be two sequences (x,”’}and (xi} such that xmfand x i never lie in the same set of { u k } and

Then, since C is a closed orbit, we can extract the subsequences (xL} and from ( x m ’ }and (x;}, respectively, so that xk, -+ x,’ and xi, xb’as mk + co ; moreover, -+

It

This inequality implies, then, that x,’ = xl.Then, since the point x,’ = xo lies in one of { or,>,say U1,the points x6, and x;, must both lie in Ul for sufficiently large mk. This contradicts the assumption on (x,’} and (xi}. Thus we see the existence of the positive number 6,, specified above. Then, from

* This means that the orbit represented by the solution abbreviation will be used hereafter.

x = p(f) lies in D . The

44

5. Moving Orthonormal Systems along a Closed Orbit

the existence of such a,, readily follows the existence of the positive number K such that, for any two points x’ and XI” on C,

1 X(X’) - X(X”)1 5 K 1 whenever 1 x’ - x” 1 < 6,.

X’

- X”

1,

(5.4)

Now put 6 = 6JM;

then, by the Schwartz inequality, for any t , and t , such that I tz we have

- t , I < 6,

5 M 21 t , - t , i2 < M2d2= 6 , 2 . On the other hand,

(5.5)

5.1. A Method to Construct a Moving Orthonormal System

45

Now, as is well known, on the unit hypersphere in n-dimensional Euclidean space, any point set of the diameter less than d is covered by a spherical cap of area less than rcd"-l, where rc is an absolute constant. Therefore, if we write the period of q(t) as w and divide the interval I = [0, w ] into N intervals of equal length less than 6 , then, from (5.6), we see that the set of points X [ q ( t ) ] on the unit hypersphere are covered by spherical caps whose total area is less than

However, this quantity tends to zero as N + 03, since n > 2. Therefore, we can choose a vector el almost everywhere on the unit hypersphere so that x [ q ( t ) ] never coincides with - e l . Q.E.D. Next we shall show how the desired moving coordinate system can be constructed along a closed orbit of the system (5.1). Let C be a closed orbit of the system (5.1) and x = q(t) be its equation. First let us consider the case where the dimension n of the phase space is greater than 2. In this case, by the above lemma, we can take a unit vector el so that x[q(t)]= X[cp(t)]/\IX [ q ( t ) ] )[ may never coincide with - e l . Starting from this e l , we construct an arbitrary constant orthonormal system { e i } ( i = 1, 2, ..., n) and put

(i = 1,2, ..., n), (5.7) denotes the transpose of the vector. From (5.7), it readily follows that e,*X[cp(t)] = cos ei

where *

n

X[cp(t)] = Ccos ei

*

e,,

i=1

from which follows

C C OeiS =~ 1, i= 1

I[

since x [ q ( t ) ] (1 = 1. From the manner of choosing vector e l , it is evident that cos el

+ 1 # 0.

(5-9)

For a moment, let us assume that cOS~,

z 1.

(5.10)

Conditions (5.9) and (5.10) say that vector X [ q ( t ) ] never coincides with vector el in their directions.

46

5 . Moving Orthonormal Systems along a Closed Orbit

Now let us rotate the orthonormal system (ei) about the (n - 2)-dimensional subspace S perpendicular to both of el and Z[q(t)]until el may coincide with x[q(t)].Let us write e, (v = 2, 3, ..., n) in the form e,

=

+ Avel + p v X ,

2,

(5.11)

and 2, is a component vector of e, in S. Then the final where X = x[q(t)], position 5, of e, after the rotation is written in the form

5,

= O,

+ &'el + pv'X.

+

Since the transformation of Avel p v x to Av'el angle 8, in the e,X-plane, it follows that

A;

= -pv

,

p,' =

(5.12)

+ pv'X is a rotation by the

A, + 2p, cos 0,.

Then, from (5.11) and (5.12), it follows that

5,

+ p v X ) - pvel + (Ay + 2pv cos 0 , ) X + pv)el + [A, + pv(2cos O1 - l)]X.

=

e, - (&el

=

e, - (a,

On the other hand, by the orthogonality of

(5.13)

e,, from (5.11), follow

el*(e, - &el - p v X ) = 0,

X*(e, - Aver - p , X ) = 0. By (5.7), these are equivalent to the following equalities:

A,

+ p, cos 0,

I, c0s el

+

=

0,

= COS 8,

Due to (5.9) and (5.10), these can be solved with respect to lows:

A,,and p, as fol(5.14)

Hence, substituting (5.14) into (5.13), we get

r, = e, - 1 +coscosevel (el + x[q(t)])

(v = 2,3,

..., n).

(5.15)

NOWlet us remove the temporary assumption (5.10) and define 5, anew by (5.15). From (5.7) and (5.9), it is evident that (, = r,(t) is continuous and periodic in t of the same period as that of cp(t). It is also easily checked that

5.1. A Method lo Construct a Moving Orthonorntal System

47

where dyP is the Kronecker delta. Thus we see that when n > 2, a desired moving coordinate system along C can always be constructed, and the simple one can be given by the orthonormal system ( x [ q ( t ) ] ,t,(t), .. ., t,(t)). When n = 2, the vector 5, given by (5.15) becomes

= -

cos 8, . el

+ cos 8 , . e,,

(5.16)

provided cos d l + 1 # 0. Then, by continuity, we may assume equality (5.16) holds also when n = 2. Equality (5.16) shows that the vector tzis a unit vector perpendicular to X [ q ( t ) ] . Hence we see that the vector 5, defined by (5.15) constitutes a desired moving coordinate system together with x [ q ( t ) ] even when n = 2, although (5.9) does not hold in this case. If X ( x ) is p-times (p 2 1) continuously differentiable in the neighborhood of the closed orbit C, then it is evident that X ( x ) satisfies a Lipschitz condition locally in the neighborhood of C. In this case, X [ q ( t ) ]isprtimes continuously differentiable with respect to t and, from (5.7), cos O i (i = 1, 2, .. ., n) are p-times continuously differentiable with respect to t . Hence, from (5.15) and (5.16), it is seen that t,(t) (v = 2, 3, .. ., n) are also p-times continuously differentiable with respect to t. This implies that if X ( x ) is p-times (p >= 1) continuously differentiable with respect to x in the neighborhood of C, then there is always a moving orthogonal coordinate system that is p-times continuously differentiable with respect to t . The results obtained in the present section are summarized as follows.

Theorem 5.1. Suppose the autonomous system (5.1), in which X ( x ) satisfies a Lipschitz condition locally in the domain D has a closed orbit C : x = q(t) lying in D. Then there is always a moving orthogonal coordinate system along C having the smoothness with respect to t the same as that of X ( x ) with respect to x. A simple moving orthogonal system is given by the orthonormal system { x [ q ( t ) ] , tz(t), t,(t)>, where x [ q ( t > ]= X [ q ( t ) ] / [x][ ~ ( t ) ]andt,(t) (V = 2, 3, n ) are the vectors given by (5.15). a a . 9

[I

..-9

5. Moving Orthonormal Systems along a Closed Orbit

48

5.2. EQUATIONS OF ORBITS WITH RESPECT TO THE MOVING ORTHOGONAL SYSTEM

In what follows, we suppose function X ( x ) of the autonomous system (5.1) is continuously differentiable in the domain D. Let C : x = q ( t ) be a closed orbit of (5.1) lying in D ; then, by Theorem 5.1 , there is a continuously differentiable moving orthonormal system { x [ q ( t ) ] ,t2(t),..., t,(t)} along C . Let C' be any orbit lying near C ; then, by Theorem 4.8, C' crosses any normal hyperplane of C in the neighborhood of C . Therefore any point x = x(z) of C' is expressed as (5.17) Here z is a time required to reach the normal hyperplane of C at the point x = q ( t ) along C' from some fixed normal hyperplane of C. Since C' is also an orbit of (5.1) the function x(z) satisfies the equation

dxo dz

(5.18)

= X[x(z)].

Then, if we put

we see that aF

-= X[x(z)],

az

aF

--

- &(t)

(v = 2, 3 ,

..., n).

(5.19)

apv

Now, if C' lies sufficiently near C, I p v I (v = 2, 3, ..., n) are small, and X ( T ) is close to q(t), as is seen from (5.17). Then, since X [ q ( t ) ] and t,,(t)(v = 2, 3, ..., n) are evidently linearly independent, from (5.19) we see that (5.20) for C' sufficiently near C,where Fi (i = 1, 2, ..., n) are the components of the vector F(z, p2, ..., pn, t). Due to (5.20), by (5.17), namely by F(z, ~

2 * *, - 3

Pn, t ) = 0,

the functions z = z(t), p y = pv(t) (v = 2, 3, ..., n) are uniquely determined, and these functions become continuously differentiable with respect to t .

5.2. Equations of Orbits

49

Substituting (5.17) into (5.18), we have

that is, (5.21) where n

= x((P +

pptp)

x

and

p=2

-'X(cp).

(5.22)

Multiplying X * on both sides of (5.21), we have then dz

-=

dt

(11

n

X

+ 1pvX*tv)/X*X'

(* = d / d t ) ;

(5.23)

v=2

consequently, multiplying rv* on both sides of (5.21), we have

( v = 2, 3,

..., n).

(5.24)

Now, from (5.22), X' can be written as

where

4)= xxr4W

(5.26)

and p is an (n - 1)-dimensional vector whose components are p v ( v = 2, 3, .., n). Hence, substituting (5.25) into (5.24), we can write (5.24) in a vector form as follows:

.

dP/dt = R(4 P) = q

q p +. 4 1 1 P 11)

(P

+

O),

(5.27)

where R(r, p) is an (n - I)-dimensional vector that is continuously differentiable with respect to p and is periodic in t, and Z ( t ) is a periodic (n - 1)by-(n - 1 ) matrix of which the ( v , p)-element ZVpis

50

5. Moving Orthonormal Systems along a Closed Orbit

-

q P ( t )= t , * ~ ( t ) < ,-

t,*t,

(v, P = 2 , 3 , .. ., n).

(5.28)

Equation (5.27) is the desired equation of the orbit with respect to the moving orthogonal system. Now, as is seen from (5.17), the vector p is the normal increment of the solution; consequently, by discarding the terms of higher order in (5.27), we have the equation for normal variation of the solution as follows: dpldt = S(t)p.

(5.29)

It is needless to say that Eq. (5.29) is nothing but the first variation equation of (5.27) with respect to y = 0. In what follows, Eq. (5.29) will be called the normal variution equation. As is seen from (5.26), the first variation equation of (5.1) with respect to the solution x = cp(t) is dy/dt = A(t)y.

Since the variation y of the solution x

=

(5.30)

q ( t ) is expressed as

let us substitute this into (5.30). Then we have

that is, (5.32) because d X [ d t ) l = A(t)X[cp(t)] dt

(5.33)

from (5.26). If we multiply t,* and X * on both sides of (5.32), then, by (5.28), we have, respectively, dP -= q t > p

dt

(5.34)

5.3. Multipliers of Solutions of the Normal Vuriation Equation

51

and

where p is an (TI - 1)-dimensional vector whose components are plr (p = 2, 3, . . ., n). As is seen from (5.31), the vector p is the normal component of the variation of the solution, and Eqs. (5.29) and (5.34) show that the normal component of the variation of the solution and the normal variation of the solution both satisfy the same differential equation.

5.3. MULTIPLIERS OF SOLUTIONS OF THE NORMAL

VARIATION EQUATION As is seen from (5.33), X[p(t)] is a solution of (5.30), which is the first variation equation of the original system (5.1) with respect to the periodic solution x = cp(t). Let y = y,(t) (v = 2, 3, ..., n) be the mutually linearly independent solutions of (5.30) that are linearly independent ofX[cp(t)]. Then, by the definition, the matrix @(t),having X[cp(t)] and y,(t) (v = 2, 3, ..., n) as the column vectors, is a fundamental matrix of (5.30). Let o be a period of cp(t); then, by Theorem 3.4, there is a regular constant matrix K such that @(t

+ o)= @(t)K.

(5.35)

By definition, the eigenvalues of K are the multipliers of solutions of (5.30). Let k i j be the (i,j)-element of the matrix K . In compliance with (5.31), let us put

then, putting t = 0 in (5.39, we have

n

52

5. Moving Orthonormal Systems along a Closed Orbit

. .., n), the equalities (5.37)

Since q(o)= q(0) and t,(o) = tp(0)(p = 2, 3, are rewritten by (5.36) as follows: n

p=2

Then, since X[q(O)], y2(0), . .., y,(O) and X [ q ( O ) ] , t2(0), ..., e n ( ( ) ) are, respectively, mutually linearly independent, we have

k l l = 1, k,, = 0

(p = 2, 3,

..., n);

(5.38)

n

(i)

QV(0)

=

c kpvop(0)+

klv9

p=2

(v, p = 2, 3,

n

(4

~ p v ( o )=

C

..., n).

(5.39)

~pKiO)kKv

K=2

The equalities (5.38) imply the matrix K is of the form

where K , is a 1-by-(n - 1) matrix and K, is an (n - 1)-by-(n - 1) matrix. Now det(p,,(O})

+ 0,

(5.41)

for, otherwise, there are numbers c,, not all zero, such that n

c CvPpv(0)

v=2

=0

Then, from (5.36), we have "

(p = 2, 3,

-.. 4. Y

5.3, Multipliers of Solutions of the Normal Variation Equation

53

Since y,(O) (v = 2, 3, ..., n) and X[cp(O)] are mutually linearly independent, the above equality implies c, = 0 (v = 2, 3, ..., n). This is a contradiction. Thus we have (5.41). Since all the column vectors of the matrix (pPV(t)) satisfy (5.34), as is shown in the preceding section, the inequality (5.41) means that matrix (pNY(t))is a fundamental matrix of the normal variation equation (5.29); and hence the equaiity (ii) of (5.39) implies the eigenvalues of K , are the multipliers of solutions of the normal variation equation (5.29). Thus, from (5.40), we have the following theorem. Theorem 5.2. One of the multipliers of solutions of thefirst variation equation of (5.1) with respect to the periodic solution x = q ( t ) is always one, and the remaining multipliers are exactly the multipliers of solutions of the normal variation equation (5.29).

This theorem will be of use when we investigate the orbital stability of the periodic solution of the autonomous system (see 6.5 of the following chapter).

6. Stability

The notion of stability comes from the study of physical motions. A ball at the bottom of a vessel is stable, while a ball at the top of a roof is unstable. Stability is related ‘to the behavior of motion after a long time. In Theorem 2.3, we have seen that the solution of a differential equation is in general continuously dependent on the initial condition in the finite interval; but this does not give any information about the stability of the solution, since stability is connected with the behavior of the solution in the infinite interval [to, m). In the present chapter, first, mathematical definitions of stability will be given, and some fundamental theorems concerning stability will be proved. Then stability for autonomous systems will be considered, and the stability of critical points and periodic solutions of autonomous systems will be discussed. References are made to Coddington and Levinson [l] and Niemytski and Stepanov [11 concerning fundamental theorems and to Urabe [4], concerning stability for autonomous systems. 6.1. DEFUVITION OF STABILITY

Consider a system of differential equations dx/dt = X ( t , x),

(6.1)

where x and X ( t , x) are vectors of the same dimension. Suppose the system (6.1) has a solution xi = q o ( t ) on the interval [to, m) and the function X ( t , x) is continuous in the domain SZ of the tx-space containing the trajectory x = PO(t). Let F be a family of solutions of (6.1) containing the solution x = qo(t). Then the solution x = q o ( t ) is said to be stable with respect to the family 9 if, for any positive number E , there are two numbers 6 and T 2 to such that for any solution x = q ( t ) E F,

I q ( t ) - qo(t) I c

E

on the interval 54

[T,a),

(6.2)

55

6.1. Definition of Stability whenever

I

Cp(t0)

- VO(t0) 1 < 6.

(6.3)

The solution x = qo(t) is said to be asymptotically stable with respect to 9 if it is stable and, in addition, lim r+m

I

- 4oo(t)

I=0

for any solution x = q ( t ) E 9 satisfying (6.3). If the solution x = qo(t) is not stable with respect to any subfamily of % containing x = qo(t),the solution x = qPo(t) is said to be unstable with respect to the family 9. For example, if there is a positive number E such that for every solution x = q ( t ) E 4 except x = qo(t), q(t') - qO(t') 2 E holds for certain t' 2 to, then the solution x = qo(t) is unstable with respect to 9. When the family 4 is the one consisting of all solutions of (6.1), the sohtion x = cpo(t) that is stable or asymptotically stable or unstable with respect to % is respectively said to be absolutely stable, absolutely asymptotically stable, or absolutely unstable. However, the term absolutely is usually omitted unless the meaning is considered ambiguous. The solution x = qo(t) is said to be conditionally stable if it is neither absolutely stable nor absolutely unstable. In other words, the solution is conditionally stable if it is stable with respect to a certain family of solutions, but it is not stable with respect to the family consisting of all solutions. The above definitions of stability are all connected with the behavior of the solutions for t 2 to. We shall add the term in the positive sense when we want to underscore the fact that t 2 to. Stability, however,$canbe considered also in connection with the behavior of the solutions for t 6 to. This stability is called stability in the negative sense, and to express this stability, the terms stable in the negative sense, unstable in the negative sense, etc., will be used. When X(t, x) of (6.1) satisfies a Lipschitz condition with respect to x locally in the domain R, the inequality (6.3) in the definition of the stability can be replaced by

1

I d t l ) - cpo(t1) 1 < 6,

I

(6.4)

where tl is an arbitrary fixed value o f t such that to 5 t l . In fact, in this case, by Theorem 2.3, the solution of (6.1) depends on the initial condition continuously; consequently, q(to)+ cpo(to)implies q ( t l ) + po(tl),and vice versa. From this, the equivalence of the conditions (6.3) and (6.4) readily follows.

6. Stability

56

6.2. FUNDAMENTAL THEOREMS CONCERNING STABILITY

To investigate the stability of the solution x

=

cpo(t),we put

and substitute this into (6.1). Then we have

where yo, Y ) = X [ t , cpo(t> + vl

- XCt, cpo(t)].

(6.7)

From (6.7), evidently Y(t,0 ) = 0,

(6.8)

and this implies y = 0 is a solution of (6.6). As is seen from (6.5), the solution x = cpo(t) o f (6.1) corresponds to the trivial solution y = 0 of (6.6), and the stability of the solution x = cpo(t) is the same as that of the trivial solution y = 0. Thus the investigation of the stability of the solution of (6.1) is reduced to that of the stability of the trivial solution of the system of the form (6.6). In what follows, we shall prove some fundamental theorems concerning the stability of the trivial solution of the system of the form (6.6). Theorem 6.1. Consider a system of direrential equations dx/dt

=

AX + X(t, x),

(6.9)

where x and X(t, x ) are the vectors of the same dimension and A is a constant matrix. Suppose (i) X ( t , x ) is continuous and satisfies a Lipschitz condition with respect to x in the region D : 1x1 < H , t o 5 t < m ( H > 0); (ii) X(t, 0) = 0 ; (iii) X(t, x ) = o( I x uniformly on the interval [to,m) us x -,0.

)I

6.2. Fundamental Theorems Concerning Stability

57

Then the trivial solution x = 0 of (6.9) is asymptotically stable if the eigenvalues of the matrix A are all negative in their real parts.

I

I

PROOF. Let x = q ( t ) be an arbitrary solution of (6.9) such that q(to)

< H. Such a solution really exists for t ( 2 to) sufficiently near to, and it is continued for larger values o f t so long as q ( t ) I c H. For such a solution,

I

evidently

holds; consequently, applying (3.38) to the above equation regarding X[t, q(t)]as b(t), we have

I

I

on the interval where the solution x = q ( t ) exists and q ( t ) < H. Now, by the remark made to Theorem 3.3, all the elements of the matrix etA are o f the forms

c e%(t),

(6.11)

i

where l i are the eigenvalues of the matrix A, and Pi(t)are the polynomials. In (6.11), the real parts of l i are, however, all negative by the assumption; therefore there is a positive constant Q such that %(Ai) c - Q. Then evidently eta

1e'"Pi(t) + o

as t + 00.

i

In other words, for any positive number Z, there is a number z 2 0 such that

I eru1eraiPi(t)I < I i

for any t 2 7. This implies the boundedness of e'"& e'"Pi(t) on the interval [0, a),since the quantity e'"& e'"P,(t) is evidently bounded on the interval [O,z]. From the boundedness o f the quantity e'"Z, e'"'Pi(t), the existence of a positive constant K (> 1) such that

I=

l e" < Ke-'" readily follows.

forany t 2 0

58

6. Stability

Now let q be an arbitrary positive number less than assumption (iii), there is a positive number 6 < H such that

r ~ . Then,

by

(6.13) on the interval [ro, co) whenever

1x15s. Let x = q(t) be an arbitrary solution of (6.9) such that

I 4 t O ) I < d / K < 6.

(6.14)

We shall prove that such a solution exists on the infinite interval [to, m) and

I q(t) I < 6

I

on the interval

[to, a).

(6.15)

I

Since q(to) < 6 < H , the solution x = q(t) certainly exists for t ( Z t , ) sufficiently near to, and this is continued for larger values of t as long as I q(t) < H. Since q(to) < 6, let us suppose that the solution x = q ( t ) exists on the interval [to, tl] and

I

I

1

I q(t) I < 6

I

on the interval

[to, tl),

(6.16)

I = 6.

tp(tl)

Then, from (6.10), by (6.12) and (6.13), we have

I q(t)1 6 Ke"('-'') for t E

[to,

t J . Let us put

(6.18)

then the inequality (6.17) is rewritten as follows:

44 6 K I d t o ) I + vlU(t>.

(6.19)

Since du(t)/dt = u(t), the above inequality is further rewritten as follows:

6.2. Fundamental Theorems Concerning Stability

This inequality can be easily integrated by multiplying Thus we have e-Vt)

s K I d t o ) I j1 (e

-vto

e-9'

59

on both sides.

- e-vr);

that is, v(t)

K

5 - 1 q(to) I . [eq(t-'o) - I]. v

Then, substituting this into (6.19), we have

1

u(t) 6 K q(to)

1

eVfr-'O).

By the first part of (6.18), we have

I q ( t ) I S K I cp(ro) I e-('-9)(t-to). This holds for any t

E

[to, t,]; consequently, by

1 &)

[ < &-(-9)('l-'d

(6.20)

(6.14), for t = t,, we have

< 6,

(6.21)

because q < IT by assumption. Inequality (6.21) contradicts the second equality of (6.16). Thus we see that the solution x = q(t) actually exists on the infinite interval [to, a),and inequality (6.15) holds on the interval [to, 00). From this result, it is evident that inequality (6.20) holds for any t E [to, a).Then, since q < IT, for any positive number E,

I q(t) 1 < E

I

on the interval [to, a)

I

if q(to) < K-' min ( 6 , ~ ) By . definition, this means that the trivial solution x = 0 is stable. (The quantity T i n the definition of stability is now equal to to). From (6.20), it also follows that

1

I

lim q(t) = 0. t-+m

This shows further that the stable trivial solution x = 0 is asymptotically stable, and completes the proof. Q.E.D. In the proof of the above theorem, the uniqueness of the solution is not used at all. As is stated in 2.1,a Lipschitz condition is not necessary for the existence of the solution. Therefore, in the assumption of the above theorem, a Lipschitz condition may be dropped. The following corollaries readily follow from Theorem 6.1.

6. Stability

60 Corollary 1.

In Theorem 6.1, replace the constant matrix A by the continuous periodic matrix A(t) and consider the system dx/dt = A(t)x

+ X(t, x),

(6.22)

where X(t, x ) satisfies the same conditions as in Theorem 6.1. The trivial solution x = 0 of (6.22) is then asymptotically stable if the characteristic exponents of the linear homogeneous system dy/dt = A(t)y

(6.23)

are all negative in their real parts. PROOF. Let w (> 0) be a period of A(t), and let @(t)be an arbitrary fundamental matrix of (6.23). Then, by Theorem 3.4, there is a constant matrix B' such that @(t+ 20) = @(t)eloB',

(6.24)

and, by Theorem 3.5, the matrix @(t)e-'B' = Pl(t) is periodic in t of period 2w. Let us now transform the given system (6.22) by the transformation x = P1(t)z.

(6.25)

Then, analogously to Theorem 3.5, we have dz/dt = B'z

+ Z'(t, z),

(6.26)

where Z(t, z) = P * - ' ( t ) x [ t , Pl(t)Z].

I

I

I

I

Since P l ( t ) is periodic, P1-'(t) and Pl(t) are both bounded on the interval (- 00, 0 0 ) ; therefore the function Z'(t, z ) also satisfies conditions (i), (ii), and (iii) of Theorem 6.1. On the other hand, by the remark made to Theorem 3.5, the eigenvalues of B' are equal to the characteristic exponents of the system (6.23) in their real parts. Therefore, by assumption, the eigenvalues of B' are all negative in their real parts. Thus, by Theorem 6.1, the trivial solution z = 0 of (6.26) is asymptotically stable. By (6.25), this implies that the trivial solution x = 0 of (6.22) is asymptotically stable because 1 Pl(t) Q.E.D. and Pl-'(t) are both bounded on the interval (- 00, 00).

I

I

I

6.2. Fundamental Theorems Concerning Stability

61

Theorem 2.1 is valid, however, even when x is a complex vector; therefore Theorem 6.1 is valid even when x, A, and X(t, x) are complex. Then, in the above proof, we may use Theorem 3.4 directly, instead of Theorem 3.5. Namely, instead of B’ satisfying (6.24), we may take a complex matrix B such that

+ w ) = @(t)euB;

@(t

and, instead of (6.25), we may use the complex periodic transformation x = P(t)z,

where P(t) = @(t)e-‘”. Using the above transformation, the original system (6.22) is transformed to the system dzldt

=

BZ + Z ( t , z),

where Z(t, z) = P-’(t)X[t, P(t)z]. Since the eigenvalues of B are the characteristic exponents themselves of the system (6.23), the corollary readily follows from Theorem 6.1. Corollary 2.

Consider a system of differential equations dxldt = X(t, x),

(6.27)

where x and X ( t , x) are vectors of the same dimension, and X(t, x) is periodic in t of period w > 0. Suppose that the system (6.27) has a periodic solution x = qo(t) of period w and that X(t, x ) is continuoudy differentiable with respect to x in the region

n = {(t, x)

I I x - qo(t) I s H,

--co < t

< m}* ( H > 0).

Then the periodic solution x = qo(t) is asymptotically stable if the characteristic exponents of thefirst variation equation of (6.27) with respect to x = qo(t)are all negative in their realparts.

*

I

I

This denotes the set of points ( t , x ) such that x - qo(r) S Hand -a < t

< CO.

62

6. Stability

then, as is shown at the beginning of the present section, the given system Is transformed into the system dyldr

=

(6.28)

Y(t,Y),

where Y(t,y ) = xrt, cPo(4

+ YI - xrt, cpo(t)l-

Let us put Y(t,r)

= KCtY

cPO(t)lY + Yl(4 Y ) ;

then

YI(C Y ) = xCt, cPo(t)

+ YI - x[t,cPo(t)I - xx[4c ~ o ( t ) I ~ - (6.29)

We shall show that the function Y,(t, y ) satisfies conditions (i), (ii), and (iii) of Theorem 6.1. Since the function Yl(t, y ) is periodic in t of period w and is continuously differentiable with respect to y in the region D:ly(SH,

--cO
the derivatives of Y,(t, y ) with respect to y are all bounded in the region D. As is well known, this implies the function Yl(ty y ) satisfies a Lipschitz condition in the region D . From (6.29), it is evident that Y,(t, y ) is continuous in the region D, and Yl(t, 0) = 0. Finally, in order to check condition (iii) of Theorem 6.1, let us rewrite (6.29) as follows: Yl(4 r) = p , C t y cpo(t)

=

jO1

+ @71Y dB - &Ct,

(KC4 cpo(t)

cpO(t)lY

+ ev1 - x x c t 9 cpo(t)l)

*

Y.

Here, by periodicity and continuity of Xx(t, x), the quantity XJt,

cpo(t)

+ eY1 - xxC4 cpO(tl1

(0

s 0 6 1)

converges to zero uniformly as y + 0. Then the above equality implies Y,(t,y ) = o( y uniformly as y + 0. This proves the validity of condition (iii) for Yl(t,y). The above results show that Eq. (6.28) is of the form of the equation considered in Corollary 1. Then the trivial solution y = 0 of (6.28) is asymptotically stable by Corollary 1 if the characteristic exponents of the equation

1 I)

dzldt

= xx[t,~ o ( t ) ] z

(6.30)

are all negative in their real parts. However, Eq. (6.30) is evidently the first variation equation of (6.27) with respect to x = cpo(t).Thus we see the validity

63

6.2. Fundamental Theorems Concerning Stability

of the present corollary, since the asymptotic stability of the trivial solution y = 0 of (6.28) implies stability of the same type for the solution x = qo(t) Q.E.D. o f the given system (6.27). Corollary 2 is frequently used for the determination of stability. Before going to the case where the eigenvalues of the matrix A in (6.9) are not all negative in their real parts, we shall prove the following lemma.

Lemma 6.1. Let B be an n-by-n matrix such that

where B , is a k-by-k matrix whose eigenvalues are all negative in their real parts, and B2 is an (n - k)-by-@- k ) matrix whose eigenvaluesare aNpositive in their real parts. Let y be an n-dimensional vector and Y(t, y ) be an n-dimensional vector function such that

(i) Y(t,y ) is continuous in the region D : I y I < H’, to 5 t < 00 (H’ > 0); (ii) Y(t, 0) = 0 ; (iii) Y(t,y’) - Y(t,y ” ) = o(I y’ - y” as yr,y” -+ 0.

I) uniformly on the interval [to,co)

Put U,(t)

=

(f” :)

and

U,(t) =

6

:tB2);

(6.31)

then, in a suflciently small neighborhood of 8 = 0, the integral equation

has one and only one solution 8 = 8(t, a) on the interval [to, co) for every a, provided I a is suficiently small. Such a solution 8 = 8(t, a) is continuous

I

64

6. Stability

with respect to t and a, and satisfies the inequality

I O(t, a) I 5 2K I a I e-'('-'O)

(6.33)

on the interval [to, m) for some positive constants K and u.

PROOF. Let us suppose the real parts of the eigenvalues of B1 are all less than -a (a > 0). Then, taking a positive number D sufficientlysmall, we may suppose that the real parts of the eigenvalues of B, are all smaller than -(a u) and the real parts of the eigenvalues of B, are all greater than 6.Then, in a similar way as for (6.12), we see that there is a positive constant K such that I U l N <= Ke-(a+b)' for t 2 0 (6.34) and

+

I

1 U,(t) I

for t

Ke"'

5 0.

(6.35)

From assumption (iii), for an arbitrary positive number E such that E

< u/4K,

(6.36)

there is a positive constant 6' < H' such that

I Y(t,Y') - Y(t,Y") I 5 & I Y' - Y" I on the interval [to, m) whenever I y' I , I y" I 5 6'.

(6.37)

Suppose

I a I < 6'/2K,

(6.38)

and consider the successive approximations Bm(t, a) (m = 0, 1,2, ...) defined recursively by eo(t, a ) = 0,

ern+l(t, a ) = v l ( t - to>a +

I[..(1-

s)y[s, orn(s, a)] ds

To begin with, we shall prove that e,(t, a) (m = 1,2, ...) are indeed constructed on the interval [to, m) and that

1 k ( t , a ) I 5 6', for t

2 to.

(6.40)

6.2. Fundamental Theorems Concerning Stability From the second of (6.39) for m O,(t,

=

65

0,

a ) = U,(t - to)a;

consequently, by (6.34),

I O l ( 4 4 I ==. Ke-(a+u)(r- I a I 5 K I a I e-a(r-fo) ro)

for t that

2 to. This proves (6.41) form =

(6.42)

1. By (6.38), from (6.42), it follows also

I el@,a ) I 5 K 1 a I < 6'/2 < 6' for t 2 to. This proves (6.40) for m = 1 . Now let us assume that O,(t, a), O,(t, a), ..., O,(t, a) are constructed on the interval [to,a)so that (6.40) and (6.41) hold on the interval [to, a)for m = 1, 2, ..., 1. Then e,(t, a ) 5 6' < H ' ;

I

1

hence, by (6.37),

I Y[S, q s , 41 I s E d ' . By (6.35) this implies the existence of the integral

p d t

- S)Y[S,

el(&

41 cis.

Then, by the second of (6.39), O,+ ,(t, a ) is indeed constructed on the interval [to, GO) and, by (6.39), (6.37), (6.34), (6.35), (6.41), and (6.36), we have successively

ds

(6.43)

66

6. Stability

This proves (6.41) for m &+I

=

1

+ 1. Then, since

- 4)f (8, - 4 - 1 )

= (4+1

+ .*. + (4- 60) + 00,

by (6.41) and (6.38), we have e,+,(t,a )

1 s K I a I e-a(t-to) 2'1 + ( 5 2K 1 a I e-a('-to' -

1

21-1 ~

+ ... +

< 6'.

(6.44)

+

T h s proves (6.40) for m = I 1. Thus we have seen that 8,(t, a) (m = 0, 1,2, .. .) are actually constructed on the interval [to, 00) and that (6.40) and (6.41) hold for m = 1,2, ... on [ t o , a). From (6.41), it follows then that

Since KI a ]/2"-' -+ 0 as m + 00, the above inequality implies the sequence (B,(t, a)) (m = 0, 1,2, ...) converges uniformly with respect to t and a on the interval [to, 00). Let lim &(t, a )

=

e(t, a ) ;

m-rm

then, from (6.44), it is evident that

I O(t, a) I S 2K I a I e-"('-'O) < 6'

(6.45)

on the interval [to, m) for every a satisfying (6.38). Now, from (6.39), it is readily seen that 8,(t, a) (m = 0, 1,2, . ..) are continuous with respect to t on the interval [to, 03) for every u satisfying (6.38). From (6.39), it is also seen that I8,(t, a')

on the interval

- B,(t,

[to, 00)

u")

I 5 2K I a'

- u"

I

(m = 0, 1, 2, ...) (6.46)

for every a' and u" satisfying (6.38). In fact, (6.46)

6.2. Fundamental Theorems Concerning Stability

67

2 2K 1 a' - u f f I for t 2 to [see (6.36)]. This proves that (6.46) holds on [to, co) for any m. The inequality (6.46) shows Bm(t, a) (m = 0, 1, 2, ...) are uniformly continuous with respect to a on the interval [to, co), provided a satisfies (6.38). From the continuity with respect to f , this implies Bm(t, a) ( m = 0, 1, 2, ...) are continuous with respect to t and a on the interval [to, 03) for every a satisfying (6.38). Then, from the uniform convergence of (eIn(t,a)> ( m = 0, 1,2, ...), we see that e(t, a) is also continuous with respect to t and a on the interval [to, a) for every a satisfying (6.38). We shall show that 0 = O(t, u) satisfies the given integral equation (6.32). Let 7 be an arbitrary positive number; then, by the uniform convergence of O,(t, u) to e(t, a), there is a positive integer m, such that

I Bm(t, a ) - e(t, a) I < 7

for any m > mo.

(6.47)

Since e(t, u) is continuous with respect to t on the interval [to, 03) and satisfies (6.45), we have successively U , ( t - s)Y[s, e(s, u)] ds

- l m U 2 ( t - s ) Y [ s , 6(s, a ) ] ds

6. Stability

68

Here T,I is an arbitrary positive number; therefore

qt,

=

u,(t - to)a + -

lo

u,(t - s)r[s, e(s, a)] as

p2(r

- s)Y[s,

e(s, a)] ds.

(6.48)

This shows that 8 = e(t, a) satisfies the integral equation (6.32), which evidently proves the existence of the solution of the given integral equation (6.32). Now we shall prove that besides 8 = e(t, a) obtained above, the integral equation (6.32) has no other solution that exists on [to, 00) and satisfies 0 5 6' there. Let 0 = B'(t, u ) be an arbitrary solution of (6.32) that exists on [to, a)and satisfies 0 6' there. Then, from (6.32) and (6.48), we have

1 1

I Is

Let A4 be

(6.49) from the above inequality, we then have

EKM I eyt, - q t , I 5 ~__ d ~

U)

U)

!M 2

[see (6.36)].

6.2. Fundamental Theorems Concerning Stability

Needless to say, this inequality holds for any t

69

1 to. Then, by (6.49), we have

2cKM 1 OSMS< -M, d 2 which implies M = 0; that is, 8’(t, a) = 8(t, a). The uniqueness of the solution of (6.32) in the neighborhood of 8 = 0 is thus proved. The continuity of the solution 8 = 8(t, a) is already shown, and the ineQ.E.D. quality (6.33) is evident from (6.45). This completes the proof. In (6.32), let ai (i = 1,2, . .., n) be the components of the vector a. Then, as is seen from (6.31) and (6.39), &(t, a) (m = 0, 1,2, ...) are dependent only on a, (K = 1,2, .. ., k).Therefore, as the limit of tl,(t, a), the solution 8 = e(t, a) of (6.32) is also dependent only on a, (K = 1,2, ..., k). Next, if we differentiate both sides of (6.48) with respect to t, we have de(t, __ a )_ -BU,(t - to) a dt

= Bqt, a)

+ q t , e(t, a ) ] .

This shows that y = e(t, a) is a solution of the differential equation dy/dt = By

+ Y(t,y).

(6.M)

Now we shall prove the following theorem, whch is concerned with the case where the eigenvalues of the matrix A in (6.9) are not all negative in their real parts.

Theorem 6.2. Consider a system of differential equations dx/dt = Ax

+ X(t, x),

(6.51)

where x and X(t, x ) are the n-dimensional vectors and A is an n-by-n constant matrix.

6. Stability

70 Suppose

(i) X ( t , x ) is continuous in the region D : x < H , to 5 t < co ( H > 0); (ii) X ( t , 0) = 0 ; (iii) X(t, x‘) - X ( t , x”) = o(( x’ - x” uniformly on the interval [to, co) as XI, x” + 0.

I

I

I)

If the real parts of the k eigenvalues of the matrix A are all negative, and the real parts of the remaining ( n - k ) eigenvalues of the matrix A are all positive, then, in the x-space, there is a k-dimensional manifold S passing through the origin such that the trivial solution x = 0 of (6.51) is asymptotically stable with respxt to the family 9 consisting of solutions intersecting S at t = to and is unstable with respect to the family 5 consisting of remaining solutions.

PROOF. From the assumption on the matrix A , by Theorem 1.2, there is a real regular matrix P such that (6.52)

where B, is the k-by-k matrix whose eigenvalues are all negative in their real parts, and B, is the (n - k)-by-(n - k ) matrix whose eigenvalues are all positive in their real parts. Let us put Px

=y

;

(6.53)

then, as is readily seen, the given system (6.51) is transformed to the system of the form

dyldt = BY

+ Y(t,Y),

(6.54)

where Y(t, y ) = PX(t, F l y ) . From conditions (i) and (ii) on X ( t , x), it is evident that Y ( t , y ) is continuous in the region

D ’ : l y l < H ’ = H / l P - ’ ) , to 5 t <

00,

and Y(t, 0) = 0. Further, from condition (iii) on X ( t , x), for any positive number E , there is a positive constant 6 < H such that

I 5 I I I p-’I I x’ - I on the interval [to, m) whenever I x’ I , I x” I 6 6 . Then, for any y’ and y” I X ( t , x‘)

-X(t,

E

X‘I

X’I)

*

6.2. Fundamental Theorems Concerning Stability

I

71

I I I 5 6‘ = 6/ 1 P-‘ I < H‘, on the interval

such that y’ , y” have

I Y(t, y’)-

Y(t, y”)

1 5 1PI

*

I q t , P-’y’)--(t,

[to,

a),we

P-ly’’)1

5+’-y’’I.

(6.55)

These results show that Y(t, y) also satisfies conditions (i), (ii), and (iii) of the theorem. For the matrix B of (6.52), let us define the matrices U,(t) and U,(t) by (6.31). Then it is evident that

etB = V , ( t )

+ U,(t)

and

dUi

-

dt

= BUi

(i = 1, 2).

Let us suppose the real parts of the eigenvalues of B, are all less than - a > 0). Taking a positive number CT sufficiently small, we may suppose that the real parts of the eigenvalues of B1 are all smaller than -(a 0 ) and the real parts of the eigenvalues of B, are all greater than n. Then, as is shown in the proof of the preceding lemma, we have (6.34) and (6.35) for some positive constant K . In the sequel, let us suppose E is chosen so that (6.36) may hold. Next, comparing (6.55) with (6.37), we may suppose 6’, chosen above, coincides with 6‘ in the proof of the preceding lemma. Now, from (6.55), it is evident that the function of the right-hand side of (6.54) satisfies a Lipschitz condition in the region

(a

+

I I S 6‘, to S t <

D,. : y

00.

1

I

Let y = q ( t ) be an arbitrary solution of (6.54) such that q(to) is sufficiently small. Then this solution exists for t (2 to) sufEciently near to, and it is continued for a larger value of t as long as q ( t ) 5 6’. For such a solution, analogously to (6.10), the equality

I

q(t) = e(‘-‘O)Bq(t,)

I

+ l f - s ) ~ Y ~qs( s, ) l cis

holds i e the interval where the solution y = q(t) exists and

I q(t) I 5 6‘.

6. Stability

72

The above equality can, however, be rewritten in terms of U,(t) and U,(t) as follows :

4)= u,(t - t o ) d t 0 )+ uz(t- t o ) c ~ ( t o )

+LUdt +p

-

s)YCs, ml ds

z ( t-N

- S ,

4+)l ds-

(6.56)

Let us now suppose that the solution y = q(t) exists on the interval [to, a),and there q(t) S 6'. Then, by (6.35),the integrals

I

I

exist for any t 2 to because

I Y[s, cp(s)] 1 = I Y[s, cp(s)l - yrs, 01 I i I d s > I 6 Ed' by (6.55). We can then rewrite the second integral in the right-hand side of (6.56)as follows:

(6.57) Substituting (6.57)into (6.56),we have

(6.58)

6.2. Fundamental Theorems Concerning Stability

73

where

Let us rewrite (6.58) as follows: UZ(t- t0)c = V(t),

(6.60)

where

By (6.31), the equality (6.60) implies ,$t-to)Bz

c

- V'(t),

(6.61)

where c' and V'(t) are the (n - k)-dimensional vectors whose components are the last ( n - k) components of the vector c and V(t),respectively. Equality (6.61) is evidently equivalent to the equality cf =

,-(t-fo)Bzyt

( ).

(6.62)

However, by (6.31) and (6.35),

I e-tBZ I = I ~ , ( - t ) I 5 Ke-"'

for

t

2 0.

Hence, from (6.62), we have

I c' I -

Ke-"('-'o)

I V'(t) I

for t 2 to.

(6.63)

Now, by (6.34) and (6.35), V ( t ) is bounded as t + co, since I cp(t) 1 S d' for t 2 to, by assumption. The boundedness of V(t)evidently implies the boundedness of V'(t). Hence, letting t + 00 in (6.63), we see that c' = 0.

(6.64)

On the other hand, from (6.59), c, = %(to)

.( = 1,2, ..*, k),

(6.65)

where cpi(to) ( i = 1, 2, ..., n) are the components of the vector cp(fo). By (6.64) and (6.65), from (6.58), we have thus

6. Stability

74

~ ( t =) U , ( t - t0)c

-

p2(t

+

6:

U , ( t - s ) Y [ s ,v(s)] ds

- S)Y[S,

cp(s)l ds.

(6.66)

This is of the same form as Eq. (6.32) in the preceding lemma. Hence, by the preceding lemma, we have ~ ( t=) O(t, Now, for the solution O (K =

OK(to,a)= U , Ov(t,,u) = - [

1 2 ( f 0

=

for t 2 to.

C)

(6.67)

O(t, u ) of (6.32), from (6.32),

1,2, ..., k),

3.

(v= k

- s)Y[s,O(s,u)]ds

+ l , k + 2,...,n),

(6.68)

where O,(t, u) (i = 1, 2, .. ., n) are the components of the vector O(t, a), and denotes the v-th components of the vector inside the brackthe notation [. ..Iv ets. Put $v(ui,

a27

...

7

Uk) =

-

[p2(f0 .)I -

(V

then (6.68) shows that the point y fold s" defined by Yv

= $v(Y1, yi,*

* *,

Y,)

=

=k

s)Y[s, O(s,

ds

I.

+ 1, k + 2, ..., n);

(6.69)

O(to, u ) lies on the k-dimensional mani-

(v = k

+ 1, k + 2,

f

* *Y

4

in the y-space. By (6.67), this implies that the point y = cp(fo) lies on the manifold s", in other words, the trajectory y = cp(t) of (6.54) intersects s" at t = to. Since O(t, 0) = 0 as is seen from (6.33), it is evident that $"(O, 0, ..., 0) = 0 (v = k + 1 , k + 2, .. ., n); that is, s" passes through the origin in the y-space. Let %' be the family of solutions of (6.54) whose trajectories intersect the manifold s" at t = to. Then our result says that any solution y = p(f) of (6.54) that exists on [to, co), and satisfies cp (t) 5 6' there, necessarily belongs to the family 9'. Clearly, this result implies that the trivial solution y = 0 of (6.54) is unstable with respect to the family consisting of solutions that do not belong to 9' (see Fig. 1). Next we shall prove that, with respect to the family F', the trivial solution y = 0 of (6.54) is asymptotically stable. Let -v= p(t) be an arbitrary solution and suppose I q(to) is sufficiently small. of (6.54) belonging to the family 9'

I

I

1

6.2. Fundamental Theorems Concerning Stability

75

Put (K =

VK(t0)= U K

1,2, ...)k);

(6.70)

then, by the assumption, we have vdt0) =

$v(al,

- - * ?

ak)

+ 1, k + 2, ..., n).

(v = k

(6.71)

I I

On the other hand, since a, (K = 1,2, ..., k) are small by assumption, we have a solution 8 = O(t, u ) of (6.32) corresponding to this a, (K = 1,2, ..., k). As is shown already, y = O(t, a) is then a solution of (6.54) and, by (6.68) and (6.69), we have e~(t07

(K =

)' = K'

O ~ ( ~ 0a) 7

=

$V(~I?

1,2, ...)k),

( v = k + 1 , k + 2,..., n).

ak)

By (6.70) and (6.71), this implies that O(t074

=

dto).

Since the solution of (6.54) satisfying the same initial condition is unique in the region Db, by (6.55), the above equality implies v(t) =

e(t,

for t

2 to.

Then, from (6.33), we have

1 q(t) I 5 2~ I a I e - " ( ' - ' ~ )

for t 2 to.

(6.72)

Here y = q ( t ) is an arbitrary solution of (6.54) belonging to 9' such that q(to) is small; therefore, by (6.70), the inequality (6.72) implies the trivial solution y = 0 of (6.54) is asymptotically stable with respect to the family 9'. The conclusions of the theorem are then easily derived from the results

I

I

6. Stability

76

obtained above. The k-dimensional manifold S for the original system (6.51) Q.E.D. is evidently the transform of s" by the transformation (6.53). Remark 1.

As is seen from the above proof, the equation of S is given by x = P-'y(a), where a = (al, a,, ..., a&)is a k-dimensional parameter vector and Yd.1

=

(K =

a,

Y v ( 4 = $v(als a,, .*

*Y

4

1,2, ..., k),

(v=k+l,k+2

,..., n).

(6.73)

Moreover, from (6.69), by (6.55), (6.35), and (6.33), we have

Pm

2&KZ

=-

u + a

14-

(6.74)

Since E is an arbitrary positive number satisfying (6.36), the above inequality implies $v(al,a,,

.*.,

a&) =

.(I

a

1)

as a -+ 0,

(6.75)

because the inequality (6.74) holds for any a satisfying (6.38), where 6' is a number depending only on E . Remark 2. In Theorem 6.2, suppose X ( t , x) is analytic with respect to x . Then Y(t, y ) in (6.54) is evidently analytic with respect to y . However, in the lemma to Theorem 6.2, the solution O = O(t, p) of (6.32) is analytic with respect to a if Y(t, y ) is analytic with respect to y . This is because, in this case, the successive approximations O,(t, a ) ( m = 0, 1, 2, ...) defined by (6.39) are all 1,2, lytic with respect to a, and the complex extensions of O,(t, a) (m = 0, ana...) converge uniformly to O(t, a). From the analyticity of O(t, a), we see then that the functions $Jul, a,, ..., a&) (v = k + I , k + 2, ..., n) defined by (6.69) are all analytic with respect to a. This implies the k-dimensional manifold 5 ' is an analytic manifold.

6.2. Fundamental Theorems Concerning Stability

77

In Theorem 6.2, we have considered the case where no eigenvalue of A is zero in its real part. For the case where some of the real parts of the eigenvalues of A may be zeros, we have the following theorem. Theorem 6.3. For system (6.51), we assume conditions (i), (ii), and (iii) of Theorem 6.2 and, in addition, we suppose the real parts of the k eigenvalues of the matrix A are all negative and the real parts of the remaining ( n - k) eigenvalues of the matrix A are all nonnegative. Then, in the x-space, there is a k-dimensional manifold S passing through the origin such that the trivial solution x = 0 of (6.51) is asymptotically stable with respect to the family F consisting of the solutions intersecting S at t = to.

PROOF. Suppose the eigenvalues of A whose real parts are all negative are all less than --a (-a > 0) in their real parts. Put = e-a(t-to)

and substitute this into (6.51):Then

we have

dyldt = BY

where B

=

A

(6.76)

Y

+ Y(t,y),

(6.77)

+ CXE,

Y ( t ,Y) = ea(t- to)^ [ , e - a ( t -

(6.78) to)Y l -

(6.79)

Equality (6.78) says that the eigenvalues of B are all greater than those of A by -a. By the assumption on A , this implies that the k eigenvalues of B are all negative in their real parts, and the remaining ( n - k) eigenvaluesare all positive in their real parts. Then, since Y ( t , y ) satisfies conditions (i), (ii), and (iii) of Theorem 6.2 as is seen from (6.79), Theorem 6.2 can be applied to system (6.77). Thus we see that in the y-space, there is a k-dimensional manifold s”: y = $(al, a,, ..., ak) passing through the origin such that the trivial solution y = 0 of (6.77) is asymptotically stable with respect to the family @ consisting of the solutions intersecting 5 at t = to. Now, by the transformation (6.76), the solutions of (6.77) belonging to the family are transformed into the solutions of (6.51) intersecting the k-dimensional manifold S : x = $(al, a2, ...,ak)

78

6. Stability

at t = to, because x and y take the same value at t = to in the transformation (6.76). Let 9 be the family of solutions of (6.51) intersecting S at t = to. Then, since x I 5 I y for t 2 to in the transformation (6.76), we see that the trivial solution x = 0 of (6.51), which corresponds to the solution y = 0 of (6.77), is asymptotically stable with respect to the family F. This proves the theorem. Q.E.D.

I

I

Under the assumptions of Theorem 6.3, it may occur that the trivial solution x = 0 is stable with respect to the larger family of solutions containing F. In illustration, consider the linear system dx/dt

(6.80)

= AX,

where -1 A = (

0 0

88

The general solution of this system is evidently x1 = c,e-',

x2 = c2,

(6.81)

x 1 = c3ef,

while the equation of the manifold S for (6.80) is evidently x1 = a,

x1 = 0,

x3 = 0

(a : parameter).

Therefore, the solutions belonging to the family intersecting S at t = 0, are x1 = ae-',

x 2 = 0,

(6.82)

F,that is, the solutions

x3 = 0.

From this it is evident that the trivial solution x = 0 of (6.80) is asymptotiHowever, as is readily seen from cally stable with respect to the family F. (6.81), the trivial solution x = 0 of (6.80) is stable with respect to the family B consisting of solutions such that x 1 = c,e-',

x2 = c2,

x3 = 0

(cl, c2 : arbitrary

constants).

Evidently B is the larger family of solutions containing 9 as the subfamily. Theorems 6.1,6.2, and 6.3 are concerned with stability in the positive sense. Concerning stability in the negative sense, however, analogous theorems can easily be derived from these theorems by replacing the time variable f by - t . For example, from Theorems 6.2 and 6.3, we have the following theorem.

