Contributions to the Theory of Nonlinear Oscillations (AM-20), Volume I 9781400882632

Table of contents :
PREFACE
CONTENTS
I. On Systems of Ordinary Differential Equations
II. Oscillation of a Third Order Nonlinear Autonomous System
III. Nonlinear Difference-Differential Equations
IV. Forced Oscillations in Nonlinear Systems
V. Singular Perturbations of a Van der Pol Equation
VI. The Existence of Forced Periodic Solutions of Second Order Differential Equations near Certain Equilibrium Points of the Unforced Equation
VII. The Construction of Periodic Solutions of Singular Perturbation Problems

Citation preview

ANNALS OF M ATH EM ATICS STUDIES Number 20

ANNALS OF MATHEMATICS STUDIES

Edited by Marston Morse and Emil Artin by P R. Introduction to Nonlinear Mechanics, by N. K Lectures on Differential Equations, by S

7. Finite Dimensional Vector Spaces, 11. 14.

aul

H alm os

ryloff

olom on

and N. B

o g o l iu b o f f

L efsch etz

15. Topological Methods in the Theory of Functions of a Complex Variable,

by

M a r st o n M orse

16. Transcendental Numbers,

by

C a r l L u d w ig S ie g e l

17. Probleme General de la Stabilite du Mouvement,

by C. A. and K . C

18. A Unified Theory of Special Functions, 19. Fourier Transforms, 20.

by

S.

Bochner

by

M. A.

L ia p o u n o f f

T r u esdell

h an d r asekh aran

Contributions to the Theory of Nonlinear Oscillations,

edited

by

S. L e f s c h e t z

by J II, by J

21.

Functional Operators,

Vol. I,

22.

Functional Operators,

Vol.

ohn

von

N eum ann

ohn

von

N eum ann

23. Existence Theorems in Partial Differential Equations,

by D

orothy

B e r n s t e in

edited by by S. B

24. Contributions to the Theory of Games,

25. Contributions to Fourier Analysis, M . M o r s e , W. T r a n s u e , and A. Z y g m u n d

A. W.

ochner,

T ucker

A. P.

C alderon ,

CONTRIBUTIONS TO THE THEORY OF NONLINEAR OSCILLATIONS J. G. WENDEL

S. P. DILIBERTO

C. E. LANGENHOP

L. L. RAUCH

A. B. FARNELL

F. H. BROWNELL

W. W ASO W

M. L. C A R T W R IG H T

EDITED BY S. LEFSCHETZ

PRINCETON PRINCETON UNIVERSITY PRESS *95°

C O PYR IG H T, 19 5 O , PR IN CETON UNIVERSITY PRESS LONDON!

G EO FFR EY

CUM BER LEGE,

OXFORD

UNIVERSITY

PRESS

The papers in this volume by Piliberto, Rauch, Cart­ wright, Langenhop, Farnell, and Wasow were prepared under contract with the Office of Naval Research, and reproduction in whole or in part for any purpose of the United States Government will be permitted.

PRINTED IN TH E UNITED STATES OF A M E R IC A

PREFACE The common theme o f t h e monographs i n t h e p r e s e n t c o l l e c t i o n i s t h e s tu d y o f n o n l i n e a r p e r i o d i c n o t i o n s :"_n d i s s i p a t i v e

system s.

In the r e l a t i v e l y

sim p le c a s e

o f one d e g r e e o f freed om t h e b a s i c e q u a t i o n f o r f r e e o s c illa tio n s

(1 )

i s o f t h e form x + p ( x ,x ) x + q(x) = o

w here d o t s i n d i c a t e tim e d e r i v a t i v e s .

The d i s s i p a t i v e

m id d le term may a r i s e from f r i c t i o n i n a m e c h a n ic a l s y s te m o r from r e s i s t a n c e i n a n e l e c t r i c a l c i r c u i t . Such system s may w e l l assume s p o n t a n e o u s ly o s c i l l a t i o n s v e r y d i f f e r e n t from t h o s e o c c u r r i n g i n t h e u s u a l ( l i n e a r ) harm onic o s c i l l a t o r s .

