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.