6.2. Fundamental Theorems Concerning Stability

79

Theorem 6.4.

Consider a system of dtflerential equations dx/dt = AX + X(t, x),

(6.83)

where x and X ( t , x ) are the n-dimensional vectors and A is an n-by-n constant matrix. Suppose

(i) X ( t , x ) is continuous in the region D:IxI
tozt>-W

(H>O);

(ii) X(t, 0) = 0, (iii) X(t, x’) - X ( t , X I ’ ) as x” -+ 0.

=

I - x” I)

o( x’

uniformly on the interval (- co, to]

XI,

Assume the real parts of the k eigenvalues of the matrix A are allpositive, and the real parts of the remaining ( n - k ) eigenvalues of the matrix A are all nonpositive. Then, in the x-space, there is a k-dimensional manifold S passing through the origin such that the trivial solution x = 0 of (6.83) is asymptotically stable in the negative sense with respect to the family 9 consisting of the solutions intersecting S at t = to. When every eigenvalue of A does not vanish in its real part, the trivialsolution x = 0 of (6.83) is unstable in the negative sense with respect to the family consisting of solutions that do not belong to 9.

If we combine Theorems 6.2, 6.3, and 6.4, then we have the following theorem. Theorem 6.5.

Consider a system of diflerential equations dx/dt = AX + X ( t , x),

(6.84)

where x and X ( t , x ) are the n-dimensional vectors, and A is an n-by-n constant matrix. Suppose (i) X(t, x ) is continuous in the region D:IxI
X(t, 0) = 0 ;

(H>O);

80

6 . Stability

(iii) X(t, x’) - X(t, x”) as x‘, x” -+ 0.

= o(

I x’ - x” I) uniformly on the interval (-

00, 00)

Assume that, in their real parts, k eigenvalues of A are negative, 1 eigenvalues of A are positive, and the remaining (n - k - 1) eigenvalues of A are zero. Then, for anyfinite to, in the x-space, there are two manifoldr S , and S - of dimensions k and 1 passing through the origin such that the trivial solution x = 0 of (6.84) is asymptotically stable in the positive sense with respect to the family %, consisting of solutions intersecting S+ at t = to and in the consisting of solutions intersecnegative sense with respect to the family 9.ting S - at t = t,. Especially when n = k + 1, the trivial solution x = 0 of (6.84) is unstable in both positive and negative senses with respect to the family consisting of solutions that belong to neither 9 , nor %-. 6.3. ORBITAL STABILITY

As is mentioned in Chapter 4, solutions of an autonomous system dx/dt = X ( X )

(6.85)

represent trajectories in the tx-space and orbits in the phase x-space. The stability in the preceding sections is, however, connected only with trajectories. Consequently, in the present section, stability connected with orbits will be discussed. Suppose the system (6.85) has a solution x = po(t) on the interval [r,, a), and the function X ( x ) is continuous in the domain D of the phase space that contains the orbit Co :x = po(t). Let F be a family of solutions or orbits of (6.85) containing the solution x = po(t) or C,. Then the solution x = po(t) or the orbit C, is said to be orbitally stable with respect to the family % if, for any positive number E , there are two numbers 6 and T 2 to such that, for any solution x = p ( t ) E 9, dist [q(t), C,] < E

for t

>= T

(6.86)

whenever

I Cp(t0) - cPo(t0) 1 < 8.

(6.87)

Here the notation dist [ ,] denotes the distance in the phase space between the two geometric objects inside the brackets.’ The solution x = po(t) or the

6.3. Orbital Stability

81

orbit Cois said to be asymptotically orbitally stable with respect to the family $” if it is orbitally stable and, in addition, lim dist [cp(t), Co] = 0 t-m

for any solution x = cp(t) E 9 satisfying (6.87). If the solution x = cpo(t)is not orbitally stable with respect to any subfamily of F containing x = cpo(t),the solution x = cpo(t)or the orbit C, is said to be orbitally unstable with respect to the family 9. On the basis of the above definitions, absolute orbital stability, conditional orbital stability, orbital stability in the positive sense, orbital stability in the negative sense, etc., are defined analogously to the definitions in 6.1. As is evident from the above definition, the solution x = cpo(t) is orbitally stable with respect to the family F if it is stable with respect to 9. However, the converse is not always valid. In illustration, consider the system

dx1 _ dt

27r

_1

+ ( X 1 2 + x22)1’2

x2,

dx2 dt 1

271

+ ( x I 2 + x2 )

2 1/2

x1. (6.88)

Putting x1 = r cos 8, x 2 = r sin 8, let us transform the coordinates (xi, x 2 ) to the polar coordinates (r, 8). Then we have the system

dr -=o, dt

d8 - -- - 27r dt 1 + r’

(6.89)

the general solution of which is

(6.90) where a and cp are the arbitrary constants. As is seen from the first of (6.90), the orbits of the system (6.88) are the concentric circles with the common center at the origin. Thus any solution of (6.88) is absolutely orbitally stable. However, any solution of (6.88), except r = 0, cannot be absolutely stable. In fact, let

27r 8 = e(t) = -t + @ 1+8

r = 6,

(8

> 0)

(6.91)

be an arbitrary nontrivial solution of (6.89), and consider the following solutions of (6.89):

r=a,=a--

1

2m

+ 1 (1 + a),

e = e,(t)

=

27r

~

1

+ a , t + 4,

(6.92)

82

6. Stability

where m is an arbitrary positive integer such that

m>

-.1

26

Then, for any positive integer k, we have d[km(l

O,[km(l

+ @, + a)] = k(2m + 1)n + @. + a)]

=

2kmn

(6.93)

c

Let and Cmbe the orbits represented by (6.91) and (6.92), respectively, and let P, Q,, and Q,' be the points such that their polar coordinates are (a, @), (am,@) and (am,@ + n), respectively. Then (6.93) says that the point at P for 1 = 0 arrives again at P moving along c after the time km(1 a), while the point at Q, for t = 0 arrives at Q , or Q,' moving along Cmafter the same time, according as k is even or odd. Since

+

Q m P + 0 and Qm'P + 2ii as m

+

co,

the above fact shows that solution (6.91) is not stable with respect to the family of solutions (6.92) (see Fig. 2).

FIG. 2

This example makes clear the difference between orbital stability and the stability of 6.1. 6.4. ORBITAL STABILITY OF CRITICAL POlNTS This section is concerned with the orbital stability of the critical point of the autonomous system of the form dx/dt

=

AX

+ X(X),

(6.94)

6.4. Orbital Stability of Critical Points

83

where A is an n-by-n constant matrix. Theorem 6.6 readily follows from Theorem 6.1.

Theorem 6.6.

In system (6.94), suppose (i) X ( x ) is continuous and satisjies a Lipschitz condition in the neighborhood of x = 0 ; (ii) X(0) = 0; (iii) X ( X ) = o(( x 1 ) a s x + 0; (iv) the eigenvalues of A are all negative in their real parts.

Then the origin x = 0 in the phase space is a critical point, and this critical point is absolutely asymptotically orbitally stable. The following corollary follows from this theorem in a way similar to that of Corollary 2 of Theorem 6.1. Corollary

Consider an autonomous system dx/dt = X ( X ) , where X(0) = 0 and X ( x ) is continuously diyerentiable in the neighborhood of x = 0. Then the critical point x = 0 is absolutely asymptotically orbitally stable if the eigenvalues of the matrix X,(O) are all negative in their real parts. From Theorem 6.2 follows the following theorem, which corresponds to Theorem 6.5.

Theorem 6.7.

In system (6.94), suppose (i) X(0) = 0; (ii) X ( x ' ) - A'("'') = o(1 x' - x" as x', x" -,0, (iii) the real parts of the k eigenvalues of A are all negative, and the real parts of the remaining ( n k) eigenvalues of A are all positive.

I)

-

84

6. Stability

Then, in the phase space, passing through the origin, there are apositive integral manijold S , of dimension k and a negative integral manifold S- of dimension (n - k) such that the critical point x = 0 is (1) asymptotically orbitally stable in the positive sense with respect to the jamily 9+ consisting of orbits lying on S , ; (2) asymptotically orbitally stable in the negative sense with respect to the family 9consisting of orbits lying on S - ; (3) orbitally unstable in both positive and negative senses with respect to the family 3 consisting of orbits lying on neither S+ nor S- .

In this theorem, by a positive integral manifold, is meant a manifold such that any orbit x = q ( t ) lies on the manifold for t 2 to whenever the point q(to)belongs to the manifold. Similarly, by a negative integral manifold, is meant the manifold such that any orbit x = q ( t ) lies on the manifold for t 5 to whenever the point q(to) belongs to the manifold.

6.7. If we take an arbitrary to, then, by Theorem 6.2, PROOFOF THEOREM in the x-space, there is a k-dimensional manifold S , passing through the origin such that the trivial solution x = 0 of (6.94) is asymptotically stable consisting of solutions in the positive sense with respect to the family F+' intersecting S + at t = to and is unstable in the positive sense with respect to the family 9'consisting of remaining solutions. As is seen from the proof of Theorem 6.2, for such S + , there is a positive number 6 such that for any solution x = q ( t ) of (6.94), (a) x

=

q ( t ) exists on [to, a),and there

I q(to)I is sufficiently small; and, conversely,

I q(t) I 5 6 if q(to)E S ,

I

(b) q(to)E S , if x = cp(t)exists on [to, m), and there q(t)

and

I 5 6.

First we shall prove that, for any solution x = q ( t ) of (6.94) such that q(to) is sufficiently small, q ( t ) E S , for any t 2 to. By (a), let us suppose

q(to)E S , and

I

I

I q(t) I 5 6

for any t 2 to.

(6.95)

Let t , be an arbitrary value of t such that t , 2 to, and consider x = q(t to + t,). Then, since (6.94) is an autonomous system, x = q ( t - to t l ) is also a solution of (6.94). In addition, from (6.95),

+

I q ( t - to + t , ) I 5 6

for any t 2 to,

(6.96)

6.4. Orbital Stability of Critical Points since t - to implies

+ t , 2 t , 2 to for

85

t 2 to. Then, by (b), the inequality (6.96)

(6.97) + t l ) E s,. such that ti 2 to, (6.97) means q ( t ) E S +

d t l ) = d t o - to

Since tl is an arbitrary value of t for any t 2 to. This evidently implies S, is a positive integral manifold in the phase space. Next we shall show that the manifold S, is fixed independently of to.

t

.

r t

t, FIG.3

Let S, ' be another k-dimensionalmanifold corresponding to another value of to, say t , (see Fig. 3). Since t , and to can be interchanged, we may suppose without loss of generality that t , 2 to.Now let x1 be an arbitrary point of S,' and let x = q(t) be a solution of (6.94) such that q ( t l ) = x , . Then, by (a), if x1 is sufficiently small, the solution x = q(t)exists on the interval [tl, co), and there q ( t ) 5 6 . Such a solution, however, by the continuity of the solutions (see Theorem 2.3), also exists on the interval [to, t , ] , and there also q(r) 5 6 if x , is sufficiently small. This implies by (b) that q ( f o ) E S,, because now x = q ( t ) exists on [to, co), and there q ( t ) S 6. Then, since S , is a positive integral manifold as has already been proved, q(to)E S, implies x , = q ( t l ) E S , . Since x , is an arbitrary point of S,', this implies S+' c S,. Next let xo be an arbitrary point of S,, and let x = q(t) be a solution of (6.94) such that cp(to) = x,. Then, by (a), if xo is SUBciently small, x = q ( t ) exists on [to, co), and there q(t) S 6. Let us consider x = q ( t - tl + to). Then, since (6.94) is an autonomous system, x = q(t - t , to) is also a solution of (6.94), and, in addition,

I I

I

I

I

I I

I

I

I

I

I 1 I

+

I q(t - t , + to) I 5 6 +

for any t

2 t,,

(6.98)

since r - r , to 2 to for r 2 t , . The inequality (6.98) implies by (b) that q ( t , - t1 to) = q(to)= xo E S,'. Since xo is an arbitrary point of S , , this implies S, c S,'.

+

6. Stability

86

Thus we have S , = Sf', which proves S , is fixed independently of to in the phase space. Since such S + is a positive integral manifold as has already been proved, conclusion (1) of Theorem 6.7 readily follows from Theorem 6.2. The existence of the ( n - k)-dimensional negative integral manifold S and conclusion (2) readily follow from the above results through the transformation t = -t'. Lastly, in order to prove conclusion (3), let us consider an arbitrary orbit C : x = q ( t ) that lies on neither S , nor S - . Let xo = q(to)be a point of C that belongs to neither S , nor S - . Then, by (b), the orbit C : x = q ( t ) cannot remain in the region x S 6 when t is increased from to. Likewise, the orbit C also cannot remain in the same region when t is decreased from to. This proves conclusion (3) of Theorem 6.7. Q.E.D.

I I

In illustration, let us consider the system d-x l - Ax1, dt

dx2 - - - px2 dt

(6.99)

+ CLXl 2 .

The general solution of this system is easily found as follows: x1 = c1 elt,

(6.100)

x1 = c1 e"',

x2

=

(c2

+ c12at)er'

(6.101) (p = 21).

If 1and p are both negative, then, Theorem 6.6 says that the origin in the phase space is absolutely asymptotically orbitally stable. This is easily checked from (6.100) and (6.101). If I and p are of opposite signs, then Theorem 6.7 says that the origin in the phase space is conditionally orbitally stable and that there are two orbits (that is, one-dimensional integral manifolds) S , and S - passing through the origin such that the origin is orbitally unstable in both positive and negative senses with respect to the family consisting of orbits differing from S ,

6.5. Orbital Stability of Periodic Solutions

87

and S - . These conclusions are easily checked from (6.100). In fact, as is seen from (6.100), the orbits S , and S - are as follows: s + : x 2=

s- : x1

a

21 - p ~

x1

2 Y

=0

1< 0 and p > 0;

when

s, : x1 = 0, S-:x,

=

a

21 - p ~

X12

when 1 > 0 and p < 0. Consequently, by (6.100), the orbits of the system (6.99) behave in the neighborhood of the origin as shown in Fig. 4. Evidently, Fig. 4 shows that the above conclusions derived from Theorem 6.7 are all correct.

6.5. ORBITAL STABILITY OF PERIODIC SOLUTIONS

In this section, we shall discuss the orbital stability of the periodic solution x = p ( t ) of the general n-dimensional autonomous system

(6.102)

dx/dt = X ( X )

using the moving orthonormal system introduced in Chapter 5. We suppose X ( x ) is continuously differentiable in the domain D containing the closed orbit C represented by the periodic solution x = p(t). Let

{~CdOl = x C 4 t ) l I I xCml

IIY

52(t)Y

- w>> * -9

88

6. Stability

be a continuously differentiable moving orthonormal system along C. Then, by 5.2, any point x = x(z) of any orbit C' of (6.102) lying near C is expressed as (5.17), and, for z and pv (v = 2, 3, ..., n), we have the differential equations (5.23) and (5.24). Equation (5.24) is written in vector form as (5.27), and here the (n - 1)-dimensional vector p denotes the normal increment of the orbit. Now, for any fixed t = to, suppose n

4zo) = d t o )

+

c

P"(tO>tV(tO>

v=2

and put

44 = Cx(z0) - d4l* C 4 z o ) - cp(Q1.

(6.103)

Then a(t) denotes the square of the distance between the point x(zo) and the point q(t)on C. Since

from (6.103), we have 4to) =

II Po

*(to)

-2X*C47(to)I

=

1I2Y

ii(t0) = -2X"(P(to)l

-

[X:x(70)

(6.104)

- cp(t0)l = 0,

+ 2 I1 XCcp(tOl1 /I2)

~x*[4+o)l Cx:X(zo)- &Oll

-

where = d/dt, and po is an (n - 1)-dimensional vector whose components are p,(to) (v = 2, 3, ..., n). As is seen from the last of (6.104), ii(to) > 0 if po is sufficientlysmall.By the second of (6.104), this implies a(t) > u(to) for t # to; that is,

I] I]

(1 Po 1 = dist

CI

CX(~O),

-

(6.105)

Since to is an arbitrary value of t, (6.105) implies

I] P(t) ]I

= dist C44t))Y

Cl.

Since p = p ( t ) satisfies the system (5.27)) by the definition of 6.3,the orbital stability of the closed orbit C is thus decided by the stability of the trivial solution p = 0 of the system (5.27). However, the system (5.27) is of the form considered in Corollary 1 of Theorem 6.1. Therefore, the stability of the solution p = 0 is decided by investigating the characteristic exponents of the normal variation equation (5.29). The characteristic exponents of (5.29) are, however, by Theorem 5.2, (n - 1) characteristic exponents of the first variation equation of (6.102) with respect to the periodic solution x = cp(t). Since,

89

6.5. Orbital Stability of Periodic Solutions

by Theorem 5.2, one of the characteristic exponents of the first variation equation of (6.102) with respect to the periodic solution is always zero in its real part, we thus have the following theorem.

Theorem 6.8. Suppose the autonomous system (6.102) has a nonconstant periodic solution x = cp(t), and X ( x ) is continuously diferentiable in the domain D containing the closed orbit C'represented by the periodic solution x = cp(t). Then the periodic solution x = q(t) is asymptotically orbitally stable i f ( . - 1) characteristic exponents of the jirst variation equation of (6.102) with respect to the periodic solution x = q ( t ) are negative in their real parts.

If the characteristic exponents of the normal variation equation (5.29) are all negative in their real parts, then, as is seen from (6.20) and the proof of Corollary 1 of Theorem 6.1, for any neighboring orbit C', there are two positive constants K and a such that

I p ( t ) I 5 Ke-"'

for t 2 0.

(6.106)

Now, for the orbit C', by (5.23), we have

x + fPVX", [r(t") - t"] - [(t') - t']

=

-

-l]di

v=2

X*X'

dt,

X'X

(6.107)

where t' and t" are the arbitrary values of t such that 0 5 t' < t" and n

u(t, P I = X(Cp

c

+ p = 2 PrSp) -X(cp)

n

-

c

X,(cp>v = 2 P V t V

As is seen from (6.106), or Theorem 6.8, the orbit C'lies in the neighborhood of the closed orbit C for large t, consequently, substituting (6.106) into (6.107), we have

6. Stability

90

I [T(t”>

t’l I 5 K , JrL-atdt

- t”] -[z(t’> -


where K , is some positive constant. Since exp[ -ut’] -+ 0 as t‘ above inequality implies the existence of the number to such that lim [ z ( t ) - t ] =

-+

CQ,

the

(6.108)

to.

1-m

Now, for C‘, - x(t

X[T(t)]

+ to)=

K

X[X(T)]dT,

(6.109)

and X [ X ( T ) ]is bounded for large T, since C‘ lies in the neighborhood of C for large T , as stated above. From (6.109), it then follows that

[ x[z(t)] - x(t

+ to) I 5 K2 I T(t) - ( t + to) I

for large I,

where K , is some positive constant. By (6.108), the above inequality implies

I

lim x[s(t)] - x(t

+ t o ) 1 = 0.

f‘W

On the other hand, from (5.17) and (6.106), it follows

I

I

Iim x[r(t)] - q(t) = 0. t+W

Hence it follows that

I

Iim x(t

+ t o ) - q(t) I = 0 ;

t’W

that is, lim I-+

I

X(T)

- q(t

-

I

to) =

0.

m

We thus have the following theorem.

Theorem 6.9. Suppose that the autonomous system (6.102) has a nonconstant periodic solution x = q(t), and X ( x ) is continuously differentiable in the domain D containing the closed orbit C represented by the periodic solution x = q(t). y ( n - 1) characteristic exponents of thefirst variation equation of (6.102) with respect to the periodic solution x = q ( t ) are negative in their real parts, then the periodic solution x = q ( t ) is not only asymptotically orbitally stable, but, as t + 00, all the neighboring solutions x = X ( T ) of (6.102) tend to the periodic solution x = q ( t ) itself, except for diflerences of the phases.

7. Perturbation of Autonomous Systems

Suppose the given autonomous system

dx/dt

=

(7.1)

X(X)

has a periodic solution x = q ( t ) . Does the given system still have a periodic solution close to x = q ( t ) when the given system is varied by a small amount? The system differing from (7.1) by a small amount is called the perturbed system of (7.1) and the original system (7.1) is called the unperturbed system. In practical applications, the above problem is very important. First, in practical problems, Eq. (7.1) is not usually exact, but has some small errors. Therefore, if the perturbed system has no periodic solution close to the periodic solution of the unperturbed system, the conclusion derived from the unperturbed system is not valid for practical problems. Second, if the perturbed system has a periodic solution close to the periodic solution of the unperturbed system, we can ascertain the character of the periodic solution of the complicated perturbed system through the periodic solution of the simple unperturbed system. In the present chapter, the foregoing problem will be studied for the case where the perturbed system is of the form

dxldt where X ( x , 0)

=

=

(7 4

X ( x , E),

X(x). 7.1. FUNDAMENTAL FORMULAS

We suppose system (7.1) has a nonconstant periodic solution x = q(t) of primitive period w and X ( x ) is continuously differentiable in the domain D containing the closed orbit C represented by the periodic solution x = q(t). For the perturbed system (7.2), we assume that X ( x , E ) is continuously differentiable with respect to x and E for x E D and E < 6 (6 > 0). Let

I I

{xCV(t)l = X[cp(t)l

I I X[cp(t)l I1 91

9

t2(%

.* t.(t>) *)

7. Perturbation of Autonomous Systems

92

be an arbitrary continuously differentiable moving orthnormal system along C, and C' be an arbitrary orbit of (7.2) lying near C. Then, as in 5.2, for any point x = X(Z) of C', we have

.and, in the same way as for (5.23) and (5.24), we have

-dr--

I x )I2 +

dt

PVX'tV

(- = d/dt),

X'X'

dt

dPv _-

i

v=2

(1 x (I2 + iP P X * t , p=2

X'X'

(7.4)

n

tV*x'- C pptv*tp

(v

=

2,3,

..., n), (7.5)

p=2

where

Put

4 0 = Xx[cp(tll and let p be an (n - 1)-dimensional vector whose components are pv (v = 2, 3, ..., n ) . Then, from the latter part of (7.6), we have

Hence, substituting (7.7) into (7.9, and writing the resulting equation in a vector form similarly to (5.27), we have dP - = R(t,P, E ) = q t ) P dt

+ EV(t) + .(I[

P

1 + I El)

(P, E

+

017

(7.8)

where Z ( t ) is the same matrix as..the one defined in 5.2 and q(t) is an 1)-dimensional vector whose components are

(E -

Vv(f) =

tv'(t) *

ax

[cp(t), 01

(v

=

2, 3 ,

. a ' ,

n).

(7.9)

Now, by the assumption on differentiability of X ( x , E ) , the function R(t, p, E ) is continuously differentiable with respect to p and E . Therefore, by

7.2. Periodic Solutions of the Perturbed System

Theorem 2.4, the solution p

= p(t,

93

c, E ) of (7.8) such that

(7.10)

P(0, c, E ) = c

is continuously differentiable with respect to c and E . Then, since p(t, 0,O) = 0 as is readily seen from (7.8), the function p(t, c, E ) can be written in the form P(t, c, E )

=

+ .(I1

G(t)c +

c

1 + I E 1)

(c, E

+

0) .

(7.11)

Here (7.12) where p,(t, c, E ) is a Jacobian matrix of p(t, c, E ) with respect to c. However, as is seen from (2.10) and (2.11), the matrix G ( t ) and the vector r(t) satisfy, respectively, the equations (7.13) and (7.14) and, in addition, as is seen from (7.12), they satisfy the initial conditions G(0) = E

and

r(0) = 0.

(7.15)

Hence, the matrix G(t) and the vector r(t) are, respectively, determined uniquely as the solutions of (7.13) and (7.14), satisfying the initial conditions (7.15). 7.2. PERIODIC SOLUTIONS OF THE PERTURBED SYSTEM

I I

As is seen from (7.4), dz/dt > 0 for C' lying near C if E is sufficiently small. Therefore the neighboring orbit C' can be closed if and only if the equality

P ( W , c, E )

(7.16)

=c

holds for a certain positive integer p . Equality (7.16) is rewritten by (7.1 1) as follows: [G(po) - E] c

+ er(pW) + o(ll c 1 + I I) = 0. E

(7.17)

94

7 . Perturbation of Autonomous Systems

Since the left-hand side of (7.17) is continuously differentiable with respect to c and E, by the theorem on implicit functions, we see that if det [G(pw) - El # 0,

(7.18)

I 1,

then, for sufficiently small E Eq. (7.17) has a unique solution c = c(E), which is continuously differentiable with respect to E and vanishes with E . Let C,l be the orbit of the perturbed system (7.2) corresponding to c = c ( E ) ; then evidently this is a closed orbit and, as is seen from (7.4), the period a,,’ of C,l is given by

0;

=

1:‘

n

c X*X[(P + c P,(G(E), (1 x \I2

+ v = 2 P”C4

4

4

9

ElX*4,

dt.

n

v=2

(7.19)

&)tV, El

I I

It is evident that 0,’ is close to pw, provided E is sufficiently small. Now, as is seen from (7.13) and (7.19, the matrix G(t) is a fundamental matrix of the normal variation equation (5.29). Hence, by Theorem 3.4, there is a constant matrix K such that G(t w ) = G(t)K. However, if we put t = 0 in this equality, from (7.15), we have K = C(w). Thus, from the above equality, we have

+

G(t

+ w ) = G(t)G(w),

(7.20)

from which readily follows (7.21)

G(pw) = GP(w).

Then, if (7.18) holds, we have det [G(w)

- El

(7.22)

# 0,

because

G(Jw) - E = GP(w)- E = [G(w)

- E]

*

+ Gp-’(w) +

[CP-‘(w)

+ G(w) + El. (7.23)

If (7.22) holds, then, as before, there is a closed orbit C,’ of the perturbed system (7.2) whose period is close to o.However, for this Cl’, as is evident geometrically, equality (7.16) holds again. Then, from the uniqueness of the solution of (7.17), C,,’ must coincide with C,’. This means that, in the

7.2. Periodic Solutions of the Perturbed System

95

neighborhood of C, there is not a closed orbit C,' differing from C,' if (7.18) holds. However, if (7.18) does not hold but (7.22) holds, then, in the neighborhood of C, there may be a closed orbit C,' differing from C,'. The periodic solution of the perturbed system corresponding to such C,' is called the subperiodic solution. Now, by (7.20), the eigenvalues of G(w) are the multipliers of solutions of the normal variation equations (5.29). Then, by Theorem 5.2, these are (n - 1) multipliers of solutions of the first variation equation of the unperturbed system (7.1) with respect to the periodic solution x = cp(t). Since one of the multipliers of solutions of the first variation equation of (7.1) with respect to x = cp(t) is one by Theorem 5.2, we have the following theorem. Theorem 7.1.

Suppose that the unperturbed system (7.1) has a nonconstant periodic solution x = cp(t) of primitive period w, and X ( x ) is continuously differentiable in the domain D containing the closed orbit C represented by the periodic solution x = cp(t). Further, for the perturbed system (7.2), suppose that X ( x , E ) is continuously differentiable with respect to x and E for x E D and I E I < 6 (6 > 0). Then, if (n - 1) multipliers of solutions of the first variation equation of the unperturbed system (7.1) with respect to the periodic solution x = q ( t ) are all different from one, the perturbed system (7.2) has a unique periodic solution of the period close to w in the neighborhood of x = cp(t),provided E is sufficiently small. If any one of (n - 1) multipliers of solutions of the first variation equation of the unperturbed system (7.1) with respect to the periodic solution x = q ( t ) is not an integral root of one, then the perturbed system (7.2) has no subperiodic solution in the neighborhood of x = cp(t), provided I E I is suficiently small. However, ifsome of ( n - 1) multipliers of solutions of thefirst variation equation of (7.1) with respect to x = cp(t)are integral roots of one, then theperturbed system (7.2) may have some subperiodic solutions.

I I

When X ( x , E ) = X ( x ) , from (7.6),

consequently, from (7.5), the right-hand side of (7.8) does not contain E. Then, by definition, the solution p = p(t, c, E ) of (7.8) does not depend on E . Hence, in the present case, Eq. (7.17) becomes

96

7. Perturbation of Autonomous Systems

[G(Pco) - E ] c

+ o(II c 11)

=0

(C

--*

0).

(7.23)

Then, if (7.18) holds, Eq. (7.23) has only a trivial solution. Thus we have the following theorem.

Theorem 7.2. Suppose the system (7.1) has a nonconstant periodic solution x = cp(t), and X ( x ) is continuously differentiable in the domain D containing the closed orbit represented by the periodic solution x = cp(t). If any one of (n - 1 ) multipliers of solutions of the first variation equation of the system (7.1) with respect to the periodic solution x = q ( t ) is neither one nor an integral root of one, then, in the neighborhood of x = cp(t), the system (7.1) has no periodic solution other than x = cp(t). By this theorem, the periodic solution of (7.1) with respect to which any one o f ( n - 1) multipliers of solutions of the first variation equation of (7.1) is neither one nor an integral root of one will be called the isolated periodic solution o f the system (7.1). Then Theorem 7.1 says that if the unperturbed system (7.1) has an isolated periodic solution x = cp(t), the perturbed system (7.2) has always a unique periodic solution in the neighborhood of x = cp(t), and the primitive period of the periodic solution of the perturbed system is always close to the primitive period of the periodic solution of the unperturbed system; in other words, no subperiodic solution exists for the perturbedsystem.

Remark. When n = 2, let P and Q be the points whose coordinates are, respectively,

x =

d o ) + P2(0)t2(0)

and

x = cp(4 = cp(0)

+ P2(4t2(4 + P2(452(0).

If P # Q,then, after the time z = ~ ( o )the , orbit C' of the perturbed system always remains inside or outside of the region bounded by the arc PQ of C' and the segment PQ on the normal of C. Hence, if P # Q,the orbit C' cannot return back to the point Q after the time z = T(w). In other words,

7.3. Stability of the Periodic Solutions of the Perturbed System

97

if P # Q, the orbit C' cannot be closed (see Fig. 5). This means that when n = 2, the perturbed system has no subperiodic solution under any conditions whatsoever.

FIG. 5

7.3. STABILITY OF THE PERIODIC SOLUTION OF THE PERTURBED SYSTEM Concerning the stability of the periodic solution of the perturbed system, we have the following theorem.

Theorem 7.3. Suppose the unperturbed system (7.1) has a nonconstant periodic solution x = cp(t), and X ( x ) is continuously direrentiable in the domain D containing the closed orbit C represented by the periodic solution x = cp(t). Further, for

the perturbed system (7.2), suppose X ( x , E ) is continuously differentiable with respect to x and E for x E D and E < 6 (6 > 0). Then, if (n - 1) characteristic exponents of the Jirst variation equation of the unperturbed system (7.1) with respect to the periodic solution x = q ( t ) are all negative in their real parts, the perturbed system (7.2) has a unique periodic solution in the neighborhood of x = q(t) for suflciently small E and this periodic solution of the perturbed system is asymptotically orbitally stable, together with the periodic solution x = cp(t) of the unperturbed system.

I I

I 1,

PROOF. By assumption, (n - 1) multipliers of solutions of the first variation equation of (7.1) with respect to x = q ( t ) are all less than one in their absolute values; therefore x = cp(t) is an isolated periodic solution of (7.1). Hence, by Theorem 7.1, the perturbed system (7.2) has a unique periodic

98

7. Perturbation of Autonomous Systems

I 1,

solution in the neighborhood of x = p ( t ) for sufficiently small E and it has no subperiodic solution. Let US denote such a periodic solution of the perturbed system (7.2) by

where c = C ( E ) is a solution of (7.17) for p = 1. In the present case, the condition (7.22) is evidently satisfied because the eigenvalues of G(o) are all smaller than one in their absolute values. By Theorem 6.8, it is evident that the periodic solution x = q ( t ) of (7.1) is asymptotically orbitally stable. In what follows, we shall prove that the periodic solution (7.24) of (7.2) is also asymptotically orbitally stable. To begin with, using the techniques of 5.1, let us construct a moving orthonormal system along the closed orbit C' represented by (7.24). Following (5.7), Put

I(

11

I

1,

where X[cp(.r, E ) , E ] = X[rp(z, E ) , E ] / X[cp(t, E ) , E ] . Then 1 - cos 81 cos 8, (v = 2, 3, .. ., n) are all small for sufficiently small E because lim C(E) = 0, and this implies X[(P(T,E ) , E ] = X[rp(t)]+ for small E The

1

I

I 1,

I I.

e-0

fact that I 1 - cos 8, I = 0 implies (cos 8, + 1) never vanishes. This enables us to construct a moving orthonormal system along C' according to (5.15) as follows : (V

I

I

= 2,3, ..., n). (7.26)

I

Since cos 8, (v = 2, 3, ..., n) are small, it is evident that t V ( z ,E ) (v = 2, 3, ..., n) are all small for sufficiently small E Since X [ q ( z , E), E ] is expressed by (7.25) as

X[&,

I I.

E), E ]

= cos 8,

. x[&)]

n

+ 1 cos 8, .r,(t), ,=2

t

The notation = expresses the approximate equality.

- t,(t) I

99

7.3. Stability of the Periodic Solution of the Perturbed System equalities (7.26) are rewritten as follows:

L ( z , 6 ) = t,(t) - cos 0, . X [ V ( t ) l -

COS

e,ys e, . (,(t)

(V= 2, 3, ..., n). (7.27) + Now, from (7.4), I dz/dt - 1 I is small for sufficiently small I E I. Therefore, from (7.25), for sufficiently small I E I, we have successively d = 2 i cos e,

(V= 2 , 3 , ..., n), (7.28)

= 0,

where Xx(x, E ) and Xx(x) are, respectively, the Jacobian matrices of X(x, and X ( x ) with respect to x . Since cos 0, = 0

(v = 2, 3,

..., n)

for small

I E 1,

from (7.27), it follows that

for small

I E 1.

Since x,[V(z,

for small

I E 1,

4,E l

from (5.28), for small

qz,E ) where E(z,

E)

=

I E 1, =

and dz/dt

=

1

E)

(7.29)

xMt)l we thus have

(7.30)

qt),

is a periodic matrix whose (vp)-elements Sv,(r, E ) are

=&,

8)

=

t,*(z, E)X,CVp(T, 4 Eltp(r.,E ) - tv*(z,E ) dt,(T, 7 4 (v, p

=

2, 3 ,

..., n).

(7.31)

100

7. Perturbation of Autonomous Systems

From (7.31), it is evident that the normal variation equation of the perturbed system (7.2) with respect to the periodic solution (7.24) is (7.32)

dp/dr = E(T, E)P.

Let G(r, E ) be the fundamental matrix of (7.32) such that G(0, E ) = E ; then G ( T ,E ) evidently satisfies the equation

Since G(t) satisfies Eq. (7.13) and the initial condition G(0) and (7.30), we see then G(w’, E ) k C(w)

=

E, from (7.29) (7.33)

I

if E I is sufficiently small. Here, as is seen from (7.24), w’ = ~ ( wis) a primitive period of the periodic solution (7.24) of the perturbed system (7.2). Since the eigenvalues of G(w) are all smaller than one in their absolute values, as mentioned in the beginning of the proof, the approximate equality (7.33) implies then that the eigenvalues of G(w’, E ) are also all smaller than one in their absolute values if E is sufficiently small. By the definition of G(r, E ) , this means the multipliers of solutions of the normal variation equation of the perturbed system with respect to the periodic solution (7.24) are all smaller than one in their absolute values if E is sufficiently small. By Theorem 6.8, this proves the periodic solution (7.24) of the perturbed system (7.2) is also asymptotically orbitally stable if E is sufficiently small. Q.E.D.

I I

1 1

I I

Theorem 7.3 says that if a periodic solution of the autonomous system having the specified smoothness mentioned in Theorem 7.3 is asymptotically orbitally stable under the condition of Theorem 6.8, it changes by a small amount, keeping stability under the small change in the original autonomous system. This property is very important in practical applications, as mentioned in the beginning of this chapter.

8. Perturbation of Fuly OsciIlatory Systems

By a fully oscillatory system is meant an autonomous system whose orbits, lying near a certain closed orbit, are all closed. Such a system appears frequently in practical problems. For example, consider the equation

X

+ g(x) = 0

(. = d/dr),

(8.1)

where xg(x) > 0 for x # 0. Equation (8.1) is evidently equivalent to the autonomous system dx,/dt = ~

2

,

dXJdt = - g ( x l ) .

(8.2)

Since the orbits of (8.2) are of the form j:L(u)du

+ +xzz = const,

it is evident that the system (8.2) is a fully oscillatory system. The system (6.88) of 6.3 is also a fully oscillatory system. The present chapter is concerned with the perturbation of such a fully oscillatory system. The problem of seeking a periodic solution of the equation

X - n(1 - x2)i

+x =0

I

for small 11 is a typical example of the problems with which we are concerned. As is evident from the definition, the periodic solutions corresponding to the closed orbits of a fully oscillatory system are not isolated (see Theorem 7.2); consequently, the results of the preceding chapter are not applicable to the problem of the present chapter. In the present chapter, we suppose that, for the given n ( 2 2)-dimensional autonomous system dx/dt = X ( X ) , 101

(8.3)

102

8. Perturbcztion of Fully Oscillatory Systems

X(x) is twice continuously differentiable in the domain D, and the orbits lying near a closed orbit Co : x = rpo(t) contained in D are all closed. For the perturbed system of (8.3) dx/dt = X ( x , E )

(8.4)

(X(X, 0 ) = X(x)),

we suppose that X(x, E ) is twice continuously differentiable with respect to x and E for x E D and E < 6 (6 0). The primitive periods of the periodic solutions of the unperturbed system (8.3) corresponding to the closed orbits lying near Co are in general different from each other. In the present chapter, we shall assume that these primitive periods are bounded when n 2 3.

=-

1 I

8.1. UNIVERSAL PERIODS As a preliminary example, let us consider the system dx I -= -

dt

1

d%

dt

-

+

2n

(XI’

+ x” + x3’ +

X4’)lI2

x21

2n 1

dx4 _dt 1

+ ( X I 2 + x” + x32 + x

4 y

R

+ (x12 + x2’ + x3’ + x4 )

2 112 x3.

The general solution of (8.5) is evidently x1 = a cos

x2 = a sin

271

(I

2n

(1

x3 = bcos (1 x4 = b sin

+ (a’ + b2)’I2 + (a’ + b2)’/’ t + e l ) ,

+ ( a 2n+ b2)1/2t + O’), n

(1

+ (a’ + b2)’/‘

8.1. Universal Periods

103

where a, b, el, and 6, are arbitrary constants. Let C, be a closed orbit represented by the periodic solution x1 = a, cos

211

~

1

+ a,

t,

.

xz = a , sin

2n

~

I

+ a, t ,

~3

= x ’ ~= 0, (8.7)

where a, is a positive constant. Then, as is easily seen from (8.6), the orbit of the system (8.5) lying near C, are all closed; consequently the system (8.5) is a fully oscillatory system. As is readily seen from (8.7), the primitive period of the periodic solution (8.7) is 1 a,. However, as is readily seen from (8.6), the primitive period of the periodic solution corresponding to the closed orbit C of (8.5) lying near C, is 1 (az + b’)’l2 or 2[1 + (a’ + b ’ ) 1 / 2 ] , according to whether or not C lies on the plane x3 = x4 = 0. This shows the primitive period of the periodic solution corresponding to the closed orbit C does not vary continuously with C. In the present case, however, 2(1 a,) is also a period of (8.7), and 2[1 + (a’ + f ~ ’ ) ~ is /also ’ ] aperiod of the periodic solution corresponding to an arbitrary closed orbit C lying near C,. Moreover such a period evidently varies continuously with the closed orbit C. The period that varies continuously with the closed orbit will be called the universal period. In the above example, then, 2[1 + (a’ b z ) 1 / 2 ]is a universal period. When the perturbation of a fully oscillatory system is discussed, the universal period must be brought into consideration because o f its continuity property. In the following theorem, we shall prove that the closed orbits of a fully oscillatory system have in general universal periods.

+

+

+

+

Theorem 8.1. For the fully oscilIatory system (8.3), the periodic solutions corresponding to the closedorbits lying near C, always have universalperiods, provided n = 2 , or otherwise the primitive periods of the periodic solutions are bounded.

PROOF. Let ll be a normal hyperplane of C, at the point lo = rp,(O). Then, by Theorem 4.8, any orbit of (8.3) lying near C,, crosses ll at a point C near lo; and, by assumption, any orbit C of (8.3) crossing ll at a point Cis closed. Let w(C) be the primitive period of the closed orbit C of (8.3) crossing ll at the point [ and put o([,)= 0,. Let {x[VO(t)l/

11 x [ ~ O ( t > l 11

7

rZ(t)

7

’*

en(‘)>

8. Perturbation of Fully Oscillatory Systems

104

be an arbitrary continuously differentiable moving orthonormal system along Co.Then, if 4 is close to coy by (5.17), any point x = x(z) of C is expressed as

..., n), we have (5.23) and (5.24).

and, for T = z(t) and pv = p , ( t ) (v = 2, 3, The relation (8.8) implies that n

I; = i o + v1 PV(0)tV(O). =2 This can be solved with respect to pv(0) (v = 2, 3, .. ., n) as follows:

(v = 2, 3,

Pv(o) = tv*(o)(C - l o )

--

-3

n).

(8.9)

For z = z(t), without loss of generality, we may suppose that (8.10)

z(0) = 0,

since T is the time measured along the orbit C of the autonomous system (see 4.1). By (8.10), from (5.23), we have then

I/ ?

= ?(t) =

X C ~ O ( S > l

]I2

+

i:

Pv(s)x"cpo(~)l~v(s)

v=2

i:

x*Ccp0(~)lxC~0(s) +

ds.

(8.11)

P V ( M S ) l

v=2

NOW,when n = 2, by the Remark in 7.2,

Since pv = pV(t) (v = 2, 3, ...)n) is a solution of the differential equation (5.24), it depends continuously on p,(O).(v = 2, 3, . .., n); consequently, by (8.9), it depends continuously on c. Then, by (8.11), ?(o0)depends continuously on 4. By (8.12), this implies that a ( [ ) is continuous with respect to C. By definition, this proves that the primitive period is itself a universal period when n = 2. NOWlet us consider the case where n 3. In this case, instead of (8.12), we have in general

4 5 ) = 4PWO)

(8.13)

8.I . Universal Periods

105

for a certain positive integer p. However, for any positive integer p, from (8.11), we have

(8.14)

where p is an ( n - 1)-dimensional vector whose components are pv (v = 2, 3, ..., n) and n

x = XC470(S>I9

X' = XC47O(S)

c

+ v = 2 PV(S)tV(S)I,

4 s ) = XX~470(41. Since C is closed, (8.14) implies

Mpwol - T K P

- 1)001)

- wo = O(ll P

11)

uniformly with respect to p as p -+ 0; that is, for sufficiently small positive number H, there is a positive number K such that

I { 4 P W O l - 4 ( P - 1>%1> - I s K . max I P I( 0 0

1 I( < H. Then, from (8.13), we

for any positive integer p whenever max p have

140 - P O 0 I S p K - m a x 1) P I(. This implies that 0 < PCWO - K . max

11 P 111 s 4 3

1) (1

if p is sufficiently small. Since o([)is bounded in the present case by the assumption, the above inequality implies that the positive integers p for which (8.13) holds are also bounded. Then we can take the least common multiple L of such positive integers p. For such L, let us put

&([)

L

L

P

P

= - w(i) = - 7(poo).

(8.15)

8. Perturbation of Fully Oscillatory Systems

106

Then, since the integrand in the integral of the right-hand side of (8.11) is periodic of period pw,, from (8.15), we have

6(l)= z

t

- *

)

pw, = ‘(LO,).

(8.16)

The T(Lw,)is, however, continuously dependent on (, as is shown for z(wo). Moreover, as is readily seen from (8.15), 6(()is a period of the closed orbit C. Hence (8.16) shows that 6(()defined by (8.15) is a universal period of the closed orbit C. This completes the proof. Q.E.D. 8.2. PRELIMINARY THEOREM

Our theory of perturbation of fully oscillatory systems is based on the following well-known theorem.

Theorem 8.2. Consider a system of diferential equations (8.17)

dxldt = EX(t, X , E ) ,

where X(t, x , E ) is periodic in t of period w (>O) and continuously differentiable with respect to x and E for ( t , x, E ) such that - 00 < t < 00, x E D ( D : a domain in the x-space), E < 6 (6 > 0). Corresponding to (8.17), consider the autonomous system

I

I

dxjdt

=

f

1

where X(X)=

(8.18)

EX(X),

0

w o

X ( t , X , 0) dt.

If the autonomous system (8.18) has a critical point x det X,(c,)

# 0,

(8.19) = co E D such

that

(8.20)

then the initial system (8.17) has a unique periodic solution of period w in the neighborhood of x = c,, provided E # 0 and E is suficiently small. In this case, if the eigenvalues of the matrix EX,(C,) are all negative in their real parts, the critical point x = c, of (8.18) is asymptotically orbitally stable and the corresponding periodic solution of (8.17) is asymptotically stable. If the autonomous system (8.18) has no critical point in the domain D,then the initial system (8.17) has no periodic solution of period w for suficiently small E > 0.

I

I I

I

8.2. Preliminary Theorem PROOF.

Let x = q ( t , c,

E)

107

be a solution of (8.17) such that (8.21)

q(0, c, E ) = c.

By Theorem 2.4, q(t, c, E ) is continuously differentiable with respect to E and, as is seen from the form of (8.17), q(t,c, 0) = c. Therefore we have

Let us rewrite this as follows:

where

lo I

q y t , c,

E) =

a’P(t, c, Oe) do.

a&

(8.23)

By (8.21), from (8.23), it is evident that q‘(0, c,

E)

(8.24)

= 0.

Now substitute (8.22) into (8.17); then we have dq’/dt = X ( t , c

+ E q ’ , E).

(8.25)

Thus we see that q’(t,c, E ) is a solution of (8.25) satisfying the initial condition (8.24). Then, by Theorem 2.4, we see that q’(t, c, E ) is continuously differentiable with respect to c and E . Now the necessary and sufticient condition that the solution x = q(t, c, E ) be periodic of period w is q(w, c, E ) = q(0, c,

E)

= c.

(8.26)

The necessity is evident. The sufficiency may be proved as follows: Since x = q(t, c, E ) is a solution of (8.17), qt(t, c, E ) = EXCf, cp(f, c, E),

where

El,

108

8. Perturbation of Fully Oscillatory Systems

Since cp,(t

+ 0,c, E )

= (a/at)cp(t

+ w , c, E ) ,

+

equality (8.27) implies x = cp(t o,c, E ) is also a solution of (8.17). However, the two solutions x = cp(t, c, &) and x = cp(t o,c, E ) take the same value for t = 0, by (8.26). Therefore, by the uniqueness of the solution, these two solutions must coincide for all values of t. This means that the function q(f, c, E ) is periodic in t of period o,which proves the sufficiency of (8.26). As is seen from (8.22), condition (8.26) is replaced, however, by

+

(8.28)

cp’(0, c, E ) = 0

if

0. Condition (8.28) can be regarded as the equation with respect to c. From the continuity of the function cp‘(t, c, E ) , Eq. (8.28) has no solution for small E if the equation E f

I I

(pyo,c, 0) =

(8.29)

0

has no solution. However, as is seen from (8.24) and (8.25), X(t, c, 0 ) dt = wX(c).

q’(0, c, 0 ) =

(8.30)

J O

Therefore Eq. (8.29) is equivalent to the equation

(8.31)

X ( c ) = 0.

I I

Thus we see that Eq. (8.28) has no solution for small E if Eq. (8.31) has no solution. This proves the last conclusion of the theorem. To prove the first conclusion of the theorem, let us suppose X(Co)

=0

(8.32)

and det Xx(co) # 0.

(8.33)

By (8.301, equality (8.32) and inequality (8.33) then imply, respectively, cp’(0,co, 0) = 0

(8.34)

and det cpc’(w,co, 0) # 0,

(8.35)

where cpc’(w, c, 0) is a Jacobian matrix of ~ ’ ( o c, ,0) with respect to c. From

109

8.2. Preliminary Theorem

(8.34) and (8.35), we see then by the theorem on implicit functions that, for sufficiently small [ E Eq. (8.28) has a unique solution c = C ( E ) such that c(0) = c,, and C ( E ) is continuously differentiable with respect to E . This proves that the initial system (8.17) has a unique periodic solution

I,

x = qo(t,E ) =

C(E)

+ EP'(t,

(8.36)

C(E), E )

I I

of period o in the neighborhood of x = c,, provided E # 0 and I E I is sufficiently small. Lastly, let us investigate the stability of the periodic solution (8.36). 'The first variation equation of (8.17) with respect to the periodic solution (8.36) is evidently &ldt = EX,[[, PO(^, E ) ,

&]YO

(8.37)

Let Q(t, E ) be the fundamental matrix of (8.37) such that Q(0, E ) = E.

(8.38)

Then, from the form of (8.37), it is evident that Q(t,

Now, for

E

(8.39)

0) = E.

# 0, let us put Q(t, E ) =

E

+ &Ql(t,E ) ;

(8.4)

then, substituting this into the equality

we have

From (8.38) and (8.40), it is evident that

QI(O,

E)

=0

(E

# 0).

(8.42)

Now, corresponding to (8.41), let us consider the matrix linear differential equation

dY/dt = X,[t, qo(t,

E), E]

+ EXx[t,

cPo(t, E ) , E I Y ,

(8.43)

and let Y = Y(t, E ) be a solution of (8.43) such that

Y(0,E ) = 0.

(8.44

8. Perturbation of Fully Oscillatory Systems

110

Then, by the continuity of X,(t, x , E ) and qo(t,E ) , the solution Y = Y ( t , E ) exists 011 the interval (- co, 00) for sufficiently small E 1 inclusive of E = 0 (see Theorem 2.2), and Y(t, E ) is continuous with respect to E in the neighborhood of E = 0 (Theorem 2.3). Since Q,(t, E ) is a solution of the differential equation (8.43) satisfying the initial condition (8.44) as is seen from (8.41) and (8.42), Ql(t, E ) = Y(t, E ) for E # 0, by the uniqueness of the solution. Then, from the continuity of Y ( t ,E ) , it follows that

1

lim #l(t,

= Y ( t , 0).

E)

&+O

This enables us to define Ql(t, 0) as follows: Ql(t, 0) = lim @,(t,E )

=

(8.45)

Y ( t , 0).

&+O

Then, from (8.39), we may suppose equality (8.40) holds even for Next, by (8.45), from (8.43) and (8.44), we have

E =

0.

and Ql(O, 0) = 0. Therefore, we have (8.46)

Now the multipliers of solutions of (8.37) are the eigenvalues of the matrix Q(w, E ) = E

+ eQ1(w, E ) .

Therefore they are of the forms 1

+ &A&),

where A,(&) are the eigenvalues of the matrix Ql(m, E ) . However, from the continuity of @,(a, E ) , Ai(E) = Ai(O),

provided

I E I is sufficiently small. Hence we have 1

- log[l w

+ &A,(&)]

1

= - [&Ai(&)

w

1

= - [&n,(o) 0

+ .(I E I)] + .( E I)] I

(E

-,0).

8.2. Preliminary Theorem

111

The A,(O) are, however, by (8.46) and (8.19), the eigenvalues of the matrix Q ~ ( w0) , =

:j

X,(t,

cO,

0) dt = wX,(co).

Therefore, if the real parts of the eigenvalues of the matrix &X,(cO)are all negative, then the real parts of (8.47)

1 I

are all negative, provided E is sufficiently small. Since the quantities (8.47) are the characteristic exponents of (8.37), this implies by Corollary 2 of Theorem 6.1 that the periodic solution (8.36) is asymptotically stable, provided E is sufficiently small. That the critical point x = co of (8.18) is asymptotically orbitally stable under the specified condition is evident from the Corollary of Theorem 6.6, and this completes the proof. Q.E.D.