A w e ll- k n o w n i n s t a n c e i s g i v e n

by t h e e q u a t i o n o f v a n d e r P ol (2)

x + M x

2

- 1 )x + X = 0.

N o n l i n e a r c o n s e r v a t i v e o s c i l l a t o r s h av e been i n ­ v e s t i g a t e d m ain ly i n c o n n e c tio n w ith c e l e s t i a l m ech an ics, and t h e i n f o r m a t i o n a v a i l a b l e f o r them i s t h e r e f o r e rath er e x te n siv e .

It

i s known, f o r e x a m p le , t h a t the'

t r a j e c t o r i e s a r e e x t r e m a l s o f a v a r i a t i o n a l p roblem , so t h a t one may b r i n g t o b e a r upon t h e p roblem M o r s e 's te c h n i q u e f o r t h e d i s c o v e r y o f c l o s e d g e o d e s i c s on m a n ifo ld s .

N o th in g o f t h e s o r t i s a t hand f o r t h e

d i s s i p a t i v e t y p e , making p r o g r e s s r a t h e r s lo w .

A re­

new al o f I n t e r e s t i n t h i s f i e l d has t a k e n p l a c e i n t h e l a s t t h i r t y y e a r s due m a in ly t o v a n d e r P o l and t h e

v

vi

PREFACE

f o l l o w e r s o f L i a p o u n o f f i n t h e US8 R

who h av e known how

t o make e x t e n s i v e a p p l i c a t i o n o f t h e c l a s s i c a l . d i s ­ c o v e r i e s o f P o i n c a r e , L i a p o u n o f f , and G. D. B i r k h o f f . I n t h i s c o n n e c t i o n se e n o t a b l y N. M in o r s k y , I n t r o d u c t i o n t o N o n l i n e a r M echanics

(David T a y l o r Model B a s i n R e p o r t ,

a l s o i s s u e d by Edwards B r o s . ,

l 9 h j ) , and A. A. Andronow

and C. E. C h a i k i n , T h e o ry o f O s c i l l a t i o n s U n iv ersity Press,

(P rin ceto n

191*9).

T hree o f t h e monographs t h a t f o l l o w , t h o s e b y D i l i b e r t o , Rauch, and B r o w n e l l , d e a l w i t h n o n l i n e a r non­ co n serva tive o s c i l l a t o r s .

D i l i b e r t o t a k e s up a number

o f g e n e r a l q u e s t i o n s c o n n e c te d w i t h P o i n c a r e ’ s e a r l y work on d i f f e r e n t i a l e q u a t i o n s .

Rauch d i s c u s s e s a

p roblem o f th e t h i r d o r d e r a r i s i n g out o f a n e l e c t r i c a l c i r c u i t w i t h vacuum t u b e s , e s t a b l i s h e s t h e e x i s t e n c e o f a d e f i n i t e o s c i l l a t i o n and s t u d i e s a number o f i t s p ro p erties.

H is work has c l o s e c o n n e c t i o n s w i t h a n

e a r l i e r p a p e r by F r i e d r i c h s on a s i m i l a r q u e s t i o n . B r o w n e ll i n v e s t i g a t e s th e o s c i l l a t o r y s o l u t i o n s o f a la r g e c la s s o f d i f f e r e n e e - d i f f e r e n t i a l eq uatio n s a r i s i n g f o r i n s t a n c e i n c o n t r o l p r o b le m s .

E q u a tio n s o f

t h i s n a t u r e a r e o b t a in e d i n p h y s i c a l system s w i t h retard ed responses to a d is tu rb a n c e . g e n e r a l l y p a r a s i t i c a l and makes i t s

The e f f e c t I s

s t u d y a l l t h e more

d e sira b le . I f a p h y s i c a l s y s te m governed by a n e q u a t i o n ( 1 ) is

s u b j e c t e d t o a v a r i a b l e e f f e c t d e p e n d in g upon t h e

t im e one must r e p l a c e (3 ) where e ( t )

( 1 ) by a n e q u a t i o n o f t h e t y p e

x + p ( x ,x ) x + q(x) = e ( t )

,

i s r e fe r r e d to as th e f o r c i n g te r m .