I I

Theorem 8.2 is one of the most important theorems in the theory of nonlinear oscillations. The method of averaging (see, e.g., Bogoliubov and Mitropolski [l], pp. 297-326; Hale [l], pp. 130-140) that is used frequently in the study of forced nonlinear oscillations is based on Theorem 8.2. The autonomous system (8.18) is called the stroboscopic image (Minorsky [11, pp. 390-415) or the averaged system (Bogoliubov and Mitropolski [l], pp. 297-326; Hale [l], pp. 130-140) of the periodic system (8.17). Concerning the generalization of Theorem 8.2 to the case where X ( t , x , E ) is almost periodic in r or to the case where the averaged system admits of a periodic solution, see Bogoliubov and Mitropolski [l] , pp. 327-406.

Remark. Theorem 8.2 is concerned only with a periodic solution of period w. However, does the system (8.17) have a subperiodic solution, namely, a periodic solution of a period that is not w but an integral multiple of w? If such a periodic solution exists, then, for such a periodic solution, . Cp(PW,

c,

6)

= c

(8.26‘)

must hold for a certain positive integer p ( # 1) instead of (8.26). The equality (8.26’) then implies (P’(PW,

c, 6) =

0

(8.28’)

112

8. Perturbation of Fully Oscillatory Systems

instead of (8.28). Now, from (8.30), (P’(PW, c,

0) = P o q c ) ;

(8.30’)

consequently, Eq. (8.28’) has no solution for small I E I if Eq. (8.31) has no solution. This proves that the initial system (8.17) has no subperiodic solution for small E I if the averaged system (8.18) has no critical point in the domain D . When the system (8.18) has a critical point x = c, E D, from (8.300, we have

I

Cp’(P0, c,, 0) = 0.

(8.34‘)

In this case, if (8.20) holds for such c,, then, from (8.30’), det qC’(pw,c,, 0) # 0.

(8.35‘)

I I,

Then, by the theorem on implicit functions, for sufficiently small E Eq. (8.28’) has a unique continuously differentiable solution c = c,(E) such that c,(O) = c,. However, in such a case, as is shown in the proof of Theorem 8.2, Eq. (8.28) has a unique continuously differentiable solution c = C(E) such that c(0) = c,. For such c = c ( E ) , evidently ( P ’ ( 0 , C(E),

&) = 0.

Then, since q’(t, c, E ) is a solution of the periodic system (8.25) satisfying the initial condition (8.24), the above equality implies that Cp’(t, c(E), E ) is periodic in t of period o [cf. (8.26)]. Then we have Cp’(PWC(&),&) = 0

for any integer p . However, in the present case, (8.34’) and (8.35’) hold; therefore we have Cp(E)

= C(&).

This proves that the initial system (8.17) cannot have any subperiodic solution in the neighborhood of a critical point of the averaged system. The above discussion shows that, under the conditions of Theorem 8.2, no subperiodic solution existsfor the system (8.17). As is seen in the above discussions, the proof is based on the relation Cp’(P0, c, 0) = P P ’ ( 0 , c, 0)

[see (8.30) and (8.30’11, which yields the equivalence between conditions (8.34), (8.35) and (8.34‘), (8.35’).

8.3. Perturbation of a Fully Oscillatory System

113

8.3. PERTURBATION OF A FULLY OSCILLATORY SYSTEM Let (X[vo(t)l = X[vo(t)l/llX[vo(t)1 11 L ( t ) ... t,(t)> be an arbitrary continuously differentiable moving orthonormal system along the closed orbit Co: x = qo(t) of the unperturbed system (8.3). Let x = x ( t ) be any point of the orbit C of the perturbed system (8.4) lying near Co; then, as is stated in 7.1, for x = x(r), we have (7.3) and, for z = t ( t ) and p y = p,(t) (v = 2, 3, ..,n), we have (7.4) and (7.5). As in 8.1, for r = z(t), we may suppose (8.10) without loss of generality. Let us write Eq. (7.5) in a vector form as follows: 7

Y

(8.48)

dp/dt = R(t, p . E).

Then, by assumption, as is seen from ( 7 . 9 , R ( f ,p, E ) is twice continuously differentiable with respect to p and E . Needless to say, for the unperturbed system (8.3), corresponding to (8.48), we have (8.49)

dp/dt = R(t, p , 0).

Let p = $(t, c ) be a solution of (8.49) such that $(O, c ) = c ; then, since (8.3) is a fully oscillatory system by assumption, $(t, c ) is periodic in f for any c, provided c is sufficiently small. Let o be a universal period of x = q0(t);then, as is seen from (8.12) and (8.16), $(t, c ) is periodic in t of period o. Now, making use of the function $(t, c ) , let us transform p to u by the transformation

1 11

P =

$(f,

.).

(8.50)

Then, from (8.48), we have

a$(r, at

+ $"(t, u ) dU - = R [ t , $(t, dr

u), El,

(8.51)

where $,,(t, u) is a Jacobian matrix of $(t, u ) with respect to u. Since

from the meaning of the function $(t, u), Eq. (8.51) is rewritten as follows:

114

8. Perturbation of Fully Oscillatory Systems

Here, however, det $,,(t, u) # 0, because, as is mentioned in 2.1, $,,(t, u) is a fundamental matrix of the first variation equation of (8.49) with respect to p = $(?,a) such that $,,(O, a) = E. Thus, from (8.52), we have

(8.53)

= EF(t, U , E ) ,

where

(8.54) The function F(t, u, E ) is, however, periodic in t of period w , since $(t, u) and R(t, p , E ) are periodic in t of period w. Further, F(t, u, E ) is continuously differentiable with respect to u and E , since R(t, p, E ) is twice continuously differentiable with respect to p and E, and $(t, u ) is also, by Theorem 2.4, twice continuously differentiable with respect to u. These remarks show that system (8.53) is of the form (8.17). Thus, by Theorem 8.2,we have the following theorem. Theorem 8.3.

IJ'the averaged system du dt

(8.55)

of (8.53) has a critical point u = uo such that (8.56)

det Fu(uo) # 0, then the perturbed system (8.4) has a unique periodic solution n

x = .(T)

= VO(0

+

c $,[h U ( f ) l L ( f )

(8.57)

v=2

in the neighborhood of the periodic solution n

x = xo(4 = cpo(t)

+ 1$"C4 LlOlLtt)

(8.58)

v=2

1 1

of the unperturbed system (8.3), provided E # 0 atid E is suficiently small. Here $v(t, u) (v = 2, 3, . . ., n ) are the components of $(t, u ) , and u(t) is a pe-

riodic solution of (8.53) of period w lying near u

=

uo.

8.4. An Example

I15

The periodic solution (8.57) is asymptotically orbitally stable if the eigenvalues of the matrix ~F,,(zi,)are all negative in their real parts. If the averaged system (8.55) has no critical point, then the perturbed system (8.4) has no periodic solution in the neighborhood of the periodic solution x = qo(t) of the unperturbed system (8.3)for suficiently small E > 0.

I I

Remark By the remark in 8.2, under the conditions of Theorem 8.3, the given perturbed system (8.4) has no subperiodic solution, namely, a periodic solution corresponding to the subperiodic solution of the transformed equation (8.53). 8.4. AN EXAMPLE

In illustration of Theorem 8.3, let us consider the equation

x + w2x = $(x,

i),

where * = d/dt, w > 0, E is a small positive parameter, and&, y ) is a function continuously differentiable with respect to x and y for any finite x and y . Replacing wt by t, we can transform the given equation to one of the same form with w = 1. Therefore, in what follows, we shall be concerned with the equation

+ x = Ef(X, i ) .

x

(8.59)

Needless to say, Eq. (8.59) is the perturbed equation of the linear equation

x+x=o. Equation (8.59) is evidently equivalent to the system i= - y

,

3

=x

- &f(x, -y).

(8.60)

The general solution of the unperturbed system x = -y,

(8.61)

j = X

is

x = acos(t

+ e),

y = asin(t

+ e),

(8.62)

where a and 0 are the arbitrary constants. The orbits corresponding to the solutions (8 62) are the concentric circles with the common center at the origin. From this, it is evident that the unperturbed system (8.61) is a fully 0scillatory system. Let C, be an arbitrary orbit of (8.61) and let its equation be

8. Perturbation of Fully Oscillatory Systems

116

y

C, : x = a, cos t,

=

a, sin t.

Since the normal of C, is of the direction of the radius, any point of the orbit of the perturbed system (8.60) is then expressed as follows:

- p ) cos t, y = y(z) = (a, - p)sin t. x = ~ ( z )= (a,

(8.63)

For simplicity, putting

(8.64)

a,-p=r,

let us rewrite (8.63) as follows: x = x(7) = r cos t,

y

= y(z) =

r sin t.

(8.65)

This is nothing but the representation of a point (x, y) in terms of polar coordinates. Now, to obtain the equation for drldt, let us substitute (8.65) into (8.60). Then we have dr - c ost dt

- rsint

=

. dz -rsint*dt '

(8.66)

dz f i s i n t + r c o s t = [ r c o s t - & j ( r c o s t , -rsint)]-. dt dt Multiplying both sides by -sin t and cos t, respectively, and adding the results, we have dz . r = [r - &f(rcos t, - r sin t) cos t] dt' that is, dz - dt

r r - cf(r cos t, - r sin t) cos t

(8.67) '

Likewise, multiplying both sides of (8.66) by cos t and sin t, respectively, and adding the results, we have dz dr - = -&j(r cos t, - r sin t) sin t . - . dt dt

(8.68)

Thus, substituting (8.67) into (8.68), we get dr dt

-=

&'

-f(r cos t, - r sin t) r sin t

r - &j(rco st,- r s i n t ) c o s t '

(8.69)

8.4. An Example

117

This is an equation of the form (8.17). By (8.19), the averaged system of (8.69) is dr

- = EF(r), dt

(8.70)

F(r) = - - j ( r cos t, - r sin t ) siii t dt.

(8.71)

where 21z~02"

Thus, by Theorem 8.3, we see that, if F(r) = 0 has a simple root r = r,, the given equation (8.59) has a unique periodic solution in the neighborhood of y = r, sin t

x = r, cos t ,

(8.72)

for sufficiently small E and that such a periodic solution is asymptotically orbitally stable if F'(ro) < 0 (' = d/dr). When f ( x , i )= (1 - XZ)k,

Eq. (8.59) is a van der Pol equation (see, e.g., McLachlan [1], pp. 41-46, 148-151). In this case, 1 F(r) = 2n 1 8

jo

= - r(4

2n

(1 - r2 cos2 t ) r sin2 t dt

- r').

(8.73)

Therefore, for the root r, of equation F(r) = 0, we have

r,, = 0, +_ 2. And, as is readily seen from (8.73),

F'(0) = 3 # 0,

p ' ( + 2 ) = -1 < 0.

(8.74)

Thus, for each of r, = 0, 2, the van der Pol equation has a unique periodic solution lying near the solution (8.72) if E is sufficiently small. However, for r, = 0, the solution (8.72) of the unperturbed system is x = y = 0, and this is also a periodic solution of the van der Pol equation. Therefore, from the uniqueness of the periodic solution of the perturbed system, the periodic solution of the van der Pol equation corresponding to ro = 0 is x = y = 0 and only this. This implies that ro = 0 does not yield a proper periodic solution of the van der Pol equation.

118

8. Perturbation of Fully Oscillatory Systems

Next let us note that r, = - 2 and r, = 2 yield the same periodic solution of the van der Pol equation. Indeed solutions (8.72), corresponding to r, = - 2 and ro = 2, represent the same closed orbit. Therefore, by the uniqueness of the closed orbit of the perturbed system, the periodic solution of the van der Pol equation corresponding to r, = - 2 coincides with the one corresponding to r, = 2. From these discussions, we see finally that the van der Pol equation has one and only one asymptotically orbitally stable periodic solution close to x = 2 cos t,

if

E

> 0 is sufficiently small.

y

=

2 sin t

9. Perturbation of Partially Oscillatory Systems

By a partially oscillatory system is meant an autonomous system that admits a certain family of closed orbits in the neighborhood of a certain closed orbit. In illustration, consider the system dx1 di

dx3 _dt

+

1

2n XI2

+

XZ2

+

x3

x2

9

-x3.

As is easily seen, the general solution of this system is x1 = a cos 0,

x 2 = a sin 8,

x 3 = be-‘,

(9.2)

where

e=-

2n

1

+ a2 log I b + (1 + a2)ef I + 8 , .

are the arbitrary constants. As is readily seen from (9.2), Here a, b, and system (9.1) admits of a one-parameter family of the closed orbits

x12 + xZ2 = a’,

x3 = 0

(a: parameter).

Therefore system (9.1) is a partially oscillatory system. But, as is seen from (9.2), system (9.1) is not a fully oscillatory system. In the present chapter, we are concerned with the perturbation of such a partially oscillatory system. A typical example is the problem of seeking a periodic solution of the system dx/dt = A x where E is a parameter such that matrix of the form

I

E

+ &f(X,

E),

I is small, and A is a direct sum of the 119

120

9. Perturbation of Partially Oscillatory Systems

and a matrix Az whose eigenvalues all differ from zero in their real parts (see, e.g., Hale [l], pp. 89-94). In the present chapter, we suppose that for the given n-dimensional autonomous system dx/dt = X(X),

(9.3)

X ( x ) is three times continuously differentiable with respect to x in the domain D, and the system (9.3) admits of an m-parameter family 9 of closed orbits in the neighborhood of a closed orbit Co : x = qo(t) contained in D. When m = n - 1, 9 consists of all orbits; consequently, in this case system (9.3) becomes a fully oscillatory system. This case has already been discussed in the preceding chapter; therefore, in the present chapter, we assume that 1 5 m 5 n - 2. By this assumption, it is needless to say that n 2 3 in the present chapter. For the primitive periods of the periodic solutions corresponding to the closed orbits belonging to 9,we assume, as in the preceding chapter, that they are bounded. Then, since the proof of Theorem 8.1 holds also for the closed orbits belonging to 9,we see that the periodic solutions corresponding to the closed orbits belonging to 9 have the universal periods. For the perturbed system of (9.3) dx/dt = X ( X ,E )

( X ( X ,0) = X(X)),

(9.4)

we suppose in the present chapter that X ( x , E ) is twice continuously differentiable with respect to x and E for x E D and E < 6 (6 > 0). The discussion of the present chapter is based mostly on the author’s paper ~ 3 1 .

1 I

9.1. THE REDUCED FORM OF THE PARTIALLY OSCILLATORY SYSTEM

1

1

Let {X[rpo(t)l = X[qo(t)] / X[qo(t)] L(t), ... , tn(t)} be an arbitrary continuously differentiable moving orthonormal system along Co; then, by (5.17), any point x = x ( t ) of the arbitrary orbit of (9.3) lying near Co is expressed as n

x = X ( T ) = Po(4

c

+ v=2 Pvtv(4;

(9.5)

9.1. The Reduced Form of the Partially Oscillatory System

121

and, for z = z(t) and pv = p v ( t ) ( v = 2, 3, ..., n), we have (5.23) and (5.24). For z = z(t), however, as in the preceding chapter (see 8.1), we may suppose that z(0) = 0.

(9.6)

Let ll be a normal hyperplane of Co at the point lo = cpo(0),and let an arbitrary point on I1. Then [ is evidently expressed as

be

n

l

+ vc CVtV(0). =2

= ro

This can be solved with respect to c, ( v = 2, 3, c, =

t,*(O) (l - r o )

(9.7)

..., n) as follows:

(v = 2, 3,

*.

*¶

4.

(94

Now let V be the intersection of the closed orbits belonging to 9 with the hyperplane n. Then, since V is an m-dimensional manifold, c = (cz, cj, ..., cn) corresponding to the arbitrary point 5 of V is expressed by (9.8) as c = c(u),

(9.9)

where u = (uz, u3, ..., urn+ is an m-dimensional parameter vector. For c = c(u), evidently, without loss of generality, we may suppose that c(0) = 0.

(9.10)

In what follows, let us assume that C ( U ) is three times continuously differentiable with respect to u for small u 11. Then we can easily see that the rank of the matrix whose elements are

11

cannot be identically smaller than m. Otherwise there would be continuously differentiable &(u) (a = 2, 3, .. ., m + 1) not all vanishing such that m+ 1

ac, C A,(u)-au,

=0

(v = 2 , 3 , ..., n).

a=z

This means that cv(u) (v = 2, 3, ..., n) are all solutions of the linear homogeneous partial differential equation (9.12)

122

9. Perturbation of Partially Oscillatory Systems

Then c,(u) (v = 2,3, ..., n) will be all the functions of (m - 1) independent solutions of (9.12). This implies that c = C ( U ) can be expressed in terms of a smaller number of parameters than m. This contradicts the fact that V is a manifold of m dimensions. Thus we see that the rank of the matrix whose elements are (9.11) cannot be identically smaller than m. On the basis of this fact, in the sequel, let us assume that the rank of the matrix whose elements are (9.11) is equal to m for u = 0. By this assumption, it is evident that the rank of the matrix whose elements are (9.11) is always equal to m for any u, provided u is sufficiently small. Let us write Eq. (5.24) for pv = p,(t) (v = 2, 3, ..., n) in vector form as follows:

1 1

dP dt

(9.13)

- = R(t, p).

Then, by assumption, as is seen from (5.24), R(t, p ) is three times continuously differentiable with respect to p . Let p = $(t, c) be a solution of (9.13) such that $(O, c ) = c ; then, by Theorem 2.4, $(t, c) is evidently three times continuously differentiable with respect to c. Let (9.14) then, for this function, we can prove the following lemma. Lemma 9.1. The function $(t, u ) and its derivative with respect to t are both three times continuously diflerentiable with respect to u and they are periodic in t of period w (> 0), where o is a universal period of the periodic solution x = cpo(t) of (9.3). In addition, the rank of the matrix whose elements are

., m

1 1

+ 1)

(9.15)

is always equal to m for any t, provided u is suficiently small. Here $,(t, (v = 2, 3, ..., n) are the components of the vector $(t, u).

U)

PROOF. Since $(t, c) is three times continuously differentiable with respect to c, and c(u) is three times continuously differentiable with respect to N, it is evident that $(t, u) = $[t, c(u)] is three times continuously differentiable with respect to u.

9.1. The Reduced Form of the Partially Oscillatory System

Since p

=

123

$(r, c) is a solution of (9.13),

which implies

-- .) - R[t, $(t,

u)].

at

Then, since R(t, p ) is three times continuously differentiable with respect to p, i t follows from the differentiability of $(t, u) with respect to u that a$(t, u)/at is also three times continuously differentiable with respect to u. Now p = $(t, u) = $ [ t , c(u)] represents an orbit of (9.3) passing through a point of the manifold V, and o is a universal period of x = p,,(o(t).Therefore, in the same way as for a fully oscillatory system, we readily see that $(t, u) is periodic in t of period w . Finally, let us prove the last conclusion. Suppose the rank of the matrix whose elements are (9.15) is smaller than m for some values of u and t . Then, 1)not all for such values of u and t, there are numbers 2,(a = 2, 3, ..., m zero such that

+

m+l

a$ CI,'=O

u=2

a@,

( ~ = 2 , ,..., 3 n).

By (9.14), this can be written as follows:

This implies m+ 1 Aa-d c p - O a=2

du,

-

( p = 2 , 3 ,..., n),

(9.16)

because det t)=(t,c) # 0, as is mentioned in connection with (8.52). Equality (9.16) implies that the rank of the matrix whose elements are (9.11) is smaller than m. This contradicts the assumption on the function ~ ( u ) Thus . we see that the rank of the matrix whose elements are (9.15) is always equal to m for any u and t , provided (1 u (1 is sufficiently small. This completes the proof. Q.E.D. Since a$(t, u)/& is continuously differentiable with respect to u by Lemma 9.1, it is evident that a$(t, u)/au, (a = 2, 3, ..., m 1) are continuously

+

9. Perturbation of Partially Oscillatory Systems

124

+

differentiable with respect to t . The (a$/au,)(t, 0 ) (a = 2, 3, ..., m l), however, by Lemma 9.1, constitute a periodic m-ple system of (n - 1)-dimensional linearly independent vectors. Hence, by the well-known Gram-Schmidt process,* we can construct a periodic orthonormal m-ple system of (n - 1)dimensional continuously differentiable vectors P,(t) (a = 2, 3, . ., m 1) so that

.

m+ 1

P,(t)

=

a$ ( t , 0) ,=1 , k,,(t) au,

(a = 2, 3,

..., m + 1)

+

(9.17)

may hold. On the basis of this m-ple system {Pa(t)}(a = 2, 3, ..., m l), we shall construct a continuously differentiable periodic orthonormal (n - 1)-ple system. This will be done using the following lemma concerning orthogonal matrices.

+

Lemma 9.2. Let A be an arbitrary N-by-Nproper orthogonal matrix; then there is an orthogonal matrix T such that cos a,

- sin a,

)BE.

(9.18)

PROOF.Let 1 be an arbitrary eigenvalue of A , and x be a corresponding eigenvector of A . Then, by definition, AX = AX.

(9.19)

Taking the transpose conjugate of the both sides, we have

+

* Let { e , } (a = 2, 3, . . ., rn 1) be an rn-ple system of linearly independent vectors. To make an orthonormal rn-ple system on the basis of {em},one determines the factors 02, 03, ... , c , + ~and the coefficients'a3,, ~ 4 2 a43, , ..., a m + l l ,am+12,... , am+l,,, successively, so that

I

e',

B3

=02 e,, =

o3(a3,z2 + e3),

......................

Frn + 1 = o m + l ( a m + 12 e',

+ . . . + a m + 1 m e'm + em + 1)

may be orthonormal. This process to make the orthonormal system {Fa} from the given system {e,} is the Grarn-Schmidt process.

9.1. The Reduced Form of the Partially Oscillatory System x*A* = AX*.

125 (9.20)

Multiplying (9.19) and (9.20) side by side and using the equality A*A = AA* = E,

(9.21)

we then have x*x =

I I 12 x*x,

which implies

( A ( = 1. Let us put

I

=

e-ia

(9.22)

and x = x1

+ ix,.

(9.23)

Then, from the equality (9.19), we have

+ ix,)

A(x,

=

(cos a - i sin a)(xl

+ ix,),

which is equivalent to Ax, = cos ci . x1

Ax,

=

- sin a

+ sin CI x,, x1 + cos a x,. *

(9.24)

*

If we take the transpose of both sides of the first of (9.24), then we have x,*A* = cos CI xl*

+ sin a

(9.25)

x,*.

Multiplying (9.25) and (9.24) side by side and using the equality (9.21), we then have

i

+

+

xl*xl = cos2 a * xl*xl 2 sin a cos a * x1*x2 sin’a xI*x2 = -sin a cos a * xl*xl (cos’a -sin2a) xl*xz

+

*

x2*x2,

+ sin a cos a - x,*x,.

These can be rewritten as follows:

Isin’

x,

11’ - 11 x, 1’) + 2 sin a cos xl*xz = 0, 1 1 ’ - 1 x, /I2) - 2 sin’ a . x1*x2 = 0. CL.

sin aa( cos a ( x2

(9.26)

When sin CI = 0, the eigenvalue I and the corresponding eigenvector x are both real and, from (9.24), it follows that Ax = Ax

(A = 1 or -1).

(9.27)

126

9. Perturbation of Partially Oscillatory Systems

When sin a # 0, from (9.26), we have

Since sina cosa

2cosa = -2 # 0, - 2sina

the equalities (9.28) imply (9.29) Since x1 and x2 can be multiplied by an arbitrary common factor, from (i) of (9.29) we may suppose

II I = 1 x1 )I x2

(9.30)

= 1.

Equality (ii) of (9.29) says that the vectors x1 and x2 are mutually orthogonal. Now take an arbitrary eigenvalue A, of A. If Al is real, then, by the preceding argument, there is a unit real eigenvector el corresponding to A1 , and it follows that

(A, = 1 or -1).

Ae, = Ale,

(9.31)

Let {e,) (v = 2, 3, ..., N ) be an arbitrary orthonormal system such that

e,*e, = O

(v = 2, 3,

..., N ) ,

(9.32)

and let T1 be an N-by-N matrix whose column vectors are el, e2, ..., eN. Then Tiis evidently an orthogonal matrix and the (i, j)-element of the matrix T1*AT1is

ei*Aej. However, by (9.31), (9.32), and (9.21),

el*Ael = Alel*el = Al = +1 or -1, 1

el*Ae, = - (Ae,)* Ae, = 0

(v = 2, 3, ..., N ) ,

e,*Ae, = I,e,*e, = 0

(v = 2, 3,

21

Thus Tl*ATl is of the form

..., N ) .

9.1. The Reduced Form of the Partially Oscillatory System

6' p4,) -

(A, = 1 or -l),

127

(9.33)

where A, is an (N 1)-by-(N - 1) matrix. Since T,*AT, is an orthogonal matrix, A , is also an orthogonal matrix. If I , is not real, then, by the preceding argument, there is a couple of unit mutually orthogonal vectors el and e, such that

-

Ae, = cos a, * el + sin a1 e,, Ae, = -sin a, * el + cos CL, e2

-

(sin a, # 0).

(9.34)

Let {e,} (v = 3,4, ..., N) be an arbitrary orthonormal system such that

e,*e, = e,*e,

=

0

(v = 3,4, ..., N ) ,

(9.35)

and let T , be an N-by-N matrix whose column vectors are el, e,, e3, ..., f?N. Then, as in the former case, T , is an orthogonal matrix and the (i, &element of the matrix T,*AT, is ei*Aej.

However, in the present case, by (9.34), (9.39, and (9.21), el*Ae, = e,*(cos a, el + sin a, * e,) = cos a , , e,*Ae, = el*(-sin a, * el + cos a1 * e,) = -sin a l , ez*Ael = e2*(cos a, el + sin a, e,) = sin a , , e,*Ae, = e,*(-sin a, el + cos a, * e,) = cos a , ,

-

i

e,*Ae, = (cos a, * A e , - sin a, * Aez)* Ae, = (cos a, * e,*A* - sin a, * e,*A*) Ae, = 0, ez*Ae, = (sin a, Ae, + cos a, * Aez)* Ae, = (sin a, * e,*A* + cos a1 * e,*A*)Ae, = 0, e,*Ae, = e,*(cos a, el + sin a1 . e,) = 0, e,*Ae, = e,*(-sin a, * el + cos a,- e,) = 0

-

(v = 3, 4,

...)N ) .

Thus T,*AT, is, in the present case, of the form cosa, sina,

-sina, cosa,

(9.36)

where A, is an (N - 2)-by-(N - 2) matrix. Since T,*ATl is an orthogonal matrix, A, is also an orthogonal matrix, as in the former case. Let us write (9.33) and (9.36) in a single form as follows: T,*AT, = Ul 0 A , ,

(9.37)

9. Perturbation of Partially Oscillatory Systems

128 where

U, = (A,)

or

cos a1 -sin a, sina, cosal

(9.38)

Since Al is an orthogonal matrix, we can continue the above process, and we can find an orthogonal matrix T,’ such that

T2’*A1T2’= U 2@ A,,

(9.39)

where

U , = (A,)

(A2 = 1 or

-1)

or

cos a2 -sin a, sina, cosa,

).

(9.40)

From (9.37) and (9.39), we can easily see that

T,*Tl*AT,T, = U1 0 U2 8 A2

3

where

T 2 = ( 0E OT,’) Such a process evidently can be continued until we have

Tp* where TI,T,,

U,= (A,)

T2*T,*ATIT2*.* Tp = U 1 @ U, @ **. @ U p ,

(9.41)

..., T pare the orthogonal matrices, and

(A, = 1 or

-1)

or

cos a, -sin a, sin a, cos a,

(I

= 1,2,

..., p).

If we put TIT2

Tp = To,

then To is evidently an orthogonal matrix, and (9.41) says that

(9.42) Interchanging the orders of the rows and columns, from (9.42), we have

9.1. The Reduced Form of the Partially Oscillatory System

129

where El and E2 are the unit matrices and T' is a product of the matrices of the form 0 0

... o i

0 1

0

(unit matrix) 8

8 (unit matrix).

1 '1 0

0

1 0

...

0 0

Since T' is an orthogonal matrix, (9.43) can be written as (9.44) where T = TOTis an orthogonal matrix. Now, by assumption, A is a proper orthogonal matrix, namely, an orthogonal matrix such that detA

=

(9.45)

1.

Then, in (9.44), the order of the matrix (-El) must be even. The matrix (-El) can then be written as cos a

--E,=E@(sin a

-sin a cos a

(a = n).

Then, from (9.44), we have (9.18). Q.E.D. Using Lemma 9.2, we shall prove the following lemma, which shows that one can construct a continuously differentiable periodic orthonormal (n - 1)-ple system containing the m-ple system {P.(t)> (a = 2, 3, ..., m 1) given by (9.17).

+

Lemma 9.3. For any given continuously diferentiable periodic orthonormal m-ple system {Pa(t)}(a = 2, 3, ..., m + l), there is a continuously differentiable periodic orthonormal (n - 1)-pie system of the same period containing the given m-ple system {Pa(t)} (a = 2, 3, ..., m + 1).

130

9. Perturbation of Partially Oscillatory Systems

PROOF. Let o ( > O ) be a period of the given rn-ple system {P,(t)} (a = 2, 3, ..., m + 1). At any point t = to of the interval [0,0], we can construct a linearly independent (n - 1)-ple system { P v }(v = 2, 3, ..., n) so that P, = P,(to)

(a = 2, 3,

...)rn +

1).

Then, by the continuity of {P,(t)}, on a sufficiently small interval containing t = to, the system {P2(t),P3(t),..., Pmtl(t),Pmt2, ..., P,} forms a continuous linearly independent (n - 1)-ple system. On the basis of this system, using the Gram-Schmidt process, on a sufficiently small interval containing t = to, we can construct a continuous orthonormal (n - 1)-ple system {Pv(t)> (v = 2, 3, ...,n) containing the given rn-ple system {P,(t)> (a = 2 , 3, ..., m 1). Let V ( t )be a matrix whose column vectors are P2(t),P3(t), ..., Pn(t).Then it is evident that V(t) is an (n - 1)-by-(n - 1) orthogonal matrix. Now, since the interval [0, 01 is covered by a finite number of the intervals on each of which V ( t ) is defined, we shall denote such covering of [0,03 by Zo < I , < Z2 < ... < Zp and denote the V ( t ) defined on (k = 0, 1, 2, ...,p) by Vk(t)(k = 0, 1,2, ...,p). Here z k < z k + means that zk+ contains t greater than any t belonging to 1,. For any t , E I, n 11,let us consider an orthogonal matrix

+

,

Ml = K ‘ ( t 1 )

(9.46)

VO(tI),

and let us define the matrix Ul(t) so that (9.47)

Equality (9.46) implies (9.48)

Vl(t1)Ml = Vo(t1);

consequently, if we denote the elements of M , by mvp (v, p then we have

(a=2,3

where P:”(t) and Pto’(t) (K = m

,..., m + I ;

+ 2, ..., n) are the Kth

=

2, 3, ..., n),

i = m + 2

,..., n),

column vectors of

9.1. The Reduced Form of the Partially Oscillatory System

131

V,(t) and V,(t), respectively. From the first of (9.49),

where ,a, is the Kronecker delta. Since P2(tl),..., Pm+i(tl),PLY2(t1), P,(')(t,) are linearly independent, (9.50) implies mDa= ,a,

+ 2, ...,n). (9.51) (9.49), the vectors P,(t,) (/? = 2, 3, ..., rn + 1) are both of PL1)(fl)and PLo'(tl) (IC = m + 2, ..., n), since (a,/? = 2,3, ..., m

mra = 0

+ 1;

...,

K = rn

In the second of all orthogonal to V l ( t l ) and V,(t,) are both orthogonal. Multiplying Pa*(t,) (a = 2,3, ..., m + 1) on both sides of the second of (9.49), we then have

man = 0

(a = 2, 3,

..., m

+ 1;

1=m

+ 2, ...,n).

(9.52)

Then (9.51) and (9.52) mean that the matrix M , is of the form

(9.53) where Em is a unit matrix of the order m. From (9.47) we thus see that U,(t) is a continuous orthogonal matrix defined on Z,uZ,, and its first m column vectors are Pa(t)(a = 2,3, ..., m 1). Repeating the above process, for any t2 E Zl n Z2, we put

+

M2

= V2(t2)402)9

and we define U2(t)so that U2(t) =

{!:t).,

for 0 6 t 5 t 2 , for t 2 6 t E 1 2 .

Then, by the above argument, we see that U2(t)is a continuous orthogonal matrix defined on Z, u Z, u 12, and its first rn column vectors are still P.(t) (a = 2,3, ..., m + 1). We repeat this process step by step; finally, after the pth step, we have a continuous orthogonal matrix Up(t)defined on [0, w], of which first m column vectors are Pa(t)(a = 2, 3, ..., m 1). Now put

+

u;'(w) UP(O)= A ;

(9.54)

then A is a proper orthogonal matrix because Up(t)is a continuous orthogonal matrix, and hence det U,(w) = det Up(0).Now the equality (9.54) is of the same form as (9.46), and the first rn column vectors of U p ( o )and Up(0) are bothP,(O) (a = 2, 3, ..., m + l), since Pa(w) = Pa(0)(a = 2, 3, .., m 1).

.

+

132

9. Perturbation of Partially Oscillatory Systems

Hence, as for M I in (9.46) A is of the form A

=

t

i).

(9.55)

Here, since A is a proper orthogonal matrix, ;j[ is also a proper orthogonal matrix. Then, by Lemma 2, there is an orthogonal matrix T such that

T*RT

=

;8 (sincos

a, u,

-sin a, cos u,

(9.56)

where E, is a unit matrix of a certain order, say 1. Since cos a, -sin a,) = exp (0, sin a, cos a,

-);

as is easily seen, let us put (9.57)

and

Em @

T = T,

(9.58)

where Om and 0, are, respectively, the zero matrices of the order m and 1. From (9.55) and (9.56), we then have

T*AT = exp (COB,), which implies A = exp (COB),

(9.59)

B = TBOT".

(9.60)

where

Substituting (9.59) into (9.54), we thus have

~ ~ (=0 Up(m)eoB. )

(9.61)

Now let us consider the matrix V ( t ) = Up(t)exp (re). As is easily seen from (9.57), (9.58), and (9.60), B is of the form

B = 0, @ 8;

(9.62)

9.1. The Reduced Form of the Partially Osciflatorysystem

133

consequently, elB is of the form exp ( t E ) = Em 8 exp (tB). Then, from (9.62), we readily see that the first m column vectors of Q(t) are P,(t) (a = 2, 3,...,m 1). Now, from (9.57),

+

Eo* = -Bo;

therefore, from (9.60), B* = TBo*T* = -TB 0 T* = -B.

Then

-

[exp (tE)]* exp ( t B ) = exp (tE*) exp (tB) = exp (- tB) exp (tB) = E, which says that ere is an orthogonal matrix. From (9.62), we then see that U ( t ) is an orthogonal matrix. On the other hand, from (9.61) and (9.62), we see that

U(0)= U(0). Therefore, if we define U(t) outside the interval [0, relation U(t

+

0) =

03

successively by the

U(t),

we have a continuous periodic orthogonal matrix U(r) of period o. Since Pa(t) (a = 2, 3, ..., m + 1) are periodic of period o,it is evident that the first m column vectors of the extended U(t) are also Pa(t)(a = 2, 3, ..., m 1). Finally, we shall show that we can get a continuously differentiable periodic orthonormal (n - 1)-ple system by modifying the last (n - m - 1) column vectors of the above U(t). Let p3,(t)(K = m + 2, ..., n) be the last (n - m - 1) column vectors of the above U(t). Since p3,(t)( K = m 2, ..., n) are continuous and periodic, they are uniformly continuous on the interval -00 < t < 00. Then, for any positive number E, we can take a positive number 6 so that

+

+

I P,(t‘) - Pl((t”)I < E (K = m + 2, ...,n) whenever I t’ - t“ I 5 S. Let us consider the vectors (rc = m + 2, ..., n); &(t) = - [-?3,(s) ds 2s

(9.63)

(9.64)

134

9. Perturbation of Partially Oscillatory Systems

from (9.63), it follows that

1 p,(t) - P,(t) I = I

- F,(t)] ds

E.

Then P2(t), ..., P,,,, l(t), p,,,+2(t), ..., b,(t)are mutually linearly independent, provided E is sufficiently small, since the determinant of the orthogonal matrix U(t)whose column vectors are P2(t),. .., p,,,+l(t), p,,,+2(t), . .., F*(t)never vanishes. In addition, from (9.64),

I-!,.(s) t+S

=

ds

26 =

R(t>

(K =

m

+ 2, ..., n)

and

These relations say that p,(t) ( K = m + 2, .. ., n) are periodic of period o and are continuously differentiable. Since {P.(t)} (ct = 2, 3, ..., m 1) is, by assumption, a continuously differentiable periodic orthonormal m-ple system of period o,by applying the Gram-Schmidt process to the (n - 1)ple system { ~ ~ ( .t..,) P , ,+ l(t), P,,,+2(t),..., P,(t)>, we can construct an orthonormal (n - 1)-ple system {PV(t)}(v = 2, 3, ..., n) so that P,(t) (K = m 2, ..., n) are periodic of period o and are continuously differentiable. In this case, it is needless to say that the first m vectors of the newly constructed orthonormal (n - 1)-ple system are the given P,(t) (a = 2,3, ..., m 1). These show that the orthonormal (n - 1)-ple system {PY(t)}(v = 2, 3, .. ., n) constructed above is a desired orthonormal system. Q.E.D.

+

+

+

In 5.1, under a Lipschitz condition, we proved the existence of a continuous moving orthonormal system along a closed orbit of the autonomous system. However, as is seen from the above proof, using the similar technique, we can prove without a Lipschitz condition the existence of a continuous moving orthonormal system along a closed orbit for any continuous autonomous system. It is clear that the method of 5.1 is far more convenient for practical application than the method of proof of Lemma 9.3. This is the reason why we adopted a slightly less general method in 5.1.

9.1. The Reduced Form of the Partially Oscillatory Systems

135

Let {Pa(t)}(a = 2, 3, ..., m + 1) be a periodic continuously differentiable orthonormal m-ple system of period w constructed so that (9.17) may hold. Then, by Lemma 9.3, we can construct a periodic continuously differentiable orthonormal(n 1)-ple system {Pv(t)} (v = 2, 3, ... , n) of the same period containing the initial {Pa(t)}(m = 2, 3, . .., m 1). Let PPv(t)(p,v = 2, 3, ..., n) be the components of the (n - 1)-dimensional vectors Pv(t)(v = 2, 3, ..., n). Then the vectors xCvo(t)l = X [ V O ( ~I ]I( X[vo(t)] 1 and C:=2Ppv(t)S,(f) (v = 2, 3, ..., n) constitute a continuously differentiable moving orthonorma1 system along C,. With regard to this moving orthonormal system, (9.5) is written as

-

+

(9.65)

consequently, comparing this with (9.5), we have

c pv,(t)a, n

pv =

(v

= 293,

-*-,

.).

(9.66)

p=2

Let us write this in vector form as follows: p = P(t)D,

(9.67)

where p and p are the (n - 1)-dimensional vectors whose components are, respectively, pv and by (v = 2, 3, ..., n) and P ( t ) is a matrix whose elements arePpv(t) (p, v = 2,3, ..., n). Since {Pv(t)}(v = 2, 3, ..., n)isan orthonormal system, it is evident that P ( t ) is an orthogonal matrix. Now, for the arbitrary orbit of (9.3) lying near C,, p = p(t), and this function satisfies the differential equation (9.13). Therefore, for such an orbit, j3 = p(t), and this function satisfies the differential equation d)/dt = R(t,’$),

(9.68)

R(t,p) = P*(t)R(t, P(t)j?) - P*(t) d P ( 0 p. dt

(9.69)

where

Now, p = $(t, u) of (9.14) is a periodic solution of (9.13); therefore, from (9.671,

p

= $(t, u) Zf P*(t)$(t, u)

(9.70)

is a periodic solution of (9.68). Then, since $(t, u) is continuously differentiable

9. Perturbation of Partially Oscillatory Systems

136

with respect to u, we have Ppv(t)

=

au4

p=

2

(v = 2, 3,

aua

..., n ; a = 2,3, ..., m + 1). (9.71)

However, in (9.17), det K ( t ) # 0

for any t ,

(9.72)

+

where K ( t ) is a matrix whose elements are ka,(t) (a, fl = 2 , 3 , ..., m l).In fact, otherwise, there are numbers xa (a = 2, 3, ..., m + 1) not all zero such that

for some t, and this implies that m+ 1

C Kapa(t) = 0,

4=2

+

which is a contradiction, since {Pa(t)}(a = 2, 3, ..., m 1) is an orthonorma1 m-ple system. By (9.72), the relation (9.17) can be solved inversely with respect to (a$/au,) ( t , 0) as follows: m+ I

a-(t,h

0) =

au4

C KUP(t)PpP(t)(p = 2, 3, ..., n ;

a = 2,3,

p=2

..., m + I).

(9.73)

Here the functions Kap(t)(a, fl = 2, 3, ..., m + 1) are all continuously differentiable and periodic of period w. Let us substitue (9.73) into the righthand side of (9.71); then we have

(v = 2,3,

..., n ;

a = 2,3,

..., m

+ 1).

(9.74)

However, since P ( t ) is an orthogonal matrix, it holds that n

C Ppv(t)Ppp(t) = a,

(v = 2, 3,

..., n ; jI = 2 , 3 , ..., m

+ I),

p=2

where 6,, is the Kronecker delta. Thus, from (9.74), we see that

!.!L (t, 0) = 0 aUa

(K =

m

+ 2, ..., n ;

tl =

2, 3, ..., m

+ l),

9.1. The Reduced Form of the Partially Oscillatory System

137

The second of these equalities implies that

a ($23 $39 ..*) $ m + 1 > a ( ~ 2 ~, 3 , u m + 1 )

# 0

for any

t,

(9.75)

“‘7

provided 11 u 11 is sufficiently small, because the matrix whose elements are K,,(t) (a, y = 2, 3, .. ., m + 1) is K - ’ ( t ) , and evidently det K - ’ ( t ) # 0 for any t . Now, making use of the function $(t, u), let us transform p to p by

$& P 2 , P31 = + ~

ba = D K

PK

$K(t,

**-*

P,+J

(a=2,3

2 ~ , 3 a O,. 9 P m + , )

(x = m

,..., m + l ) , + 2, ..., n).

(9.76)

From (9.79, the Jacobian of the right-hand side of (9.76) never vanishes for any t, provided p is sufficiently small; consequently, formula (9.76) indeed defines a transformation of j!, to P in the neighborhood of p = 0. Since $(t, u) is a solution of (9.68),

11 1

~-

- R [ t , $(t, u ) ] ;

(9.77)

at

(9.78)

Owing to (9.75), the equalities (9.78) can be rewritten in the form

dpldt = R( t, p),

(9.79)

where R(t, p ) is periodic in t of period o and is twice continuously differentiable with respect to p. In addition, as is readily seen from (9.78), we have

R(t, p2, ..., p m + l ,o,o,

..., 0) = 0

(9.80)

9. Perturbation of Partially Oscillatory Systems

138

for any t and p z ,

..., pm+

Equality (9.80) means that

pm = const. p, = 0

(a = 2, 3, (K =

m

..., m

+ l),

+ 2, ..., n )

(9.81)

is always a solution of (9.79). By (9.80), K(t, p ) is expressed as

therefore, if we put

(ti =

rn

+ 2, ..., n),

(9.82)

we can write (9.79) as follows: (9.83) Here, as is readily seen from (9.82), R.,(t, p ) (K = m + 2, . .., n) are continuously differentiable with respect to p and are periodic in t of period w . The result obtained is stated in theorem form as follows.

Theorem 9.1.

If the three times continuously diferentiable partially oscillatory system (9.3) admits of an m-parameter family F of closed orbits in the neighborhood of a closed orbit C,, and c(u) of (9.9) for a closed orbit belonging to F is three times continuously direrentiable, then the equation of any orbit of (9.3) in the normal hyperplane of C, is reduced to an equation of the form (9.83) by the periodic transformation (9.76) when a moving orthonormal system along C, is suitably chosen. In (9.83), R.,(t, p ) (ti = in + 2, ..., n ) are continuously diferentiable with respect to p and periodic in t of period w where w > 0 is a universal period of the closed orbit C., That (9.81) is always a solution of (9.83) implies (1)

The mangold V defined by

p,

=

O

(K

=m

+ 2, ..., n)

139

9.2. Perturbation of a Partially Oscillatory System

in the phase space is an ( m + 1)-dimensional integral manifold, namely, a manifold such that the orbit passing through any point of the manifold is contained completely in the manifold;

(2) Any orbit contained in the integral manifold V is always given by

pa = const.

(a = 2, 3, ..., m

+ 1).

9.2. PERTURBATION OF A PARTIALLY OSCILLATORY SYSTEM

1

/I

Let (x[rpo(t)l = X[rpO(t)l/ X[cpo(t)I M t ) , * * t&)> be an arbitrary continuously differentiable moving orthonormal system along the closed orbit Co : x = rpo(t) of the unperturbed system (9.3); then, as is mentioned in 7.1, any point x = X ( T ) of an arbitrary orbit of the perturbed system (9.4) lying near Cois expressed as (7.3) and, for z = z(t) and py = p,(t) (v = 2, 3, ..., n), we have (7.4) and (7.5). Let us write Eq. (7.5) in vector form as follows: 9

dpldt = R(t, P, 8 ) ;

(9.84)

then, as is seen from (7.9, by assumption, R(t, p, E ) is twice continuously differentiable with respect to p and E . It is clear that R(t, p, E ) is periodic in t of period w , where w is a universal period of the periodic solution x = cpo(t) of the unperturbed system (9.3). Since X(x, 0) = X ( x ) holds between the perturbed system (9.4) and the unperturbed system (9.3), we may suppose that the relation R(t, P, 0) = R(t, P )

(9.85)

holds between R(t, p , E ) of (9.84) and R(t, p ) of (9.13). Now let us transform p to P by formula (9.67). Equation (9.84) is then transformed into the equation d$/dt = R(t, 6, E),

(9.86)

where

R(t, p, E )

= P*(t)R[t,P(t)$,

€1- P*(t)[dP(t)/dtJ$.

(9.87)

From (9.87), it is evident that R(t, P, E ) is periodic in t of period w and is twice continuously differentiable with respect to fi and E . By (9.85), from (9.69) and (9.87), it is also evident that R(t, P, 0)

=

R(t, P).

(9.88)

9. Perturbation of Partially OsciIIatory Systems

140

Let us further transform b to p by the formula (9.76). Then, substituting (9.76) into (9.86) and making use of the identity (9.77), we have

(9.89)

+ C -a$, (t,p).$ #=?

( ~ = m + ,..., 2 n).

dF

aua

Owing to (9.79, equalities (9.89) are rewritten in the form

dt

=

R ( f ,p , E),

(9.90)

and K ( t , p , E ) is periodic in t of period o and is twice continuously differentiable with respect to p and E. If we compare (9.89) with (9.78) and make use of relation (9.88), then we readily see that (9.91)

K(t, p, 0 ) = R(t, p)

holds between R(t, p , E ) of (9.90) and K(t, p ) of (9.79). Now R(t, p , E ) is expressed as q t , p, E ) = R(t, p )

+ H(t, p ;

E)E,

(9.92)

where H ( t , p ; E ) = Jo

aK

(4 p,

do.

(9.93)

Hence, from (9.83), we have n

q t , p, E ) =

1

W.,(t,

p)p,

+ EH(t, p ; 8).

(9.94)

K=m+2

+

Here, by Theorem 9.1, R.,(t, p ) (K = m 2, ..., n) are periodic in t of period o and continuously differentiable with respect to p and, as is seen from (9.93), H(t, p ; E ) is periodic in t of period o and is continuously differentiable with respect to p and E.

9.2. Perturbation of a Partially Oscillatory System

Now let

p

=

141

~ ( ta,; E ) be a solution of (9.90) such that

X(0,a ; E )

=

(9.95)

a.

Then, by Theorem 2.4, the function ~ ( ta;, E ) is twice continuously differentiable with respect to a and E . Let a, ( v = 2, 3, .. ., n) be the components of the vector a, and let a’ and a” be, respectively, the vectors whose components are a,(a = 2, 3, ..., m 1) and a, (K = m 2, ..., n). We may then write X(t, a ;&)asX ( t , a‘, a ” ; 8). In a similar way, by ~ ’ ( a’, t , a”; E ) and X”(t, a‘, a”; E), we shall denote the vectors whose components are, respectively, X,(t, a’, 2, ..., n), where a”;e)(a = 2, 3, ..., m 1) and X,(t, a’, a”; E ) (K = m ~ , ( ta’, , a”; E ) (v = 2, 3, ..., n) are the components of the vector X(t, a’, a”; E). Since p = X(t, a’, a”; 0 ) is a solution of (9.83) as is seen from (9.91), by the remark on solution (9.81), it is easily seen that

+

+

+

+

X‘(t, a’, 0; 0) = a’, X”(t,

(9.96)

a’, 0 ; 0 ) = 0.

Let us now put

-(r, &V

aD,,(t, a’)

(v = 2, 3,

..., n ;

&(t,a’,O;O)

= r,(f,a’)

(v

a’ , 0; 0) =

aa,

a&

K =

=

m

+ 2, ..., n),

2,3, ..., n),

(9.97) (9.98)

and let us denote, (1) by W(t, a‘), the m-by-(n - m - 1) matrix whose elements are @,(f, a ’ ) ( a = 2 , 3,..., m + 1 ; ~ = m + ,..., 2 n);

(2) by @”(t, a‘), the (n - rn - 1)-by-(n - m - 1) matrix whose elements are @ , , (t, a’) (p,K = m + 2, .. ., n);

(3) by r’(t, a’), the m-dimensional vector whose components are ra(t,a‘) ..., m 1);

(a = 2, 3,

+

(4) by r”(t, a’), the (n - m - 1)-dimensional vector whose components are rrr(t,a’)(K = m 2, ..., n).

+

Let R,,(t, p ) ( v = 2, 3, ..., n; K = m + 2, ...,n) be the components of the vectors K.,(t, p), and let H,(t, p ; E ) (v = 2, 3, ..., n) be the components of

9. Perturbation of Partially Oscillatory Systems

142

the vector H(t, p ; E ) . Let us denote: (1) by R'(t, a'), the m-by-(n - m - 1) matrix whose elements are BaK(t, 1; K = m + 2, ..., n); a,, ..., a,+l, 0, ..., 0) (u = 2, 3, ..., m

+

(2) by W"(t, a'), the (n - m - 1)-by-(n - m - 1) matrix whose elements are RPK(t,a2, ...,a,+1, 0, ..., 0) (p, K = m 2, ..., n);

+

a,,

(3) by H'(t, a'), the m-dimensional vector whose components are Ha(t, ..., a,,,, 0, ..., 0) (a = 2, 3, ..., m + 1);

(4) by H"(t, a'), the ( n - m - 1)-dimensional vector whose components are HK(t,a2, ..., a,+l, 0, ...,0) (K = m 2, ..., n).

+

Then we can prove the following lemma.

Lemma 9.4. The matrix W(t, a') is a fundamental matrix of the linear homogeneous periodic system dy/dt

=

R"(t, a')y,

(9.99)

and it satisfies the initial condition (D"(0, a') = E.

(9.100)

For W(t, a'), r"(t, a'), and r'(t, a'), there hold @(t, a') =

sd

r"(t, a') = cD"(t, a')

R'(s, a')W'(s, a') ds,

(9.101)

J:

Cp"-l(s, a')H"(s, a')ds,

(9.102)

+ H'(s, a')] ds.

(9.103)

[R'(s, a')r''(s,

a')

PROOF. From (9.95), it is evident that @VK(O,

")

'VK Y

rv(0,a') = 0

(v = 2, 3, ..., n;

K =

m

+ 2, ..., n),

(9.104)

9.2. Perturbation of a Partially Oscillatory System

143

where 6,, is the Kronecker delta. Conditions (9.104) are written in matrix form as follows: 0,

(9.105)

@”(O, a’) = E ,

(9.106)

r’(0, a’) = 0,

(9.107)

r”(0, a’) = 0.

(9.108)

@’(O, a’)

=

Now, since p = x(t, a’, a”; E ) is a solution of (9.90) satisfying the initial condition (9.99, by (2.10) and (2.1l), we have

2 ,n), ( v = 2 , 3 ,..., n ; ~ = m + ,... d axy(t, a’, a ” ; 8 ) n aK, = -=- [ t , x(t, a’, a“; E), dt a& P = 2 aP& -

c

aR +,

a&

[ t , x(t, a’, a ” ; E),

El

El

ax,,(t, a’, a” ;6 )

(9.109)

a& (v = 2, 3,

..., n).

If we put a”

= E = 0 in (9.109) and make use of (9.94) and (9.96), then, by (9.97) and (9.98), we have

2 n), ( v = 2 , 3 ,..., n ; ~ = m + ,...,

+H,(t,

a27 *.-,

am+,,

0,

0;0) (v = 2, 3,

..., n).

Equations (9.110) are written in a matrix form as follows : d@‘(t,a‘) = R’(t, a’)”‘(t, a’), dt

(9.111) (9.112)

dr’(t, -- a’) - K’(t, a’)r’’(t,a’) dt

+ N ( t , a‘),

(9.113)

9. Perturbation of Partially Oscillatory Systems

144

(9.114) From (9.106) and (9.112), follows the first conclusion of the lemma that @”(t, a’) is a fundamental matrix of (9.99) satisfying (9.100). Equation (9.101) readily follows from (9.105) and (9.1 11). Since @”(t, a’) is a fundamental matrix of (9.99), as is shown above, applying Theorem 3.6 to (9.114), we obtain (9.102) owing to the initial condition (9.108). Equality (9.103) readily follows from (9.107) and (9.1 13). These complete the proof. Q.E.D. For @’(t,a’), @”(t, a’), r’(t, a’), and r”(t, a’), we have the following lemma.

Lemma 9.5. For any positive integer p , it holdr that @“(pa,a’) = @:p,

(9.115)

+ Q0” + + @tP-’)r”(a, a‘), @’(pa, a’) = @’(a,a’)(E + (Do” + ... + @tP-’ >,

(9.1 16)

r”(pa, a‘) = ( E

- a -

(9.117)

r’(pa, a’) = pr’(a, a‘)

+ [@)’(a, a’) + a’(20.1,a‘) +

*-.

+ @‘((p-l)m, a’)] ~ ” ( c oa’), ,

(9.118)

where @: = @”(a,u‘).

PROOF. Since @”(t, a’)is a fundamental matrix of the linear homogeneous periodic system (9.99), by Theorem 3.2, there is a constant matrix K such that

However, if we put t = 0 in the above equality, from (9.100), we see that K = @”(a, a’) = @ .: Thus we have @”(t + o,a’) = a”(?,a’)@:, from which (9.115) readily follows.

(9.119)

9.2. Perturbation of a Partially Oscillatory System

If we replace t by t r“(t

145

+ w in (9.102), then, by (9.1 19), we have successively

+ o,a‘) = @”(t, a)@,” +

j;+k‘’-’(s,

[:j

@”-‘(s,

1

a‘) H”(s, a’) ds

= @”(t, a’) r”(w, a‘)

j:

@”-‘(s

x @:

a’) H”(s, a’) ds

+ ~ ” ( t a, )

+ o,a’) H”(s, a’) ds

= ~ ” ( ta’) , r”(w, a ‘ )

+ r”(t, a‘).

(9.120)

From this (9.1 16) follows. For W ( t , a), from (9.101) and (9.119), we have successively @’(t

+ O, a’) = W(W, a’) +

j:”

R’(s, U’)@”(S,

= W(w, a‘)

+ [>’(a,

= @’(a,a’)

+ @’(t, a’)@”’.

a’)@’’(s

a‘) ds

+ w, a’) ds

From this (9.1 17) readily follows. For r’(t, a’), from (9.103) and (9.120), we have successively

r‘(t

+ w , a’) = r’(w,a‘) + [:+bR‘(s,

a’)r‘’(s,a’)

+ H’(s, a’)] ds

[R‘(s, a’)r‘‘(s+ o,a‘) =

r’(w,a’)

+

+ H’(s, a‘)] ds

sd

[R’(s, a’)@”(s, a’)r”(w,a’)

+ R’(s, a’)r”(s, a’) + H’(s, a’)] ds = r’(o, a’) + @’(t,a’)r’‘(w,a’) + r’(t,a’)

(see (9.101)).

From this, (9.1 18) readily follows. This completes the proof. Q.E.D. Now we shall seek a periodic solution of the given perturbed system (9.4). Theperiodicsolution of (9.4),which is close to the periodic solution x = %(t) of the unperturbed system (9.3), represents a closed orbit in the phase space, and such a closed orbit is represented by a periodic solution of (9.84) of a

146

9. Perturbation of Partially Oscillatory Systems

period that is an integral multiple of w. Such a periodic solution of (9.84), however, corresponds to a periodic solution of (9.90) of the same period by the transformation described in the beginning of the present section. Since p = ~ ( ta ,; E ) is an arbitrary solution of (9.90) satisfying (9.95), the problem is thus reduced to one of finding a such that ~ ( ta,; E ) may be periodic in t of a period that is an integral multiple of o. Since p = x(t, a ; E ) is a solution of the periodic system (9.90), the function x(t, a ; E ) is periodic if and only if (9.121)

X(PW a ; E ) = a

for a certain positive integer p [see the proof of (8.26)]. Condition (9.121) can be written in terms of X’(t, a’, a“; E ) and ~ ” ( ta‘, , a ” ; E ) as follows: (i)

~ ’ ( p oa’, , a ” ; E ) = a‘,

(ii) ~ ” ( p oa’, , a”; E)

(9.122)

= a”.

By (9.96), condition (9.122) is satisfied identically in a’ when a” = E = 0.

On the other hand, by (9.97), the Jacobian matrix of x ” ( p 0 , a’, a“; E )

- a”

with respect to a” is equal to

@”(PO, a’) - E for a’’ = E = 0. Therefore, if det [@“(PO, a’) - E l # 0, then, for sufficiently small

I E 1,

(9.123)

we can get a” = fp(a’,E ) ,

(9.124)

solving (ii) of (9.122) with respect to a”. Since X(t, a ; E ) is twice continuously differentiable with respect to a and E , fp(a‘,E ) is, of course, twice continuously differentiable with respect to a’ and E. Then, since fp(a’, 0) = 0

as is evident from the derivation offp(a’, ten as follows:

fP(d E)

E),

(9.125)

the function &(a‘, E ) can be writ-

= E f P ’ ( d E),

(9.126)

9.2. Perturbation of a Partially Oscillatory System

where j-pl(a’,

8)

lo2

147

1

1

= -fp(a‘,E ) = E

(a’, e E > de.

(9.127)

Since fp(a’,E ) is twice continuously differentiable with respect to a’ and E, it is evident that f,’(a’, E ) is continuously differentiable with respect to a’ and E. From (9.127),

f,‘ (a’, 0) = af, - (a’, 0).

(9.128)

a&

This quantity is, however, determined in the following way. In fact, since (9.124) is a solution of (ii) of (9.122) with respect to a”, it follows that X”[PW,

a’,f,(a’, 8 ) ; E l = fp(a’,6 ) .

Then, differentiating both sides with respect to E and putting that, by (9.97), (9.98), (9.125), and (9.128), we have

E

= 0 after

@“(PO, a’)fpl(a’,0 ) + r ” ( p o , a’) = fpl(a’,0); that is [@”(pw,a’) - E l fp’(a‘, 0) = - r ” ( p o , a’).

By the condition (9.123), this can be solved with respect to fp’(a’,0), and we obtain &‘(a’, 0) =

- [@“(pw, a‘) - El-’

r”(po, a‘).

(9.129)

Now, let us substitute (9.124) into (i) of (9.122). Then we have x “ p 0 , a‘,fp(a‘, &);

E]

- a’ = 0.

(9.130)

By(9.125) and the first of (9.96), this is valid for any a’ if E = 0. Moreover, the left-hand side of (9.130) is twice continuously differentiable with respect to a‘ and E . Hence, in a similar way as in (9.126), we can write Eq. (9.130) as follows :

where gp(a’,E )

1

= -{x“po, &

a’,f,(a’,8 ) ; E ] -a‘>.

(9.132)

9. Perturbation of Partially Oscillatory Systems

148

Equation (9.131) is equivalent to the equation

(9.133)

gp(a” 8 ) = 0,

provided E # 0. However, as for&’(a’, E), the function gp(a’,E ) is continuously differentiable with respect to a’ and E . Therefore, Eq. (9.133) cannot have a solution for sufficiently small [ E unless the equation

I

(9.134)

g p ( d 0) = 0

has a solution. On the contrary, if Eq. (9.134) has a solution a’ = up’,then, by the theorem on implicit functions, Eq. (9.133) has a unique solution a’ = up)(&)in the neighborhood of a’ = up’ for sufficiently small E provided the Jacobian of gp(a’,0) with respect to a’ does not vanish for a’ = a;. However, from (9.132), by (9.97), (9.98), (9.125), and (9.128),

I I,

gp(a’,0) = @’(PO, a’)fp’(a’,0)

+ r’(pO, a’).

By (9.129), we have then gp(a’,0) = - @’(PO, a’) [@”(pw,a’) - E ] - ’ r ” ( p o , a‘)

+ r’(pw, a’).

(9.135)

In order to simplify the right-hand side, let us substitute (9.115)-(9.118) into the right-hand side of (9.135). Then we have gp(a’,0) =

- @’(O,a’) ( E + @: + .*.+ @ : p - ’) - E)-’(E

x

+ CD: +

* a *

x

+ @:”“)~”(o, a’)

+ pr’(o, a’) + [@’(a,a’) + a’(20, a’) +

e . 1

+ @’((p - l ) ~a’)]r’’(w, , a’).

(9.136)

However, since (E

+ @: +

+ @:‘-I)

(@: - E )

=

a:” - E

= @”(PO, a’) - E ,

from (9.123), it holds that det (0:- E) # 0

(9.137)

and E

+ @: + ... + @tP-’ = (a:” - E ) ( @ : - E ) - ’ .

(9.138)

Hence, substituting (9.138) into (9.136), and making use of relation (9.1 17), we have successively

9.2. Perturbation of a Partially Oscillatory System

gp(a',O) = - @'(a,a ' ) ( E

+ @: + ... + @gp-')(@:

- E)-'r"(a,

149 a')

+ pr'(w, a') + @'(a, a') [ E + ( E + @): + ) ] p "2 (a, + ( E + 0; + + @ y *.*

=

- @'(a, a') ( E + @;

a')

+ *.. + @gp-') (a:

- E)-'/'(w, a')

+ pr'(0, a'). - E ) + (@:2 - E ) + - E)] (a: - E)-lr"(a, a')

+ @'(a, a') [(@: +

(@;P-'

= @'(a, a') [

+ (@: x

(@:

- (E

+ @: +

- E) + - E)"r"(a,

- E)

+ @gP-'

+ +

) - El1

(@:'-I

a')

+ pr'(a, a') =

- p@'(a, a') (@:

- E)-'r"(a, a') + pr'(a, a').

This shows that g p ( d

0) = pg,(a', 0) =

p { - @'(a, u')[@''(w, a ' ) - E]-'l;''(o, a')

+ r ' ( ~a')}. ,

(9.139)

Evidently this implies Eq. (9.134) is equivalent to g&', 0) = -@'(a, a') [@''(a,Q ' ) - E ] - ~ ~ ' ' ( O a'),

+

a') = 0. (9.140)

Now let us suppose that Eq. (9.140) has a solution a' = a,' for which the Jacobian A(a') of gl(a', 0) with respect to a' does not vanish for a' = q', and that the condition (9.123) holds for a' = al' and a certain positive integer p. Then, since a' = a,' is also a solution of (9.134) for which the Jacobian of gp(a', 0) with respect to a' does not vanish, Eq. (9.133) has a unique continuous solution a' = aP'(&)such that aP'(e)+ a,' as E + 0. For such a' = a,,'(&), from (9.124) we have continuous a" = a;(&) = fp[a,,'(~), E ] such that a;(&)+ 0 as E + 0. Since a' =

Up)(&),

a" = a;(&)

is a solution of (9.122) for sufficiently small solution jj

(9.141)

I E I, we thus have a periodic

= i g t ) = x(t, apt(&), a;(&);8)

(9.142)

9. Perturbation of Partially Oscillatory Systems

150

I E I, and (9.142) is a unique periodic solution

of (9.90) for sufficiently small of the period pw such that

x’(0, U p ’ ( & ) , a;(&); E )

= U p ’ ( & ) + a,’,

X”(0, UP’(&), a;(&); E ) = a;(&) --+ 0

as E

-+

(9.143)

0. In the present case, however, condition (9.123): det [@“(PO, a,’) - E l # 0

(9.144)

implies the condition (9.137): det [@“(w, a,‘) - E l # 0,

(9.145)

as has been shown already. Then, in quite the same way as for p = pp(t), for Eq. (9.90), we have a unique periodic solution of the period o

p

=

(9.146)

p,(t) = x(t, Ul‘(E), a;‘(&);E )

such that X’(0, a,’(&),a;’(&);E ) = a,’(&)

X”(0, u , ’ ( E ) , U ; ’ ( E ) ;

E) =

-+

a,’,

a:(&) + 0

(9.147)

as E + 0. However, for the periodic solution (9.146), x(pw, U l ’ ( E ) , a;(&);E )

=

X(0,4(&),a:(&);E )

evidently holds for any integer p . This implies x’(po, a,’(&),a:(&); 8) x”(p0,

Ul’(E),

= a,’(&),

(9.148)

a;’(&);E ) = a;(&).

On the other hand, in the present case, Eq. (9.122) has a solution (9.141) and, because of conditions (9.144) and A(a,’) # 0, the solution of (9.122) such that a’

-+

all,

a” + 0

as

E

+0

is unique. Hence, from (9.148), we see that Up’(&)

= a,’(&),

a;(&) = a;(&);

namely, by (9.142) and (9.146), we see that

pp(4

=

ijl(t).

(9.149)

This implies that the given perturbed system (9.4) has no subperiodic solution

9.2. Perturbation of a Partially Oscillatory System

151

(see 7.2) of a period close to p w if p is a positive integer for which (9.123) holds. Now, by (9.147), the periodic solution (9.146) of Eq. (9.90) converges to

p

= X(t, 4

' 9

(9.150)

0;0)

as E + 0. By (9.96), equality (9.150) says that

p'

= a,'

and

p"

=

(9.151)

0,

where p' and p" are, respectively, the vectors whose components are pa (ci = 2, 3, ..., m + 1) and ?,(K = m 2, ..., n). By (9.76), the equalities (9.151) imply

+

b

=

w,%'),

which implies by (9.67), (9.70), and (9.14) that

Thus, by (9.5), the periodic solution (9.150) of Eq. (9.90) with (9.83) ,yields the periodic solution

E

= 0, namely

(9.152) v=2

of the unperturbed system (9.3) belonging to 9. This means that the closed orbit of the perturbed system (9.4) corresponding to the periodic solution (9.146) of Eq. (9.90) lies in an arbitrary neighborhood of the closed orbit represented by the periodic solution (9.152) of the unperturbed system (9.3) belonging to 9,provided E is sufficiently small. The results obtained are summarized in the following theorem.

I I

Theorem 9.2. We suppose that the partially oscillatory system (9.3) admitting of an mparameter family 9of closedorbits of boundedprimitiue periods in the neighborhood of a closed orbit Co is perturbed as in (9.4) and that X ( x ) of (9.3) is three times continuously diferentiable with respect to x , X ( x , E ) of (9.4) is twice continuously dixerentiable with respect to x and E , and C(U) of (9.9) for closed orbits belonging to 9is three times continuously dixerentiable with respect to u. If Eq. (9.140) has a solution a' = a,'for which the Jacobian A (a') of gl (a', 0) and det [@''(a, a') - El [see (9.123)] do not vanish, then, for sufficiently small E >O, the perturbed system (9.4) has a unique periodic solution such

I I

9. Perturbation of Partially Oscillatory Systems

152

that the period is close to w and the closed orbit represented by this periodic solution lies in an arbitrary neighborhood of the closed orbit (9.152) of the unperturbed system (9.3). Moreover, in this case, the perturbed system (9.4) has no subperiodic solution if det [@"(PO, al')

- E] # 0

(9.153)

for any positive integer p. If Eq. (9.140) has no solution even though (9.123) holdr for any a' and for any positive integer p , then the perturbed system (9.4) cannot have any periodic solutionfor small E >0.

I I

93. STABILITY OF THE PERIODIC SOLUTION OF THE PERTURBED SYSTEM

In the present section, we shall investigate the stability of the periodic solution of (9.4) whose existence has been affirmed by Theorem 9.2. In our investigation, the following lemma on the eigenvalues of matrices will be used.

Lemma 9.6. Ifdet B(0) # 0, and I E square matrix of the form

I is suficiently small, then the eigenvalues of the (9.154)

which is continuous with respect to E, are of the form E

where

[A

+

0(1)]

or

p

+ o(1)

(E

-,O),

(9.155)

A and p are, respectively, the eigenvalues of the matrices A,(O) - A(0) B-'(O)B, (0)

and

B(0).

PROOF. Let p be an arbitrary eigenvalue of the matrix (9.154). Then (9.156)

where El and E2 are the unit matrices. Since the eigenvalues are continuously dependent on the elements of the matrix, letting E + 0 in (9.156), we have

9.3. Stability of the Periodic Solution of the Perturbed System

153

(9.157) where po = lim p. Equality (9.157) implies that po

=

0, or po is an eigenvalue

&+O

of the matrix B(0). In what follows, we shall prove that the eigenvalues of the matrix (9.154) suchthat p + 0 as E + 0 are of the form &[A + o(l)]. Let p be an arbitrary eigenvalue of the matrix (9.154) such that lim p = po = 0. Then, since p is an &+O

eigenvalue of (9.154), there is a nonzero vector (f) such that EAI(E)x EB1(E)X

+ A ( E )=~ px, + B(E)y = p y .

(9.158)

The second of the above equations is, however, written as follows: [B(E) - pEz]y =

(9.159)

-EB1(E)X.

Then, since det [B(E)- pE,] = det B(0) # 0 for sufficiently small

I E I by assumption, from (9.159), we have y = -E[B(E) - pEz]-'B1(E)X.

(9.160)

Substituting this into the first of (9.158), we have E(Al(E)

- A(E)[B(E) - p~z]-'B,(E))x

= px,

(9.161)

which, by substitution (9.162)

P = &Pi.

is rewritten as follows: {A&) - A(E)CB(E) - PEzI-'B,(E))x

= PlX

for E # 0. Here, however, x # 0, because, otherwise, (9.160) implies (): which is a contradiction. Then, putting x

/

1x1 =

(9.163) =

0,

(9.164)

x1

from (9.163), we have Xl*(Al(E)

- A(E)CB(E)- pEzI-'B1(E))X1

= P1

(9.165)

9. Perturbation of Partially Oscillatory Systems

154

1

11

1

1

because x1 = 1 by (9.164). Since x1 = 1, equality (9.165) implies p1 is bounded for sufficiently small E Then, since

I I.

) plEl} det {A1(&) - A(E)[B(E)- p E , ] - l B 1 ( ~ -

from (9.163), letting satisfy

E

-, 0, we

=

0

see that any limiting value p 1 of p1 must

det([A,(O) - A(O)B-’(O)B,(O)] - p l E , )

=

0.

This shows that PI must be an eigenvalue of the matrix A,(O) A(O)B-’(O)B,(O), which proves the eigenvalue p of (9.154) such that p -+ Oas E + 0 must be of the form &[I. o(l)], thus completing the proof. Q.E.D.

+

Now let us investigate the stability of the periodic solution (9.146) of Eq. (9.90). By Corollary 2 of Theorem 6.1, the solution (9.146) is asymptotically stable if the characteristic exponents of the first variation equation of (9.90) with respect to the solution (9.146) are all negative in their real parts. However, since P = x(t, a ; E ) is a solution of (9.90) satisfying (9.99, by (2.10) and (2.12), it follows that

(W) X,(k a ; E )

= R,(t, X(t, 0 ;E ) ,

E)

X,(t, a ; E )

(9.166)

and X,(O, a ; E )

=

(9.167)

E,

where X,(t, a ; E ) is a Jacobian matrix of ~ ( ta,; E ) with respect to a, and R, ( t , i j , E ) is a Jacobian matrix of R(t, P, E ) with respect to p . Since (9.168)

d y l d t = R,(h X ( 4 a ; E)’ E)y

is a first variation equation of (9.90) with respect to the solution i j = ~ ( ta ,; E ) , equalities (9.166) and (9.167) show that x,(t, a ; E ) is a fundamental matrix of (9.168). Then, since x(t, al’(&),a;)(&);E ) is periodic in t of period w, by Theorem 3.2, there is a constant matrix K such that X,(t

+ 0,al’(E),a;)(&);E ) = x.(t,

However, if we put t

=

Ul’(E),

a;)(&);E)K.

0 in the above equality, then, by (9.1671, we have K

= X,(W

a l ’ ( ~ )a;(&); : E).

Thus, by definition, we see that the eigenvalues of the matrix x,(w,

al’(E),

9.3. Stability of the Periodic Solution of the Perturbed System

155

a:(&); E ) are the multipliers of solutions of (9.168). Then, since the characteristic exponents of (9.168) are related as (3.26) with the multipliers of solutions of the same equation, by Corollary 2 of Theorem 6.1, the periodic solution (9.146) is asymptotically stable if the eigenvalues of the matrix x,(w, a,’(&), a?(&); E ) are all less then one in their absolute values. In order to calculate the eigenvalues of x,(w, a,’(&), a:(&); E ) , let us write the matrix x,(t, a ; E ) as follows:

(xl!(t,

&(t, a‘, a”;E )

x,(t, a ; E )

=

a’, a ” ;

E)

),

x:+,

a’, a”;8 ) x:’..(t, a‘, a ” ; E )

(9.169)

where Xi.(t, a’, a ” ; E ) is a Jacobian matrix of ~ ’ ( ta’,, a”; E ) with respect to a’, and &(t, a’, a ” ; E ) , x;(t, a’, a ” ; E), x:!,(t, a’, a”; E ) are the similar Jacobian matrices. For the periodic solution (9.146), al’(&)is, however, a solution of (9.133) with p = I ; therefore a,’(&) is continuously differentiable with respect to E . Then, since a;’(&) = f,[a,’(~),E ] , a?(&) is also continuously differentiable with respect to E , and

where (dfl/aa‘)(a’,E ) is a Jacobian matrix of fl(a’, However, by (9.125),

E)

with respect to a’.

(afllaa’)(a’, 0) = 0, and, by (9.128) and (9.129), (afl/a&)(a,’(O), 0) =

fl’@l’, 0)

= - [ ~ ’ ( w a,’) ,

- El-

r”(w, a,’).

Hence, from (9.170), we have

On the other hand, for the periodic solution (9.146), by (9.96),

x:.(w, a’, 0; 0)

=

E

and

Therefore, by (9.97) and (9.98), we have

x::(w, a’, 0; 0)

=

0.

9. Perturbation of Partially Oscillatory Systems

156

ax, [w, a,+),

a,”(+

E]

8% = d,,

+

E

{i-

+--asaa,

(0,a,‘,

e=O

0;0)

+ o(1)

+dr, (w, a , ’ ) + o(1)

(a, p = 2,3,

( E -+ 0)

8%

..., m +

(9.172)

.

where a;:(&) (K = m + 2, . ., n) are the components of the vector a:(€) and [ (K = m + 2, . ., n) denotes the rcth component of the vector inside the brackets. Let us write (9.172) in matrix form as follows:

- - -IK

.

{

+E -

x:.[o, alp(&), a:(&); E ] = E

E

~

(0,al‘)

[ ~ ( wal’) ,

- ~ ] - ‘ r ’ ’ ( wal’) ,

0).

(9.173)

}

+ a (0’a1’) + o(l> ar’

In a similar way, we have xi:[w, a,’(&), a:(&); E ] = E

{ - z: ~

ar“ +7 aa

(E

-P

ul’) [ ~ ( m ul’) , - ~ ] - ‘ r ’ ’ ( oal’) ,

(0,

(0’ a , ‘ )

+ o(l)}

(E

+

(9.174)

0).

The following equalities are evident from (9.97) and (9.98):

El &[o,a,’(&),a’;(&);E l

x:..[o,a,’(&),a’;(&);

+ o(1) = @”(w, q ’ )+ o(1) = @’(a, a,’)

o),

(9.175)

(E + 0).

(9.176)

(E

-P

Thus, if we put X 0 ( W a,’(&),a‘+);

by (9.169) and (9.173)

-

E) =

(9.176), we have

E

+ y,

(9.177)

9.3. Stability of the Periodic Solution of the Perturbed System

157

where

am‘ w _ - a-(@,al’), ad

aa’

dr’

ar‘

Q’’ = @”(co,al’),

- = - (w, al‘),

aa’

da’

awl -aa’

-

~

aa’

r” = r ” ( u , a l ’ ) ,

@’ = W(co,al’),

ar” ar” - = - (w, al’). aa‘ aa’

(w, a l ’ ) ,

Now let us suppose that the eigenvalues of the matrix @”(o,a,’) are all less than one in their absolute values. Then evidently det [W’(w, a,’)

- El

(9.179)

# 0,

because, otherwise, some eigenvhlue of Q”(c0, al’) must be one, which contradicts the assumption. Since the matrix Y is of the form (9.154), let us apply Lemma 9.6 to the matrix Y. Owing to (9.179), Lemma9.6canindeed beapplied to the matrix Y and hence, for sufficiently smal.1 E we see that the eigenvalues of Y are approximately equal to the eigenvalues of the matrix

I I,

@‘’(a, a,’) - E

(9.180)

and those of the matrix

e

{-

(W- E)-*r” + art aa

a

= E - [-

ad

= &J(a,‘)

a’(@’’ - E)-’r’’

+ r’]

CseeTheorem 1.61 [see (9.14011,

(9.18 1)

158

9. Perturbation of Partially Oscillatory Systems

where J(a’) is a Jacobian matrix of g,(a’, 0) with respect to a’. Then, by (9.177), the eigenvalues of the matrix ~ , , ( w a, , ’ ( E ) , a:(&);E ) are approximately equal to the eigenvalues of cD”(o,a,’) and those of the matrix E + EJ(a,’). The eigenvalues of E EJ(a,’) are, however, of the form 1 + &(a + ip), where a + ip is an eigenvalue of the matrix .I(.,’). Since

+

I 1+

E(CL

+ ip) 1 = [I + 2 ~ a+ &’(a2 + p’)]”’ = 1+ + -+ o), Ec!

0(E2)

(E

the eigenvalues of the matrix E + E J(a,’) are all less than one in their absolute values for sufficiently small E if the eigenvalues of eJ(a,’) are all negative in their real parts. Thus, by the assumption on the eigenvalues of @“(w, a,‘), we see that the eigenvalues of the matrix xa(w, all(&),a?(&);E ) are all less than one in their absolute values for sufficiently small E if the eigenvalues of &J(a,’) are all negative in their real parts. This proves the periodic solution (9.146) of (9.90) is asymptotically stable under the specified conditions. As is easily seen from (9.76) and (9.67), the asymptotic stability of (9.146) implies, however, the asymptotic orbital stability of the periodic solution of (9.4) corresponding to the periodic solution (9.146) of (9.90). Thus we have the following theorem.

1 I

I I

Theorem 9.3. Under the same conditions on (9.3) and (9.4) as in Theorem 9.2, we swppose that equation (9.140) has a solution a‘ = a,’ such that the eigenvalues of the matrix W ‘ ( q a,’) are all less than one in their absolute values and the eigenvalues of the matrix EJ(a,’) are all negative in their realparts, where J(a‘) is a Jacobian matrix of g,(a’, 0). Then, .for sufficiently small E I > 0, the perturbed system (9.4) has a unique periodic solution such that the period is close to o and the closed orbit represented by this periodic solution lies in an arbitrary neighborhood of the closed orbit (9.152) of the unperturbed system (9.3). Moreover, if E 1 is sujJciently small, this periodic solution of the perturbed system (9.4) is asymptotically orbitally stable, and the perturbed system (9.4) has no subperiodic solution.

I

I

In this theorem, the first conclusion on the unique existence of a periodic solution of the perturbed system is included in Theorem 9.2, since the assumpa,’) implies (9.179), as is shown already, and the assumption on tion on @”(o, d ( a , ‘ ) implies det J(al’) = A(a,’) # 0,

9.4. An Example

159

as is readily seen. The last conclusion on the non-existence of a subperiodic solution is also included in Theorem 9.2, since the assumption on W’(o,al’) also implies (9.153) for any positive integer p [note that the eigenvalues of @”(pa,a,’) = @“’(o, al’)are thepth powers of the eigenvalues of @“(o,a,‘)]. The second conclusion on the stability of the periodic solution of the perturbed system is a new conclusion, and this is a result of the present section. 9.4. AN EXAMPLE

We shall illustrate our results by the perturbed system of the partially oscillatory system (9.1). As is seen from (9.2), system (9.1) admits of a one-parameter family of periodic solutions x, = a cos

2n

~

1

2n

x2 = a sin-

+ az t ,

x3 = 0 ( a : parameter).

t,

l + a (9.182) Let a, be an arbitrary positive value of a, and C, be a closed orbit represented bY 2R . 271 X I = a, cos t, x2 = a, sin t, x3 = 0. (9.183) 1 a, 1 a,

+

~

+

~

The closed orbit C, evidently has a moving orthornormal system consisting of the tangent vector

2n

t,

(-sin

cos-

2n

1

+ a,

2 4

0)

and the two normal vectors

2n

-sin

t,

2n

1). With respect to this moving orthonormal system, any orbit of (9.1) is expressed as 271 x i = X,(Z) = (a, - p2)cos1 + ao2t , (

~2

x3

0

= x2(7) = (ao = x&)

0

9

= p3.

- p2) sin

9

2x ~

1

+ ao2

t9

160

9. Perturbation of Partially Oscillatory Systems

Hence, by (5.23) and (5.24), we have (9.184)

d pz - 0, dt

(9.185)

d p , = - 1 + (a0 - PZlZ + P 3 P3 * dt 1 aoz

+

Equation (9.185) is of the form (9.83). Therefore, in the present case, transformations of the forms (9.67) and (9.76) are not necessary. According to (9.92), let

dp, = dt

PZY P 3 ;

EH2(t7

&)¶

(9.186)

be the perturbed system of (9.185). For this system, by the notations of 9.2, evidently K’(t, a’) = 0, K”(t, a‘) = -

1

+ (ao - a’)2 1

+ ao2

’

H’(t, a’) = H2(t, a’, 0 ; O), H”(t, a’) = H3(t,a’, 0 ; 0).

Consequently, by Lemma 9.4, we have 1

+ ao2

a’(?,a’) = 0,

(9.187)

[?+

r y t , a’) = fexp 0

r’(t, a’) = J>z(s,

1

- a’)2 ( S - t ) ] H,(s, a‘, 0 ; 0) ds, + ao2

a’, 0 ; 0) ds.

In the present case, as is readily seen from (9.182) and (9.183) 0

=1

+ aoZ.

9.4. An Example

161

Therefore, from the first of (9.187), we have

I (D”(o,a‘) I = exp { - [I + (ao - a‘)’]}

< 1.

(9.188)

Now, substituting the second and the last of (9.187) into (9.140), we have

jo

1 +no2

g l ( d 0) =

H 2 ( t , a’, 0; 0) dt.

Thus, if the equation 1 +a02

jo

H 2 ( t , a’, 0; 0) d t = 0

I I

has a simple root a‘ = q’,then, by Theorem 9.2, for sufficiently small E > 0, the perturbed system has a unique periodic solution such that the closed orbit represented by such a periodic solution is close to the following closed orbit of the unperturbed system: X I = (a0

- ul’) cos

2n

+

~

1

2

t = (ao - a,’)cos

a0

2R

x2 = (ao - a,’) sin ___ t = (ao

1

x3

=

+ a0

- ul’) sin

271

1

+ (a0 - a,’)’

7,

2n

1

+ (ao -

*,

0;

and, by Theorem 9.3, such a periodic solution of the perturbed system is asymptotically orbitally stable for sufficiently small E if

I I

The above assumptions on H2(t, p z , p 3 ; 0) are the same as the conditions on X(t, x, E ) in Theorem 8.2 if p3 is put at zero in H,(t, p z , p 3 ; E). On the other hand, as is shown below, inequality (9.188) is the condition that the integral manifold x 3 = 0 of the unperturbed system (9.1) is asymptotically stable; in other words, any orbit of (9.1) not lying on x3 = 0 approaches this indefinitely as t -+ co. The latter is seen in the following way. In fact, the integral manifold p3 = 0 of (9.185) corresponds to the integral manifold x3 = 0 of (9.1). However, for any solution of (9.185), we have P2

= a’,

(9.189)

162

9. Perturbation of Partially Oscillatory Systems

where a‘ is an arbitrary constant. For the second equation, the first variation equation with respect to p3 = 0 is then -d y= -

dt

1

+ (ao - a’)’ 1

+ ao2

Y.

Hence, if we regard the second of (9.189) as the periodic system of period o = 1 + at, then, by Corollary 2 of Theorem 6.1, inequality (9.188) implies that the trivial solution p3 = 0 of the second of (9.189) is asymptotically stable. This evidently implies that x3 + 0 as t+ 00 for any orbit of (9.1) not lying on x3 = 0, which proves the desired fact.

10. Analysis of Two-Dimensional Autonomous Systems

In the present chapter, the general theories of the preceding chapters are applied to the two-dimensional system, and formulas convenient for practical application will be derived. Next, on the basis of these formulas, the analytic two-dimensional autonomous system will be analyzed in more detail. Let us write the given two-dimensional autonomous system as dx dt

-=

dY dt

- = y(x, Y )

x ( x , Y),

(10.1)

and its perturbed system as dx - = q x , Y ) + &H(X,y, E), dt

2 = Y ( x , y ) + E K ( x ,y, dt

(10.2) 8).

For (10.1) and (10.2), we assume (1) X ( x , y ) and Y ( x , y ) are continuously differentiable with respect to x and y in the domain D of xy-plane; (2) H(x,Y,E) and K ( x , y, E ) are continuously differentiable with respect to x, y, and E for (x, y ) E D and < 6 (6 > 0); and (3) the system (10.1) has a periodic solution

IE 1

x

=

444,

Y

=

w

(10.3)

of primitive period o (> 0) such that the closed orbit C represented by (10.3) is contained in the domain D. The discussion of the present chapter is based mostly on the author’s paper [41.

10.1. FUNDAMENTAL FORMULAS The moving orthonormal system along C consists of the unit tangent vector 163

10. Two-Dimensional Autonomous Systems

164

X

(10.4)

and the unit normal vector (10.5)

where

x = XCq(0, $(t)l,

y

=

YCq(t),$(Ol.

For convenience, adopting the upper ones of the double signs in (10.5)’ we suppose that the unit normal vector constituting the moving orthonormal system is Y

(5

+ Y 2)1 1 2 ’

= - (x2

q =

(x2f

Y

).

y

(10.6)

By (7.3)’ we have then

for any point ( x = x (T), y = y(z)) of an arbitrary orbit of (10.2) lying near C.

For (10.6), we have, however, dr - X(SY - X q ) dr (Xz+ Y 2 ) 3 / 2’

(10.8)

_ -- Y ( A Y - X Y ) dt (Xz + Y 2 y ’ dq

where d

= - X [ q ( t ) , $(t)] = x,x dt

+ XYY,

d Y = - Y [ q ( t ) ,$(t)] = Y,X dt

+ YyY.

x

Here X, means Xx[q(t),$(t)] and X,, Y,, Yyhave similar meanings. BY(10.8)’

165

10.1 Fundamental Formulas

corresponding to (7.4) and (7.5), we have the following equalities for system (10.2) : Irr7

.

r.7\

( Y 2 X ”- X 2 Y , )

.

--

XX‘

dt

+ YY’

~

+ X Y ( X , - Y,) ~-

and dP dr X Y ‘ - Y X ’ - = R(t, p, 6 ) = - ’ dt dt ( X z + Y 2 ) 1 1 2 ’

where

X’= X(cp + P5, Y’ = Y(cp + P L Here, by (7.8), R(t, p,

E)

R(t, p, E ) = [ ( X .

*+ *+

P?)

P?)

(10.10)

* *

+ Wcp + Pt, + PI, 4, + 4 c p + P5, + P?, 8).

is of the form

+ r,) - ddt log (x2+ y2)l12]p

where Hcl = H[cp(t), *(+ 03,

KO = K[P(t), * ( t ) , 01. be the solution of (10.10) such that p(0, c, 8 ) = c. Then,

Let p = p(t, c, E ) by (7,11), p(t, c, E ) is of the form

(10.12)

(10.13)

166

10. Two-Dimensional Autonomous Systems

and the functions X , Y, H,, and KO under the integral sign are, respectively,

x = X[V(S)?

y

4+)19

H , = H[V(S), $(S),

= Y[V(S)I

$(S)l?

KO = K[V(S), $(S), 01.

01,

10.2. STABILITY OF A PERIODIC SOLUTION OF THE UNPERTURBED SYSTEM As is readily seen from (lO.ll), the normal variation equation of the unperturbed system (10.1) is

9 = [(x. + Y,) dt

-

d log (XZ dt

-

+ Y2)1’2

1

p;

(10.14)

consequently, as is seen from (10.12), the multipliers of solutions of (10.14) is eh(m) . This implies 1

h = -0 h(

4

(10.15)

is a characteristic exponent of (10.14). Then Theorem 6.8 implies that the periodic solution (10.3) of the unperturbed system (10.1) is asymptotically orbitally stable if h < 0. Let us consider the case where h > 0. In this case, since p(w, c, o)/c = eoh

+ o(1)

(c + 0)

by (10.12), we have p(w, c, O)/C 2 K > 1

for any c such that 0 (10.16) implies

-= I c I 5 c,

(10.16)

if c, is sufficiently small. The inequality

p(w, c, 0) 2

KC

for any c > 0,

(10.17)

6

KC

for any c < 0.

(10.18)

and p ( o , c, 0)

For c > 0, from (10.17), we have p ( 2 q c, 0)

2 ICp(w, c, 0) 2 lczc

if p ( o , c, 0) S c,. Such a process can be continued further, and p ( n o , c, 0)

hKnC

(n = 1 , 2 , . . .)

10.4. Perturbation of a Fully Oscillatory System

167

hold so long as p(no, c, 0) s c o

(n = 1,2,

. . .).

(10.19)

Then, since K > 1, we see that, for any c > 0, the inequality (10.19) cannot hold for all positive integers n. Likewise, starting from (10.18), we see that, for any c < 0, the inequality c, 0)

2 - co

cannot hold for all positive integers n. These results imply that the periodic solution (10.3) of the unperturbed system is orbitally unstable. Summarizing the above results, we see thus that the periodic solution (10.3) of the unperturbed system (10.1) is orbitally asymptotically stable or unstable according as h < 0 or > 0. 10.3. PERTURBATION The quantity h of (10.15) is a characteristic exponent of the normal variation equation of (10.1). Therefore Theorems 7.1 and 7.3 imply that, i f h # 0 and E is sufficiently small, the perturbed system (10.2) has a unique periodic solution such that the closed orbit C represented by such a periodic solution lies i n the neighborhood of C and that such a periodic solution of the perturbed system is asymptotically orbitally stable if h < 0. The period of the closed orbit C' is necessarily close to o,since the two-dimensional autonomous system has no subperiodic solution, as is shown in Remark of 1.2. When h > 0, the characteristic exponent of the normal variation equation of the perturbed system with respect to C' is also positive, as is seen from the proof of Theorem 7.3. Hence, by the result of the preceding section, this implies that the periodic solution of the perturbed system corresponding to C is orbitally unstable, together with the periodic solution (10.3) of the unperturbed system. The orbital stability of the periodic solution of the perturbed system is thus the same as that of the periodic solution of the unperturbed system when h # 0.

I I

10.4. PERTURBATION OF A FULLY OSCILLATORY SYSTEM The present section is concerned with the case where the given unperturbed system (10.1) is a fully oscillatory system. In accordance with Chapter 8, for systems (10.1) and (10.2), we assume (1) X(x, y ) and Y(x, y ) are twice continuously differentiable with respect to

168

10. Two-Dimensional Autonomous Systems

x and y in the domain D of the xy-plane; and (2) H(x, y, 8 ) and K(x, y, E ) are twice continuously differentiable with respect to x, y , and E for (x, y ) E D and E <6 (6 > 0). Following the method of Chapter 8, we can then easily establish a scheme to find a periodic solution of the perturbed system (10.2). In the present section, however, in order to establish an explicitformula to find a periodic solution of the perturbed system, we shall proceed in a different way. To begin with, we consider an orthogonal trajectory

I I

l- : x = d a ) , y = $,(a)

of a family of closed orbits of the unperturbed system (lO.l), and we suppose that a is an arc length of r. Then we may suppose that x = tp,(a), y = $,(a) is a solution of the differential equations

YtX. , v\ [XZ(X,y)+ Y”X, y)]l’z’

dx

- = -

“ 1

da

\lU., ”A

xtx. v)

dv

20)

The functions q,(a) and +,(a) are evidently three times continuously differentiable with respect to a. Now, by x = Cp(t; a),

Y = $ ( t ; a),

(10.21)

let us denote the solution of the unperturbed system (10.1) such that q(0; a) = p1(a),

$(O;

4 = $da)-

(10.22)

Then, by Theorem 2.4, &; a), $(ti a) E c,: *

(10.23)

By assumption, the solution (10.21) is periodic with respect to t for every a, and it represents a closed orbit C,in the phase space. By o (a), we shall denote the primitive period of the closed orbit C,. To make a moving orthonormal system along C,,let us put q t ; a) =

- [XZ(t; a)Y+( t ;Ya)2 ( t ;a)]”z ’

(10.24)

169

10.4. Perturbation of a Fully Oscillatory System

where X ( t ; a) = X [ q ( t ; a), $(ti a)],

Y ( t ; a) = YCq(t;a), $(ti

41.

Then clearly the vector ( ( ( t ; a), q(t; a)) is a unit normal vector of Cayand this indeed constitutes a moving orthonormal system along C,, together with a unit tangent vector

With regard to such a moving orthonormal system along C,, any orbit C' of the perturbed system (10.2) lying near C, is expressed as = p(t; a)

+ p ( ( t ; a),

Y = Y ( T ) = $ ( t ; a)

+ pq(t;a);

x =

X(T)

(10.25)

and, for z = z(t) and p = p (t), we have ---= T ( t ,p,

dz dt

E;

a),

(10.26)

9 = R(t, p, dt

E;

a),

(10.27)

where T(t, p, E ; a) and R(t, p, E ; a) are, respectively, the functions obtained from the right-hand sides of (10.9) and (10.10) by replacing (cp(t), $(t)) and ({(t), q(t)) by ( p ( t ; a), +(t; a)) and ( ( ( t ; a), q(t; a)), respectively. From (10.9), (lO.lO), and (10.23), for T(t, p, E ; a) and R(t, p, E ; a), T ( t , P, 8 ; a), R(t, P, 6 ; a) E Clp,e,a

evidently holds. Let p = p(t, c, E ; a) be a solution of (10.27) such that p(0, Then, owing to (10.28), by Theorem 2.4, p ( 4 c, E ; a) E Cl,,,,, *

(10.28) C, E ; a) = c.

(10.29)

However, as is seen from (10.27), a p @ is a solution of the differential equation

170

10. Two-Dimensional Autonomous Systems

satisfying the initial condition

9 (0, c, a&

E;

a) = 0.

(10.31)

Hence we have further (10.32) because

FIG. 6

Since C, crosses r orthogonally (see Fig. 6.), by Theorem 4.8, the neighboring orbit C' of the perturbed system also crosses at a point [pl(a AM), $1(~ + Aa)], and, by (10.25), we have

+

If we regard (10.33) as the equation with respect to Au and t, then this is satisfied by Aa = t = 0 for c = 0 and, for c = Au = t = 0, the Jacobian of the left-hand sides of (10.33) with respect to t and Au is

= -

(Xoz

+ Yo2)1'2# 0

[see (10.20) and (10.27)],

10.4. Perturbation of a Fully Oscillatory System

171

where

xo = x[q(o;a), $(O;

a)],

Yo = YCq(0;a), $@;41.

Hence, by the theorem on implicit functions, Aa and t are uniquely determined so that (10.34) ALY= f(c, E), t = t(c, E ) E C & ,

I I

for sufficiently small c and

1 E 1,

f(0, E)

and = t(0, E ) = 0.

(10.35)

If we differentiate both sides of (10.33) with respect to c after substituting (10.34), then, for c = 0, we have

Hence we see that

(10.36) Owing to the latter of (10.36), the relation Aa = f ( c , E )

can be solved with respect to c so that c = g(Aa, E ) E cia,,

(10.37)

and

Now, for the primitive period o ( a + Aa) of the closed orbit C.+A., namely, the primitive period of the closed orbit of the unperturbed system (10.1) passing through the point ( q l ( a i- ha), G1(a Act)), from (10.26), we have

+

o(a

+ Aa) =

T [ t ,p ( t , c, 0;a), 0; a] d t . !o'(a)

Then, by (10.28) and (10.29), we see that o ( a that o(a

+ Aa) is a function of c such

+ Aa) = F(c) E Ccl.

10. Two-Dimensional Autonomous Systems

172

Then, substituting (10.37) into F(c), we have o(a

+ Aa) = F[g(Aa, O)]

E Cia.

This implies that the primitiveperiod o(a) of the closed orbit C, is a continuously dijerentiable function of a. Now let us suppose the perturbed system (10.2) has a closed orbit C. Then, as is mentioned above, C' crosses r at some point A(pl(a), JI1(a)) (see Fig. 7). Let C, be the closed orbit of the unperturbed system passing

FIG. 7

through A at t = 0; then, with regard to the moving orthonormal system along such C,, which was introduced at the beginning of the present section, we have @(E;

a) = p[o(a), 0,E ; a] = 0.

(10.38)

Hence the problem of seeking a periodic solution of the perturbed system (10.2) is reduced to one of finding a solution a = a(&)of Eq. (10.38) for sufficiently small I E However,

I.

@(O; a) = 0

identically in a. Therefore, by (10.29), @(E;

and, for

E

# 0,

a) =

@(E;

a) can be written in the form

E @ ~ ( E ; a);

(10.39)

Eq. (10.38) becomes equivalent to a) = 0.

(10.40)

10.4. Perturbation of a Fully Oscillatory System

173

Here a) =

=

J1"

a&

(Be; a) dB

lo1:

[w(a), 0,Be; a] do;

(10.41)

therefore, by (10.32), @1@;

a) E

c:,,9

(10.42)

since w(a) E C,', as has already been proved. Then Eq. (10.40) can have a solution a = a(&)for sufficiently small e only when the equation

I

I

(10.43)

has at least one real root; and, conversely, when Eq. (10.43) has a real simple root a = ao, for sufficiently small E Eq. (10.40) has indeed one and only one solution a = a(&) such that a(0) = uo and a(&) E Cel. These results imply that the perturbed system (10.2) has no periodic solution for small e > 0 if Eq. (10.43) has no real solution and that if Eq. (10.43) has a real simple root a = ao, for sufficiently small e > 0, the perturbed system (10.2) has one and only one closed orbit in the neighborhood of the closed orbit C,, of the unperturbed system' corresponding to a = a,,. To investigate the stability of the closed orbit C' of the perturbed system corresponding to the solution a = a(&)of (10.40), let us consider an arbitrary neighboring orbit C" of the perturbed system and let B{ql[a(&) 1.1, $l[a(e) r ] } be the point where such a neighboring orbit C" of the perturbed system intersects with r. By A, let us denote the point { q l [ a ( ~ ) ] , + l [ a ( ~ ) ] } ; then, evidently (see Fig. 7),

I 1,

I I

I

I

+

+

h

(10.44)

A B = r.

Let Q and R be, respectively, the points where C" intersects first with r and then with the tangent of r at B after leaving point B. Put n

(10.45)

BQ = Ar,

and let Ca(e)+r be the closed orbit of the unperturbed system passing through point B. Since BR is a normal of C,(e,+r,we have = p[w(a(&)

+ r), 0, e; a(&) + r] = @[&;

a(&)

+ r],

(10.46)

174

10. Two-Dimensional Autonomous Systems

and, by the latter of (10.36), we have

.

Ar BR

lim

-

ER-o

=

1.

(10.47)

Then, by (10.44), (10.45), (10.46), and (10.39), we have successively

BR a=-_= A3 r + Ar r

=1+

@[E;

Ar 1+--.= r BR

a(&)

r

+ r] . Ar _

-

BR

GI(&)+ r ] . Ar _ .

=1+&

r

BR

(10.48)

However, a(&)] = 0.

Therefore, by (10.42), (10.49) Hence, if &

8% (0; ao) < 0

~

aa

I I

and E is sufficiently small, then, by (10.48), (10.49), and (10.47), there is a constant K such that

AS

I I. Likewise, if

for any sufficientlysmall r

and

I 6 [ is sufficiently small, then there is a constant

I I.

JC'

such that

for any sufficiently small r As is seen from the discussion in 10.2, these results imply that the periodic solutions of the perturbed system (10.2)

10.4. Perturbation of a Fully Oscillatory System

175

corresponding to the solution a = a(&) of (10.40) is orbitally asymptotically stable or unstable for sufficiently small E according as

I

a@, -(0;a,) au

I

< 0 or > 0.

(10.50)

Let us now derive the explicit expression for @,(O, a). By (10.43), let us first derive the expression for (ap/aE)(t, 0, 0; a). By (10.30) and (10.31), (ap/&)(t, 0, 0 ; a) is a solution of the differential equation aR aR 2 = -(t, 0, 0 ; a) y + -( t , 0, 0 ; a) dt ap a&

(10.51)

satisfying the initial condition

2 a& (o,o, 0; a) = 0.

(10.52)

However, from (lO.ll), aP

d ( t , 0,o; a) = (x, + Y,) - -log dt

(x2+ y z)1 / 2,

X K , - YH, -( t , 0, 0 ; a) = (XZ + Y2)1/2' a& aR

(10.53)

where

x = X ( t ; a), Y = Y ( t ; a); x , = x,(t;a) = xx[q(t;a),$ ( t ; a)], Y, = q(t; ). = Y,[q(t; a), $ ( t ; a ) ] ; H, = ~ , ( t a) ; = ~ [ q ( ta), ; $ ( t ; a), 01, KO = K , ( t ; a) = K [ v ( t ; a), $(t; a), 01. Hence, by (10.12), we have

9 a& ( t , 0,o; a) where

h(t; a).= !:[X,(s; a)

+ Y,(s; u)] ds.

(10.55)

10. Two-Dimensional Autonomous Systems

176

For the fully oscillatory system (10. l), however, h[o(a);a] = 0,

(10.56)

because, otherwise, by 10.2, any closed orbit of the fully oscillatory system becomes orbitally asymptotically stable or unstable, and this yields a contradiction. Thus, by (10.43) and (10.54), we have 1

Q1(0;a) = [x’(o;a) + ~ ’ ( 0a)]’’’ ; I(4

(10.57)

where

As is easily seen, Eq. (10.43) is then equivalent to the equation (10.59)

I(a) = 0,

and condition (10.50) for the root a = a. of (10.43) becomes equivalent to the condition &I’(%)< 0 or

>0

(‘ = d/da)

(10.60)

for the root a = a. of (10.59). Thus, summarizing the results obtained, we have the following theorem.

Theorem 10.1. Suppose the unperturbed system (10.1) is a fully oscillatory system and, in systems (10.1) and (104,

x(x,Y ) , Y(XY Y ) E c,,[D],

q x , Y , &),

q x , Y , 8)

E

cx’,,,e for

(x, Y ) E D

and

I & I -c 6 (8 > O),

where D is a domain in the xy-plane. Let : x = Cpl(4, Y = $1(a)

be an orthogonal trajectory of a family of closed orbits of the unperturbed system (lO.l), and suppose that a is an arc length of r. If we denote by x = Cp(t;a),

Y =

the solution of (10.1) such that Cp(0; a) =

Cpl(E),

$(O; a) = Jll(4

177

10.4. Perturbation of a Fully Oscillatory System

and define the quantity I(a) by (10.58), then, for suficiently small

IE I

> 0,

(1) the perturbed system (10.2) has no periodic solution when Eq. (10.59) has no real root; (2) when Eq. (10.59) has a real simple root a = ao, theperturbed system (10.2) has one and only one periodic solution close to x = q ( t ; ao),

Y = $(ti a,,),

and such a periodic solution of the perturbed system is orbitally asymptotically stable or unstable according as d‘(ao)

-= 0

or

> 0.