The

i n t e r e s t i n g c a s e , from t h e s t a n d p o i n t o f o s c i l l a t i o n s

PREFACE i s when e ( t )

is p e rio d ic ,

lo o k f o r o s c i l l a t i o n s

vii

say o f p e r i o d T.

One w i l l

o f t h e same p e r i o d T (harm onic

r e s o n a n c e ) o r o f p e r i o d kT(k )> 1 ;

subharm onic r e s o n a n c e ) .

N ote w o rth y work has b e e n done on t h e s e q u e s t i o n s o f l a t e by C a r t w r i g h t and L i t t l e w o o d and by Norman L e v i n s o n . Miss C a r t w r i g h t ’ s c o n t r i b u t i o n i s b a sed upon a s e t o f l e c t u r e s on f o r c e d o s c i l l a t i o n s g i v e n a t P r i n c e t o n i n t h e s p r i n g o f 19^9, and d e a l s w i t h t h e g e n e r a l e q u a t i o n (3).

The same t o p i c i s d e a l t w i t h i n t h e p a p e r by

W endel, but h i s mode o f a t t a c k i s g e n e r a l l y d i s t i n c t from t h a t o f Miss C a r t w r i g h t .

In t h e i r p a p e r Langenhop

and F a r n e l l c o n s i d e r a s p e c i a l f o r c e d o s c i l l a t i o n p roblem and by new m ethods, a p p l i c a b l e t o o t h e r problem s as w e ll;

t h e y su c c e e d In " l o c a l i z i n g " p e r i o d i c

i n c e r t a i n r e g i o n s o f t h e p h ase p l a n e .

so lu tio n s

F i n a l l y Wasow,

in h is p aper, d is c u s s e s the p e r io d ic s o lu t io n s in a s y s te m d e g e n e r a t i n g when a c e r t a i n s m a ll p a ra m e te r te n d s t o z e r o , and t h i s has c o n n e c t i o n s w i t h th e p roblem d e a l t w i t h by Miss C r r t w r i g h t and by W endel. S. L e f s c h e t z P rin ce to n U n iv e r s ity Mpy 19^9

«

CONTENTS I.

On System s o f O r d in a r y D i f f e r e n t i a l E q u a tio n s

.

1

By S. P. D i l i b e r t o II.

O s c i l l a t i o n o f a T h ir d O rder N o n l i n e a r Autonomous S y s t e m ........................................................... 39 By L . L. Rauch

III.

N o n lin ear D i f f e r e n c e - D i f f e r e n t i a l E q u a tio n s.

.

By F . H. B r o w n e ll IV .

Forced O s c i l l a t i o n s

i n N o n l i n e a r System s

.

.. 1 ^ 9

By M. L. C a r t w r i g h t V.

S i n g u l a r P e r t u r b a t i o n s o f a V an d e r P o l E q u a t i o n .................................................................................. 2 bj> By J . G. Wendel

V I.

The E x i s t e n c e o f F o r c e d P e r i o d i c S o l u t i o n s o f Second O rder D i f f e r e n t i a l E q u a tio n s n e a r C e r t a i n E q u i l i b r i u m P o i n t s o f t h e U n forced E q u a t i o n .................................................................................. 291 By C. E. Langenhop and A. B. F a r n e l l

V II.

The C o n s t r u c t i o n o f P e r i o d i c S o l u t i o n s o f S i n g u l a r P e r t u r b a t i o n P r o b le m s .............................. 313 By W o lfg a n g Wasow

ix

89

I.

ON SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

By S te p h e n P. D i l i b e r t o 1

§1 . I n t r o d u c t i o n . New r e s u l t s c o n c e r n i n g t h e f o l l o w i n g t h r e e problem s a r e e s t a b l i s h e d :

I.