In illustrating Theorem 10.1, consider the system

(10.61)

where 1 is an arbitrary non-negative constant. The unperturbed system of (10.61) is evidently

dx dt

-=

-

1 1 + L(x’

+ y’)

y,

(10.62)

which is a fully oscillatory system such that all closed orbits are concentric circles with the center at the origin. We may then take the negative x-axis as the orthogonal trajectory of a family of closed orbits of (10.62) (see (10.20)). The equation of r is then evidently x = ql(a) = a, y = $l(a) = 0 (a < 0) (see Fig. 8). Hence, for the solution x = q ( t ; a), y = $ ( t ; a ) of (10.62) such that q(0; a) = a, $(O; a) = 0, we have

t

x = q ( t ; a) = a cos ___

1

+ La”

y = $(t; a) = a sin

t

1 + La’‘

~

Then the primitive period o(a) of such a periodic solution is evidently o ( a ) = 2n(1

+ La’).

Now, for system (10.62), K(X,

Y ) + Y,(X, Y ) = 0

178

10. Two-Dimensional Autonomous Systems

Y t

-%

FIG. 8

identically in x and y ; and hence, by (lo.%), h(t; u ) = 0.

Thus, by (10.58), we have

+ K sin

)dr,

1 + la2

~

(10.63)

where H =H K

=

K

UCOS--

t 1 + Au2’

UCOS-

t 1 + Au2’

( (

For simplicity, let us put t

)

+ K sin 1 + l a 2 ~

dt.

(10.64)

Then, by the restriction that u < 0, the equation Z(U) =

0

is equivalent to the equation J(a)

(10.65)

= 0.

Moreover, for a root CI = uo of Eq. (10.65), Z‘(uo) =

-

1

+uoAuo

2 J’(u0)

(’

=

d/du).

179

10.5. Fundamental Formulas for Analytic Systems

Hence, by Theorem 10.1, we see that if Eq. (10.65) has a simple negative root a = ct,, then the perturbed system (10.61) has one and only one periodic solution close to

x

= a, cos

t 1 + Auo2’

y = uo sin

t 1 + Auo2

I I

for suffciently small E > 0, and such a periodic solution is orbitally asymptotically stable or unstable according as cJ’(aO)< 0

or

> 0.

If

rl = 0,

H(x,y ) = 0

and

K ( x , y ) = - f ( x , -y)

in (10.61), then system (10.61) is reduced to system (8.60). In this case, by (10.64),

J(a) = -

cos t , - a sin t) sin t dt.

{02’f(ct

Therefore the above conclusion coincides with the results of 8.4.

10.5. FUNDAMENTAL FORMULAS FOR ANALYTIC SYSTEMS In the present and succeeding sections, we shall discuss in more detail the case where X ( x ,y ) , Y(x,y ) , H(x, y , E), and K(x, y , E ) are all analytic with respect to x,y , and E for (x,y ) E D and E < 6 (6 > 0). In this case, as is readily seen from (10.9) and (lO.lO), the solution p = p(t, c, E ) of (lO.lO), such that p(0, c, E ) = c, is analytic with respect to c and E. Since p(t, 0, 0) = 0 as is seen from (lO.lO), the function p(t, c, E ) is thus expanded in a power series of c and E aS follows:

1 I

p(t, C, E ) = p l ( t ,

C, E )

+ p2(t, C, E ) + .*. + p,(t,

C, E )

+

*.-,

(10.66)

where pm(t, c, E ) (m = 1, 2, . . .) are mth degree homogeneous polynomials of c and E . If we substitute (10.66) into (10.10) and compare the terms of the same degrees with respect to c and E , then, by (10.1 l), we have dP 1 = [(X,+ dt

~

y,)

-

d

- YH, + Y2)1’2]p1 + E XK, (XZ y2)W

log (x2

+

(10.67)

(m = 2, 3, ...), (10.68)

180

10. Two-Dimensional Autonomous Systems

. .) are Polynomials in p17 ..., P m - 1

where R,(t, pl, ..., P,-~, E ) (m = 2, 3, and E. Since p,(o,

pz(0, C,

C, E ) = C,

E) =

**.

= p,(o,

=

C, E )

-**

=o

by p(0, c, E ) = c, Eq. (10.67) and (10.68) are easily integrated as follows:

(10.69)

( m = 2, 3, ...).

Here the notations of 10.1 are used, and R,,,[s, pl(s, c, E), (m = 2, 3, ...) are abbreviated as R,(s, p17 ..., P m - 1 7 8).

(10.70)

..., p m - l ( S ,

C, E), E]

10.6. STABILITY OF A PERIODIC SOLUTION OF THE ANALYTIC UNPERTURBED SYSTEM For any point ( x = x(z), y = y(z)) of an arbitrary orbit of the unperturbed system (10.1) lying near C, we have (10.7) and, for p = p(t), we have dP/dt

=

R(4 P),

(10.71)

where (10.72)

R(4 P) = R(t, p7 0).

Let p = p ( t , c) be the solution of (10.71) such that p(0, c ) = c; then, from (10.72),

p ( t , c ) = p ( t , CY 0).

Consequently, from (10.66), we have p(t, c) = C P l ( t )

+ c2pz(t)+ + crnpm(t)+

**a,

(10.73)

where p,(t) = p,(t, c, O)/cm

(m = 2,3,

...).

(10.74)

10.6. Stability of a Periodic Solution of the Analytic Unperturbed System 181

By (10.69) and (10.70), it is then clear that

First let us consider the following case.

Case I, where h ( o ) # 0. In this case, from (10.76), we readily see that if c is sufficiently small,

I I

0 < )*

IK < 1 for some constant

K

2 - K’> 1 for some constant

K’

when h ( a ) < 0

C

and C

when h ( o ) > 0.

Hence, by the results of 10.2, we see that the periodic solution (10.3) of the unperturbed system (10.1)is orbitally asymptotically stable or unstable according LIS h ( o ) < 0 or > 0. By (10.15), this result coincides with the conclusion of 10.2.

Next let us consider the following case.

CaseII, where h ( o ) = 0 and p 2 ( o ) = In this case, equality (10.76) becomes

.*.

= pm-l(o)

= 0, p m ( 0 ) f 0. (10.77)

I I

Consequently, if c is sufficiently small,

O<- do,c) < 1, C

(10.78)

182

10. Two-Dimensional Autonomous Systems

whenever ~ ' " - ~ p , , , ( o<) 0, and P(W,

c)/c

> 1,

(10.79)

whenever c m - l p m ( o ) > 0. When the inequality (10.78) holds for any sufficiently small c > 0, we can prove that the periodic solution (10.3) is orbitally stable. In fact, let E be an arbitrary positive number. Then, since p ( t , 0) = 0, by Theorem 2.3, there is a positive number 6, such that

I I

(10.80) I p(tt c ) I < 8 on the interval [0, 03 whenever I c I < 6 , . For c such that I c I < 6,, however,

by (10.78),

I P(0, c) I 5 I c I (The equality occurs when c

=

(10.81)

< 6,.

0.) Then, since

P[t, P ( 0 ,

c)1 = P(t + 0,c),

from (10.81) and (10.80), we have

144 c) I < E on [o,201 and, from (10.78), we have

I P(2W c) I 5 I P(W 4 I < 6,. This process can be continued indefinitely, and thus we have

144 c) I < E

I I

for any t 2 0 whenever c < 6,. This proves that the periodic solution (10.3) is orbitally stable. However, we can prove further that the periodic solution (10.3) is orbitally asymptotically stable when inequality (10.78) holds. In fact, in this case, put (c # 0)

c, = p ( w , c )

and

c , + ~ = p ( o , c,)

(n = 4 2 , ...);

then we have either c

> c, > c2 >

>0

or

c

<

c1

< c2 <

1..

< 0.

In either case, lim c, = co exists and, from the continuity of p ( o , c), we n+m

have co = P(W,

CO).

10.6. Stability of a Periodic Solution of the Analytic Unperturbed System This is a contradiction, however, unless c,

=

183

0, because

holds by (10.78) if c, # 0. Thus we have lim c, = 0. f l - t 03

By Theorem 2.3, this implies lim p(t, cn) = 0

(10.82)

n'm

on the interval [0,

01. However, since

formula (10.82) implies lim p(t

+ no,c) = 0

n- m

for any t

E

[0, 01.This evidently implies

lirn p(t, c)

=

0,

i-m

which shows that the periodic solution (10.3) of system (10.1) is orbitally asymptotically stable. When, instead of (10.78), inequality (10.79) holds for any sufficiently small c > 0, we can prove that the periodic solution (10.3) is orbitally unstable. In fact, inequality (10.79) implies

I I

C

> 1,

that is,

0c

P(-W,

P( - WY c)

4 < 1.

C

Therefore, if we put c - ~= p ( - o , c)

and

c - ( , + ~ ) = p(-w, c-,)

(n = 1,2, ...),

then, by the preceding result, we have either c

> c - ~> c - > ~ . .. +

+0

or

c

c

c-

c c - ~< ... + -0.

Let c, be an arbitrary nonzero number close to zero, and suppose that c-,

2 c, > c-(,+

0

or

c-,

5

c,

c

c-(,+~)

< 0.

10. Two-Dimensional Autonomous Systems

184

Since p(w, c) = c

+ c"pm(w) +

**.

I I, it

is monotonously increasing with respect to c for sufficiently small c then follows that c-.+~

2 p(o,co) > c-, > 0 or

c-,+~S

p ( o , co) < c-, < 0. (10.83)

This process can be continued further, however, and finally the inequality p(o, c) 2 p[(n + l)o,co] > c > 0 or p(w, c) 5 p [ ( n + l)w, co] < c < 0 is obtained. This means 1 p(t, co) surpasses an arbitrary ] c for whatever small co if t is taken sufficiently large. This shows the periodic solution (10.3) is orbitally unstable. If the results obtained above on the stability of the periodic solution (10.3) of the unperturbed System (10.1) are classified according to the signs of c and p,(o), then Table I is obtained.

I

I

I I

TABLE I

m

Sign of p,(o)

-

Orbital stability of the periodic solution asymptotically stable absolute stability

odd

+ -

unstable asymptotically stable for c>O unstable for c
even

+

unstable for 00 asymptotically stable for c
Lastly let us consider the following case. Case III, where h(o) = 0 and pz(w) = p 3 ( o ) =

equality (10.76) turns to p(o,c) = c

for any c.

-*-

= 0. In this case,

10.7. Perturbation of Analytic Systems

185

This means that any orbit of the unperturbed system (10.1) is always closed. Therefore, in this case, the given unperturbed system (10.1) is a fully oscillatory system, and the periodic solution (10.3) of this system is orbitally stable but not asymptotically orbitally stable.

10.7. PERTURBATION OF ANALYTIC SYSTEMS For the solution p

= p(t, c, E )

qc,

of (10.10) such that p(0, c,

E) =

p ( 0 , c,

E)

E)

= c, let us put

(10.84)

- c.

Then, by (10.66) and (10.69), we have @(c,

E)

= (eh(o) - I) c

+ (xo2+E Y

I 0 y

(10.85) where

I

= ~ome-h's)(XKo -

YHo)ds.

(10.86)

As is seen from (10.84), the perturbed system (10.2) has a periodic solution if and only if the equation q c , E) = 0

has a solution for sufficiently small

(10.87)

I I. E

First let us consider Case I. Case I, where h ( o ) # 0. In this case, Eq. (10.87) has evidently a unique solution c = c(E), which is of the form

where [&I2denotes the sum of the terms of the second and higher orders with respect to E . By (10.7), this implies the perturbed system (10.2) has a unique periodic solution corresponding to the closed orbit x =

C' :

X(Z)

=

&) -

I 2 1/2 P [ t , C(E>,EI,

(XZ+ Y )

(10.89)

186

10. Two-Dimensional Autonomous Systems

As is seen from the discussion in 10.6, the stability of the above periodic solution of the perturbed system (10.2) is decided by examining whether

O<

P [ W , C(E)

+ E l - 44 < 1 r9

for sufficiently small

P[W

or

r

C(E)

+ r, 4- C ( E )

>

r

I I (see Fig. 9). The quantity Y

pro,

+ r,

C(E)

E]

-

C(E)

r

p[d, C(E)fI,EI-C~E) FIG. 9

is, however, estimated by means of (10.84) as follows : P [ W , C(E)

+ r, E l - C(E) r

=1+

@[C(&)

+ r, E l r

am

+ - [c(E) + Or, E ] ac aQ = 1 + - (0,o). =

1

(0

< 8 < 1) (10.90)

dC

Since

am -(0, 0) = eh(")- 1 ac

b y (10.85), we then have

do,dE)+ r,

- c(E) .-- ek(w)

r Hence the periodic solution of the perturbed system corresponding to the closed orbit C' is orbitally asymptotically stable or unstable according as h ( o ) < Oor > 0.

10.7. Perturbation of Analytic Systems

187

By (10.15), the results obtained above concerning the existence and stability of the periodic solution of the perturbed system coincide completely with the results of 10.3. Next let us consider Case 11. Case II, where h ( o ) = 0 and p z ( o ) = = pm- t(m) = 0, pm(m) f 0. In this case, let us put a * *

Prn(0) = am;

(10.91)

then, as is seen from (10.85), by Weierstrass' preparation theorem (see, e.g., Bochner and Martin [l], pp. 183-190), the function @(c, E ) is expressed for sufficiently small c and E as follows:

1 I

@(c,

1 I am[cm+ k ( E ) C m - l +

E)

+ l ( ~ ) ][l + W(C, E ) ] ,

a * -

where k ( ~ ).., ., I(&) are analytic functions vanishing with analytic function vanishing with c and E . Let us put Ql(c,

E) =

E,

(10.92)

and Y(c,

+ k(E)C"'-l + + I(&),

c"'

E)

is an

(10.93)

0 . .

then, evidently Eq. (10.87) is equivalent to @,(c,

= 0.

E)

(10.94)

Let us consider Subcase (i). Subcase (i), where m is odd. In this case, Eq. (10.94) has at least one real solution, and the function iD,(c, E ) is expressed as follows: @1(C,

E ) = [C

-

C1(&>IP' [C

-

c~(E)J'*

[C

-**

-

Ck(E)IPk @o(C, E),

(10.95)

where c,(E), c ~ ( E ) ., .., ck(e)are the real-valued distinct analytic functions of the fractional powers of E vanishing with E, and Oo(c,E ) is a real polynomial of c whose coefficients are continuous in E and @,(c, E ) > 0 for sufficiently small c and E except for c = E = 0. In (10.95),p,,p~,...,pkare positive integers, and their sum p 1 + p z + ... + p k is an odd number not greater than m, since m is odd at present. Clearly equality (10.95) implies that the perturbed system (10.2) has k distinct periodic solutions corresponding to the closed orbits

I I

I I

Y

r

x = x(.) = p(t)

- (xz

Y

+ ( x 2+X Y 2)

Ci': = Y(T) =

$(f)

+

yl).,~

I \

-. -

c

(i 112

4

~

)

&I

7

=

1, 2,

..., k).

10. Two-Dimensional Autonomous Systems

188

Now let us suppose that cl(EO)

< c2(E0) <

I I ; then, for any

for a certain sufficiently small 0 > E > e,,, we have C1(&)

< Ck(&O)

**'

E~

< c&) <

E

such that 0 < E < 6 or

< Ck(&),

(10.96)

since c1(&), c2(&),..., ck(&)are all the power series of the fractional powers of E. Let us now investigate the stability of the periodic solutions of the perturbed system corresponding to the closed orbits Ci'. If we substitute (10.95) into (10.92), then @(c, &) = am[c - C1(&)IP1 ..' [ c -

c k ( & ) ] p k @O(c,

6)

[I

+ Y(c,& ) I ;

therefore we have =

@"Ci(&),&]

= pi! U m [ C i ( & )

@ ( P [ ) [ C i ( E ) , &I

x

=

a/dc;

I I,

Then, for sufficiently small r @'[c~(E)+ r, E ]

= (Pi

[Ci(&)

. * a

[ci(&)

= 0,

- ci-l(&)]p+-I

7 Ck(E)IPk

(10.97)

E])

i = 1,2, ..., k).

we have

1

- I)!

(E), 81

. {1 + ' Y [ C i ( & ) ,

E]

@o[Ci(&),

(' ?.Pi-

- C1(&)]PI

[ci

- c i + ~ ( & ) ] ~ ' +* "~

[Ci(&)

x

... = @ p i - 1

E]

@(P')[ci(&),

+ O(rp')

( i = 1, 2, ..., k).

Hence, substituting these into (10.90), we have

+ O[(eir>"'] > o

(0 c

ei c

1; i = 1, 2,

..., k)

(10.98)

I I

as r + 0. From these, it follows that if r is sufficiently small,

O< and

+

r

r'

'1 - C i ( E ) < 1

whenever

rPt-l @(")[ci(&),

&]

< 0,(10.99)

10.7. Perturbation of Analytic Systems

189

TABLE I1 pt

odd

even

Sign of W"()[C,(E),E]

Orbital stability

-

asymptotically stable

+

unstable

-

asymptotically stable for r>O unstable for r
+

unstable for r>O asymptotically stable for r
absolute stability

conditional stability

From (10.99) and (10.100), we have Table I1 on the orbital stability of the closed orbit Ci' (i = 1, 2, ..., k). Suppose now a,,, < 0. Then, for C1', (D(Pi)[~l(~), E] < 0

or

+

>6

+

+

according as p1 is odd or even (note that p1 pz Pk is odd). As is seen from Table 11, this implies C1' is always asymptotically orbitally stable in the negative side (the side that contains the orbits such that r < 0). Likewise, we readily see that c k ' is always asymptotically orbitally stable in the positive side (the side that contains the orbits such that r > 0). In the present case, however, by Table I, the closed orbit C of the unperturbed system is asymptotically orbitally stable. Thus we see that, in the present case, the stability of C,' in the negative side and that of Ck' in the positive side coincide with that of the closed orbit C of the unperturbed system. This result is also valid, however, when a,,, > 0, as is easily seen. Thus we see that the above result is indeed always valid, whether the sign of a,,, is negative or positive. Let us now investigate the relation between the stability of the adjacent closed orbits. As is seen from (10.97), (D(Pi)[cI(E),E] and (D(Pi+l)[ci+(E),E ] are of the opposite signs if and only if pi+ is odd. Suppose C,' is asymptotically orbitally stable in the positive side; as is seen from Table 11, (D@")[c,(E), E ] < 0. Then, ifpi+, is odd, (D(pi+l)[ci (E), E] >O; and, as is seen from Table 11, this implies Cf+ is orbitally unstable. Ifpi+ is even, then (D"N+l)[ci+ (0; and, as is seen from Table 11, this implies Ci + is orbitally unstable in the negative side. These results mean that C;+ is always orbitally unstable in the negative side. In like manner, it is easily seen that Cl+ is always asymptotically orbitally stable in the negative side if C,'is orbitally unstable in the positive side.

,

,

,(&),&I

190

10. Two-Dimensional Autonomous Systems

These results imply that the stability of the adjacent closed orbits is always reversed in the region bounded by both closed orbits. The stability of each C i is then automatically determined by means of the results obtained above, if the multiplicities p i of the roots ci(&)and the stability TABLE 111 Orbits Cl ’ “2; c3,

c4

Orbital stability Negative side

pi

asymptotically stable asymptotically stable unstable unstable

2 3 4 2

Positive side unstable asymptotically stable asymptotically stable asymptotically stable

of C of the unperturbed system are known. In illustration, suppose that p 1 = 2, p 2 = 3, p 3 = 4, and p 4 = 2, and that C is asymptotically orbitally stable. Then we have four closed orbits C1‘, C2‘, C3’, C4’ of the perturbed system, and we obtain Table I11 describing their orbital stability (see Fig. 10).

+

+

+

P k is odd; therefore, at least one of Now, in (10.95), p 1 p 2 ..,Pk is odd. Suppose p l yp 2 , .. ., p i - are all even and p i is odd. Then Cl‘,C,’, .., Ci- are all conditionally stable, and Ci‘ is absolutely stable. Therefore Ci’ has the same orbital stability as that of C1’in the negative side. However, as is shown already, the orbital stability of C1’ in the negative side is the same as that of the closed orbit C of the unperturbed system. Thus the orbital stability of Ci’ is the same as that of the closed orbit C of the unperturbed system in both sides. This implies that, in the present case, the perturbed system (10.2) has at least one periodic solution having the same orbital stability as that of the periodic solution (10.3) of the unperturbed system.

p l yp z ,

.

.

10.7. Perturbation of Analytic Systems

191

In what follows, let us consider the following special case. Before proceeding to the perturbation problem, a property concerning the integral I will be derived. Let tl be an arbitrary value of t , and put

The special case, where Z # 0.

and e - h l ( S ) ( X K, YH,) ds.

I, =

(10.102)

Then, from (lO.lOl), h,(t) = h(t) - h(tlj; consequently, from (10.102),

(10.103) In the present case, however, h(t + o)= h(t), since h(w) = 0 by assumption. Hence, from (10.103), it follows that

I, = eh(rl)I.

(10.104)

This is the desired property concerning the integral I. Now let us return to the perturbation problem. If we compare (10.92) with (10.85), we find that 1(E)

=

1

a,(Xo2

+ Yo2)1'2E + O(E2).

(10.105)

Therefore, if we put (10.106) we can rewrite @,(c,

E)

cp,(c,

as follows: E)

= c'"

+ k,(a)c"-' + .*. + qa>,

(10.107)

192

10. Two-Dimensional Autonomous Systems

where

By (10.105), clearly I1(g) is of the form

r&)

=

-Q

+ O(a2).

Hence, if we substitute c =

+ U(Q)]

aly-1

(u(0) = 0)

into (10.107), we see that the roots of Eq. (10.94) are all of the forms c = u""[1

+f(U"")],

(10.108)

where u""' is an arbitrary mth root of u (which may be complex), andf(a"") is a real analytic function of ulfmvanishing with d f m The . roots (10.108) of Eq. (10.94) are real, however, if and only if oilm is real. Since m is odd at present, by (10.106), we thus see that Eq. (10.94) always has only om real simple root for an arbitrary value of E . By the results proved for the general case, we see then that in the present special case, the perturbed system (10.2) always has one and only one periodic solution with the same orbital stability as that of the periodic solution (10.3) of the unperturbed system. Let C' be the closed orbit corresponding to the above unique periodic solution of the perturbed system (10.2). Then we can prove that C' does not meet

0

c'

the closed orbit C corresponding to the periodic solution (10.3) of the unperturbed system, provided E I > 0 is suffciently small. In fact, suppose C' meets C at the point P1[q(tl),$(tl)]. Then, replacing point Po[q(0),$(O)] by point P,,we have Q(0, E ) = 0 (see Fig. 11). By (10.85), this implies

I

E

(Xi2 +

YIZ)1/2

1,

+ p2'(tl +

W,

0,E )

+ ps'(t1 + W , 0, E ) +

= 0,(10.109)

10.7. Perturbation of Analytic Systems

193

where Xl = X[cp(tl), $(tl)], YI = Y[cp(tl),$(t1)1, and Pz'(t, C, E), P 3 ' ( t , C, E), are the functions derived from p2(t, c, E), p 3 ( t , c, E), ..., by replacing 1 = 0 with t = t i . Equality (10.109) implies

...

Zl + 0

as

E

-,0.

(10.1 10)

However, by (10.104), this is impossible for our system, because Z # 0 by assumption, and h(t,) is bounded, since h(t) is periodic in t . Hence we see that C' cannot meet C if E > 0 is sufficiently small. This implies the perturbation yields a closed orbit in the inner side or in the outer side of the closed orbit of the unperturbed system. Next let us consider Subcase (ii).

I I

Subcase (ii), where rn is even. In this case, Eq. (10.94) may not have any real root. In such a case, the perturbed system (10.2) has, of course, no periodic solution. However, when Eq. (10.94) has some real roots, the function Q),(c, E ) can be again expressed as (10.95). In such a case, the arguments in the preceding subcase are valid once more, and hence, similar conclusions concerning the distribution and orbital stability are obtained for the closed orbits of the perturbed system. However, in the present case, in (10.95), the sum p 1 p z .-- Pk is even since m is even. Therefore the stability in the negative side of C,' and the stability in the positive side of ck' coincide, respectively, with those of the closed orbit C of the unperturbed system; but the stability in both sides of C is mutually different, contrary to the conclusion in the preceding subcase. This is the sole difference between the conclusions in both subcases. Thus, for example, forp, = 2, p z = 3 , p 3 = 4, p , = 1, and a,,, > 0, we have four closed orbits C1', C2', C3', C,' of the perturbed system, and we obtain Table IV describing their orbital stability (see Fig. 12).

+

+ +

TABLE IV Orbital stability

Pi

Orbits

Negative side ~

~

Cl'

2

"2;

3 4

'3,

1

c4

~~

Positive side

~

asymptotically stable asymptotically stable unstable unstable

unstable asymptotically stable asymptotically st able unstable

194

10. Two-Dimensional Autonomous Systems

FIG. 12

In what follows, let us consider the special case below. The special case where Z # 0. In this case, by substitution (10.106), we have (10.107), as in the preceding subcase. Consequently, the roots of Eq. (10.94) are also of the forms (10.108). However, in the present case, m is even; consequently Eq. (10.94) can have a real root only for E such that dla,,, < 0; and, for such E , Eq. (10.94) indeed has a pair of real simple roots. This means that the perturbed system (10.2) has a periodic solution only for E such that Ella,,, < 0, and it indeed has a pair of periodic solutions for E such that Ella,,, < 0. As is seen from (10.108), one of the roots of Eq. (10.94) is positive and the other is negative. In addition, since I # 0, the closed orbits of the perturbed system corresponding to these roots do not meet the closed orbit C of the unperturbed system, as is shown in the preceding subcase. Therefore, one of the above closed orbits of the perturbed system lies completely in the inner side of C, and the other lies completely in the outer side of C. The stability of these closed orbits of the perturbed system is then, as is seen from the results of the general case, the same as that of C in the sides containing the closed orbits of the perturbed system. The last case remaining is Case 111. = 0. In this case, the given Case III, where h(w) = p2(w) = p3(w) = unperturbed system (10.1) is a fully oscillatory system, as is stated in 10.6. The perturbation of an analytic fully oscillatory system will be discussed later in 10.9, however, since conditions are considerably different from the previous ones in the case of a fully oscillatory system. Remark. If the vector ( X + EH, Y + E K )is a rotated vector of ( X , Y) by an angle E , then X

+ EH = Xcos E - Ysin E,

Y + E K = Xsin E

+ Ycos

E.

195

10.8. Multiplicities of Closed Orbits

Therefore, X K - YH = ( X 2 + Y') sin E / E ,

from which follows

q x ,y ) K(x, Y,0) - Y(X,Y ) H(x, Y , 0) = X 2 b , Y) + Y2(X,Y). This implies that, under the rotatory perturbation, integral I defined by (10.86) is always positive. Hence, for the rotatory perturbation, the results for the special case where Z # 0 are all valid. This shows that the results of the present section include a11 the results of Uno [ l ] and Duff [l] as a special case. 10.8. MULTIPLICITIES OF CLOSED ORBITS

The results of the preceding section lead to the introduction of multiplicities to closed orbits. Namely, let

c :x

=

q(t),

y = $(t)

be an arbitrary closed orbit of the analytic system (lO.l), and let : x = cp,(@),

Y

=

$,(.I

be an arbitrary analytic curve crossing C at the point A : [q(O), $(O)] = [ql(0),+l(0)]. Let us suppose that a is an arc length of r (see Fig. 13). Now

FIG. 13

let us consider an arbitrary orbit C' of (10.1) lying near C. Then, by Theorem 4.8, C' crosses r at a point P[ql(a), $,(a)] near A . Let Q [ q l ( u Au), $,(a Am)] be the point where C' intersects first with r after leaving point P. The quantity Aa is then a function of a, and this function Au = Y ( u ) evidently vanishes for u = 0 corresponding to the closed orbit C. The multi-

+

+

196

10. Two-Dimensional Autonomous Systems

plicity of the closed orbit C is then defined by the multiplicity of the root a = O of the equation Y ( u ) = 0.

We shall show that the multiplicity of the closed orbit is independent of the choice of the curve r. Let L be the normal of C at A , and let P’and Q’be the intersections of C‘ with L such that @‘ and are small. With regard to the moving orthonormal system along C, C‘ is expressed as (10.7) and, for p = p ( t ) , corresponding to (lO.lO), we have

PQ’

(10.1 11)

dPIdt = R(t, P ) ,

where R(t, p) = R(t, p, 0). By the analyticity of system (lO.l), q ( t ) and $(t) are analytic with respect to t ; hence R(t, p ) is analytic with respect to t and p. Let p = p(t, c ) be a solution of (10.111) such that p(0, c ) = c ; then, by Theorem 2.7, p(t, c ) is analytic with respect to t and c. Put AP‘ = c ; then, for point P, by (10.6) and (10.7), we have

If we regard (10.1 12) as the equation with respect to t and a, then this is satisfied by a = t = 0 for c = 0 and, for c = a = t = 0, the Jacobian of the lefthand sides of (10.112) with respect to t and a is

since r does not contact C at A. Hence, by the theorem on implicit functions, t and a satisfying (10.1 12) are uniquely determined so that t = t(c),

a =f ( c )

(10.1 13)

I I, and

may be analytic with respect to c for sufficiently small c t(0) = f ( 0 ) = 0.

Now, if we differentiate both sides of (10.112) with respect to c after substituting (10.113), then, for c = 0, we have

X[Cp(O),$(0)] Y[Cp(O),

*

t’(0) + t(0) - Cp,’(O)f’(O) = 0,

*

t‘(0) + 1(0) - $I’(O)f’(O)

=

0,

(10.114)

10.8. Multiplicities of Closed Orbits

197

since

Equality (10.1 14) implies (10.1 15)

f'(0) # 0, because, otherwise, we have

t(0) = - t'(0) * XCdO), @)I, 1(0) = - "(0) YC4q9 @)J

-

which is a contradiction. By (10.115), we can solve the latter of (10.1 13) with respect to c so that c =g(4

may be analytic with respect to g(0) = 0,

c1

(10.1 16)

I I, and

for sufficiently small a

(" = d/da).

g'(0) # 0

(10.1 17)

Let 0 and m be, respectively, the primitive period and the multiplicity of the closed orbit C . Then, since AQ' = p ( 0 , c), by (10.116) and (10.113), we have successively def

@(c) = p ( 0 , c)

-c

+ Aa) - g(a) = y(CL)gf(a + e ~ a ) = amy',(a)g'(a + e AU) = g(a

(0 <

e < 1)

(Yl(0) # 0) = C " [ ~ ' ( ~ ~ C > ] ~ Y ~ ( C ~ e) ~ACL) ' ( C ~ (0 < el < 1).

+

(10.118)

Since

+

[.r(elc)lmwl(a)gy~ e A a ) z

1

o

I

for sufficiently small c by (10.115) and (10.117), expression (10.118) says that m is the multiplicity of the root c = 0 of the equation @(c) = 0.

Since @(c) is independent of the choice of the curve r, this shows that the multiplicity of the closed orbit is independent of the choice of the curve r. When the multiplicity is one, the closed orbit is called a simple closed orbit; otherwise, it is called a multiple closed orbit.

198

10. Two-Dimensional Autonomous Systems

When the solution p = p(t, c) o f (10.111) such that p(0, c) as (10.73), by (10.76), we have

=

c is expressed

Hence, if h ( o ) # 0, the multiplicity of the closed orbit is one; and, if h ( o ) = pz(w)

=

... = p m - l ( w ) = 0

and

p,(o) # 0,

the multiplicity of the closed orbit is m. Then, from the results of 10.6, we see that the closed orbit of the odd multiplicity has absolute stability and the closed orbit of the even multiplicity has conditional stability. I f we use the term “multiplicity” of a closed orbit, we can state the results obtained in the preceding section briefly in the following way: In Case I, that of a simple closed orbit, the perturbation always yields one and only one simple closed orbit having the same orbital stability as that of the original closed orbit. In Case II, that of multiple closed orbit, there are two subcases. In Subcase (i), that of a closed orbit of odd multiplicity m, theperturbation always yields at least one closed orbit having the same orbital stability as that of the orginal closed orbit. The perturbation, however, yields in general many closed orbits whose multiplicities do not surpass m in total. Thelinnermost and outermost of these closed orbits have, respectively, the same orbital stability as that of the original closed orbit in the inner and outer sides; and the stability of each closed orbit yielded by the perturbation is successively determined so that the stability of the adjacent closed orbits may always be reversed in the region bounded by both closed orbits. In the special case where I # 0, however, the perturbation yields only one simple closed orbit that does not meet the initial closed orbit. In Subcase (ii), that of a closed orbit of the even multiplicity m , the perturbation may or may not yield closed orbits. When the perturbation yields closed orbits, the sum of their multiplicities does not surpass m, and the stability of each closed orbit is successively determined in the same way as in Subcase (i). In the special case where I # 0, however, the perturbation does not yield any closed orbit if ella, > 0 and does yield a couple of simple closed orbits i f Ellam < 0. In the latter case, two closed orbits yielded by the perturbation lie completely on the opposite sides of the original closed orbit, and these closed orbits have the same orbital stability as that of the original closed orbit in the sides containing the closed orbits of the perturbed system. The perturbation in Case I1 is usually called the bifurcation of a multiple closed orbit.

10.9. Perturbation of Analytic Fully Oscillatory Systems

199

10.9. PERTURBATION OF ANALYTIC FULLY OSCILLATORY SYSTEMS

In the present section, we are concerned with perturbation of an analytic fully oscillatory system. Hence we suppose the unperturbed system (10.1) is a fully oscillatory system and, for (10.1) and its perturbed system (10.2), we assume: X(x, y ) and Y(x, y ) are analytic with respect to x and y in the domain D of the xy-plane; H(x, y , E ) and K(x, y, E ) are analytic with respect to x, y , and E for ( x , y ) E D and I E < 6 (6 > 0). Under these assumptions, evidently the discussions in 10.4 are all valid. In addition, in the present case, by the assumption on analyticity, the functions cp,(a), +,(a), q ( t ; a), and + ( t ; a) are all analytic with respect to a and t (see Theorem 2.7). Hence, by (10.9) and (lO.lO), the functions T(t, p, E ; a) and R(t, p, E ; a) in (10.26) and (10.27) become analytic with respect to t, p, E , and a. Then, from its definition, the solution p = p(t, c, E ; a) of (10.27) is evidently analytic with respect to t, c, E , and a. As is seen from (10.33), this implies that the functions Aa = f ( c , E ) and t = t(c, E ) in (10.34) are both analytic with respect to c and E . Then the function c = g(Aa, E ) in (10.37) becomes analytic with respect to Aa and E ; hence the primitive period ~ ( aof) C, becomes analytic with respect to a (see the proof in 10.4 that ~ ( a E) C,'). By (10.38), this implies that, in the present case, the function @ ( E ; a) of (10.38) is analytic with respect to E and a. Now, as is mentioned in 10.4, the periodic solution of the perturbed system (10.2) corresponds one to one to the solution a = a(&) of

I

(10.1 19)

@(&; a) = 0.

Let p be the multiplicity of the solution a(&) of (10.119); then we see that (i) p is also the multiplicity of the closed orbit represented by the periodic solution of (10.2) corresponding to a = a(&); and (ii) the orbital stability of the periodic solution of (10.2), corresponding to a = a(&), is decided as in Table I1 in its dependence upon the multiplicity p and the sign of a(&))(' =

a/aa).

In fact, for Ar of (10.45), by (10.34) and (10.35), we have

Ar = f[@(~; a(&) = @ ( E ; a(&)

+ r),

E]

af + r) ac

[o@(E;

a(&)

+ r), E ]

(0 < 0 < 1)

200

10. Two-DimensionalAutonomous Systems

=

1 A D ( ~ ) ( E ; a(&)+ 6,r) af [ e m ( & ; P! ac

a(&)

+ r),

(0 < el < 1)

E]

(10.120)

and @(p) (E;

a(&))

q o , E) ac

z 0, [see (10.36)].

# 0

Hence we see that r = 0 is a p-ple root of the equation Ar = 0. This proves (i). To prove (ii), let us consider the ratio of (10.48). Then, by (10.48), for sufficiently small r we have

I 1,

n

@[E;

g=1+ AB = 1+

a(&)

r

+ r ] .-Ar BR

rP- 1

[ @ ( p ) ( ~ ; a(&))

~

P!

+ O(r)]

*

[l

+ o(l)]

[see (10.47)]

From this, (ii) follows in a similar way as in 10.6. By (10.39), the function a ( & ; a) is of the form @(E;

(10. 2

a) = E @ ~ ( E ; a);

and, by (10.41), (P1(&; a) is analytic with respect to for E # 0, Eq. (10.119) is evidently equivalent to

E

and a. From (10.121), (10.122)

a) = 0.

Hence we see that Eq. (10.119) can have a solution a = a(&) for sufficiently small E > 0 only when

I I

(10.123)

@,(O; a) = .O

has a real root. On the other hand, by (10.38), (10.85), (lo.%), and (10.58), the function a) is of the form a 1 ( & ; a) =

1 I(a) ~’(0;a)]”’

[ x ~ ( a) o ;+

+ &(a) +

* * a

f ErnIrn(OL)

+ ..., (10.124)

10.9. Perturbation of Analytic Fully Oscillatory Systems

20 1

where

Hence Eq. (10.123) is equivalent to (10.126)

I(u) = 0.

Thus we see that the perturbed system (10.2) has no periodic solution for sufficiently small E > 0 if Eq. (10.126) has no real root. Let us now consider the case where Eq. (10.126) has a real root a = cq,. Since I(a) is analytic with respect to a, as is seen from (10.58), there is a positive integer m such that

I I

I(%)

=

I'(a,)

=

0 . .

= Z("'-')(a0) = 0

and Z("')(a,) # 0

(' = d/da), (10.127)

provided I(a) does not vanish identically. When (10.127) holds, by Weierstrass',preparation theorem (e.g., see Bochner and Martin [l], pp. 183-190), a) is expressed for sufficiently small a - a, and E the function as follows:

I

a) =

1

*

I

I I

1 P(a,)

+ ~ ' ( 0 ;a0)]l" m! [(a - ao)m+ k(E) (a - ao)m-l + ... + I(&)] [1 + Y(G 41,

[x'(o;a,) x

x

(10.128)

where k ( ~ ).., ., I(&) are the analytic functions vanishing with E , and " ( 6 ; a) is an analytic function vanishing for E = 0 and u = a,. Since expression (10.128) is of the same form as (10.92), the discussions in Case I1 of 10.7 hold mostly in the present case due to (ii), mentioned above. Hence, by (i), we have the following conclusions : If. = a, is an m-ple real root of (10.126) and m is odd, then the perturbed system (10.2) has at least one closed orbit of absolute stability in the neighborhood of

c,

: x = q ( t ; a,),

I

Y = $(ti a,)

I

for sufficiently small I E > 0. However, for sufficiently small I E > 0, in the neighborhood of C,,, the perturbed system has in general many closed orbits whose multiplicities do not surpass m in total. The innermost m d outermost of these closed orbits have the same orbital stability, respectively, in the inner

202

10. Two-Dimensional Autonomous Systems

and outer sides; and, in these sides, they are orbitally asymptotically stable or unstable according as &I("')(a0)< 0 or > 0. The stability of each closed orbit of the perturbed system is arranged so that the stability of the adjacent closed orbits may always be opposite in the region bounded by both closed orbits. In the special case where m = 1 or Il(a0) # 0, the perturbed system (10.2) has ,,, and this is orbitally only one simple closed orbit in the neighborhood of C asymptotically stable or unstableaccording as E I ( ~ < ) (0 ~ or ~ > )0 (see Fig. 14).

FIG.14

If. = a. is an m-ple real root of (10.126) and m is even, then the perturbed system (10.2) may or may not have closedorbits in the neighborhoodof C, for sufficiently small E > 0. I f the perturbed system has closed orbits in the neighborhood of C ,,, then the sum of their multiplicities does not surpass m, and the closed orbit c' represented by the periodic solution of (10.2) corresponding to the least solution a = a'(&) of (10.122) is orbitally asymptotically stable or unstable in the negative side-the side where a < a'(&)- according as EZ("')(U~) > 0 or < 0 ; and the closed orbit C" represented by the periodic solution of (10.2) corresponding to the largest solution a = a"(&) of (10.122) is orbitally asymptotically stable or unstable in the positive side - the side where a > a"(&)-according as &Z("')(aO)< or > 0. The stability of each closed orbit of the perturbed system is arranged in the same manner as in the case where m is odd (see Fig. 15). In the special case where Il(uo) # 0, the perturbed system (10.2) has no closed orbit in the neighborhood of C,, if&Z1(aO)/I("')(aO) > 0, and it has a pair of simple closed orbits in the neighborhood C ,, if EI1(aO)/I("')(aO) < 0 (see Fig. 16). The phenomenon described in the above conclusion can be regarded as the bifurcation of the closed orbit C ,, o f the unperturbed system.

I 1

10.9. Perturbation of Analytic Fully Oscillatory Systems

203

FIG.15

FIG.16

Next let us consider the case where Z(a) vanishes identically in a. In this case, there is a positive integer n such that

I(a) = I,(@)3

... = I n - l ( a ) = 0

and

provided @ ( E ; a) does not vanish identically in from (10.121) and (10.124), we have a)(&; a) = En+l@n+l(&;

E

In(.) $0,

(10.129)

and a. When(10.129) holds,

a),

where Qn+l(E;

a) = In(a)

+

EZn+l(U)

+

E21n+Z(U)

+ ....

As is seen from (10.124), function a ) n + l ( ~a) ; is of the same form as a), and here I,,(.) does not vanish identically in a. Hence, in the present case we have conclusions similar to those for the preceding case concerning the existence and stability of the periodic solutions of the perturbed system. When @ ( E ; a) vanishes identically in E and a, the perturbed system (10.2) is again a fully oscillatory system. In such a case, the perturbation does not yield any feature different from the unperturbed system.

11. Numerical Computation of Periodic Solutions

In the present chapter we shall describe a numerical method to compute a periodic solution of the autonomous system. The method is based on the author’s paper [S] and, in principle, it is based on the results of Chapter 5 and the Newton method for solving equations numerically. The method can be simplified when the given autonomous system has some symmetric character. In illustration of such a technique, a method to compute a periodic solution of the autonomous van der Pol equation will be described. Some important characteristics of periodic solutions of the autonomous van der Pol equation will be found from the computed results. 11.1. A METHOD TO COMPUTE A PERIODIC SOLUTION

Consider

dx/dt = X(X),

(11.1)

where x and X ( x ) are the n-dimensional vectors and X(x) is continuously differentiable in the domain D of the phase space. Let

c, : x = qo(t)

(11.2)

be an approximately closed orbit of (11.1) lying in D. By approximately closed we mean that when C, is followed from the point qo(t,) (tz 5 0) to the point q o ( t l ) ( t l > 0), the distance between these two points is small (see Fig. 17). In the present section, a method to compute a closed orbit of (11.1) lying near Co will be described. First we construct a moving orthogonal system { X [ q o ( t ) ] ,t,(t),..., t,(t)> along C, by the method of 5.1. As is seen from 5.1, this is possible for the arc of C, bounded by points qo(tz)and q,,(t,), since the lemma 5.1 holds for such an arc of C,. Next we consider an arbitrary hyperplane n, which crosses Co a t the point 204

11.1. A Method to Compute a Periodic Solution

205

FIG.17

is evident from the configuration, we may suppose without loss of generality that the point cpo(t2) is also on the hyperplane R. By 5.2, one can express any orbit C of (1 1.1) lying near C, as

cpo(t,). As

n

x =

and, for pv

= pv(t)

= cpo(t)

.(TI

c

+ v=2

(11.3)

Pvtv(9;

(v = 2, 3, . .., n) and z = z(t), one obtains n

dPv _ -

p=2

&,*XI -

X*X'

dt (a

= d/dt;

c Prtv*tp

p=2

v = 2, 3, ..., n)

(11.4)

and

-dz_ -

1 x [I2 +

i

v=2

X*X'

dt

Pv

x*ev (11.5)

,

whereX = X [ p o ( t ) ] ,and X' = X[cpo(t) + X t = 2 p v t v ( f ) ] .In the sequel, let US suppose z(0) = 0. Then z = z ( t ) is the time required to reach a normal hyperplane of C, at the point cpo(t) from the one at the point cpo(0) along C . Let US Put pv(0) =

(V =

C,

then one can write pv = pv(t) (v

=

2, 3, ..., n);

2, 3, ..., n) and z

(11.6)

= z(t) as

(11.7)

1 I . Numerical Computation of Periodic Solutions

206

and pv(t, c) (v = 2, 3, ..., n) and ~ ( tc), become continuously differentiable with respect to c, (v = 2, 3, ..., n) for sufficiently small c, (v = 2, 3, ..., n), since py = pY(t,c) is a solution of (1 1.4) satisfying the initial condition (1 1.6) and

I I

n

cis.

T ( t , C) = v=2

Let us write x

as

= x(7)

(11.8) then this is a solution of dx/dz

=

(11.9)

X(X)

satisfying the initial condition n

d o , c)

+ vc CYSY(0). =2

= cPo(0)

Let the equation of the hyperplane n be P * ( X - cpo(t1)) = 0

( 1 P I( = 1);

(11.10)

then the positive time Tl(c) required to reach n along C from the point q(0, c) is determined by the equation P*{Cp[T,(C), c] - cpo(t1))

= 0.

(11.11)

However,

P*

acp (tl,

0) = P*X[cp(tl, 011 =

P*X[qoO(tl)]z 0,

since n crosses Co at the point qo(tl).Hence, by Eq. (1 1.1l), the positive time T,(c) is uniquely determined and is continuously differentiable with respect to c, (v = 2, 3, ..., n) for sufficiently small c, (v = 2, 3, ..., n). Evidently, for sufficiently small c, ( V = 2, 3, . .., n), T , ( c ) = t , . In a similar way, for sufficiently small c, (v = 2 , 3, . . ., n), the negative time T,(c) required to reach n along C from the point q(0,c ) is uniquely determined by the equation

I 1

1 1 I 1

P*{(PCT2(C)?

cl

- rpO(tl)> = 0,

(1 1.12)

l I . 1 . A Method to Compute a Periodic Solution

207

and it is continuously differentiable with respect to c, (v = 2, 3, ..., n). Now let ( p v } (v = 2, 3, ..., n) be an orthonormal system lying in n. Then the necessary and sufficient condition that C may be closed can be written as follows : F , ( c ) ~p,*(cp[T~(~), C] - cp[Tz(c), c]}

=

0

(V = 2, 3,

..., n).

(11.13)

In fact, if C is closed, cp[T,(c), c] = cp[T,(c), c], and this evidently implies (1 1.13). Conversely, if (11.13) holds, then P,*{(cp[Tl(C), c] - cpo(t1)) - ( r P [ M C ) Y

CI

- cpo(t1))~ = 0 (v = 2,3, ..., n),

(11.14)

and, from (11.11) and (11.12)

Equalities (1 1.14) and (1 1.15) imply

that is,

This proves the closedness of C. Condition (11.13) can be regarded as the equation for c, (v = 2, 3, . ,n). If this equation can be solved numerically with respect to c, (v = 2, 3, ...,n), then the closed orbit C can be obtained by computing the solution x = q(z, c) of (1 1.9) by a step-by-step method starting from the initial point

..

We shall show that Eq. (1 1.13) can be solved numerically by the Newton method. To apply the Newton method, it is necessary to know the approximate value of JFv(c)/Jc, (v, A = 2, 3, ., n). Since F,(c)(v = 2, 3, ..., n) are continuously differentiable with respect to cA (A = 2, 3, ., n) and cA ( A = 2, 3, ...,n) are supposed to be small, it sufEices to know the value of aF,(c)/ac, (v, A = 2, 3, ..., n) for c, = 0 ( A = 2,3, ...,n).

..

..

I I

208

11. Numerical Computation of Periodic Solutions

Now

= X(Cp[T,(C), c]}

)-*a

ac,

+ ac, a Cp [T,(c), c]

(A = 2,3, ..., n). (11.16)

(1 1.17)

Hence, substituting (1 1.17) into (1 1.16), we have

(11.18)

If we put (11.19)

then we can rewrite (11.18) as follows:

n

+ 1tp(fl)GJtJ

(A = 2,3, ..., n).

(11.20)

p=2

As is seen from (11.19) the matrix (GPA(t))is a fundamental matrix of the first variation equation of (1 1.4), namely, a fundamental matrix of the linear homogeneous equation E, (f)pA

__ dpp =

dt

1=2

( p = 2,3,

..., n),

(11.21)

11.1. A Method to Compute a Periodic Solution

209

where

(a,

is the Kronecker delta) as is seen from (1 1.19), the quantities G,,,(t) (p,A = 2,3, ..., n) are then easily computed by solving (I 1.21) numerically by a step-by-step method starting from the initial values (1 1.23). Now let us put p*x(x) = X'"'(X), P*t,(t) = tF)(t)>

p,*X(x) = X(")(X), P,*t,(t) = try0

( v , p = 2, 3,

..., n).

(11.24)

Then, from (1 1.1 1) and (1 1.20), we have

+

t ~ ) ( t l ) G , , ( t l= ) 0

(A = 2. 3, ..., n),

u=2

from which follows

x

C tF)(tl)GpA(tl)(A = 2, 3, ..., n),

(11.25)

p=2

since X(o)[qo(ti)]= p*X[qo(tl)] # 0. If we substitute (1 1.25) into (11.20), then we have

n

( v , A = 2, 3,

..., n).

(11.26)

11. Numerical Computation of Periodic Solutions

210

In a similar way, we have

(v, I

=

2, 3,

..., n).

(11.27)

However, C , is approximately closed; in other words, rpo(t2) = cpo(t1).

+

Consequently, X[cpo(t2)] X[cpo(tl)], and so, by (5.15), Hence, from (1 1.27), we have

t,(t2)= tp(tl).

n

. -

c ~x'O"cpo(tl)ltl"(~l>

p=2

x""cpo(tl)ltlP'(tl)~GpL(f2) x'o"cpo(~ l)] -

(v, I = 2, 3,

..., n).

(11.28)

Then, from (1 1.13) and (1 1.26), we have

n

c

{x'"'Ccpo(~l>ltl"(tl)- X'"~[cp0(~1)]tl"'(~l)) {G,A(tl) .- p=2

X'O"cpo(t

- GCl(f2))

l)]

(v,

1 = 2, 3, ..., n).

(11.29)

This is the desired formula that gives the approximate values of aFv(c)/acA (v, I = 2, 3, ..., n). For formula (1 1.29), it is easily proved that det {X'o)[cpo(tl)]t~'(tl) - X ( v ) [ c p o ( r l ) ] t ~ ) ( t l#) }0. In fact, if the above determinant vanishes, then there are constants all zeros such that

c (X'O)tF) - X(v)t;))Ky

(11.30) K,

not

n

v=2

=

0

( p = 2, 3,

..., n).

(11.31)

11.1. A Method to Compute a Periodic Solution

211

This implies that n

(11.32) v=2

since X(O) # 0. Evidently,

Therefore, combining this with (1 1.32), we have

c n

x(v)Ic, =

0,

Ic, =

0

(v = 2, 3, ..., n),

(11.33)

v=2

because

#O

since, by (1 1.24), the row vectors of the above determinant are, respectively, the orthogonal vectors X [ q ( t l ) ] , t 2 ( t l ) ,. .., gn(tl), expressed with respect to the orthogonal system ( p , p z , ..,pn).Equality (1 1.33) contradicts the assumption that IC,(v = 2, 3, ..., n) are not all zeros. Thus we have inequality (11.30). Then, from (1 1.29), we readily see that

.

(11.34) for sufficientlysmall I c,

I (A = 2, 3, ..., n) if and only if (11.35)

det {G,A(tl> - G,dt2)1 # 0.

.

Let us assume (1 1.35), and let us denote by H,, (v, I = 2, 3, . ., n) the approximate values of W V ( c ) / d c , ( v , , I= 2,3, ..., n) computed by formula (1 1.29). Then det ( H J # 0; consequently, there is an inverse matrix (KvA)of the matrix (HvA). Then the

212

11. Numerical Computation of Periodic Solutions

Newton method to solve Eq. (1 1.13) numerically is to carry out the iterative process :

cK n

c,( " + I ) = c$")

-

F

(11.36)

vlr lr(c'"')

p=2

( v = 2 , 3 ,..., n ; m = 0 , 1 , 2

where c,(') (v = 2, 3,

,...),

..., n) may be chosen so that c:')

=0

(V

= 2, 3,

..., t ~ ) .

In the iterative process (1 1.36), the solution x = q(r, c(")) of (11.9) can be computed by a step-by-step method, and the quantities T,(c'")) and T2(c(")) can be determined by solving Eqs. (11.11) and (11.12) numerically by the Newton method using an interpolation formula. Once the solution c, (v = 2, 3, ..., n) of Eq. (1 1.13) is computed, the desired closed orbit can be computed by integrating the given equation (1 1.1) numerically by a step-bystep method starting from the initial value n

rPo(0)

c

+ v = 2 CVtY(0)'

Asis well known (e.g., see Theorem A. 1 and A.2 in the Appendix), the iterative process (1 1.36) converges if inequality (11.35) holds and if \\ q o ( t l ) - %(tl) \ is sufficiently small. This implies that if inequality (11.35) holds and qo(tl) - qo(t2) is sufficiently small, then the exact closed orbit exists and it is actually computed by the iterative process (1 1.36).

[I

[I

Remark 1.

I I I I,

The rI and t2 are usually chosen so that t , = t2 because this will in general reduce the errors caused by step-by-step numerical integration of differential equations.

Remark 2. Put

a(t) = (5,Xr))

and

G(t) = (GPA(r));

then, by (11.21), G(t) is a fundamental matrix of the linear homogeneous equation dpldt = a(t)p.

11.2. The Two-Dimensional Case

213

Then, for any w, G(t + w)G-'(w) is a fundamental matrix of the linear homogeneous equation dp/dt = q t

However, for

0

= tl

+ 0)p.

- tz, qt

+ 0)= q t ) ,

as is seen from (11.22), since the orbit Co is approximately closed. Since G(O

+ w)G-'(o)

=

E = G(O),

we have G(t

+ o)G-'(w)

k G(t).

For t = t,, this implies that G(t1)

+ G(tZ)G(tl - t z ) ,

and inequality (11,35) becomes equivalent to the inequality det [G(tl - t z ) - E l # 0.

(11.37)

This shows that condition (11.35) is independent of the choice of t,.

Remark 3.

As is readily seen from (11.29), H,, = G,,(tl) - Gv,(t,)

if the hyperplane n is chosen orthogonally to Co at the point cpo(tl) and the vectorsp, (v = 2, 3, ..., n) are chosen so that p, = r,(tl) (v = 2, 3, ..., n).

Remark 4. For finding the approximately closed orbit C,, the perturbation techniques of Chapters 7-10 are frequently used with success. For example, see 123. 11.2.

THE TWO-DIMENSIONAL CASE

In the two-dimensional case, let us write the given system (11.1) as dx/dt = X(X, y),

dyldt = Y(x, Y),

(11.38)

1 I . Numerical Computation of Periodic Solutions

214

and equation (1 1.2) of the orbit Co as x =

qoo(9,

Y = $o(t!.

According to (10.6), let us take the unit normal vector

(t(t),q(t)) of Co as

X

@)= - (XZ +

Y2)1/2

x = X[PO(l),

Il/o(t)],

= (x2

+

(11.39)

y2)1/2'

where

y

= Y[Po($

v90(0].

Then, by (10.12), the fundamental matrix (GJ?)) of the linear homogeneous equation (11.21) becomes a scalar (x02

+

eh(t)

yOz)1/2

( x 2+ Y

y

(1 1.40)

'

In the two-dimensional case, 7~ becomes a straight line. By (cos 8, sin O), let us denote the direction cosines of the line z. Then the direction cosines of the normal of 7~ may be supposed to be (sin 8, -cos 0). According to (1 1.3) and (1 1.8), by x = P(t, c),

Y =

4q4 c),

let us denote the solution of (11.38) passing through the point YO

xo

)

(11.43)

at t = 0. Then, according to (ll.ll), (11.12), and (11.13), for the closed orbit of (1 1.38), we have

-

sin 8 {cp[~,(c), c] - cpo(tl)} - cos 0 {$[T(c), c] - $o(tJ} = 0; (11.44) sin 8 * {cp[~,(c),c] - cpo(tl)} - cos O { $ [ T ~ ( Cc] ) , - $o(tl)} = 0; (11.45)

11.2. The Two-Dimensional Case

cos

F(c)'

+sin

' {q[T1(c)7

'3

- (P[T2(c),

215

c]}

e - {I@"(C),c] - +[Tz(c),

c ] } = 0.

(11.46)

Now,put

then, according to (1 1.29), by (1 1.40), we have the following formula for the derivative of F(c):

+ XI sin e) + ( X , cos 0 + Y, sin 0) (Y, sin 8 + X , cos e)]

[(x,sin e X

- y, cos e) (- y, cos e

(x,sin 8 -

1 Y, cos e) (x12 +Y ~ ~ ) ~ / ~

Since Xz k X, and Yz = Y17the above quantity is nearly equal to (11.47)

Thus, according to (1 1.36), by (1 1.46) and (1 1.47), we now have the following iterative process :

where do)= 0. When n is orthogonal to Co at the point (qo(fl), $o(fl)),

216

11. Numerical Computation of Periodic Soktions

11.3. PEWODIC SOLUTIONS OF THE AUTONOMOUS YAN DER POL EQUATTON

In the present section, the method of the preceding section will be illustrated with the autonomous van der Pol equation dZx/dt2- A(l

- x')

dx/dt

+x

=

0

( A 1 0).

(1 1.49)

Equation (1 1.49) is evidently equivalent to the first order autonomous system dxldt = y

dy/dt = - x

+ A(1

- x2)y

(= X(X, Y)), (= Y(x, y)).

(1 1S O )

Before computing the periodic solutions, we shall investigate the behavior of the orbits. Let A be the curve

-

y = (1/A> [x/(l

- x2)Ji

then evideptly A consists of three branches A,, A+, and A- (see Fig. 18). Let Z, ZZ, ..., V, VZ be the domains of the phase plane such that

I :the domain bounded by A-, A,,, and the.negative x-axis, Z I :the domaiq bounded by the upper half of A,-, and the positive x-axis, ZIZ :the domain bounded by A+, &, and the positivtt x-axis, ZV :the domain bounded by the lower half of A,, and the negative x-axis, V :the domain consisting of points lying above A-, VI :the domain consisting of points lying below A+.

11.3. Periodic Sohtions of the Autonomous van der Pol Equation

217

Y

FIG.18 TABLE V

Domain Sign of dx/dt

Sign of dyldt

I

I1

111

IV

V

VI

+ +

+

-

-

+

-

-

-

+

-

+

Then, as is seen from (1 1SO), along the orbits of (1 1SO), we have Table V. From this table, we can easily see the behavior of the orbits of (11.50). In particular, we see that any closed orbit, if it exists, must enclose the origin, because it must cross the curve A, and the x-axis, respectively, at two points lying in the opposite sides of the origin. Assuming its existence, let C be a closed orbit of (1 1.50) (see Fig. 19).

Y

218

11. Numerical Computation of Periodic SoIutions

Then it must cross the y-axis at two points B, and B2 lying in the opposite sides of the origin 0, since C encloses the origin and it behaves as in Fig. 18. Let x = cp(t),

Y

=

+(t)

be the equation of C . Then, as is seen from (1 1SO), x =

y = -+(t)

-cp(t),

(1 1.51)

is also a solution of system (1 1.50); consequently, this also expresses a closed orbit. The closed orbit C' represented by Eq. (11.51) is symmetric with C with respect to the origin, however. Therefore C crosses the y-axis at two points B,' and B2', which are, respectively, symmetric with B, and B, with respect to the origin. Then we must have ~-

OB,

=

(11.52)

OB2.

Otherwise, one of the points B,' or B,' lies in the interior of the domain bounded by C , and the remaining one lies in the exterior of that domain. Then the different orbits C and C' must intersect each other, and this contradicts Theorem 4.1. Thus we have (1 1S2). Conversely, if (1 1.52) holds for any orbit C, then C must be closed. For, if we make a symmetric orbit C by ( l l S l ) , then we can connect both orbits C and C at the points B, and B2 by assumption (1 1.52), and thus we have a closed orbit consisting of the arc of C lying in the right half-plane and of the arc of C' lying in the left half-plane. By Theorem 4.1, this proves that C is closed. By the above results, to seek a closed orbit of the system (1 1SO), it suffices to seek an orbit such that the intersections with the y-axis are equidistant from the origin. This simplifies the general method of the preceding section for system (1 1.50). Since every orbit of system (1 1.50) crosses the x-axis orthogonally, let us denote by x = cp(t, c),

y

=

+(t, c)

(1 1.53)

the solution of (1 1.50) passing through the point (c, 0) (c > 0) at t = 0. Let B , and B2 be, respectively, the points where the orbit represented by (1 1.53) crosses the negative y-axis and the positive y-axis. Then the coordinates of the points B, and B2 are, respectively, (0, +[T,(c),c]) and (0, $[T2(c),c]), and T,(c) (> 0), T,(c) (c 0) are, respectively, determined by the equations

( 11.54)

11.3. Periodic Solutions of the Autonomous van der Pol Equation

219

These equations correspond to Eqs. (11.44) and (11.45), and the line II is now the y-axis. As has already been shown, the orbit represented by (1 1S 3 ) is closed if and only if F(c)

dz$[T,(c), c] + $[Tz(c),

c] = 0.

(11.55)

This equation with respect to c corresponds to Eq. (1 1.46). Since 8 = 4 2 in the present case, comparing (1 1.47) with (1 1.26), we have (11.56)

=

II[:J

- cp2(s, c)] ds,

(11.57)

and

If we substitute (1 1.58) into (1 1.56), we then have (1 1.59)

,

Now let c = co be the first approximation of the root of Eq. (11.55) and Put (1 1.60)

Then, since

we have

11. Numerical Computation of Periodic Solutions

220

Thus, corresponding to (1 1.48), we have the following interative process : p + 1 )

=

a p )- -.

~(c'")) c, ehl - ehz

(m

= 0,

I,

...; do)= c,).

(11.61)

In order to find the first approximation c,, we start from the case where 1is small. When A is small, in 8.4, by the perturbation method, we have seen that system (1 1SO) has a unique periodic solution close to x = 2 cos t,

y =

-2 sin t.

This implies c, = 2 can be adopted for small 1. After the closed orbit for small 1 has been computed, the value of 1is increased step by step by a small amount, and the computed value of c in the preceding step is used as the first approximation c, of c in the succeeding step. The value of c is evidently an amplitude of the periodic solution (11.53) of system (1 1SO). For the periodic solution (1 1.53), the equality

holds, because

+

x = ~ ( t T l , c) Y = $(t Tl, c )

+

and

x = -q(t

Y

= -$(t

+ T,, c ) +L C )

are both solutions of system (1 1SO), and both solutions coincide with each other for t = 0, by (1 1.54) and (1 1.55). From (1 1.62), it then follows that (11.63) '(2Tl - T2, c) = -'(TI, c) $(2Ti - Tz,C) = -$(Ti, C)

=

0,

=

$(G,c);

'(2T1 - 2T2, C) = -'(Ti - T,, C) = C, $(2T1 - 2T2, C ) = -$(Tl - T2, C) = 0.

(11.64) (11.65)

The above equalities imply that the period o of the periodic solution (11.53) is =

~ [ T , ( c-) T,(c)].

(11.66)

11.3. Periodic Solutions of the Autonomous van der Pol Equation

221

In addition, from (1 1.57), by (1 1.62)-(11.65), we have successively

= h(T1)

+ l J O [l

- q"t, c)] dt

+1

[l - y"t, c)]

dt

T2

As is mentioned in 10.2, the quantity (11.68) is the characteristic exponefit of the normal variation equation (1 1.21), and the computed periodic solution (1 1.53) is asymptotically orbitally stable if h < 0. The computed values of the amplitude a = c, the period w , and the characteristic exponent h are shown in Table VI and Fig. 20. Figures 21 and 22 show q(t, c). the shape of the closed orbits and the graph of the function x

-

222

11. Numerical Computation of Periodic Solutions TABLE VI

1

a

w

h

0 1 2 3 4 5 6 8 10 20

2.000 2.009 2.0199 2.0235 2.0231 2.0216 2.0199 2.0169 2.0145 2.0077

6.283 6.687 7.6310 8.8613 10.2072 11.6055 13.0550 16.0740 19.1550 34.7103

0.000 -1.070 -2.3822 -3.9409 -5.6194 -7.3630 -9.13 16 -12.7494 -16.3775 -34.4544

In the computation of Table VI, for step-by-step integration of the given system (1 1.50), the 5-step corrector with B- # 0 (see Appendix) was used, together with the 5-step Adams predictor. For computation of h(t), Simpson’s rule was used. For details of the computation, see Urabe [2.99]; Urabe, Yanagiwara, and Shinohara [l]; and Yanagiwara [l]. From Fig. 20 and Table VI, we see that (1) When 1is increased from 0, the amplitude a increases monotonically until about 1= 3, and it decreases monotonically after 1= 4. The maximum value of the amplitude is about 2.0235, and it is attained between 1= 3 and 1 = 3.5. According to Yanagiwara [2], the maximum of the amplitude is 2.0235, and it is attained for 1= 3.2651. (For computation of the maximum of the amplitude, see Yanagiwara [2].) (2) The period w increases monotonically with A. (3) The periodic solution is asymptotically orbitally stable for every value of 1;moreover, the stability becomes stronger as the value of A increases. In 1947, Dorodnicyn [l] gave the asymptotic expressions for a and w for large value of 1.They read u 16 log1 a = 2 + -1 413--3 27 l2

1 + -91 (3bo - 1 + 2 log 2 - 8 log 3) A2

+ qn-9,

(1 1.69)

11.3. Periodic Solutions of the Autonomous van der Pol Equation

223

--0

= (3

- 2 log 2)1 + 3u2-'l3

22 log 1 --

9

1

+ (3 log 2 - log 3 - -61 + bo - 2d) -11 + O(A-4/3), where u, b,, and d are the transcendental constants such that

a = 2.338107,

bo = 0.1723,

d = 0.4889.

(1 1.70)

224

I I . Numerical Computation of Periodic Solutions

5

I -5.0

I

-100

*10 45.0

I I .4. Remarks

225

However, the minor errors in the manipulations leading to formula (1 1.70) were found by the author [6,9] and by Ponzo and Wax [1,2]. The corrected forinula for the period o replacing (1 1.70) reads 0=

(3 - 2 log2)A

+ 3crA-'/3

-

2 log1 -3 1

+ (3 log 2 - log 3 - -3 + bo - 2d) -1 + O(A-4/3).

(11.71)

1

2

If we compare our computed values with those obtained from (1 1.69), (1 1.70), and (11.71), we then have Table VII. TABLE VII a

0

1 By us

By (11.69)

By us ~

By (11.70)

By (11.71)

~~

10

2.0145

2.0138 (-0.0007)

19.1550

18.831 ( 4 . 324)

20

2.0077

2.0077

34.7103

34.4925 (-0.2178)

19.1069

...0.25 % 34.6921 . (-O.Ql82) .

...p.os%

Table VII shows that the values computed by (11.69) and (1 1.71) are quite accurate even for A = 10. Consequently, for values of 1 larggr than 10, more accurate values will be obtained by these two asymptotic expressions, as is checked for 1= 20. Then it will be unnecessary to continue the computation of periodic solutions for further values of 1over 10 unless greater accuracy is required. 11.4. REMARKS

In Urabe [ l l ] and Urabe apd Reiter [I], the author has shown that the Galerkin procedure can be used successfully for numerical computation of periodic solutions of periodic differential systems. Let dxldt = X ( t , X)

(11.72)

be the given periodic differential system. Let w (> 0) be a period of X ( t , X) with respect to I;then, replacing t by (w/2r)t, we have the system of the same

226

11. Numerical Computation of Periodic Solutions

form as (1 1.72) of the period 271. Hence, in (1 1.72), without loss of generality, we may suppose that the period of X ( t , x) with respect to r is 2n. To compute a periodic solution of (1 1.72) of period 2x, we first consider a trigonometric polynomial N

xN(f)

= a,

+ C (a, cos mt + b, sin mt)

(11.73)

m= 1

with indeterminate coefficients (a,, a , , b,, ..., aN,bN),and then we consider a Fourier expansion of X [ t , xN(t)]:

X [ f , xN(t)]

-

m

A,

+ C (A,,,cos mt + B, sin mt).

(1 1.74)

m= 1

Then A,, A , , B,, ... are all functions of (ao, a,, b,, ..., aN,bN). Hence we determine the indeterminate coefficients (Q,, a,, b,, ..., aN,bN)so that d-X-~ ( f )- A , dt

+ c ( A , cos mt + B," sin mi), N

m= 1

that is,

A , = 0, A , = mb,,

B, = - m a ,

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

(1 1.75)

holds. If we can determine the indeterminate coefficients (u,, a , , b,, ..., aN, bN) by solving the system of Eqs. (11.75), then we can obtain a periodic approximate solution of the given system which is given by (1 1.73). This is the Galerkin procedure. For details, see Urabe [1 11and Urabe and Reiter [11. The Galerkin procedure can also be applied to the autonomous system. Let

dx/dt = X ( x )

(1 1.76)

be the given autonomous system and suppose that a closed approximate orbit

c :x = q ( r )

(1 1.77)

of (11.76) is known in advance. By a closed approximate orbit, we mean, however, that, for a certain o > 0,

+ w ) = cp(4

(1 1.78)

)d*

(1 1.79)

and = X[q(t)]. dt

11.4. Remarks

227

By the method of 5.1, we can construct a moving orthogonal system {@(t), t 2 ( t ) , ..., rn(t))(- = djdt) along C. Then, with respect to this moving orthogonal system, any orbit of (1 1.76) lying near C is expressed as (11.80)

and, for p v = pv(t) (v = 2, 3, ..., n) and z = z(t), similar to (1 1.4) and(ll.5), we have

n

-

c

u=2

(v = 293, ...,).

Ppev*4,

(11.81)

and

(11.82) where

] [ q ( t ) ] # 0 for any orbit lying near C and, Now, by (1 1.79), @*XI = X*[ ~ ( t ) X by (1 1.78), the right-hand sides of (11.81) and (11.82) are all periodic in t of period w. Evidently the closed orbit of the given system (11.76) lying near C corresponds to the periodic solution of (11.81). Hence one can obtain the closed orbit of the given system (11.76) by computing the periodic solution of the periodic system (11.81) by the Galerkin procedure. The periodic system whose periodic solution corresponds to the periodic solution of the original autonomous system may be obtained in some simpler way, however. For example, for system (11.50), a desired periodic system can be obtained in the following way. If we introduce the polar coordinates (r, 6) by

x = r cos 0,

then system (1 1.50) turns to the system

y

=

r sin 0,

228

11. Numerical Computation of Periodic Solutions

dr - = A(I - rz cos' dt

e) r sin'

8,

From this, the desired periodic system is obtained as follows: A(I - r' cos' e) r sin' e dr -d o - - 1 + A(I - r2cos2e)sin8cose'

(11.84)

Of the methods of 11.1 and the present section, which one is preferable? This depends on the form of the given autonomous system. However, in general, the method of 11.1 seems to be more convenient than the present one. This is because the reduced periodic system is usually of the complicated form, and so one usually has to take a trigonometric polynomial of very high order as the approximation of the periodic solution of the reducedperiodic system, and this usually makes it difficult to determine the coefficients of such a trigonometric polynomial.

12. Center of the Autonomous System

In the present chapter, the canonical form of the autonomous system in the neighborhood of the center will be derived. By the center, however, we mean the critical point such that every orbit passing through any point in the neighborhood of the critical point is always closed. The discussion of the present chapter is based on the paper by Urabe and Sibuya [l]. 12.1. FIRST REDUCTION

Let a given autonomous system be

+ X(X),

dx/dt = AX

(12.1)

where A is an n-by-n constant matrix. Let us suppose (i) A is regular; (ii) X ( x ) is defined in the neighborhood of the origin x = 0 and

I - x"

~ ( x ') ~ ( x " )= o( x'

[)

as

x', x"

-,0;

(12.2)

(iii) the origin is a center of system (12.1); (iv) the primitive period w(t) of the orbit passing through an arbitrary point x = 5 near the origin is bounded. Let x = q(t, t) be a solution of system (12.1) such that q(0, () =

5

for small

It I .

(12.3)

Then, from (12.1) and (12.3), we have rp(4

t) = t +

1:

{A&

t) + x[(P(s,t)]) ds

for any t. By assumption (12.2), we then have (12.4) 229

230

12. Center of the Autonomous System

for some positive constant L. Put (12.5) then, from (12.4), for t

1 0, we have duldt S Lu

+ I t 1,

that is, duldt - Lu S

I t I.

(12.6)

Multiplying both sides by e-IL and integrating both sides from 0 to t, we have

that is,

Substituting this into (12.6), from (12.5), we have

I q(t, t) 1 5 e t LI t I

for t 2 0.

However, by assumption, q(t, t) is periodic in t of period o(t) and, moreover, the period w(t) is bounded. Hence, for any t and any small 5 we have

I I,

I d t , 0I 6 L1 I 5

(12.7)

I 3

where L , is a positive constant such that

L1 > = eLa(r)

for any small

I t 1.

Now, similar to (6.10), for x = q ( t , t), we have q(t, t) = etAt +

j;

e(f-S)AX[(p(s, t)] ds.

(12.8)

Hence, from the periodicity of q ( t , t), we have

[ E - ea(t)-4]5 =

j

mu(<)

0

eC43-s1A.

x

[cp(s7

t)l ds.

(12.9)

12.1. First Reduction

However, by (12.2), for any positive number 6 such that

I X(x) I 5 E I x I

whenever

E,

23 1

there is a positive constant

I x 1 5 6.

(12.10)

Hence, when

I t I 5 SlLi,

(12.1 1)

from (12.7), (12.9), and (12.10), we have

I [ E - em(c)A]tI 5 O ( ~ ) K E LIt, 1.

(12.12)

Here K is a positive constant such that

I etA I 5 K

for 0 5 t

5 a(<).

Such a constant K evidently exists, since a(<)is bounded by assumption. In (12.12), put 1 = tl 15

1;

(12.13)

then we have (12.14)

Let us rewrite this as follows:

where 1 FA(t) = ; ( E - etA) = - ( A

+ -tA A 2 + -t2A 3 + ...). 2!

3!

(12.16)

Now let us make I t I + 0 for an arbitrary fixed I, and let a, be any one of the limits of w(t). Then, since o(t) is bounded, a, is finite and, from (12.15), we have

I F A ( a O ) l I 5 &=I, which implies that h ( a o ) Z = 0,

(12.17)

since E is an arbitrary positive number. Since I is a unit vector, equality (12.17) implies det FA(ao) = 0.

(12.18)

12. Center of the Autonomous System

232

By (12.16), this clearly implies 0 0

z 0,

(12.19)

since det A # 0 by assumption (i). Equality (12.18) is equivalent to det (emoA- E ) = 0.

(12.20)

Now let

(12.21)

be the Jordan canonical form of the matrix A ; then the A's are evidently the eigenvalues of A, and, from (12.20), we have emoa -1

o

O...

=0

det (emoAo-E ) = TI

(12.22)

(see (1.12)). From (12.22), it follows then that em"*

=1

for at least one of the eigenvalues rl of the matrix A . Since any rl is not zero and a(<)is bounded, the possible values of wo are then finite in number; and the equations of the form (12.17) for all possible values of wo are finite in number. However, an equation of the form (12.17) holds for an arbitrary unit vector 1. Hence, for at least one of the possible values of wo, it must be that (12.23)

233

12.1. First Reduction

othemise, the solutions of each equation of the form (12.17) cohstitbtg 4 linear manifold of the dimensian less than n ; consequently, there are uqlf vectors 1 that do not satisfy any one of the equations of the form (12.17) for all passihle values of 0,. Equality (12.23) implies ewoA= E.

(12.24)

Then, by [12.21), we have pOa

a

.

= E,

(i2.25)

t

for every eigenvalue A of A. By (12.19) the above relation implies that the Jordan canonical form A. of A i s of diagonal form and that every eigenvalue of A is pure imaginary. Sinae is a real regular matrix, the order n of A must then be even, and Ab must be of the form (2 m

=

n),

(12.26)

k= 1

where, without loss of generality, we may suppose that &>O

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

(12.27)

As is seen from the proof of Theorem 1.2, matrix A then becomes similar to the matrix (12.28) Hence, if we put A l = T-'AT,

(12.29)

234

12. Center of the Autonomous System

then, by the transformation x = Tx', the given system is transformed to a system of the following form: dXk/dt =

-ekxk*

dXk,/dt =

ekXk

-k x k ( x ) ,

( k = 1, 2,

+ xr(X)

..., m).

(12.30)

Here, for simplicity of description, the primes of x are all omitted. It is evident that equality (12.2) holds also for the present X(x). Let x = q(t, C;) be a solution of system (12.30) such that q(0, C;) = e. Then, similar to (12.8), we have q(t, C;) =

erA1t+

sd

e('-S)AIX[q(s,

C;)] ds.

(12.31)

However, in the present case, by (12.28),

(12.32) Hence, from (12.3l), we have :

I I + 0, because, by (12.7) and (12.10),

uniformly with respect to t as 5

I C; I ) uniformly with respect to t and the direction of C; as I C; I

(12.34)

J:e('-s)AIX[q(s,C;)] ds = o(

+

0.

12.2. THE UNIVERSAL PERIOD OF THE ORBlT

By (12.29) and (12.32), equality (12.24) implies that

(12.35) for some positive integers O,, a, ..., 0,.The above relation' evidently implies that el, 02, ..., 0, are relatively commensurable. Now, since o(C;)is the primitive period of the periodic solution x = q(t,C;), from ('12.31), we have

(12.36)

12.2. The Universal Period of the Orbit

235

Here let us put p =

It I

and

t

(12.37)

= pl;

then, from (12.36), we have [ea(pl)AI

- EJ

+

lo

d P U $‘“pl)-sl”l

1

- X[cp(s, pl)] ds = 0 .

(12.38)

P

However, from (12.34),

uniformly with respect to I as p

-+

0. Therefore, from (12.38), we have

[eo(pl)”l- 1311 + 0

(12.39)

uniformly with respect to 1 as p + 0. Now, in the phase space, let us consider the cones V,, V,, sisting of the lines passing through the origin such that 11,

I+

111,

...,

I I , I + I 11, I >= l/m I < l/m, 112 I + I 1,’ I >= l/m

. . . . . . . . . . . . . .

I I , I + I 1, . .., I z ~ -I ~+ I z(,-,), I < I/m, I I,,, 1 + I I,. I L I/m

V, confor V , , for V, , for

v,.

Since 11

I = 14 I + 14,I + 11’ I + 11,. I + ’.’ + I I m I + I, ,z

I

=1

for any line passing through the origin, the phase space is evidently covered by the cones V,, V,, ..., V,. Then, for any point t = pl in the phase space, there is a cone Vk containing the point [ = pl and, from (12.39) and (12.32), for such a point [ = pl, we have [cos 8kw(pl) - 11 l k - sin 8kw(pl) ’ lr + 0, sin 8kw(pl) * l k + [cos 8kw(pl) - 11 ’ l k , + 0 uniformly with respect to I as p

-+

+

0. From (12.40), we then have

{[cos 8ko(pl) - 11’ sinZ8ko(pl)} l k {[cos 8kw(pl) - 13’ sin2 8ko(pl)} l k , uniformly with respect to 1 as p {[cos 8kw(pi)- 11’

+

+

(12.40)

0

0, Then,

+ sin’ ekw(pi))( I l k I + I lk. I ) + o

(12.41)

12. Center of the Autonomous System

236

uniformly with respect to 1 as p -, 0. Since

I l k I + I lk’ I 2 l/m in

vk,

expression (12.41) thus implies [cos 8 k o ( p l ) -

11’

+ sin2 &o(p1) -, o

(12.42)

uniformly with respect to I as p + 0. Let wo be any one of the limits of o(p1); then, from (12.42), we have (cos ekwo - 1)’

+ sin’ @#* = 0.

This implies wo must be the number such that (12.43) where n k is a positive integer. However, since w(C) is bounded, the positive integer n k in (12.43) is bounded. This implies the limiting values of w(p1) are finite in number. Then, from the uniform convergence of (12.42), we readily see that, in v k , w(p1) converges uniformly to one of the values of the form (12.43) as p -+ 0; in other words, for sufficiently small p, w(p1) corresponding to any point of v k lies in the disjoint neigborhood of the numbers of the form (12.43). Then, if we take the least common multiple Nk of all n,‘s appearing in (12.43), we can take a positive integer &(pi) corresponding to any point of v k SO that 27T Lk(Pz)O(pz)

(12.44)

k

uniformly with respect to 1 as p -, 0. Now, as is proved already, O,, 8,, . . ., 8, are relatively commensurable; therefore we can take relatively prime positive integers MI, M2,. ., Mm SO that

.

Then, if we put

from (12.44), we see that

12.2. The Universal Period of the Orbit

237

uniformly with respect to I as p + 0. Expression (12.46)is evidently valid for any point in the neighborhood of the origin, and there the positive integer kfkLk(p1) is uniquely determined corresponding to point ( = p l in the neighborhood of the origin. Hence, putting = K(pl),

MkLk@z)

we can write (12.46) as follows:

K ( p l ) o ( d ) --b

(12.47)

6 0

uniformly with respect to I as p + 0. Expression (12.47) means that the period K(()o(() of the orbit lying in the neighborhood of the origin is continuous at the origin. In that follows, the period K(()o(c) will be called the universal period of the orbit. The above result shows that in the neighborhood of the center, the orbit always has a universal period continuous at the origin although the primitive period itself of the orbit may not be continuous at the origin. In illustration, consider the system

-d x-, - dt

211

+ (X12 + x:. + x2, + x:.) = - 2nx1, + X,(X), 1

2n

dx,.

-- dt

+ (x12 + x:. + xz2 + x:.)"' = 2nx, + x,,(x), 1

-dx2 --dt

X1'

n

+ (x12 + x:. + X t 2 + x:f" = - nxz' + x,(x), 1

dx,. -- dt 1

lc

+ (xlZ+ x:, + x22 + xit)"' = nxz + x,.(x).

The general solution of (12.48) is evidently x1 = acos

2n

1

+ a' + bZ

(12.48)

12. Center of the Autonomous System

238

7L

x2 = bcos

1

x2, = b sin

(12.49)

+ u2 + b2 7L

+ u2 + b2

(1

where a, by a l , and ct2 are arbitrary constants. As is readily seen from (12.49), the primitive period of the orbit is 1 u2 b2 or 2( 1 u2 + b2), according as b = 0 or # 0. Thus, in the present example, the universal period of the orbit is

+ +

2(1

+ u2 + b')

=

2(1

+

+ xI2 + x:, + x2' + x:.).

Remark. When n = 2, system (12.30) can be rewritten as follows:

+

dxldt = - By X(x,y), dyldt = Ox Y ( x , y).

+

(12.50)

If we introduce polar coordinates ( r , cp) by y = r sin cp,

x = r cos cp,

then, from (12.50), we have: drldt = X ( r cos cp, r sin cp) cos cp

+ Y ( r cos cp, r sin cp) sin cp, dcpldt = 8

(12.5 1)

+ 1r [- X ( r cos cp, r sin q ) sin cp -

+ Y ( r cos cp, r sin cp) cos cp]. Since

I I

X(r cos cp, r sin cp), Y ( r cos cp, r sin cp) = o( r )

( Ir

I

--*

0)

(12.52)

by (12.2), from the second of (12.51), we have dcp/dt

I I.

=

8>0

for sufficiently small r Then, in a similar way as in Remark of 7.2, we see

12.3. The Canonical Form in the Neighborhood of the Center

239

that the closed orbit of (12.50) is represented by the solution of the differential equation dr d'

+

-=

8

+

X ( r cos cp, r sin cp) cos cp Y(r cos cp, r sin cp) sin cp ,(12.53) 1 - [- x ( r cos cp, r sin cp) sin cp Y(r cos cp, r sin cp) cos cp] r

+

and the primitive period o of such a closed orbit is given by w =

lo2= 8

+

-

dcp

[ - X ( r cos cp, r sin cp) sin cp

+ Y(r cos cp, r sin cp) cos cp] (12.54)

r

for r = r(rp),which is a solution of (12.53). Now, by (12.7), r = r(cp) converges to zero uniformly as the closed orbit approaches the origin. Hence, from (12.54), by (12.52), we readily see that w+-

2.n 8

as the closed orbit approaches the origin. This implies that when n = 2, the

primitive period is itself a universal period.

12.3. THE CANONICAL FORM OF THE AUTONOMOUS SYSTEM IN THE NEIGHBORHOOD OF THE CENTER

Let w(5) be the universal period of the closed orbit of (12.30) passing through the point x = c. Then, from the definition of the universal period, a(<)is of the form

w(T) = wo

+ 41)

as

c -,0,

(12.55)

and, from (12.49, wo is of the from 2n0, - 2n0, 81 82

fJjo=---

where 0,, 02,. . ., 0,are the positive integers. Let us now consider the functions

(12.56)

240

12. Center of the Autonomous System

( k = 1,2, ..., m), where x = q(t, t) is a solution of system (12.30) such that q(0, t) = t. Since

by (12.55) and (12.56), from (12.33), we then have

Likewise we also have

Hence, summarizing these, we have

W )= r + (. I t I )

(t

+

0).

(12.58)

Now, from the definitions of o(t) and q(t, t), it holds that o[q(t, q[t, tP(%

t)] = w(t), t)] = tp(t + s, t).

(12.59)

(For the latter, see Theorem 4.4). Hence, from (12.57), we have successively

12.3. The Canonical Form in the Neighborhood of the Center

241

Formulas (12.60) say that by the transformation Y = F(4,

(12.61)

the solution x = q(t, 9 ) of system (12.30) is transformed to the function y = $(t, q) such that

where q = F(<)

and

8(q) = ~ ( 5 ) .

(12.63)

From the first of (12.59) and the second of (12.63), it is clear that Q[$(t,

41 = .“)(?>.

(12.64)

242

12. Center of the Autonomous System

Hence, differentiating (12.62) with respect to t, for y = $(t,

ti‘),

we have

namely, 2x0, -dyk- - - _ _Y k ’ dt

9

(k

G(y)

dy,. 2n0, - -dt

G(y)

=

1,2, ..., m).

(12.65)

’”

Since $(O, q) = q, the above result shows that, by the transformation (12.61), the solution x = q ( t , 9 ) of system (12.30) such that q(0, t) = 5 is transformed to the solution y = $(t, q) of the systems (12.65) such that $(O, q ) = q. This means that system (12.30) is itself transformed to system (12.65). System (12.65) is the desired canonical form in the neighborhood of the center. In system (12.65), G(y) is evidently an invariant of the differential system, as is seen from (12.64). This means that the dzferential system (12.65) is linear for the solution of the system. System (12.65) becomes a true linear system when G(y) is a constant. This means that the initial given nonlinear system (12.1) is transformed to the linear system of the form 27101, dt

Yk,

9

On

by a suitable transformation of the variables if the universal period of the orbit of the given system is constant. The general solution of system (12.66) is evidently

12.3. The Canonical Form in the Neighborhood of the Center

243

where a,, ak (k = 1, 2, . . ., m) are arbitrary constants. The primitive period of the orbit such that

a,

=

...

,

,

... = a ,

= ak. = ak+ =

=0

and

ak # 0

is then wo/@k. Now, when the primitive periods of the orbits are all equal, the primitive period is itself a universal period. Hence, in such a case, we have wo - - = wo

0,

... = wo = O o ,

0 2

e m

that is, 0 , = o2=

... = 0,

(12.68)

= 1.

From this, we readily see that when the primitive period of the orbits are all equal, the given nonlinear system (12.1) is transformed to a linear system of the form dyk -dt

2x

-Yk' On

9;

(1 2.69)

by a suitable transformation of the variables. In this case, from (12.56), it is readily seen that 0, = g ,, =

... = e m def ( = 0)

and wo =

271

-. e

(12.70)

Remark. When the universal period of the orbit is constant, the function F(5) defined by (12.57) is continuously differentiable with respect to 5, provided X ( x ) of the given system (12.1) is continuously differentiable with respect to x . If the function X ( x ) of system (12.1) is analytic with respect to x , one can prove that the universal period w(5) becomes analytic with respect to 5. Hence, .in such a case, the function F(5) defined by (12.57) also becomes analytic with respect to <.For the details of this case, see Urabe and Sibuya [l] and Sibuya [l].

13. Inverse Problems Connected with Periods of Oscillations Described by x + g(x) = 0

13.1. PRELlMlNARES The equation d2x dt2

+ g(x) = 0

(13.1)

is evidently equivalent to the first-order differential system dx _ -- Y , dt (13.2) dY = - g ( x ) . dt

In the present chapter, we suppose that g(x) is continuous in the neighborhood of x = 0. If xg(x) > 0

for x # 0,

(13.3)

then system (13.2) evidently admits of a family of closed orbits y2

+

jox

g(u) du = const.,

(13.4)

and hence the origin becomes a center of system (13.2). Let (0, R), (a, 0), and (6,0) be the points where the closed orbit (13.4) cuts the positive y-axis, the positive x-axis, and the negative x-axis, respectively. Then, to the closed orbit (13.4), there corresponds a periodic motion described by Eq. (13.1), and R, a, and b express, respectively, its maximum velocity, maximum positive displacement, and minimum negative displacement (see Fig. 23). The amplitude of such a periodic motion is evidently given by A = +(a - b). 244

(13.5)

13.1. Preliminaries

245

Y

t FIG. 23

In what follows, a and b will be called, respectively, the positive halfamplitude and the negative halflamplitude. Let z1and z2 be, respectively, the time required to reach the point (0, - R) from the point (0, R) through the point (a, 0) and the time required to reach the point (0, R) from the point (0, - R) through the point (by 0). In the sequel, these z1 and z2 will be called, respectively, thepositive-side half-period and the negative-side half-period. Let w be the primitive period of the closed orbit (13.4); then it is evident that = 71

+

72.

(13.6)

Clearly w, zl,and z2 are functions of R, or a, or b. These functions will be called the period functions. Our concern in the present chapter is to find the relations between the period functions and the function g(x). Put (13.7) then Eq. (13.4) of any closed orbit of system (13.2) is written as follows: $yZ

+ G(x) = G(a) = G(b) = $R2.

From the first of (13.2), we then have

(13.8)

246

13. Inverse Problems for x -

z2 = Z2(b) = J 2 jbo

o = w(R) = 2

jir

+ g(x) = 0

dx [G(b) - G(x)]””

(13.9)

dx [ R 2 - 2G(x)]”’ ’

................................. where a(R) is a positive branch of G-’(R2/2),and b(R) is a negative branch of G-’(R2/2). When g(x) is given in advance, the function G(x) is easily calculated; hence, by means of (13.9), one can easily find the period functions. However, when g(x) is not known in advance, it is not easy to determine the function g(x) by means of (13.9) for a specialized period function given in advance. We are concerned here with finding the relations between the period functions and the function g(x) that enable us to determine the function g(x) for a given period function. Our problems are, consequently, the inverse problems connected with the various periods and the problem of isochronism, that is, the problem of finding g(x) for which o(R)= const. is a special case of our general problem. The results of Chapter 12 might be applicable to our present problems; but, in reality, it is not convenient to apply them here because the given system is of the particular form of (13.1) and not the general form of Chapter 12. Such being the case, in the present chapter, the problems will be investigated independently of Chapter 12. The approach is based on the author’s paper [lo].

13.2. LEMMAS In this section, some lemmas necessary to the subsequent discussion will be proved.

Lemma 13.1. Let g(x) be a continuousfunction differentiable at x = 0. Then the necessary and suficient condition that any orbit of the system (13.2) may be closed around the origin with a bounded primitive period is that xg(x) 0 for x # 0 and g’(0) # 0 (‘ = d/dx).

=-

PROOF. The necessity of the condition will be proved first. By assumption, the origin is an isolated center of system (13.2). Therefore

13.2. Lemmas

247

it must be that g(0) = 0 and g(x) # 0 for x # 0. Then it must be that xg(x) > 0 for x # 0, because otherwise there will be a nonclosed orbit of system (13.2) of the form +y2 + G(x) = 0.

To prove g’(0) # 0, let us introduce polar coordinates (r, 8) by x = r cos 8 ,

y = r sin 8.

Then, from (13.2), we have - cos 8

dr dt

- r sin 8 - = r sin 0,

d8 dt

dr - sin 8 dt

+

I

d8 cos 8 - = -g(r cos8); dt

hence,

+ g(r cos 8) cos e) .

d0 = -(sin28 dt

r

Then the primitive period o of any closed orbit is given by

d8 X

d8

sin 2 8

+g ( 4 cos2 8 X

d8

= 2j;

sin2 8

g(x> cos2 8 ’ +X

(13.10)

where x = r cos 8. Since g(x)/x > 0 for x # 0, CIJ

> 2j:-’

d8 sin2 8

+g ( 4 cos2 8 X

for any small positive number E . Then, if g’(0) = 0, g(x) /x -, 0 as x + 0, namely, as the closed orbit converges to the origin. In the limit, we then have

248

13. Inverse Problems for x

+ g(x) = 0

This contradicts the requirement that the primitive period be bounded, because cot E can be arbitrarily large if E is taken sufficiently small. Thus we haveg’(0) # 0. The sufficiency of the condition readily follows from (13.4) and (13.10). Thus we see the validity of the lemma. Q.E.D. Let g(x) be a continuous function differentiable at x = 0 such that xg(x) > 0 for x # 0 and g’(0) # 0. Then g’(0)

> 0,

(13.11)

and, by Lemma 13.1, any orbit of system (13.2) is closed around the origin. Let w be a primitive period of such a closed orbit and wo be a limit of w as the closed orbit converges to the origin. By (13.10), we then have

(13.12)

This fact will be used later. The following two lemmas are concerned with the integral equations that will appear in the subsequent discussion.

Lemma 13.2. Consider the integral equation T(X)

loR (R2 - X2)l12

dX

=f(R)

( R > 0).

(13.13)

Whenf(R)is continuous for R 2 0, the solution T ( X )of (13.13) continuous for X 2 0, if it exists, is determined uniquely and is given by (13.14)

for X > 0.

13.2. Lemmas

249

Conversely, fi the function T ( X ) defined by (13.14) is continuous for X 2 0, then this is indeed a solution of (13.13) for f ( R ) continuous for X 2 0. Whenf ( R ) is continuously diflerentiable with respect to R for R 2 0, equation (1 3.13) always has a unique continuous solution which is given by

+

T(X) = 2 -f(O) 71

(1 3.15)

for X 2 0. Solution (13.15) is the one derived from (13.14) by integration by parts.

PROOF. If we put X2 =

5

R2 = x ,

and

Eq. (13.13) is transformed to

(13.16) This is the special case of the equation of the following form:

Equation (13.17) is, however, similar in form to Abel's integral equation

(see, e.g., Whittaker and Watson [l], p. 229). Hence, in a way similar to that for Abel's integral equation, by making use of the identity

s' e

dx (z - x)'-"(x

71

- 5)"

=--

sinp

(t <

4

9

(13.18)

(see, e.g., Whittaker and Watson [l], p. 229), one can find the solution of Eq. (13.17). Before solving Eq. (13.17), let us note that the integral

is continuous with respect to x for x 2 0, provided u ( t ) is continuous for

250

13. Inverse Problems f o r x

+ g(x) = 0

2 0. In fact, by the mean-value theorem and identity (13.18), for x > 0, we have

for some q such that 0 S u(Q, we have

r]

5 x. Then, letting x

-+

0, from continuity of

(13.19)

Because of this property, we shall define the value of I ( x ) for x = 0 as follows: I(0)

=

u(0).

7 7c

sin p n

Expression (13.19) then shows that I ( x ) is continuous at x = 0. To prove the continuity of Z(x) at an arbitrary point xo > 0, let us consider the difference

Supposing x > xo, let us rewrite the above difference as follows:

(13.20)

For the integrals on the right-hand side, by the mean-value theorem and identity (13.18), we have successively:

13.2. Lemmas

251

Hence, from (13.20), we have

Z(x) - Z(x,)

+

as x

0

+

+ 0.

(13.21)

xo - 0.

(13.22)

xo

Interchanging x with x,, we see also that as x

I(x) - I(xo) + 0

+

Expressions (13.21) and (13.22) imply that

Z(X) - Z(xo) + o

as x

-,x,,

which proves that Z(x) is continuous at x = x, > 0. From the continuity at x = 0, which was proved first, we thus see the integral

is continuous for x 2 0. Now let us proceed to the solution of Eq. (13.17)First, suppose Eq. (13.17) has a solution u(t;) continuous for 5 2 0 corresponding to given q(x) continuous for x 2 0. Then, from (13.17), we have

for any z > 0. On the left-hand side, we can invert the order of integration by Dirichlet’s formula (see, e.g., Whittaker and Watson [l], p. 77), and thus we have

jz

0 t’-p

dc

fi

(z

dx

- x)’-”(x

- ty

=

s’ 0

’(’) dx. x)’-”

(z -

(13.24)

13. Inverse Problems f o r x

252

+ g(x) = 0

By (13.1S), we then have (13.25) which implies (13.26)

r

for > 0. This proves that any solution u(5) of (13.17) continuous for 2 0, if it exists, must be the one given by (13.26). Second, suppose the function u(5) defined by (13.26)is continuous for 5 2 0. Then, from (13.26), we readily obtain (13.25) because

due to the continuity of q(x). By (13.18), equality (13.25) implies (13.24) and hence (13.23) by Dirichlet’s formula. Equality (12.23) can be written in terms of

as follows:

Here I ( x ) and q ( x ) are both continuous for x 2 0. Then, making use of Dirichlet’s formula and identity (13.18) again, for any 5 > 0, we have successively

Namely, we have (13.27)

13.2. Lemmas

253

for any > 0. Since Z(x) - q ( x ) is continuous for x evidently implies Z(x) = q ( x ) for x 2 0.

2 0, equality (13.27) (13.28)

By the definition of Z(x), this proves that the function u(<) defined by (13.26) for 5 0 is indeed a solution of Eq. (13.17) if u(5) is continuous for 5 2 0 and p ( x ) is continuous for x 2 0. Now let us apply the obtained results to Eq. (13.16). For this equation,

=-

p = -

1 2’

.(t)

=

T(Jt),

P(X) = 2 f ( J x ) ;

therefore, for Eq. (13.16), corresponding to equality (13.26), we have

(13.29) for

5 > 0. Put

JF

=

x

and

la f(G) 0

(t - x),’,

d x = F(<);

then equality (13.29) is written as follows:

2 T(X)= -X

*

F’(X2)

71

1

d dX

= -.-F(X2). 71

(‘

=

d / d 5) (13.30)

Since

equality (13.30) is equivalent to

This is nothing but (13.14). Thus, from the results obtained for Eq. (13. we obtain the first two conclusions of the lemma. When f ( R ) is continuously differentiable for R 2 0, by integration

13. Inverse Problems f o r

254

x + g(x) = 0

parts, equality (13.14) is transformed to equality (13.15). The function defined by (13.15) is continuous for X 2 0, however, because the integral

can be rewritten as JE’2 f ’ ( X sin cp) d q , and f’(R) is continuous for R 2 0, by assumption. Hence, from the first two conclusions of the lemma, the final conclusion readily follows. This completes the proof. Q.E.D.

Lemma 13.3. Consider the integral equation

T(X)=

X -

27c

1 + T(R) ( X 2 - R2)1/2

jo

(1

1

+ T(u))du

loR

d R , (13.31)

where oois a positive constant, and F(a) is a given function defined on I[O, I] ( I > 0) satisfring a Lipschitz condition

I F(a’) - F(a”) I s L I a‘ - a”

I

( L > 0)

(13.32)

for every a’, a“ E I . Then the integral equation (13.31) has one and only one continuous solution T ( X ) on J[O, a] (a> 0), provided a is suficiently small.

PROOF. Let M be a positive number such that

I

M 2 F(a)

I

for every a € I,

(13.33)

and let k be an arbitrary positive number less than one. We choose the positive number a so that

and we shall prove that Eq. (13.31) has one and only one continuous solution T ( X )on J[O, a]. To begin with, we consider the iterative process:

T o p ) = 0,

x F

[zjoR

(1 + T,,(u))d u ] d R

( t ~=

0,1, 2, ...). (13.35)

255

13.2. Lemmas

First, we shall prove that this iterative process can be continued indefinitely on J[O, a ] , that T,,(X)( n = 0, 1, 2, . . .) are continuous on J, and that

I T,,(X)I 5 1

(n = 0, 1,2,. . .).

on J

(13.36)

For n = 0, evidently T,,(X)is continuous and it satisfies inequality (13.36) on J . Let us assume that the iterative process (13.35) is continued up to the nth step, that T,,(X)is continuous on J, and that

1 T,,(X)\S

1 on J .

(13.37)

Then, by (13.37) and (13.34), we have

05

0.1 R

2n

for

[l

+ Tn(u)]du S 00u S

1

(13.38)

71

0

R E J. Consequently, by (13.33), we have

1 loR+ (1

F

T,,(u))du]

1

for R E J.

S M

Then, by (13.35), (13.37), and (13.34), we have

+

for X EJ. This proves (13.36) for n 1. Now the integral on the right-hand side of (13.35) can be rewritten as follows :

x 271.

I

nI2

[l

+ T,(X sin cp)]

Lloxsin'+

F 2

(1

T',(u)) du

1

dq.

Consequently, the continuity of T,,+l(X) follows from the continuity of T,(X) and F(a). This completes the induction, and thus we see that the iterative process (13.35) is continued indefinitely on J[O, u] and that Tn(X)(n = 0, 1,2, . . ,) are continuous on J and satisfy inequality (13.36) on J . Second, we shall prove that the sequence (T,(X)} (n = 0, 1,2, . . .), obtained above, converges uniformly on J[O, 1x1.In fact, from (13.35), we have

13. Inverse Problems for x

256

-F

[" 271

loR(1

+ g(x) = 0

+ Tn-'(u)) d u ]

)

dR.

Hence, by (13.33), (13.32), (13.36), and (13.34), we have

S k hmax I T'(X)

-

T',-'(X)

I

XEJ

for X E J. This implies

I

1

I

max T , + , ( X )- T,(X) 5 k * max T,(X) - T,-,(X) XEJ

X d

I

(n = 1,2, ...). (13.39)

Since k < 1, this proves the uniform convergence of the sequence {Tn(.Y)} (n = 0,1,2, . .) on J[O, a].

.

The existence of a continuous solution of the given integral equation (13.31) on the interval J follows from the uniform convergence of {Tn(X)} on J. In fact, by the uniform convergence of {Tn(X)}on J, the continuous function

T(x) = lim T,(x)

(13.40)

n- m

exists on J and, by (13.36),

I T(x)~s 1

on J .

Then, similar to (13.38), we see that the integral

(13.41)

13.2. Lemmas

257

exists for X E J a n d , by (13.35), we have

-

Here the first difference in the right-hand side is estimated as for Tn+,(X) T,(X). Hence we have

loR+

T(u))du d R - T ( X )

6 k - max I T ( X ) - Tn(X)I +

I Tn+ ( X ) - T ( X ) I.

(1

XEJ

1

1 (13.42)

Since the sequence {Tn(X))(n = 0, 1,2, . . .) converges unifomly to T ( X ) on J , the right-hand side of (13.42) tends to zero as n -+ 00. This implies that

X

% JO

+

1 ‘TC(R) ( X z - R2)”’

(1 joR

1

+ T(u))du

d R = T ( X ) , (13.43)

which shows that T ( X ) is a solution of the given integral equation (13.31), thus proving the existence of a continuous solution of (13.31) on J. Last, we shall prove the uniqueness of a continuous solution of (13.31) on J. Before going to the proof, however, let us note that

I T(X)I < 1

on

J’[o, u’],

(13.44)

where u‘ is an arbitrary positive number less than a. In fact, from (13.35), (13.36), and (13.34),

1 for X

E J and,

I 7lo u’M

Tn+l(x)

5

U‘M dR -< 1 (xz- R2)1/2 - 2

by (13.40), this implies

13. Inverse Problems for x

258

+ g(x) = 0

on J‘, which proves (13.44). NOWwe shall prove the uniqueness of a continuous solution of (13.31) on J. Let ?(X) be an arbitrary continuous solution of (13.31) defined on 3[0, d ] (4 > 0). Then, from (13.31), we have

T(x)= 2nSo (Xz- TR2)’’* ( R ) F P 27l2O R ( 1+ i ( u ) ) d u ] d R -

=

5 271

+

[I

] [;)

+ T ( X sin q)

F

(1

-

+ T(u))d u ] d q

for X E 3. Therefore, letting X + 0, we see that ?(O) = 0. From the continuity of T ( X ) , there is then a positive number P 5 min (or‘, &) such that

I T(x) I < 1

[o,p].

on

(13.45)

Then, as for T,+,(X) - T,(X), we have

that is,

(1 - k) max [ T ( X ) - T ( X ) I 5 0. W>81

Since k < 1, this implies

T ( x ) = T ( x ) on

[O,P].

Then, by (13.44),

I T(P)I

< 1.

Since p ( X ) is continuous on [0, min (a’, a)], we can then replace the number fl by a larger number if P < min (a’, a). This implies that

I T(x) I

<1

[O, min(a’, a)]

on

and hence that

T(x)= T ( x )

on

[O, min(a’, d)].

When d < a, equality (13.46) implies that F(X) = T ( X ) on

[O,

a]

(1 3.46)

(1 3.47)

because a‘ can be chosen so that d < or’ < a. When d 2 a, equality (13.46) implies

T(x) = T(x)

on

[0, a’].

(13.48)

13.3. The Period Function Associated with the Maximum Velocity

259

However a' can be chosen arbitrarily close to a ; therefore, by the continuity of ? ( X ) and T ( X ) , from (13.48), follows ?(x)= T(x) on [o,a]. (13.49) Equalities (13.47) and (13.49) show that, on the interval J[O, a ] , there is no continuous solution of (13.31) other than T ( X ) . This proves the uniqueness of a continuous solution of (13.31) on J and completes the proof of the lemma. Q.E.D. 13.3. THE PEFUOD FUNCTION ASSOCIATED WITH THE MAXJMUM VELOCITY Concerning the period functions w we have the following theorem.

=

8(R), t 1 = z",(R),and

t 2=

z",(R),

Theorem 13.1.

Let g(x) be a continuous function diferentiable at x = 0. If any orbit of system (13.2) is closed around the origin with a bounded primitive period w, then (1) there are an even continuous function T ( X ) vanishing with X and an odd continuous function S ( X ) such that 2n X (1 3SO) g (1 S ( U ) T( u ) )du = wo 1 s(x) T ( X ) '

[zjox+

1

+

- +

+

where wo is the limit of w as the closed orbit of (13.2) converges to the origin; (1 3.5 1)

(1 3.52)

where R is the maximum velocity and t 1and t2 are, respectively, the positiveside half-period and the negative-side half-period; (3) the functions 6 ( R ) ,z",(R),and f 2 ( R )are all continuous for R 2 0 and

q o ) = 0 0 > 0,

q ( 0 ) = QO)

def

=

w,/2 (=

> 0;

to)

(13.54)

260

13. Inverse Problems f o r x

(4)

1 d T(X)= oo* dX

s(x)= 22, .-dX -

+ g(x) = 0

G(R) - 0 0 ( X z - R 2)112

10

dR,

- f2(R)RdR (X’ - R2)11’

(13.55) (1 3.56)

for X > 0 ; (5)

(1 3.55 ’) (13.56‘)

S(X) =

for X 2 0 i f G ( R ) and f , ( R ) - f 2 ( R )are both continuously differentiable. Conversely, consider an arbitrary continuous function G(R) such that G(0) = W O > 0. ZfG(R) is continuously differentiable or the even function T ( X ) satisfying (13.55) for X > 0 is continuous in the neighborhood of X = 0, then, for an arbitrary continuous odd function S(X), there is determined uniquely the continuous function g(x), differentiable at x = 0, for which any orbit of system (1 3.2) is closed around the origin with theprescribedprimitiveperiod w = G(R). Such g(x) is determined by (13.50) for the continuous even function T ( X ) satisfving (13.55) or (13.55‘), respectively, for X > 0 or X 2 0. In case there are given continuous functions F,(R) and f2(R)such that fl(0) = fz(0)= to > 0, if f , ( R ) and f 2 ( R )are both continuously differentiable, or if the even function T ( X ) satisfying (13.55) for X > 0 (in this case G(R) = ?,(R) ?,(R)) and the odd function S ( X ) satisfying (13.56) for X > 0 are both continuous in the neigborhood of X = 0, then there is determined uniquely the continuous function g(x), differentiable at x = 0, for which any orbit of the system (13.2) is closed around the origin with the prescribed half-periods ~1 = f , ( R ) and T~ = f z ( R ) . Such a g(x) is determined by (13.50) for the continuous functions T ( X )and S ( X ) such that T ( X )is an even functionsatisfying (13.55) or (13.55‘), respectively, for X > 0 or X 2 0, and S ( X ) is an odd function satidying (13.56) or (13.56’), respectively, for X > 0 or X 2 0.

+

PROOF. To begin with, we shall prove the first half of the theorem. By Lemma 13.1, g(x) is of the form g(x) = c’[x

+Wl,

where c > 0 and I(x) = o(x) as x + 0. Then, by (13.7),

(13.57)

13.3. The Period Function Associated with the Maximum Velocity 261

[ + 3 :j

!f

G(x) = 2 x2 1

l(u) d u ]

(1 3.58)

and

3 jox

l(u) du

+0

as x

0.

--t

(13.59)

Hence, if we put +X2 = G(x)

(13.60)

and assume x X > 0 for x # 0, then we have X = X(x) = cx [I

+f

lox l(u) d u ] ' 2

(1 3.61)

which is continuously differentiable with respect to x in the neighborhood of x = 0. Clearly,

X(0)= c > 0

(' = d/dx).

(13.62)

Therefore (13.61) can be inversely solved with respect to x so that x = x ( X ) may be continuously differentiable with respect to X in the neighborhood of X = 0. From (13.62), it is evident that

(' = d/dX).

~ ' ( 0=) l/c

(1 3.63)

In terms of ( X , y), let us now rewrite system (13.2). Then we have dXldt

=

X

dyldt = - h ( X ) ,

y,

(1 3.64)

where (13.65) The function h(X) is clearly continuous and differentiable at X = 0. From (13.57) and (13.63), it is evident that h'(0) = c > 0

(1 3.66)

(' = d/dX).

Then the function X / h ( X ) is continuous in the neighborhood of X = 0, and hence it is expressed as follows: X

-=

![I

h(X)

c

1

+ s(x)+ T ( X )

,

(13.67)

262

13. Inverse Problems for x

+ g(x) = 0

where S ( X ) is a continuous odd function and T ( X ) is a continuous even function vanishing with X . As is readily seen, the orbit of system (13.64) is given by

X2

+ y’

=

R2,

(13.68)

where R is the maximum velocity of the corresponding orbit of the initial system (13.2). Substituting (13.67) and (13.68) into the first of (13.64), we then have 1 s(x) + T(X) (R’ - X 2 ) 1 / 2 d X ,

+

(13.69)

from which readily follows (13.70) 71

- 72

“SR

= -

c

0

(13.7 1)

( R 2 s(x) - X’)l/’d X *

From (13.69) and (13.70), it follows then that n/2

= fi(R) =

c o

[1

+ S(R sin ‘p) + T(R sin q)] d q ,

[I - S(R sin ‘p)

r2 = f 2 ( R ) = -

[l

+ T(R sin cp)]

dq,

(13.72)

+ T ( R sin q)] d q ,

which implies that the period functions G(R),?,(R), and f 2 ( R )are all continuous with respect to R and that G(0) = ?l(O)

0 0

= 2n/c,

= f 2 ( 0 )=

70

= n/c = 00/2.

(13.73)

These prove conclusion (3) of the theorem. Thefirst of (13.73)is coincident with (13.12). Now, from(13.73), we have (13.74)

13.3. The Period Function Associated with the Maximum Velocity 263

Hence, substituting this into (13.69), (13.70), and (13.71), we have (13.51), (13.52), and (13.53). This proves conclusion (2) of the theorem. Let us now rewrite equality (13.52) as follows:

Then, applying Lemma 13.2 to the above equality, we immediately obtain (13.55). Equality (13.56) follows, likewise, from (13.53). This proves conclusion (4). Conclusion (5) follows in the same way from (13.52) and (13.53), by Lemma 13.2. To prove conclusion (l), let us look at relation (13.60). From this relation,

g

=---

dx

- h(X)

x ’

X

therefore, substituting (13.67) into the above equality, we have

j X

x =

1 c o

(1

+ s(u)+ T(u))du,

which is, by (13.74), equivalent to x =

w x 2 Jo [l + S(U) + T(u)]

du.

(13.75)

On the other hand, from (13.65), by (13.67) and (13.74), we have

44 = h ( X ) =C’

1

2n -- - .

X

+ s(x)+ T ( X )

wo 1

X

+ s(x)+ T ( X ) ‘

Substituting (13.75) into the above equality, we thus have (13.50). This proves conclusion (1) of the theorem and hence completes the proof of the first half of the theorem. Now let us proceed to proof of the second half of the theorem. Let us assume there is given a continuous function 6(R)such that G(0) = wo > 0 and either G(R)is continuously differentiable, or the even function T ( X ) satisfying (13.55) for X > 0 is continuous in the neighborhood of X = 0.

13. Inverse Problems for x

264

+ g(x) = 0

When G(R)is continuously differentiable, we determine the even function T(X)so that it satisfies (13.55') for X 2 0. Then, for such T ( X ) ,

WO

holds for X 2 0. Consequently, as is readily seen, T(0) = 0 and T (X ) is continuous in the neighborhood of X = 0. When G(R)is not continuously differentiable, we determine the even function T ( X ) so that it satisfies (13.55) for X > 0. In this case, such T ( X ) is continuous for X 2 0 by assumption, Therefore we have

1 T(O) = lim T ( X ) = - lim

coo X + + O

x-+o

_ 0 0

-

-

lim x-+o

[yz

&(R) - o0 2 112 R d R Jox (Xz- R )

[G(X sin 'p) -oO1sin q d'p

= 0.

Hence, also in this case, T(0) = 0, and T ( X )is continuous in the neighborhood of X = 0. For S(X), it is evident that S(0) = 0, because S ( X ) is a continuous odd function. Now let us put x =

gJo[l + o

x

S(u)

+ T(u)] d u ;

(1 3.76)

then, for the function g(x) determined by (13.50), we have

2n

g(x) =

X

001 + s(x)+ T(X)'

(13.77)

For the function x = x ( X ) defined by (13.76), we have, however,

dx dX

-=-

WO

2n

for X = 0;

(13.78)

therefore equality (13.76) can be solved inversely with respect to X in the neighborhood of x = 0 so that X = X ( x ) may be continuously differentiable with respect to x in the neighborhood of x = 0. Then, substituting X = X ( X ) into the right-hand side of (13.77), we see that g(x) is continuous in the neighborhood of x = 0 and g(0) = 0. Further, by (13.78) we have

13.3. The Period Function Associated with the Maximum Velocity 265

=

2n

-

lim

=(g) 00

x+o Z

1

+

1 S(X)

>o.

lim + T ( X ) - x+o

X

-

x

(13.79)

This implies that xg(x) > 0 for x # 0 in the neighborhood of x = 0. Thus, by Lemma 1, we see that, for such g(x), any orbit of system (13.2) is closed around the origin with a bounded primitive period. In the sequel, the primitive period of such a closed orbit will be denoted by o. Now, for g(x) determined by (13.50), from (13.76) and (13.77), we have g(x)dx = X d X .

Then, integrating both sides, we have (13.60). Since x and X are of the same sign in the neighborhood of x = 0 as is seen from (13.76), X = X ( x ) derived from (13.76) thus coincides with X ( x ) of (13.61). Then, comparing (13.65) with (13.77), we have

2n h ( X ) = -* 0 0

1

+

X S(X)

+ T(X)'

that is,

- - - ""[l h(X)

272

+ S(X) + T ( X ) ] .

This is of the same form as (13.67); therefore, for the primitive period o, from (13.70), we have

(13.80) The function T ( X ) is, however, the continuous even function satisfying (13.55) or (13.55'), respectively, for X > 0 or X 2 0. Therefore, by Lemma 13.2,

holds for R > 0. Then, substituting this into (13.80), we have o = &(R).

This proves that for g(x) determined by (13.50), any orbit of system(l3.2)

266

13. Inverse Problems for x

+ g(x) = 0

is closed around the origin with the prescribed primitive period w = &(R). The uniqueness of such g ( x ) is evident from the first half of the theorem. The proof for the case where both z",(R)and ?,(R) are given is quite similar. Thus we see the validity of the latter half of the theorem. Q.E.D. Remark 1. As is seen from (13.76) and (13.77), relation (13.50) can be regarded a s a parametric representation of the function g(x) if S ( X )and T ( X ) are the known functions. Needless to say, in this case, X is the parameter. Remark 2. When only the function 6(R)is given, the function g ( x ) is not determined uniquely until a continuous odd function S ( X ) is given. In other words, in this case, there is the freedom left to choose S ( X ) arbitrarily. Corollary 1. Let g ( x ) be a continuous function differentiable at x = 0 and, f o r such g(x), suppose that any orbit of the system (13.2) is closed around the origin with a bounded primitive period. Then a necessary and sufficient condition that g(x) may be odd is that S ( X ) = 0 in (13.50).

PROOF. If g(x)is odd, then the function X ( x ) given by (13.61) is odd; consequently, the function h ( X ) defined by (13.65) is odd. Then the function X / h ( X )is even; consequently, by (13.67), S ( X ) 3 0. Conversely, suppose S ( X ) 3 0. Then, from (13.50), we have

If we put

2 jo[l + T(u)] o

x

g(x) =

-

x =

du,

(13.82)

then, from (13.81), we have 2TI wg

However, from (13.82),

X T(X)'

. -____ 1

+

(13.83)

13.3. The Period Function Associated with the Maximum Velocity 267

[l

Yn

+ T ( u ) ] d u = -2

[l

JOX

+ T(u)] du = -x.

Therefore, from (13.83), we have g(-x)

271

= -. 0 0

1

+

-X T(-X) -

_-.2n 0 0

1

+

X = -g(x). T(X)

This proves that g(x) is odd. This completes the proof. Q.E.D. Corollary 2.

Under the same assumptions as in Corollary 1, a necessary and sufficient condition that g(x) may be odd is that both half-periods are always equal to each other. By Corollary 1, this is evident from (13.53) and (13.56). Corollary 3.

Let g(x) be a continuous function differentiable at x = 0. Then, for such g(x), a necessary and suficient condition that any orbit of system (13.2) may be closed around the origin with a constant primitive period is that (13.50) holds for T ( X ) = 0, S ( X ) being an arbitrary continuous odd function.

This is evident from (13.55) and (13.52). Corollary 3 is an answer to the problem of isochronism. However, in the problem of isochronism, one can drop the condition of differentiability of g(x) at x = 0. This was shown by Levin and Shatz [l]. For a different solution of the problem of isochronism, see Koukles and Piskounov [l]. Corollary 4.

Let g ( x ) be a continuous function diyerentiable at x = 0, and, for this g(x), suppose that any orbit of system (13.2) is closed around the origin with both constant half-periods. Then both hawperiods must be equal to each other, and g(x) must be of the form

where

T~

is a positive constant to which both half-periods are equal.

This corollary readily follows from (13.54) and (13.50) by means of (13.56) and Corollary 3.

13. Inverse Problems for x

268

+ g(x) = 0

As is seen from the definition, each half-period is connected with the halforbit lying in a half-plane corresponding to the half-period under consideration and it has no connection with the half-orbit lying in the opposite half-plane. In other words, the positive-side or the negative-side half-period is connected, respectively, only with the half-branch of g(x) corresponding to the nonnegative x or with the one corresponding to the nonpositive x. However, this character of the half-periods does not appear clearly in the expressions of Theorem 13.1. Hence, here we shall reform Theorem 13.1 so that the above character of the half-periods will appear explicitly. In what follows, for brevity, the half-branch of g(x) corresponding to the nonnegative x will be called the positive-side half-branch of g(x), and the one corresponding to the nonpositive x will be called the negative-side half-branch of g(x). Theorem 13.2. Suppose g(x) satisfies the conditions:

(i) g(x) is continuous for x 2 0 ; (ii) g(x) > 0 for x > 0 and g(0) = 0 ; (iii) g(x) has a nonuanishing right-derivatiue at x = 0. Then, for such g(x), any orbit of system (13.2) has a positive-side halfperiod z, and (1) z is a continuous function of the maximum velocity R and, for this function z = .?(R),

f ( 0 ) = zo > 0; (2)

there is a continuous function V ( X ) vanishing with X such that gr:

1,'

(1

+ V(u))du

71

X

(13.84)

for X 2 0 ;

(3) 22,

z = f(R)= -

10

1

+ v(x)d X

,/R2 - X 2

( R > 0);

(13.85)

(4) 1 v(x)= zo

*

d f ( R ) - to R dR d X 1 0 J X 2 - R2

-

(13.86)

13.3. The Period Function Associated with the Maximum Velocity 269

for X 2 0 if?(R)is continuously diferentiable. Conversely, consider an arbitrary continuous function ?(R) such that f ( 0 ) = zo > 0. If ?(R)is continuously direrentiable, or if the function V ( X ) defined by (13.86) for X > 0 is continuous for X 2 0, then there is determined uniquely the continuous positive-side half-branch of the function g(x) satisfying (i), (ii), and (iii),for which any orbit of system (13.2) has theprescribedpositiveside half-period z = ?(R).Such g(x) is determined by (13.84) for the continuous function V ( X )satisfving (13.86) or (13.86’), respectively, for X > 0 or X 2 0. I f conditions (i), (ii), and (iii) are replaced by the conditions : g ( x ) is continuousfor x 5 0 ; (ii’) g(x) < 0 for x < 0 and g(0) = 0 ; (iii’) g(x) has a nonvanishing left-derivative at x = 0 ; then the same conclusions are valid for a negative-side half-period and for the negative-side half-branch of g(x) except for equality (13.84). The equality substituted for (13.84) is (i’)

g [ - “r(1 + V ( U ) ) ~ U=]- -.7t

zo 1

7 1 0

X

+ v(x)

(13.84‘)

for X 2 0.

PROOF. Let h(x) be a function such that h(x) =

-g(-x)

for x for x

2 0,

s 0;

(13.87)

then h(x) is evidently continuous in the neighborhood of x = 0 and is differentiable at x = 0. Moreover, by assumptions (ii) and (iii), xh(x) > 0 for x # 0 and k’(0) # 0. For such h(x), let us consider the system

dx/dt

=

y,

dy/dt = -h(x).

(13.88)

Then, by Lemma 13.1, any orbit of system (13.88) is closed around the origin with a bounded primitive period. In addition, h(x) is odd; therefore the continuous odd function S(X) in (13.50) corresponding to system (13.88) vanishes identically by Corollary 1 to Theorem 13.1. Then, by replacing

13. Inverse Problems f o r X

270

+ g(x) = 0

T ( X ) with V ( X ) in Theorem 13.1, we readily obtain the first part of the present theorem. Conversely, when ?(R)is given in advance, let us take the continuous function T2(R)so that t 2 ( R ) = ?(R).Then, by Theorem 13.1, corresponding to these ?(R)and f 2 ( R ) ,the continuous function g ( x ) is uniquely determined, differentiable at x = 0, for which any orbit of system (13.2) is closed around the origin with the prescribed half-periods z1 = ?(R)and tt = f 2 ( R ) . Such g ( x ) evidently satisfies conditions (i), (ii), and (iii). In addition, in the present case, the continuous odd function S ( X ) in (13.50) vanishes identically by Corollaries 1 and 2 to Theorem 13.1. Hence, replacing T ( X ) by V ( X )in Theorem 13.1, we readily get the converse part of the present theorem. When conditions (i), (ii), and (iii) are replaced by conditions (i’), (ii’), and (iii’), using techniques similar to those used in the previous case, from Theorem 13.1, we have

S(X) = 0

and g [ ~ ~ o x +( lT ( u ) ) d u

x

X

for X 5 0.

Replacing T ( X ) by V ( X ) and X by - X , we then have (13.84) for X 2 0. From this, the last part of the theorem readily follows, and this completes the proof. Q.E.D. Equalities (13.84) and (13.84‘) express explicitly that the positive-side and negative-side half-branches of g ( x ) are determined, respectively, only in connection with the positive-side and negative-side half-periods.

13.4. THE PERIOD FUNCTIONS ASSOCIATED WITH THE AMPLITUDE AND THE HALF-AMPLITUDES First, we shall prove the following theorem, which gives the relation of the maximum velocity R to the positive half-amplitude a, the negative halfamplitude b, and the amplitude A .

Theorem 13.3. Let g ( x ) be a continuous function diflerentiable at x = 0 and, f o r such g(x), suppose that any orbit of system (13.2) is closed around the origin with a bounded primitive period o. Then, in terms of the continuous even function

13.4. The Period Functions Associated with the Amplitudes

271

T ( X ) vanishing with X and the continuous odd function S ( X ) that are associated with g(x) by (13.50), the relations of a, b, and A to R are given by (13.89) (13.90)

(13.91)

PROOF. In (13.50), put (13.92) then, from (13.50), we have 2n g(x) = WO

X - 1 + s(x) + T(X)'

(13.93)

From (13.92) and (13.93), it follows that

g(x) dx

=

XdX.

Integrating both sides, by (13.7), we then have

G(x) = + X z .

(13.94)

G(u) = G(b) = +RZ,

(1 3.95)

However, from (13.8),

and, in addition, a 2 0, b

5 0, R 2 0. Since

[dG(x)]/dx = g(x) # 0

for x # 0,

comparing (13.94) with (13.95), we thus have

The former is (13.89) itself, and the latter is easily transformed to (13.90). Relation (13.91) then readily follows from (13.89), (13.90), and (13.5). This completes the proof. Q.E.D.

13. Inverse Problems for x

272

+ g(x) = 0

By means of Theorem 13.3, the following theorem concerning the period function w = @ ( A ) readily follows from Theorom 13.1. Theorem 13.4.

Let g(x) be a continuous function differentiable at x = 0. If any orbit of system (13.2) is closed around the origin with a bounded primitive period w, then (1) w is a continuous function of the amplitude A and, for this function w = w(A), we have

wkl0

R

(1

+ T(u))du] = 8(R),

(1 3.96)

where T ( X ) is the continuous even function vanishing with X associated with g(x) by (13.50); (2)

&lo loR+ X

T(X) = 0 0

-

(1

{w

T(u))du

( X z - R2)-'"R d R (13.97)

for X > 0 ; (3)

X 1 + T(R) T(X)= l,z 2zl0 (Xz - R )

lo + R

0'

(1

T(u))du] d R

(' = d/dA) (13.98)

for X 3 0 i f w ( A ) is continuously differentiable. Conversely, consider an arbitrary continuously differentiable function w(A) such that w(0) = wo > 0. If the derivative of o ( A ) satisfies a Lipschitz condition, then, for an arbitrary continuous odd function S(X), there is determined uniquely the continuousfunctiong(x), differentiable at x = 0,for which any orbit of system (13.2) is closed around the origin with the prescribedprimitive period w = w(A). Such g(x) is determined by (13.50)for the continuous even function T ( X ) that satisfies the integral equation (13.98) for X 3 0.

PROOF. From (13.91), A is a continuously differentiable function of R; moreover, A vanishes with R and dA/dR = (wo/2n) [l

+ T(R)] > 0

13.4. The Period Functions Associated with the Amplitudes

273

for sufficiently small R 2 0. Therefore relation (13.91) can be solved with respect to R so that R = R(A) may be continuously differentiable with respect to A for sufficiently small A 2 0 and R(0) = 0. Then, by conclusion (3) of Theorem 13.1, o becomes a continuous function of A and, for this function o = @(A), o(0) = 8(0) = w,

> 0,

and, by (13.91), one has (13.96). This proves conclusion (1). Conclusions (2) and (3) then readily follow from (4) and (5) of Theorem 13.1. If we regard relation (13.98) as the integral equation with respect to T(X), then, by Lemma 13.3, it has a unique continuous solution for sufficiently small X 2 0. Hence, by means of Theorem 13.3, the converse part of the present theorem readily follows from Theorem 13.1. This completes the proof. Q.E.D.

In illustration of Theorem 13.4, the function g(x) for which the primitive period of the closed orbit of (13.2) varies linearly with the amplitude will be calculated. In the present example, o ( A ) = 0,

+ cA;

therefore the integral equation (13.98) to be solved is now C

T(X) = -A' 4

+ 211 X

IOx

T(R)2

(X2 - R )

dR

(X 2 0).

This can be easily solved, say, by the iterative process or by calculating a formal solution. The even function T ( X ) satisfying the above equation for X 2 0 is thus found as follows:

where

274

13. Inverse Problems for X

+ g(x) = 0

Hence, putting T ( u ) d u = -*-2.n

x [ e x p ( 16n L x'> - 1 1 1x1

(&x) - x,

+ 9q*2 by (13.50), we have g(x) =

-.2n

X S(X)

+ + T(X), x =2.n x + v(x)+ J O x S ( u ) d u ] , wo 1

"I

FIG.24

(13.100)

13.4. The Period Functions Associated with the Amplitudes

-*

FIG. 25

FIG. 26

215

276

13. Inverse Problems for x

+ g(x) = 0

c=5 -

-x

FIG.

27

- x

FIO.

28

13.4. The Period Functions Associated with the Amplitudes

277

The maximum velocity R is also a continuously differentiable function of a, as is seen from (13.89), and it is a continuously differentiable function of 6, as is seen from (13.90). Hence, by Theorem 13.1, the primitive period w also becomes a continuous function of a and, at the same time, it becomes a continuous function of b. For these functions, we can obtain theorems similar to Theorem 13.4 if we generalize Lemma 13.3 slightly. For details, see Urabe [lo]. Now, corresponding to Theorem 13.2, we shall prove the following theorem. Theorem 13.5.

Suppose g ( x ) satisfies conditions (i), (ii), and (iii) of Theorem 13.2. Then, (1) for the maximum velocity R and the positive half-amplitude a, there holris

+ V ( U ) ]du,

a = 2 ['[l

n o

(1 3.101)

where V ( X )is a continuous function vanishing with X associated with g(x) by (1 3.84) ; (2) the positive-side half-period z1 of the orbit of system (13.2) is a continuous function of a, and, for this function z1 = z,(a), it holds that Z1(0)

and

["[

R

71

v(x)= - - zo d X

(1

n o

IOx[" {z~

= To

(1 3.102)

>0

+ V(u))du]

=

(1 3.103)

?,(R);

+

s R ( l V ( U ) )du] - z 0 ] ( X 2 - R2)-1'2R d R n o (1 3.104)

for X > 0 ; (4)

v(x)= n

1'

(x2

- R2)1/2

' T1

s'

n o

(1

+ V ( U ) )du] d R

('

= d/da)

(13.105)

for X 2 0 ifr,(a) is continuously direrentiable.

13. Inverse Problems for x

278

+ g(x) = 0

Conversely, consider an arbitrary continuously diflerentiable function rl(a) such that ~ ~ ( = 0 ro ) > 0. If the derivative of 71(a)satisfies a Lipschitz condition, then there is determined uniquely the continuous positive-side half-branch of the function g(x) satisfying conditions (i), (ii), and (iii) of Theorem 13.2,for which any orbit of system (13.2) has the prescribed positive-side half-period z1 = Ti(.). Such g(x) is determined by (13.84) for the continuous function V ( X ) which satisfies the integral equation (13.105)for X 2 0. Zf conditions (i), (ii), and (iii) of Theorem 13.2 are replaced by conditions (i’), (ii’) and, (iii’) of Theorem 13.2, then we have: (1) for the negative half-amplitude b, there holds (13.101’)

where V ( X ) is a continuous function vanishing with X associated with g(x) by (13.84‘); (2) the negative-side half-period r z of the orbit of the system (13.2) is a continuous function of b, and, for this function r2 = r2(b), we have r2(0)= Tl(0) = 70 > 0

and

l

[

R

r2 - 3 (1 n o

V(X)= 1 d

{rz

a

70

dX

o

+ V(u))du]

[-5JoR+ (1

(1 3.102)

= f2(R);

(13.103’)

V(U))du] - T ~ ] ( X ’ - R 2 ) - l I 2 R dR

x

(13.104‘)

for X > 0 ; (4) 721

[-: loR+ (1

1

V(u)) du dR (‘ = d/db)

(13.105’) for X 2 0 ifr2(b)is continuously diferentiable. Conversely, if there is given an arbitrary continuously diferentiable function r2(b)such that rz(0)= ro > 0, and the derivative of z,(b) satisfies a Lipschitz condition, then conclusionssimilar to thosefor thefirst case holdfor the negativeside half-branch of the function g(x).

13.4. The Period Functions Associated with the Amplitudes

279

PROOF.Relations (13.101) and (13.101') are derived, respectively, from (13.84) and (13.84'), analogously to (13.89) and (13.90). The remaining parts of the theorem then follow from Theorem 13.2 in a way similar to that of Theorem 13.4. This completes the proof. Q.E.D. As is seen from Theorem 13.5, if there are given continuously differentiable functions zl(a) and ~ , ( b )such that z,(O) = z2(0) = zo > 0, and the derivatives of rl(a) and z,(b) both satisfy a Lipschitz condition, then there is determined uniquely the continuous function g(x), differentiable at x = 0, for which any orbit of system (13.2) is closed around the origin with the prescribed half-periods zl = z,(a) and z2 = z2(b). The positive-side halfbranch of g(x) is determined by (13.84) for V ( X ) satisfying (13.105) for X 2 0, and the negative-side half-branch of g(x) is determined by (13.84') for V ( X )satisfying (13.105') for X 2 0. The uniqueness of g(x) for given zl(u) and z,(b) was proved by Opial [l] under weaker conditions, but the method for determining g(x) from given zl(a) and z2(b)was not given by him. For details, see his paper [13.

Appendix The Newton Method and Step-by-step Methods for Ordinary Differential Equations.

The numerical method of Chapter 11 is based on the use of the Newton method to solve equations numerically and the step-by-step method to solve the Cauchy problem for ordinary differential equations numerically. These methods will be explained in this appendix taking recent developments into account. In numerical computation, it is more convenient to use the norm of a vector defined by the quantity 1x1 = m a x I x i I i

for an arbitrary vector x = (xi) rather than the one defined by

I x I = C I xi I. Evidently, the quantity I x 1 satisfies condition (1.1) for norms, and hence it really defines a norm of a vector. Corresponding to this norm of a vector, by (1.2), the norm I A I is defined for an arbitrary matrix A = (aij). For I A I, similar to (1.4), it is easily seen that

IAI = maxzIaij(. i

j

In this appendix, for convenience, the norm

I *.. I.

280

I I will be used instead of

A.I. The Iterative Method to Solve Equations Numerically

281

A.l. THE ITERATIVE METHOD TO SOLVE EQUATIONS NUMERICALLY

For the numerical solution of nonlinear equations, we have many methods, but most of them, including the Newton method, are based on the following iterative method. First we rewrite the given equation in the form x =f(x),

(A. 1)

where x andf(x) are vectors. Next, starting from a suitable approximate solution x = x,, we compute x,, (n = 1,2, . . .) successively by x,,

= f (x,)

(n = 0, 1,2, . . .).

(A4

Then we obtain an approximate solution of the given equation, using x,, for suitable n. Such a method is successful when the sequence {xn}is convergent. Whether {x,,} is convergent or not depends on the character off ( x ) and, at the same time, on the choice of the starting value x,. A practically convenient condition for f ( x ) and x, is given by the following theorem due to Collatz [l].

Theorem A.l. Suppose that f ( x ) is defined in a region D and that, for a constant KOsuch that

0 s KO < 1,

(A.3)

f ( x ) satisfies a Lipschitz condition

for every x’ and x“ belonging to D. Let x, be an arbitrary point belonging to D such that the set S of all x satisfying the inequality

is contained in D.

Then the sequence {x,} (n = 0, 1,2, . . .) is constructed in D by (A.2), and it converges to a certain K belonging to D. This E is a solution of Eq. (A.l) and, moreover, this is a unique solution of (A.l) in the region D.

282

Appendix

PROOF.To begin with, let us prove that the sequence {x,,} (n exists and that

=

(n = 1,2,. . . .), I x , , + ~- x, I 5 KO" I x1 - xo I (n = 1 , 2 , . . .). X,ES

.

0, 1,2, . .)

(A4 (-4.7)

As is seen from (A.5), x1 =f(xo) E S c D ;therefore x2 =f(xl) is constructed and, by (A.4), we have

I x z - x1 I

= lf(X1)

-f(xo)

I 5 KO I x1 - xo I.

These show that xl,x2 are actually constructed, and (A.6) and (A.7) are valid for n = 1. To prove the assertion by induction, let us assume that xo, xl, . . ., x,, x,+ are constructed, and (A.6) and (A.7) are valid for n = 1,2, . . ., m. Then, from (A.7), we have

I

xm+1

-

I5I

xm+1

- (KO" S

- xrn I + I x m - x m - 1

+ K r - ' + + KO) I .**

I+

+ I xz - x1 I

- XO

XI

I

since 0 5 KO < 1 by (A.3). The above inequality implies x , + ~E S. Since S c D, x,,~ ED,thenx,,, = f ( ~ , + ~is)actually constructed and, by (A.4),, we have 1xm+z - xm+l I = If(xrn+l) -f(xm) I

5 KO I xm+1 - xm 1, which implies

I xm+z - xm+1 I s K:+' I x1 - xo I by (A.7) for n = m. These results show that x , + ~is actually constructed, and (A.6) and (A.7) are valid for n = m 1. This completes the induction and proves the beginning assertion. Let us now prove that the sequence {x,,} (n = 0, 1,2, . . .) is convergent. From (A.7), for any m and n such that m < n, we have

+

1

xn

- xm 1 5

I

xn

-

xn-11

(K",-'+ K",'

+ I xn-1 - xn-2 J + ... + I x m + 1 - xm I

+

S - KOrn 1x1 - X o I . 1 - KO

+ KO") I

- XO

XI

I

A.1. The Iterative Method to Solve Equations Numerically

283

Since [ KO < 1, this implies

IX,-X,,,)-P~

as n , m + c o ,

which means that the sequence {x,} is a fundamental sequence, that is, is convergent. Put X = lim x,; n-rm

then, from (A.6), Z E S c D. Thenf(X) is defined and, by (A.4), we have

1~ - f ( ~I)S 1 x - xn+l I + If(4 - f ( X ) I 5 I x - xn+i I + KO 1 - z 1, xn

from which, letting n

-P

00,

we have

X

(A4

= f(X).

This proves that X is a solution of Eq. (A. 1). Finally let us prove that the above X is a unique solution of Eq. (A.l) in region D. Let 2’ be another solution of (A.l) in region D. Then

X’ = f(X’).

(A.9)

Subtracting (A.8) from (A.9), we then have, by (A.4),

I X’ - X I = If(-

X’)-f(X)I

SK,IX’-XI,

from which follows (1 - K0)I X’ - 21 5 0. This implies X’ = 3, since 1 - KO > 0 by (A.3), and proves X is a unique solution of (A.l) in region D. This completes the proof. Q.E.D. The condition for x,, namely, the condition that S c D, is related to the accuracy of the starting approximate solution x,. In fact, if xo is accurate, then the quantity (A. 10) I f ( X 0 ) - xo I = I xi - xo I will be small; consequently, the set Swill be a sphere of small radius, and this will be contained in D provided x, is an inner point of D. Thus the condition that S c D is always satisfied if x, is an accurate approximation lying inside D. The fact that Eq. (A.l) has only one solution in region D implies the solution X obtained by the iterative process (A.2) is independent of the choice of the starting approximate value x,.

284

Appendix

However, in Theorem A.l, errors arising in the practical computation of f (x) are not taken into account. Therefore the conclusions of Theorem A.l are not valid in general for practical computation, where, the sequence obtained by the iterative process (A.2) is not the exact sequence {x,,}described in Theorem A.l, but a somewhat different sequence {x,,*}.Let us suppose xi+l =f’ (x,,’)

(n

=

0,1,2, ...; xb = xo)

(A.ll)

is the practical process of process (A.2) and that

If*(XI - f (4I s E .

(A. 12)

Here, needless to say, f * ( x ) is the practical evaluation of f ( x ) , and E is the bound of errors arising in the practical evaluation. Concerning the practical iterative process (A.l l), we have the following theorem, which is due to the author [l]. Theorem A.2.

Supposef (X)satisfiesthe conditions of TheoremA.1 and let xo be an arbitrary point belonging to D such that the set X of all x satidying the inequality

is contained in D. Then thepractical iterative process (A.11) can be continued indefinitely in D and, a f e r afinite number of repetitions, it attains the state of numerical convergence, in which the sequence {x,,’} oscillates, taking some finite number of values. (In the sequel, this state will be called “the state ONC,” anabbreviation of the “state of oscillatory numerical convergence”). The desired approximate solution of Eq. (A.l) is given by any one of x,’ in the state ONC, and its error bound is given by (A.14)

PROOF.To begin with, let us prove that the set S of Theorem A. 1 is contained in the set Zo consisting o f x such that (A.15)

A . I . The Iterative Method to Solve Equations Numerically

285

In fact, let x be any point contained in S. Then, by (A.5) and (A.12), we have successively

Ix -

f(x0)

1 5 I x - f ( X 0 ) I + If ( x 0 ) - f ( X o ) I I --KO If(x0) - xol + 1 - KO &

By (A.15), this shows that x E Co. Since x is an arbitrary point belonging to

S,this implies

s c c,.

(A.16)

Since I:, c I: as is readily seen from (A.13) and (A.l5), by assumption, we thus have

S c Z,C Z c D .

(A.17)

The relation S c D shows that the assumptions of Theorem A.1 are all satisfied in the present case. Thus,by Theorem A.l, we see that Eq. (A.l) has a unique solution x = I in D, and the sequence {x,,} (n = 0, 1 , 2 , . . .) is constructed by (A.2) so that Es

(n = 1,2, . . .)

(A.18)

x,+X

as n-, co.

(A. 19)

X"

and Since xl* = f' (xo) E Xo c I: as is readily seen from (A.15) and (A.17), * I let us assume that x1 , x2 , . . ., xm*are constructed in C.Then, since X,*E

=f'(x,,,')

Z c D,

is indeed constructed, and we have successively

I x1' - x1 I = If' (xo) - f(x0) I 5 I x2I - I = lf'(.I'> - f ( X d I s lf'(.l'> - f ( x J I + I f ( X l * ) - f(x1) I s + KO I x1* - x1 I s (1 + KO)&, &,

x2

&

286

Appendix

(A.20) Since x,+~ E S by (A.18), x:+~ lies in the So-neighborhood of S. By (A.16), lies then in the So-neighborhood of Xo, that is, Z. This proves by induction that the sequence (x,') (n = 1,2, . . .) is indeed constructed in E c D. The x,' obtained in practical computation are, however, the numbers represented by a finite number of digits. Therefore the values that x ,' can take cannot be infinite, since all x,* lie in the finite set Z. This implies that it necessarily happens that xLo = x:, for a pair of certain different finite integers mo and no. Then, in the iterative process (A. 1 l), we have

for any positive integer n. This implies that if mo > no, the state

is repeated after the noth step in the iterative process (A.l l), which proves the iterative process (A.11) always attains the state ONC after a finite number of repetitions. Now let x,* be an arbitrary value in the state ONC. Then, for a certain positive integer m,we have = x,*

( i = 1, 2,

...).

However, by (A.20), IX;+im

- Xn+irn I s

60.

Therefore, for any positive integer i, we have

Ixn* - Xn+irn I S

60-

Then, letting i + oc), from (A.19), we have

I x,'

- x I 5 So.

This proves that the error bound of x,' is given by So, which completes the proof. Q.E.D.

A.1. The Iterative Method to Solve Equations Numerically

287

As is seen from Theorem A.2, when we compute the solution of Eq. (A. 1) by the iterative process (A. 1I), we usually obtain many approximate values, and we have no theoretical grounds for choosing a particular one of these many values. In other words, we can take any one of the values in the state ONC as the approximate solution of the given equation. In illustration, let us consider the equation 8 Zni/3

z = ;e

(A.21)

z,

where z is complex. Let us suppose all computations are carried out up to three decimal places. Then i e 2 n i / 3is replaced by-0.444 + 0.770i. If we start the iterative process from z, = 0.010, then we have the results shown in Table VIII. TABLE VIII n

.Z"*

0 1

- 0.004 + 0.008i

8

-0.001 -0.003i

9 10 11 12 13 14

0.002 -0.001 +O.O02i - 0.002 - 0.002 0.003 - 0.001i +O.O02i - 0.002- 0.001i

0.010

n

zn*

15 16 11 18 19 20 21 22 23 24

0.002 - 0.002 0.001 0.003i - 0.002 0.001 - 0.002i

+

0.002f0.002i -0.003 +O.OOli -0.002i 0.002 +0.001 i -0.002+0.002i - 0.001 0.003i

-

As is seen from Table VIII, zi4 = zg*. This implies that, in the present I * iterative process, z g , z 9 , . . ., z i 3 are repeated, and these are in the state ONC. Hence, we can take anyone of zg*,z9*,. . ., z;~as the approximate solution of Eq. (A.21). In this example. E

=

J2x

103

K, = 819; *

*

The true errors of zg , z 9 , . . ., 2 *2 3 are JE x 2 x . . ., $ x and the maximum of these is JE x lo-' = 3.2 x The error bound 6, is therefore a little larger than the exact error bound. therefore 6,

= 9JZ

x

288

Appendix

A.2. THE NEWTON METHOD In the present section, the Newton method will be explained on the basis of the author's paper [8], taking into account the round-off errors arising in practical computation. The Newton method is a method that transforms an arbitrary given equation of the form

dx)=0

(A.22)

into the equation of form (A.l) which satisfies the conditions of Theorem A.1. Let q be an arbitrary approximate solution of Eq. (A.22) and h be its correction, that is, the quantity such that X = x, h is an exact solution of (A.22). Then, if q ( x ) is continuously differentiable with respect to x, we have

+

cp(%

+ h) % q(x,) + J(x,)h + 0,

(A.23)

where J(x) is a Jacobian matrix of q ( x ) with respect to x . From (A.23), we then have h

+ - J-'(xo)q(xo),

(A.24)

provided det J(x,) # 0. Relation (A.24) suggests that x1 = xo -J-'(xo)'p(xo)

will be an approximate solution more accurate than the initial x,. Thus, repeating the above process, the iterative process x n + i = x,

- J-'(x,)q(x,)

(n = 0, 1,2, . . .)

(A.25)

is considered. This process is nothing other than the Newton method. However, in practical computation, J-'(x) is often replaced by its approximation H ( x ) . Therefore, in the present monograph, the iterative process xn+ 1

= xn

- H(Xn)q(xn)

(n = 0 , 1 , 2 , * * .),

(A.26)

where H ( x ) is an approximation of .J'-'(x), will be also called the Newton method. If it is necessary to distinguish process (A.25) from process (A.26), the former process will be called the proper Newton method, and the latter will be called the genera2 Newton method. It will be assumed in the sequel that H(x) satisfies a Lipschitz condition

IH(x') - H(x") I 5 L I x' - x" I

(A.27)

or every x' and x" belonging to, the region of definition. Let us suppose that p(x) is defined in a convex closed bounded region D

A.2. The Newton Method

289

and is twice continuously differentiable with respect to x in D. Then, for every x’ and x” belonging to D, we have q(x’)

- q(x”) = J ( X ” )

(x’

+ R(x’,

- XI’)

(A.28)

XIf),

where

I R(xf,x”) I 5 Mo I x’ -

I

2.

(A.29)

- H(x)rp(x);

(A.30)

XIf

Now let US put =x

f(X)

then, from (A.28), for every x f and x f fbelonging to D, we have

f ( x ’ ) - f ( x ” ) = x‘ - x” - H(x”) [rp(x’) - rp(x”)]

- [H(x’) - H(x”)]cp(x’) - H(x”) J(x”)] (x‘ - x”) - H(x“)R(x’, x”) - [H(x‘) - H(x”)]~(x’).

= [E

Therefore, by (A.27) and (A.29), we have

)’.(fI

-f(x“)

I I{ I E

+

If we put

+

H(x”)J(x”)I Mo L I q ( x ’ ) I } I x’ - X f ‘ I. -

I X’

- X”

I I H(x”) I (A.31)

max 1 E - H(x)J(x)1 = K , xeD

(A.32)

then, from (A.31), it follows

If(.’)

- f ( x ” ) I 5 K(X‘, x”) I x’

where

K(x’, x”) = K

- x” I,

(A.33)

+ MI x f - x” I + L Icp(x’)I.

(A.34)

Let us now suppose

K(x’, x”) 5 KO < 1

(A.35)

for every x f and x“ belonging to D. This is not an onerous restriction however, because inequality (A.35) is always satisfied if H ( x ) is chosen sufficiently close to J-’(x) and D is chosen sufficiently small in the neighborhood of the solution of the given equation (A.22). Inequalities (A.33) and (A.35) show that the iterative process (A.26) satisfies the condition of Theorem A.l. This implies that the conclusions of Theorems A.l and A.2 are all valid for the Newton method.

290

Appendix

The error bound for the approximate solutions computed by the Newton method is given by the following theorem. Theorem A.3. Let E be the bound of errors arising in the computation of function (A.30). Then, when the solution of (A.22) is computed by the Newton method (A.26), namety, by

( n = 0, 1,2, ...),

=f'(xJ

(A.36)

the error bound for the approximate solution x,,' in the state ONC is given by 1 . 1 6= ((1 2M

K)

~

- [(I - K

) ~-

EM]''^}

+ __ l - K E

9

(A.37)

where K and A4 are the constants defined in (A.32). Before going to the proof, let us note that inequality (A.35) implies det H(x), det J(x) # 0

for any x

E D.

(A.38)

In fact, if we put H ( x ) J ( x ) = E - U(X),

(A.39)

then, from the first of (A.32),

I U(x) I 5 u

for any x E D,

(A.40)

and, by (A.34) and (A.39, lc

If det [ H ( x ) J(x)] = 0 for some x

< 1.

E D,

(A.41)

from (A.39) we have

det [ E - U(x)] = 0

for some x E D.

Then there is a nonzero vector u such that

[ E - U(x)]u = 0, that is, u = U(x)u.

Then, by (A.40), we have

1.1

5

+I,

A.2. The Newton Method

29 1

that is, (1

- .)I

6 0.

u1

Since K: < 1 by (A.41), this implies I u 1 = 0, namely, u = 0, which is a contradiction. Thus we have (A.38). Now we shall prove Theorem A.3. x,, , x,,+~,. . ., xi+,-i be the values in the state ONC, and suppose x:+, = xn*. By Theorem A.2, we then have

PROOF.Let

*

I

&

I x : + i - x l s s , = - -1-- KO

( i = 0, 1,2, ..., m

- l),

(A.42)

where X is an exact solution of the equation x = f(4,

which is evidently equivalent to Eq. (A.22) since det H ( x ) # 0 by (A.38). Now, by (A.18) and (A.19), X E S ; therefore, by (A.16), X E Zo (see Fig. 29).

FIG. 29

Then, since Z is the a0 -neighborhood of Zo, the set V, consisting of x such that

Ix-xl~a, is included in Z. By (A.42), this implies x:+i E

v1 c z

(i = 0,1,2, ..., m

- 1).

Let us put

K l = SUP K ( 3 , x); xev,

then, by (A.34) and (A.35),

(A.43)

292

Appendix

In this inequality, xn* can be clearly replaced by any one of x:+,, Thus we have

. . .,

x:+,-,.

Ix:+i

- X I 5 61,

where

6, = -. 1 - K1 &

(A.45)

If Kl = KO,eviL:ntly 6, = 6,, and (A.44) does nc give an error bound more precise than (A.42). In this case, from (A.43), we have, however,

therefore, substituting this into (A.37), we have

8 = 6,. This proves the validity of the theorem in the present case. If Kl < KO,then 6, < 6,, and (A.44) does give an error bound more precise than (A.42). In this case, if we put

K, = u + M6,-,,

then we have

(A.46)

A.2. The Newton Method

293

and

- E I s 6,

I

( p = 0, 1,2, ...).

(A.48)

The equality signs in (A.47) occur only when M = 0. Anyway, by (A.47),

8=

lim 6,

I? = lim

and

Kp

P-m

P'W

exist and, by (A.48), 1.:+i

holds. For 8 and

-q 58

(A.49)

R, from (A.46), it holds that

R=K+M$

b=-

and

&

1-R'

Then, eliminating I? from these equalities, we have

b=

&

(A.50)

(1 - K) - M8'

that is, M6z - (1 - K ) b

+

&

(A.51)

= 0.

When M = 0, evidently

8=-.

E

(A.52)

1 - K

Since (1 - K)' M-0

-

EM)"']

=

2M

& 1 - K'

inequality (A.49) and equality (A.52) prove the theorem in the present case. When M # 0, the quadratic equation (A.51) is solved as follows:

8 = 12M[ ( 1 - K) & ((1 - ')K - 4eM)li2]. However, from (A.47),

6<S0=-

&

1 - KO'

(A.53)

294 This implies that

Appendix

8

=

O(E)as

E + 0.

Thus, from (A.53), we see that (1 - K)'

- 4eM

2M

1

.

>"'

By (A.49), this proves the theorem. Q.E.D.

As is seen from (A.32), for the proper Newton method, IC = 0; therefore, from (A.37), the error bound for x,,* in the state ONC is nearly equal to e. On the other hand, from the meaning of E , we cannot hope to obtain the approximate solution accurate within error bound e. This shows that the proper Newton method is the one that gioes the approximate solution of the highest accuracy. Up to now, only the x,,*'s in the state ONC have been considered. However, when the computation is carried out on an electronic computer, it is not convenient to continue the iterative process until the state ONC is detected, because, to do so, it is necessary to store every x,," (n = 0, 1,2, . . .) obtained by the iterative process. Hence, when the computation is carried out on an electronic computer, the iterative process is often stopped by the criterion of the form (A.54) where a is a positive number suitably chosen. If x,,' is in the state ONC, then, by Theorem A.3,

therefore

Hence, if a 2 28, there is only a finite number of x,,*'s that do not satisfy (A.54). This implies that if. 2 28, the criterion (A.54) can indeed be used effectively for stopping the iterative process (A.36). However, if a < 28, there may not exist x,,' satisfying (A.54), because it may happen that

for every x,,' in the state ONC. In such a case, the criterion (A.54) does not work effectively for stopping the iterative process. From this discussion, we see that when we want to stop the iterative process by the criterion of the form (A.54), we should choose a so that a

2 28.

(A.55)

A.2. The Newton Method

295

When the iterative process (A.36) is stopped by the criterion (A.54), on obtained at the last iterative step, we have the the error bound for following theorem.

Theorem A.4.

In Theorem A.3, suppose that the iterative process (A.36) is stopped by the criterion (A.54). Then the error bound for x:+ is always given by E

+ KoU.

(A.56)

l-Ko’

but, when

(A.57) the error boundfor x:+ is given more precisely by E

3-

R’tl E + KU R‘ . 1 - K L

1-

+-(1MU2 -

(AS)

K)j’

where

(1 - K)2 - 4M(a

+ E)

)‘”I

.

(A.59)

PROOF. For x,’ satisfying (A.54), by (A.33), we have

I X I - X I 5 I x,* - x:+ I + If*(X”*) - f(X”*) I + If(Xn’) - f(X) I (A.60) 5 tl + + K(X, x,*) I x,* - X I, 1

c

8

where X is an exact solution of the equation x =f(x),

which is equivalent to Eq. (A.22), as mentioned already in the proof of Theorem A.3. From (A.60), by (A.35), readily follows

I x,*

-E

s

a+&

- 1 - K(X, x,*)*

(A.61)

However, by (A.34) and (A.39,

K(?,xn*)= Ic

4- M I X , *

- XI 6 KO < 1.

(A.62)

296

Appendix

Therefore, from (A.61), we have

(A.63) Now let us consider the case where (A.57) holds. In this case, if we put

6,‘ = ___ 1 - KO

and

+

K1’ = K

+ Ma,’,

then, from (A.62), (A.63) and (A.57), we have K(Z, x,,’)

5

+ M6,’

K

= K,’

< KO.

Then, from (A.61), it follows

I x.’

-xI

s S,’,

(A.64)

where

6,’ =

U + E ~

1 - K1’

< 6,‘.

(A.65)

The above process can be continued further. Thus, if we put K,’ = K

6,

‘

=-

+ M6;-,, U + &

(P = 192, ...),

(A.66)

...).

(A.68)

1 - K,’

then we have

and

I

..* xl s -

( p = 0,1,2,

6,’

Then, by (A.67),

8‘ = lim 6,’

and

2’ = lim K,’

exist and, by (A.68), we have

I x.’ - z I 5 8’.

(A.69)

For 8’ and R’, however, from (A.66), it holds that

R’= K + M8’

and

u+e 8’ = ____ 1 - R”

(A.70)

297

A.2. The Newton Method

Therefore, eliminating 8' from these equalities, we have R ' - K =

+

M(u E ) 1-i2"

that is,

- (1 - K) (R' - K) + M ( a +

(K'- ')K

E)

= 0.

This quadratic equation i s readily solved as follows: (1 - K)'

- 4M(u + E ) )

1.

112

(A.71)

However, from the first of (A.70),

0 which implies we see that

R'

R' - K

- K = MS' 5 M60'

=

O( I u

+

M(u 8 ) l-K/

+ E I ) as I a + E I -+0. Thus, from (A.71), (1

=

=

1 [(I + K) - ((1 2

- ')K

- 4M(M

- K)z - 4M(a

+

+

]

112

E))

E)

.

(A.72)

Now, for x:+ satisfying (A.54), from (A.33), it holds that

I x:+ i - 3 I 5 I f ( ~ n * ) - f ( ~ n * )I + If(xn*) - f(3)I 4 E + K(X, X"*) I xn*- 5 I.

(A.73)

Therefore, from (A.62) and (A.63), we always have (A.74) When (A.57) holds, by (A.69), (A.62), and (A.70), from (A.73), we have

I

- XI 5

E

+ R'&

which is rewritten by the second of (A.70) as follows:

lx;+l - 31 5 However, by (A.72),

E

+ R'.-1a-+R&'

=

e+aR' -.1 - R '

(A.75)

298

Appendix

( l + K ) - ( l - K )

=

(E

M + KCL) + _ _ u(u + E ) + uO((u + E)’) 1 - K

(A.76)

and 1

--

1 -

K’

-

2 (1 - K )

-- 1

1 -K

+ ((1

- K)’ -

+-(1 -

(a K)3

4M(@.+ &))”’

+ E ) + M Z *O((a + E)’).

(A.77)

Therefore, substituting (A.76) and (A.77) into (A.79, we have

=--

1 -

K

+

(1 -

K)3

(A.78) Inequalities (A.74) and (A.78) prove the theorem.

Q.E.D.

In practical computation, E is small and u is chosen small. Therefore we may suppose that inequality (A.57) usually holds in practical computation. Then, comparing the error bounds (A.37) and (A.58), we see that the approximate solution having accuracy similar to those in the state ONC will be obtained by stopping the iterative process by the criterion of the form (A.54) i f u is chosen SO that E

6

Ci

6

&‘I2, EIK.

(A.79)

In (A.79) a 6 b means that a = o(b) as b + 0, and the condition E -4 o! secures requirement (A.55). In practical computation, K is usually small; therefore it is usually possible to choose u so that (A.79) may hold. This means that in practical computation, the approximate solution having the best possible accuracy, that is, the accuracy of the approximate solutions in the state ONC can usually be obtained by stopping iterative process (A.36) by the criterion of the form (A.54), choosing u suitably.

A.3. A Numerical Illustration of the Newton Method

299

Remark. From (A.73) and (A.34), it follows that

I

-

x I 5 & + (K + M I x,*

This implies that, when E and

I

K

- x 1 )I X"*

- x 1.

are small,

-

x I 5 M I X"'

- xI

(A.80)

approximately. On the contrary, when the iterative process described in Theorem A.l is employed, we have

- x) s

I

&

+ K O I xn* - q.

This means that Ixi+i - 3 1 6 K o ( x n * - X (

(A.81)

approximately. Hence, comparing (A.80) with (A.81),we see that when the Newton method is employed, the state ONC is attained more quickly than when the iterative process described in Theorem A.l is employed. This fact is confirmed by comparing (A.58) with (A.56). In fact, when the iterative process described in Theorem A.l is employed, by (A.56) and (A.14), can be in the state ONC only if I xz+ - x,* I = o(&), while, when the Newton method with small K is employed, by (A.58)and (A.37), xi+ can be in the state ONC if( x ; + ~- x i 1 = o ( E " ~ s ) o(e). A.3. A NUMERICAL ILLUSTRATION OF THE NEWTON METHOD Let us illustrate the theory of the preceding section by the system of equations def

+ 6xy2 - 4~ - 3.304 = 0, - 6x2y - 3y3 + 36y - 0.323 = 0.

q1(x, y ) = 3x3 - 3x2y ~ Z ( X ,y )

def

=x

3

(A.82)

This system is a modification of Eq. (1 1.75) in the Galerkin procedure for the nonlinear equation ii

+ u3 = sin t

(. = d/dt),

and the x and y are the coefficients of a trigonometric polynomial of the form x sin t y sin 3t. From the meaning of system (A.82), an approximate solution of (A.82) will be obtained by solving the single equation

+

300

Appendix cpl(x, 0) = 3x3

- 4x - 3.304 = 0.

Since this equation has only one real root close to 1.5, we may suppose

x = 1.5,

(A.83)

y =0

is an approximate solution, and that this will serve as a starting value for the Newton method. Taking (A.83) as a starting value, we experimented the Newton method on the CDC 1604 computer. Table IX shows the results obtained by the proper Newton method, and Table X shows the results obtained by the general TABLE IX R

xn*

yn*

0 1 2 3

1.5oooO00000 1.4049740082 1.4000777297 1.4oooO 00047 1.4oooO00000 1.4oooO00000 1.4oooO00000

-0.10713 66469 - 0.0999931486 -0.1oooOoooO6 -0.loooO00000 -0.loooO00000 -0.1oooOo0O00

0 1 2 3 4

1.5oooO00000 1.4049740082 1.4002040864 1.4000206557 1.4oooO 11088 1.4000001106 1.4oooO 00060 1.4oooO oooO6 1.4oooO oooO1 1.4oooO 00000 1.4000000000 1.4oooO00000 1.4oooO 00000

5

6 7 8 9

: [;; c

12

( x i j * . ~ 1 3 * )= ( ~ 1 0 ~Y ~3 O * )

An

0.00000 00000

0.00000 00000 -0.10713 66469 -0.0997508574 -0.10003 17877 -0.0999987458 -0.1ooOO 01651 -0.0999999937 -0.1oooOoooO9

-0.loooO00000

-0.1oooOm -0.loooO00000

-0.loooO00000 -0.loooO00000

0.10713 66469 0.0071434983 O.oooO7 77250 0.0000000047 0.00000 00000 0.00000 00000

0.10713 66469 0.00738 57895 0.0002809303 O.ooOo3 30419 0.00000 14193 0.0000001714 0.00000 00072 0.00000 00009 0.00000OOOOl 0.00000 00000 0.00000 00000 0.00000 00000

A.3. A Numerical Illustration of the Newton Method

30 1

Newton method such that H ( x ) of (A.30) is a constant matrix J-'(x0). In the present case, evidently x, = (1.5, 0). In the sequel, this general Newton method will be called the simplified Newton method. The CDC 1604 computer has 36 binary digits in the mantissa. Therefore, in our computation, the bounds of round-off errors in evaluation of rp,(x, y) and q 2 ( x ,y) are, respectively,

3.544 x TJ2 and 4.654 x 2-34

1,

I

I + I

for x and y such that x- 1.4 y 0.1 S loe6. From this, after elementary calculation, we see that for the computed values lying in the region D: x - 1.4 y 0.1 6

I

1, I + K

1

M < 2.473,

= 0,

E

= 0.83 x lo-''

(A.84)

if the proper Newton method is employed, and K

< 0.1388,

M < 2.473,

E

= 0.79 x lo-''

(A.85)

if the simplified Newton method is employed. When the proper Newton method is employed, as is seen from Table IX, (xm*,yn*)( n = 4, 5, 6) are in the state ONC. [Note that (x.*, yn*) (n = 4, 5, 6) are not exactly equal to each other, since the numbers stored in the computer are all binaries and the numbers printed out are the rounded numbers of the ones in storage.] Since x = 1.4,y = -0.1 is an exact solution of (A.82), the values in the state ONC have no errors in their rounded numbers, while the error bound given by (A.37) for these values is, by (A.84),

8

= 0.83 x lo-''.

This shows that the error bound given by (A.37) is sufficiently precise. When the iterative process (A.36) is stopped by the criterion (A.54) with u = from Table IX, we see that the iterative process (A.36) stops at (xl,y l ) . This shows that the value of the best possible accuracy is obtained even when the iterative process is stopped by the criterion (A.54) with u = lo-'. The error bound given by (A.58) for (x:, y:) is now, by (A.84), 0.83 x lo-''

+ 2.473 x (48 x

10-10)2 = 0.83 x lo-",

(A.86)

since

as is seen from Table IX. Expression (A.86) shows that the error bound given by (A.58) is also sufficiently precise.

302

Appendix

When the simplified Newton method is employed, as is seen from Table X, (xn*,7"') (n = 10, 11, 12) are in the state ONC. These values have no errors in their rounded numbers, while the error bound given by (A.37) for these values is, by (A.85), A

dk---0*79 x lo-'' 0.8612

c 0.92 x lo-''

This shows that the error bound given by (A.37) is also sufficiently precise for the simplified Newton method. When the iterative process (A.36) is stopped by the criterion (A.54), from Table X, we have Table XI, which shows that the error bound given by (A.58) is also sufficientlyprecise for the simplified Newton method. Table XI also TABLE XI

a

Process stops at

Error bound given by (A.58)

True errors ~

2 x lo-'' 10-~

10-~ 10-6

(X*~,Y~*) (xg*,yg*)

(x7*,y7*) (x6*,y6*)

12.69

X

lo-''

276.33

X

lo-''

(= 0.28 x

~~

0.5 x lo-'' 1.5 x

1.24 x lo-'' 2.53 x lo-''

9.5 x lo-" 63.5 x lo-'' ox

(+

shows that a should be chosen so that a = lo-" or lo-("-') in order to get the approximate solution accurate to n decimal places. However, as is mentioned concerning criterion (A.54), GL = lo-" is in danger of failure for stopping the iterative process when n is close to the computer's limit of of decimal places. In practical computation, this should always be kept in mind. Table IX and X show clearly that the proper Newton method is preferable to the general Newton method as long as H(x,,) cp(x,,) = J-'(x,,) cp(xn) in (A.26) is computable without much difficulty. However, in practical computation, sometimes there appear cases where the computation of J(x,,) is very difficult, or the computation of J - ' ( x n ) cp(x,,) at each step needs an enormous amount of time. In such cases the suitable general Newton method becomes preferable to the proper Newton method. The method described in Chapter 11 is clearly based on the general Newton method for this reason.

303

A.4. Step-by-step Methods for Ordinary Diflerential Equations A.4. STEP-BY-STEP METHODS FOR ORDlNARY DIFFERENTIAL EQUATIONS

Let (A.87)

dxldt = X(t,x) be a given vector differential equation and X(t0) = xo

(A.88)

be a given initial condition. The step-by-step method for computing the solution x = x(t) of (A.87) satisfying the initial condition (A.88) is the method for computing the values x(t,) ( n = 0, 1,2, . . .) successively. Usually, points tn (n = 0, 1,2, . . .) are equidistant, and the quantity

h = t,+l - t,,

(n = 0, 1,2,. . .)

is called the step-size of the method. The method is called the one-step method if only the value x(t,,) is required for computation of the value of x(t,,+l), while the method is called the multi-step method if not only the value of x(rn) but also a certain number of the preceding values x(tn-l), x(t,,-2), . . . are required for computing the value of x(t,,+J. Of various one-step methods, the most popular one is the Runge-Kutta method. In the Runge-Kutta method, the value x,,+~of x(t,,+J is computed from the value x,, of x(tn)according to the following formulas: xn+i = xn

+ i(k1n + 2k2n + 2k,n + khn),

(A.89)

where k,n k2n k3n k,,,

hX (tm x n ) , hX (tn + +h, xn + +kin), = hX (2. + j h , xn + +kzn), = hX (t,, h, x, + k3,,). =

=

(A.90)

+

When X(t, x ) E C:x, the value x,,+~given by (A.89) coincides with the value of cp,,(t,, h) within the error 0(h4), when x = p,(r) is a solution of (A.87) such that cp,,(t,,) = x,,. This suggests the solution x = x(t) of (A.87) satisfying the initial condition (A.88) is indeed computed accurately by the Runge-Kutta method, provided h is chosen sufficiently small. The difference p,,(t, + h) - x,,+~is called the local truncation error. For proof that

+

I I

pp.(tn

+ h) -

Xn+1

= 0(h4)

(h -,O),

see, for example, Kopal [1], pp. 195-213, or Henrici [l], pp. 118-123.

304

Appendix

Of the multi-step methods, a typical one is a predictor-corrector method. It is based on the formula of the form xn+k

+ h(b-lxn+k+l

= xn+k-l

+ box,+, + p l x n + k - l +

* * *

+ bkxn),

(A.91)

where (i = II, II

Xi = X(t,, xi)

+ 1, . . .,n + k + 1).

Formula (A.91) is a k-step Adams predictor when /3-1 = Po = 0. By means of the Adams predictor, the value Xn+k of x(t,,+k) can be computed directly from the preceding values x , , + k - 1 , X n + k - 2 , . . ., x,, of x ( t n + k - l ) , x(tn+k-2),

-

*

-9

.(tn)*

When /3-1 = 0 but #lo# 0, formula (A.91) is a (k + 1)-step Adams corrector. The Adams corrector is regarded as an equation of form (A.l) with respect to &+k, since the preceding values x,,+k-l, X,,+k-2, . . ., x,, are known in advance in practical application. By Theorems A.l and A.2, this equation can be easily solved numerically by the iterative method if X(t, x ) satisfies a Lipschitz condition and I h 1 is chosen sufficiently small, since the starting approximate value of Xn+k can be readily found by the Adams predictor. The Adams corrector is used for correcting the values obtained by the Adams predictor, since the corrector is more accurate than the predictor for the same number of steps, as will be shown later. has been devised by the'author (Urabe, Formula (A.91) with nonzero Yanagiwara, and Shinohara [11;Urabe [7]). This is a corrector of the (k 2)steps and is used simultaneously with the (k + 2)-step Adams predictor

+

%+k+l

= xn+k

+

+ P Z ' X n + k - l + + /%+lXn + B;+ZXn-l).

h(bl'Xn+k

"'

(A.92)

Namely, when the preceding values x,,+k-i, X , , + k - p , . . ., x,, x,,- are known in adance, the value x,,+k is computed by solving numerically the simultaneous equations (A.91) and (A.92) with respect to x , + p and x n + k + l. The SimUltaneous equations (A.91) and (A.92) are equivalent to the single equation xn+k

= %+k-1

+ h{B-lX

+ pOX(fn+k,

xn+k)

[fn+k+l,

+

"'

Xn+k

+ h(Bl'Xn+k +

*'*

+ b;+2xn-l)]

+ /3kXn>,

and this is of the form (A.1). Therefore, by Theorems A.1 and A.2, the simultaneous equations (A.91) and (A.92) can be easily solved numerically with respect to x,,+k and X,,+k+l by the iterative method if X ( t , x ) satisfies a Lipschitz condition and I h I is chosen sufficiently small, since the starting approximate values of x,,+k and X,,+k+l can be readily found by the Adams

A.4. Step-by-step Methods for Ordinary Diyerential Equations

305

predictor. The corrector of the form (A.91) with nonzero P- is used instead of the Adams corrector, since, as will be shown later, the former is more accurate than the latter for the same number of steps for k 2 3. The Adams predictor and corrector are well known; the formula (A.91) with nonzero PUl, however, is not popular. Hence a method to calculate coefficients /3-1, Po, /I1, . . ., Pk of (A.91) will now be explained. Put

- Ik - 1 ,

p(I)

= 'I

.(A)

= p-1Ikk"

+ P O I k + PIIk-' +

..a

+

Pk-lA

+

(A.93) Pk;

then, from (A.91), we have

M E ) - h4VIx(tn) = Tn,

(A.94)

where T,, is a local truncation error, D = d/dt, and E is a translation operator such that

Ex(t) = x(t

+ h).

(A.95)

Formula (A.91) must be applicable to a wide class of differential equations; therefore it should be applicable for arbitrary analytic differential equations. This implies (A.94) should hold for arbitrary analytic x(t). For analytic x(t), we have, however, by Taylor's theorem,

Ex(t) = x(t

+ h) = ek'x(t),

(A.96)

provided I h I is sufficiently small. Then, for sufficiently small 1 h I, we can rewrite (A.94) as follows:

[p(ehD)- cr(ehD)* hD]x(tn)= T,,.

(A.97)

This implies T,, is of the form

+

T,, = [ C h P + l D p + l

] x(t,,).

(A.98)

Then, comparing the both sides of (A.97), we have

because (A.97) should hold for arbitrary values of x(tn), %(t,,),Z(t,,), = d/dt). Relation (A.99) is equivalent, however, to the relation

p(e5) - g(e4 6 = C ~ P ++ '

... ,

where 5 is an arbitrary small number. Let us now put

e':= 1 + c ;

. . .,

(A.lOO)

306

Appendix

then, from (A. loo), we have p(l

+ 5) - a(l + 5).log(l + 5) = ClogP+'(1 + i)+

-*-

(A.lO1)

Since log(1

+ r) = r(1 - 5 + 7 c2 - .-) -

2

for

I 5 I < 1, we can rewrite (A.lO1) as follows: (A. 102)

Now, by the first of (A.93),

m

=

1 c:5:

(A.103)

r=O

where

Since the integer p should be as large as possible as is seen from (A.98), equality (A. 102) implies that k p =k k

p-l

when when when

+1

+2

p-,

p-l

= =

Po

0, # 0,

=

0,

(A.105)

and, for such p , a(1

+ 5) =

c C,k ir.

P- 1

r=O

Equality (A. 106) implies

rcc:( "-1

.(A)

=

r=O

O-t

r

- 1 + A).

(A. 106)

A.4. Step-by-Step Methods f o r Ordinary Differential Equation

307

Thus, from the second of (A.93), we have p - 1 -s

(-1y

=

pk-s

r=O

r:r)c,k+r

(s = 0,

1, 2,

p - 1).

(A.107)

From this, coefficients p-', Po, PI,. . ., p k are easily calculated. By (A.102), equality (A.106) also implies

c = Cpk.

(A.108)

This enables us to compare the accuracy of the different multistep formulas. In fact, by (A.104) and (A.105),

jo 1

C = C,k =

C = Ci-'

+ k - 1) ... (u + 1).

(1.4

= '/o'(u

k!

L!

C = C:-z = -

1

Jo

(U

du for the k-step Adams predictor,

+ k - 2)...(u + l)u(u

1)du for the k-step Adams corrector,

-

+ k - 3) ... (U + ~ ) u ( u- 1) (U - 2) du for the k-step corrector with P-

# 0. (A. 109)

However, as is easily seen, 0<

c;-z < I c;-* 1 < c;

for k 2 4. Hence, by (A.98), we see that, of the formulas of form (A.91) with the same number of steps, the formula with nonzero /3-1 is the most accurate, and the Adams predictor is the most inaccurate, provided the number of steps is not less than 4. By (A.96), relation (A.99) implies that [ p ( E ) - ha(E)D]x(t) = C h p + l ~ ( p + ( t'))+ O(hP+')

(A.110)

+

for every ( p 1)-times continuously differentiable function x(t). By (A.108) and (A.94), relation (A.110) implies that the local truncation error of formula (A.91) is of the form (A.lll)

+

for any ( p 1)-times continuously differentiable solution x = x(t). Among the multi-step formulas, the 5-step formulas seem to be most convenient for practical computation since the number of steps is moderate and, in addition, these are more accurate than the Runge-Kutta method,

308

Appendix

which is a typical one-step formula. By (A.107) and (A.109), the 5-step formulas are 5-step Adams predictor: = xn+4

x,+S

T -

95 -

- 288

h + -((19O1xn+4 720

h6P(t,)

- 2774x,+,

+ 2616X,+2

- 1274X,+ 1

+ 251X,),

(A.112)

+ o(h6);

5-step Adams corrector: Xn+4

= x,+3

h + -(251Xn+4 + 646Xn+3- 264X,+2 + lO6X,+, - 19Xn), 720

(A.113)

T, = - - h6x ( 6 )(t,) 160

+ o(h6);

5-step corrector with p-, # 0: Xn+3

=

Xn+z

..

11

h +( - 19X,+4 + 346X,+3 + 456X,+2 - 74X,+, + llXn), 720

+

(A.114)

T, = __ h6x@)(t,) o(h6). 1440 Formulas (A.112) and (A.114) were used for numerical computation of periodic solutions of the van der Pol equation in 11.

Remark. When we want to use the multi-step methods for numerical integration of differential equations, to start the computation, it is necessary to know in advance some values of the solution in the neighborhood of to. These are computed frequently by means of Taylor expansions of the solution and sometimes by means of Runge-Kutta method of the smaller (say, half) step-size. When the step-size can be chosen sufficiently small, the Runge-Kutta method might be preferable to the multi-step method ;otherwise the multi-step method will be preferable, since its local truncation error can be made considerably smaller than that of the Runge-Rutta method.

A.5. Convergence of the Runge-Kutta Method

309

AS. CONVERGENCE OF THE RUNGEKUTTA METHOD Let x, be the computed values of x(t,) by the step-by-step method. Then the step-by-step method is said to be convergent if x, -+ x(t’) for any fixed t’ = nh as h + 0. In other words, the step-by-step method is convergent if and only if it enables one to obtain an approximate solution over a given finite interval as accurately as one desires by choosing the step-size sufficiently small. As is readily seen from this definition, the step-by-step method is of practical use if and only if it is convergent. In the present section, convergence will be proved for the Runge-Kutta method under some general conditions. The assumptions on the given differential system (A.87) are as follows: System (A.87) has a solution x = x ( t ) over the finite closed interval I = [to, to + I ] or [to - I, to],and X(t, x ) is continuous with respect to t and x in the region V : t E I, I x - X ( t ) I 5 p ( p > 0) and satisfies a Lipschitz condition

I X(t, x’) - X ( t , x”) I s L I x’ - x” I

(A.115)

for every (t, x’), ( t , x”) E V. The conclusion of the present section is that the Runge-Kutta method is convergent if all round-ofl errors arising in practical computation are uniform infinitesimals of the step-size h of the order higher than one. The proof is as follows. According to (A.89) and (A.90), for the exact values I, = x(t,), put

(A. 116) and

kin = hX(tn, X n ) ,

+ + h, 2, + + + h, 2, + = hX(t, + h, I, + ft3,). hX(tn

Z2n

=

R3n

= hX(tn

R,,

+ft,n),

(A.117)

+R2n),

Here evidently t, = to + nh (n = 0, 1,2, . . .). Since X [ t , x(t)] is uniformly continuous on the interval I,

In+l= x(tn

+ h) = In+ hX(t,, 2,) + o(h)

uniformly as h

+ 0.

On the other hand, since X ( t , x ) is uniformly continuous and bounded in the region V, from (A. 117),

310

Appendix

Ll,, EZn. E3,, E,,

+ o(h)

= h ~ ( t , 2,) ,

uniformly as h

+ 0.

Hence, from (A.116), we have

T,

=

o(h) uniformly as h + 0.

(A. 118)

Now let x , and (kin, k,,, k3,,k4”)be, respectively, the computed values by formulas (A.89) and (A.90). Then, because of round-off errors, we have

x,+1

+ @ l n + 2k2, + 2k3, + k4,) + R ,

=

x,

=

hX(tn xn)

(A.119)

and kin

+ rln

3

(A.120) k,,

=

hX(t,

+ h, x, + k3,) + r4,,

where R,,rln, r,,, r3, and r,,, are the round-off errors. In order to prove the convergence of the Runge-Kutta method, we consider the extension x) of the function X ( t , x) such that x ) coincides with the original X ( t , x) in the region V and, when t E I and I x - x ( t ) I > p,

s(t,

s(t,

q t , x) = q t , a),

(A.121)

where (A.122) The function 8 ( t , x) is then defined in the region

v,

: t€Z,

1x1 coo,

and 8 ( t , x ) is uniformly continuous and bounded in the region V,. Moreover, 8(t,x) satisfies a Lipschitz condition

I q t , x’) - q t , x”) I s 2L I xf - x” I for every ( t , x‘), (t, x”) E V , . In fact, if I x’ - x(t) I , I X” - x ( t ) I

then

P

9’= x(t)

+

R”

+ I XIf -P X(t) I [XIf - x(t)];

= X(t)

I xf - X(4 I

[x’ - X(t)],

(A. 123)

=- p, put

A.5. Convergence of the Runge-Kutta Method

A’ - 2”

=

I x’ -

-

P

I [x‘

I

I x’ +

- X(t)] -

{[x’ -

X(t)]

(I x’ P x ( t ) ) - I x“

I x” +p’

- x(t) I

P - X(f)

I x”

-

[XI’

- x(t)I

I

- x(t)]

- x(t)]}

)

- x‘ - X ( t )

I [x”

31 1

[XI’ - x(t)]

I [XI’

- X(t)].

~x‘-x(t)l*lx’’-x(t)l

Therefore

From this, by (A.121) and (A.l15), it readily follows

I q t , x ’ ) - q t , x ” ) I = I X ( t , 2 ’ ) - X ( t , 3“) I 5 L 12‘ - A” I 5 2L I x’ - x” which is the inequality to be proved. If I x’ we have

-

therefore we have

I XI - X(t) I

1,

- x ( t ) I > p but I X‘’

{[x‘ - x(t)]

- [x” - x ( t ) ] }

- x(t) I 5 P ,

Appendix

312

5

2p

Ix’ -

I

1 x’ - x“ I

2 I x’ - x”

I.

This is of the same form as (A.124). Hence we again have (A.123). When x’ - x ( t ) x” - x(t) I 5 p , inequality (A.123) is evident from (A.115). Thus we see that inequality (A.123) holds always for every ( t , x’), (t, x”)

1

E

I,I

v,. Now let us consider the differential system dx

-=

dt

(A.125)

q t , x).

Then x = x(t) is evidently a solution of this system since 8 ( t , x ) coincides with X ( t , x ) in region V. For the present, let us suppose the Runge-Kutta method (A.89) (A.90) is applied to system (A.125) instead of the original system (A.87). Since 8 ( t , x ) is defined in the region V,, x, (n = 0, 1,2, . . .) are indeed obtained for every t,, E I. Put x,

- Zn = en

(n = 0,1,2, . . .),

(A.126)

and let us suppose that

where

K

=~

Q

= r(i

+ I + h + 3 Ih + p I L + 3 1 h I’C+ + p + R + T.

( 22 h I L

p3),

13~3)

From (A. 129), by induction, it readily follows

(A. 130) (A. 131)

A.6. Convergence of the Multi-Step Methods

=(1

+ I h ( K ) ” ( e o I+ ( l + ) h J K ) ” - l Q

(n

313

=

1, 2, ...).

lhlK

+ exp(n I I K , - 2

(TI

= 1, 2,

...).

(A.132)

. lhl

K

I I 5 1, the above inequality implies

Since n h

e

~ % ~ 5 K l ~ e o ] + ~ 2 (n . - = I , 2, ...) (A.133) lhl for some constants K l and K,. Now, for round-off errors R,,, rln, r,,,, r3,,,and r4,,, we may suppose by assumption that

R = o(h),

r = o(h)

as h

0

(see (A.127)). Then, by (A.118) and (A.131), it holds that

Q

= o(h)

as h

3

0.

Hence, from (A.133), we see that Ie,,I+O

( n = 1 , 2 , ...)

as h + O

I I

1 i

(A.134)

if I e, 1 0. By (A.126), this implies that if h and eo are sufficiently small, points (tn,x,,), (tn f h , x,, !A,,), (t,, f h , x,, t k ~ n ) , and (t,, h, x,, k3,,)all belong to region V for every t,, such that f,, t,, h E I. Since z(t,x ) coincides with X(t, x ) in region V, this means that x, ( n = 0, I , 2, . . .) obtained by applying the Runge-Kutta method to the system (A.125) are nothing else but the values obtained by applying the RungeKutta method to the original system (A.87). Then the property (A.134) means that the Runge-Kutta method is indeed convergent.

+

+

+

+

+

+

+

A.6. CONVERGENCE OF THE MULTI-STEP METHODS

In the present section, for the differential system (A.87) satisfying the same conditions as in the preceding section, it will be proved that the multi-step methods mentioned in (A.4) are also all convergent i f all round-off errors

314

Appendix

arising in practical computation are uniform injinitesimals of step-size h of the order higher than one. To prove the convergence of the multi-step methods, as has been seen in the preceding section, it suffices to prove it for the case where the region of definition of the function X ( t , x) is V,. Therefore, in the present section, this will be assumed, and a Lipschitz condition (A.115) will be assumed for every ( t , x'), ( t , x") E v,. As in the preceding section, the computed values and the exact values of x(t,) will be denoted, respectively, by x , and Z,, and the errors of x, will be denoted by en,that is, equality (A.126) will be set. According to (A.91), let us put gn+k

=

where

*

=

+ PO;n+k + P l % + k - l +

+ h(P-l'n+k+l

In+,-,

'*'

+ pknn> + Tn,

(A.135)

d/dt. It then holds that

T, = o(h)

uniformly as

h

(A. 136)

+ 0.

In fact, since x ( t ) is continuously differentiable in the closed finite interval

I,it holds that T , , + ~ ~ x(t, =

X7n+i

= $(t,

uniformly as h

+ ih) = T, + ihx', + o ( ~ l ) , + ih) = ;, + ~ ( l )

+0

P-1

(i = 0, 1, ..., k

+ 1) (A.137)

and, by (A.93), (A.106), and (A.104), we have

f

+ + + p1

*..

Pk

= a(1) =

c,"

(A.138)

= 1.

Hence, substituting (A.137) into (A.135) and making use of (A.138), we readily obtain (A.136). First the convergence will be proved for the Adams predictor and corrector. According to (A.135), let us put

and xn+k

=

xn+k-l

h[PoX(tn+k,x,+k)

".

PkX(fn, Xn)]

+ Rn

9

(A.140)

where R, is a round-off error. Subtracting (A.139) from (A.140), we then have en+k =

where

en+k-I

+ b ( t w en+k, ..-)en) + Rn -

Tn

9

(A.141)

A.6. Convergence of the Multi-Step Methods

315

Now let us consider the vector

then, from (A.141), we have (A.144) where

l 0

E

0

o...o-

. 0. . E * O .. . .. O . . . . ..

0

A =

. .

0

0

...

0

0

...

. .

0

0

E

0

E-

( E : unit matrix),

( A .145)

(A.146)

Appendix

316

(A.147)

Corresponding to A , let us consider the matrix

For this matrix, any eigenvalue

IC

and any corresponding eigenvector

satisfy the relation u2

=

KU1,

u3

= Kuz,.

u2

=

. .,

uk

=

ICuk-1,

uk

=

KUk.

(A.148)

This implies = IC 2u1,

K U 1 , u3

. . ., uk

=

ICk-l

u1

(A. 149)

and (Kk

- ICk-')U1

= 0.

Since u1 # 0 from (A.149), we then have Kk

- ICk-1

= 0,

which implies that the eigenvalues of A' are

(A.150)

A.6. Convergence of the Multi-Step Methods

317

k- 1

0,O ,..., 0

andl.

However, by (A.149), each different eigenvalue of A' can have only one eigenvector. Hence the Jordan canonical form of A' must be of the form

.o

0 .

0

-

s o 0

s 7

. . s

o...

0

0

-

.o

0 .

(A.151)

1

where '6 is an arbitrary nonzero constant. Now let T' = ( t i j ) be a matrix such that T'-'A'T' = A';

then, for the matrices (A. 152)

and

A =

0

0

iE

0

...

. .S E .

0

.

0 0

0

(A.153) 0

SE

.o

0 . from (A.145) and (A.151), we have

T-'AT = A.

Then, putting en = Te,'

318

Appendix

in (A.144), we have eA+l = Ae,‘

+ hq’ (l,,e:,,, en’)+ R,‘,

(A. 154)

where q ’ ( t n , e:+1, en’) = T-’ q(tm

T ~ A + 1,

(A.155)

Ten’),

R , = T-’R,.

(A. 156)

Now 6 has been an arbitrary nonzero constant. Therefore let us choose 6 so that 0 < 6 < 1 . By (A.153) we then have \A\=’.

(A.157)

For round-off errors R,, let us suppose that

R, = o(h) uniformly as h

0.

-+

(A. 158)

Then, by (A.136), (A.147), and (A.156), it follows that

R,’ = o(h) uniformly as h

-+

0.

This implies that there is the quantity r(h) such that

I R,’I s r(h)

and

r(h) = o(h)

as

h

+ 0.

(A.159)

The p’(rn,e:+ en’)in (A. 154), however, by (A. 143), (A.146), and (A. 155), satisfies the inequality

I q ’ ( t n , e:+,, en’)I

2

~1

I

%+I

I + Kz 1 en‘ I

(A.160)

for some positive constants K , and K,. Thus by (A.157), from (A.154), we have eb+l I 5 en’ I + h ( ~ I1eL+l I + ~2 en’ ) + r(h), (A*’‘’)

I

I I

I

I I

from which it readily follows

for any h such that I h I

5 h,, provided h, is sufficiently small. Since

I I

it is evident that for any h such that h S h,, (A.163)

A.4. Convergence of the Multi-Step Methods

319

where

By (A.163), inequality (A.162) can be then rewritten as follows:

This is of the same form as (A.129). Therefore, by (A.132), we have

I I

1 and, by (A.159), r(h)/l h

Since n h implies

Ie,’

I 1

provided eo’ + 0 as h implies that

-+

I -,o

as h

+ 0,

0. By the definition of the vectors en, this evidently

le,1+0

I

I I, I 1,

I + 0 as h + 0, inequality (A.164)

h + ~ ,

as

I

provided e, el ... , e k - l -+ 0 as h + 0. This proves the convergence of the Adams predictor and corrector. Next we will prove the convergence of the multi-step formula with P- # 0. When the multi-step formula with P- # 0 is used for practical computation, by (A.92), we have %+k

+ h[P-l

= %+k-l

+ + **’

%+k+l

=

P k x(zn,

+ h[Pl’

%+k

X(tn+k+l,

gn)]

gn+k+l)

+ P O X(tn+k,

zn+k)

+ K,

(A.165)

X(tn+k, gn+k) f

**‘

+ P;+Z

X(fn-l,

zn-l)]

+ Tn’

for the exact values 2, of x(tn)and %+k

=

&+k-l

+ $n+k+l

=

*‘*

xn+k

+ h[P-l + Pk X(tn,

X(zn+k+l, xn)]

+ Rn

%+k+l)

f P O X(fn+k, %+k)

(A.166)

9

+ h[P1’ x ( t n + k , % + k ) +

*”

+ P;+Z

X(zn-l,

xn-l)]

+ Rn‘

for the computed values xn of x(tn), where T,, T,,’ are the local truncation errors, R,,R,’ are the round-off errors, and R n + k + l are the subsidiary values

Appendix

320

appearing in the course of computation. Subtracting (A. 165) from (A.166), we then have

where

and

By (A.115),it is evident that

1

'%fk+l,

%+k,

5 L( I b - 1

I

$(tn,

%+k,

**.Y

II

**.7

en)

&n+k+l

I I -t

I

en,

I

5 L( b1' I I % + k

I+

"'

+

I P ; + l ( ( e n l +

lb;+zllen-ll)*

Equalities (A.167) can be rewritten, however, by eliminating follows : en+&= e n + k -

+ R,'

+ h$(tn, - Tn:en+k,. . ., en] + Rn - T ,.

1

+ hq[tn,

en+k

en+k,

--*3

&+k+

1

as

en-1)

(A.169)

A.6. Convergence of the Multi-Step Methods

321

This is similar in form to (A.141). Therefore, as in the previous case, we consider the vector

en =

Then, as before, we can rewrite equality (A.169) as follows:

where A , cp,,, and R,,are, respectively, of the same forms as (A. 145), (A. 146), and (A.147). As in the previous case, equality (A.170) can be further rewritten as follows: e:+l = Ae,,‘

+ hqn’(tn,

4 + 1, en’)

+ Rn’,

(A.171)

where A is the matrix of the form (A.153). In the present case, however, from (A.169) and (A.170), we have IVA (Le’n+l,en’) ]

5 Ki ] e:+i

I + K , ] en‘ ] + K3 I K’ - Tn’ ]

(A.172)

I 1 5 ho for an

for some positive constants K , , K,, and K 3 , provided h trary positive constant ho. Let us now suppose that

R,,’= o(1) and R,,= o(h)

uniformly as h

arbi-

+ 0.

Then, since

T,,,T,,‘ = o(h)

uniformly as h

-+

0

by (A.136), we have

1 h I K~ [ R,,’- T,,’I + IR,,’I = o(h)

uniformly as h

+

0.

This implies that there is the quantity r(h) such that

I h I IRn‘ - T n ’ [ + [ R,’ I 5 r(h) K3

(A.173)

and

r(h) = o(h)

uniformly as h

+ 0.

(A.174)

322

Appendix

Now, as in the previous case, let us suppose that the S in (A.153) is a positive constant less than 1. Then we have (A.157). Hence, from (A.171), by (A.172) and (A.173), we have

I e:+ 1 I 5 I I + I h I (Ki I %+ 1 I + K , I en’ I ) + r(h). en’

(A.175)

This is of the same form as (A.161). Thus we have the inequality analogous to (A.164) and, from this, we see that (e,1+0

I 1, I I

I I

as h - 0 ,

provided e,, el ..., e, -+ 0 as h -+ 0. This proves the convergence of the multi-step formula with B- # 0. The proof of this section is quoted from the author’s paper [7].

Bibliogruphy Bellman, R. [ l ] “Introduction to Matrix Analysis,” New York, 1960. Bochner, S., and Martin, W. T. [ l ] “Several Complex Variables,” Princeton Univ. Press, Princeton, New Jersey, 1948. Bogoliubov, N. N., and Mitropolski, Y. A. [I] “Asymptotic Methods in the Theory of Nonlinear Oscillations” (In Russian), Moscow, 1958; English translation, New York, 1962. Coddington, E. A., and Levinson, N. [l] “Theory of Ordinary Differential Equations,” New York, 1955. Collatz, L. [l ] Einige Anwendungen funktionalanalytischer Methoden in der praktischen Analysis, 2. Angew. Math. Phys., 41, 327-357 (1953). Diliberto, S. P., and Hufford, G. [ l ] Perturbation theorems for non-linear ordinary differential equations, “Contributions to The Theory of Nonlinear Oscillations,” Vol. 3, pp. 207-236. Princeton Univ. Press, Princeton, New Jersey, 1956. Dorodnicyn, A. A. [ l ] Asymptotic solution of van der Pol’s equation (in Russian), Prikl. Mat. Meh., 11, 313-328 (1947). Duff, G. F. D. [ l ] Limit-cycles and rotated vector fields, Ann. ofMath., 57, 15-31 (1953). Hale, J. K. [ l ] “Oscillations in Nonlinear Systems,” New York, 1963. Halmos, P. 111 “Finite Dimensional Vector Spaces,” New York, 1959. Henrici, P. [l ] “Discrete Variable Methods in Ordinary Differential Equations.” New York, 1962. Kopal, Z. [ l l “Numerical Analysis,” London, 1961. Koukles, I., and Piskounov, N. [ l ] Sur les vibrations tautochrones dans les systemes conservatifs et non conservatifs, C.R. Acad. Sci., URSS, 17,471-475 (1937). Levin, J. J., and Shatz, S. S. [l] Nonlinear Oscillations of Fixed Period, J. Math. A n d . Appl., 7, 284-288 (1963). McLachlan, N. W. [ l I “Ordinary Non-linear Differential Equations in Engineering and Physical Sciences,” Oxford Univ. Press, London and New York, 1950. Minorsky, N. 111 “Nonlinear Oscillations,” New York, 1962.

323

324

Bibliography

Niemytski, V. V., and Stepanov, V. V. [l]“Qualitative Theory of Differential Equations” (In Russian), Moscow, 1949; English translation, Princeton Univ. Press, Princeton, New Jersey, 1960. Opial, 2. [I] Sur les pbiodes des solutions de Equation differentielle x”+g(x) = 0, Ann. Polon. Math., 10, 49-72 (1961),. Ponzo, P. J., and Wax.N. [l] On the periodic solution of the van der Pol equation, IEEE Trans, Circuit Theory, CT-12, 135-136, (1965). [2] On certain relaxation oscillations: asymptotic solutions, J. SOC.Indust. Appl. Marh., 13, 740-766 (1965). Sibuya, Y. [l] Remarques sur la theorie des centres aux dimensions supkrieures, J. Math. SOC. Japan, 8, 1 - 6 (1956). Uno, T. [l] On some systematic method for finding limit cycles, Proc. 1st Japan National Congress for Applied Mechanics, Tokyo, 1951, pp. 513-516. Urabe, M. [l]Convergence of numerical iteration in solution of equations, J. Sci. Hiroshima Univ. Ser. A, 19, 479489 (1956). [2]Periodic solutions of Van der Pol’s equation with damping coefficient A = O(0.2)l.O. J . Sci. Hiroshima Univ. Ser. A, 21, 193-207 (1958). [3] On the nonlinear autonomous system admitting of a family of periodic solutions near itscertain periodic solution, J . Sci. Hiroshima Univ. Ser. A, 22,153-173 (1958). [4] Geometric study of nonlinear autonomous oscillations, Funkcial. Ekvac., 1, 1-83 (1958). [5]On a method to compute periodic solutions of the general autonomous system, J. Sci. Hiroshima Univ. Ser. A , 24, 189-196 (1960). [6] Remarks on periodic solutions of van der Pol’s equation, J. Sci. Hiroshima Univ. Ser. A, 24, 197-199 (1960). [7]Theory of errors in numerical integration of ordinary differential equations, J. Sci. Hiroshima Univ. Ser. A-I Math., 25, 3-62 (1961). [81 Error estimation in numerical solution of equations by iteration process, J. Sci. Hiroshima Univ. Ser. A-I Math., 26., 77-91 (1962). [9]Numerical study of periodic solutions of the van der Pol equation, “International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics,” New York, 1963, pp. 184-192. 1101 Relations between periods and amplitudes of periodic solutions of x+g(x) = 0, Funkcial. Ekvac., 6, 63-88 (1964). [I11 Galerkin’s procedure for nonlinear periodic systems, Arch. Rational Mech. Anal., 20, 120-152 (1965). Urabe, M., and Reiter, A. [1I Numerical computation of nonlinear forced oscillations by Galerkin’s procedure, J. Mafh. Anal. Appl., 14, 107-140 (1966). Urabe, M., and Sibuya, Y. 111 On center of higher dimensions, J. Sci. Hiroshima Univ. Ser. A , 19, 87-100 (1955).

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Urabe, M., Yanagiwara, H., and Shinohara, Y. [I] Periodic solutions of van der Pol’s equation with damping coefficient 1= 2- 10, J . Sci. Hiroshima Univ. Ser. A , 23, 325-366 (1960). Whittaker, E. T., and Watson, G. N. [lJ “A course of Modern Analysis,” Cambridge Univ. Press, London and New York, 1935. Yanagiwara, H. [ l ] A periodic solution of van der Pol’s equation with a damping coefficient 1= 20, J . Sci. Hiroshima Univ.Ser. A , 24, 201-217 (1960). [2] Maximum of the amplitude of the periodic solution of van der Pol’s equation, J . Sci. Hiroshima Univ. Ser. A-I Math., 25, 127-134 (1961).

Subject Index A Abel’s integral equation, 249 absolute asymptotic orbital stability, 83, 86 asymptotic stability, 55 orbital stability, 81, 198, 201 stability, 55 unstability, 55 Adams corrector, 304, 307, 308, 314 predictor, 304, 307, 308, 314 amplitude, 244, 270 of periodic solution of van der Pol equation, 220, 221 analytic function, 21 analytic manifold, 76 analytic system, 179 approximately closed orbit, 204 associated equation, 25 asymptotic expression for amplitude, 222 for period, 223, 225 asymptotic orbital stability, 89, 90, 97, 106, 115, 158, 166, 167 in negative sense, 84 in positive sense, 84 with respect to family of solutions, 80 asymptotic stability, 57, 60, 61, 106 in negative sense, 79, 80 in positive sense, 80 with respect to family of solutions, 55, 70, 77 autonomous system, 34 averaged system, 111, 114, 115

B bifurcation of closed orbit of fully oscillatory system, 202 of multiple closed orbit, 198

C canonical form of autonomous system in neighborhood of center, 239, 242 canonical form of matrix, 3 center, 229, 244 characteristic exponent, 30, 60, 61, 89, 90, 97, 166, 167 for periodic solution of van der Pol equation, 221 closed approximate orbit, 226 closed orbit, 39 of even multiplicity, 198 of odd multiplicity, 198 complex extension, 21 conditional orbital stability, 81, 86 stability, 55, 198 continuity of orbit, 39 convergence of multi-step method, 313 of Runge-Kutta method, 309 of step-by-step method, 309 corrector formula with nonzero /?-I, 304, 307, 308, 319, 322 criterion for stopping iteration, 294, 295, 298, 301, 302 critical point, 37, 82, 83, 84, 106, 114, 115

321

Subject Index

328

D dependence of solution on initial condition and parameter, 19 derivative of X - l ( r ) , 15 differential operator D, 305 differentiation and integration of matrix, 15 direct sum of matrices, 7 Dirichlet’s formula, 251, 252

E elementary divisor, 3, 4 element-wise convergence, 6 operation, 15 equation with respect to moving orthogonal system, 50 error bound for approximate solution, 290, 295, 301, 302 even function T(X),259,270,272, 273 exp X,9 exp (tA), 15, 26 F first variation equation, 20, 50, 89, 90, 96, 97 fully oscillatory system, 101, 167, 176 function V ( X ) , 268, 277, 278 fundamental existence theorem, 17, 18 fundamental matrix, 25 5-step Adams corrector, 308 Adams predictor, 222, 308 corrector with nonzero B- 308

G Galerkin procedure, 225,226 general Newton method, 288, 300 Gram-Schmidt process, 124

H half-orbit, 268 h, 166 h(t), 165

I integral equation, 63, 248, 254. 272, 273, 278 integral manifold, 139 inverse problem, 244, 246 isolated periodic solution, 96 iterative method for numerical solution of equation, 281 Z, 185 Z(a), 176 Z(x), 249, 251

J Jordan canonical form, 3, 4

L Lipschitz condition, 17, 254, 272, 278, 281, 288, 304, 309, 310 constant, 18 local Lipschitz condition, 18 local truncation error, 303, 305, 307, 308, 319 logarithm of matrix, 10 log ( E X ) , 9

+

M manifold, 70, 77, 79, 80 mantissa, 301 maximum of amplitude of periodic solution of van der Pol equation, 222 maximum positive displacement, 244 velocity, 244, 259, 270, 277 method of averaging, 111 method of variation of constant, 33 minimum negative displacement, 244 moving orthogonal coordinate system, 47 orthonormal system, 42, 47, 138 multiple closed orbit, 197, 198 multiplicity of closed orbit, 195, 196, 199 multiplier of solution, 29 of first variation equation, 53, 95, 96 of normal variation equation, 53 multi-step method, 303, 313

329

Subject Index

N negative half-amplitude, 245, 270, 278 negative integral manifold, 84 negative-side half-branch of g(x), 268,269, 270, 278 half-period, 245, 259, 269, 270, 278 Newton method, 204, 207, 212, 280, 288, 290 norm, 1 I2 II,2 280 normal component of variation of solution, 51 variation of solution, 51 normal variation equation, 50, 89, 166, 167

... I

I ... ...I,

0 odd function S(X), 259, 271, 272, 274 one-step method, 303 orbit, 34, 37, 80 orbital asymptotic stability, 167, 177, 181, 184, 186, 189, 201, 202 orbital stability, 80, 167, 198, 201 in negative sense, 81 in positive sense, 81 of critical point, 82 of periodic solution, 87 with respect to family of solutions, 80, 81 orbital unstability, 167, 177, 181, 184, 186, 189, 201, 202 in negative sense with respect t o family of solutions, 84, 86 in positive sense with respect to family of solutions, 84, 86 with respect to family of solutions, 81 ordinary point, 37 orthogonal trajectory, 168, 176 orthonormal m-ple system, 129 (n - 1)-ple system, 129

P parametric representation of Ax), 266

partially oscilIatory system, 119, 138, 151 period function, 245, 246 associated with amplitude, 270, 272 associated with half-amplitude, 270 associated with maximum velocity, 259 period of closed orbit, 39 period of periodic solution of van der Pol equation, 220, 221 perturbation, 91, 167 of analytic fully oscillatory system, 199 of analytic system, 185 of fully oscillatory system, 101, 113, 167 of partially oscillatory system, 119, 139 perturbed system, 91 phase, 90 phase space, 34, 43, 80 positive half-amplitude, 245, 270, 277 positive integral manifold, 84 positive-side half-branch of Ax), 268, 269, 270, 278 half-period, 245,259,268,269,270,277, 278 predictor-corrector method, 304 primitive period, 38, 95, 96, 103, 229, 239 problem of isochronism, 246, 267 proper Newton method, 288,294, 300

R real analytic differential system, 21, 23 real canonical form of matrix, 4 real logarithm, 11 rotatory perturbation, 195 round-off error, 309,310, 313, 318, 319 Runge-Kutta method, 303, 308 S

sequence and series of matrices, 6 simple closed orbit, 197, 198, 202 simplified Newton method, 301, 302 Simpson’s rule, 222 stability, 54 in negative sense, 55 in positive sense, 55 of periodic solution of analytic system, 180

330

Subject Index

of periodic solution of perturbed system 97, 152 with respect to family of solutions, 54, 78 state ONC (state of oscillatory numerical convergence), 284, 290, 301 step-by-step method for ordinary differential equation, 280, 303 step-size, 303, 308, 309, 314 stroboscopic image, 111 subperiodic solution, 95, 96, 97, 112, 115, 152, 158, 167 successive approximation, 22, 64

U uniform Lipschitz condition, 19 universal period, 103, 122, 138, 234, 237, 238, 239 unperturbed system, 91 unstability in negative sense, 55 in negative sense with respect to family of solutions, 79, 80 in positive sense with respect to family of solutions, 80 with respect to family of solutions, 55, 70

V T time required to reach point, 36 trajectory, 18, 34 translation operator E, 305 two-dimensional autonomous system, 163

van der Pol equation, 34, 117, 204, 216, 308

Y 2 +Ax) = 0, 244