The r e d u c t i o n , by l i n e a r

t r a n s f o r m a t i o n s , o f system s o f f i r s t o r d e r l i n e a r d i f f e r e n t i a l e q u a tio n s w ith v a r i a b l e c o e f f i c i e n t s to d i a g o n a l o r t r i a n g u l a r form (theorem s 1 , 2 , 3 ) . II.

G eo m etric c r i t e r i a f o r s t a b i l i t y o f p e r i o d i c

so lu tio n s

( c l o s e d t r a j e c t o r i e s ) o f system s o f f i r s t

o rd e r n o n lin e a r d i f f e r e n t i a l eq uatio n s 6).

III.

(theorem s b, 5,

Bounds on t h e number o f p e r i o d i c s o l u t i o n s

o f a s ys tem o f f i r s t

o rd e r d i f f e r e n t i a l eq u a tio n s w ith

p o ly n o m i a l f u n c t i o n s

(theorem s 7 ) .

Theorems 1 and 2 a r e used a s t o o l s f o r d e v e l o p i n g a d i r e c t t r e a t m e n t o f t h e main r e s u l t s on t h e ( g e n e r a l ­ i z e d ) c h a r a c t e r i s t i c exp onent ( L i a p o u n o f f )

[ M .L .] ,

[0 .P .2 ]. The c o n n e c t i n g l i n k b e tw e en t h e r e s u l t s p r e s e n t e d is c o r o l l a r y 1.1

w h ic h shows t h a t twc d im e n s io n a l

v a r i a t i o n a l e q u a tio n s a re I n t e g r a b le by q u a d ra tu re s. K P r i n c e t o n U n i v e r s i t y and t h e U n i v e r s i t y o f C a l i ­ fo rn ia . The a u t h o r ’ s t h e s i s , " R e d u c tio n Theorems f o r System s o f O r d in a r y D i f f e r e n t i a l E q u a t i o n s " , P r i n c e t o n U n i v e r s i t y 1 9 ^7 , done u nder p a r t i a l s p o n s o r s h i p o f O .N .R . , NR 014-3-9^2, c o n s t i t u t e s a b o u t h a l f o f t h e r e s u l t o f t h is paper.

2

S . P. DILIBERTO

R e f e r e n c e s t o th e b i b l i o g r a p h y a r e i n d i c a t e d by

[ ].

We ack n ow led ge o u r in d e b t e d n e s s t o P r o f e s s o r L e f s c h e t z f o r t h e g e n e m u s amounts o f tim e he has sp en t d i s c u s s i n g t h e s e problem s w it h u s , h i s m a th e m a t ic a l c r i t i c i s m s , and h i s c o n s t a n t en co u ra g em en t. S3 . N o t a t i o n s . We s h a l l u se b oth caps and s m a ll l e t t e r s

to

d e n o te v e c t o r s , but o t h e r w i s e s ta n d a r d n o t a t i o n f o r m a t r i c e s * v e c t o r d i f f e r e n t i a l e q u a t i o n s , v e c t o r norms and i n n e r p r o d u c t s . We r e c a l l some o ld n o t a t i o n s and a s s o c i a t e d fo r m a l p ro p erties:

l e t B = ( b j j ) t h e n B^ and B^ stand r e s p e c t ­

i v e l y f o r t h e j - t h column and i - t h are

row o f B.

(Thus t h e y

v e c t o r s ) . L e t B = A C and l e t I be t h e i d e n t i t y

m atrix ;

we t h e n have

(P 1 )

th e f o l l o w i n g p r o p e r t i e s

B J’ = ACJ';

Bj_ = A._C

(P2 ) b i;j = (A1 -CJ ) (P 5 )

1^ 8i j

- A “ - A -5 = A (A _1 )-5; ^ = (Ai 1

= A^A_1

= A 7 1A

' A y dt

- -i

az

1 - !> j

= - ^ i ^ r*=i s=i dt _ 1 _£ d f . r=l

Thus c^ . = 0 i f

d f. — ^ Y r^ dt ? )

- - 1 £ d> r=i

-1 f i S in c e B i s o r t h o g o n a l (B^ ) = B ci j 1J

^

3=1 aL

ps

and so

« r s (Bi 1 -BS ) rs 1

s=i i > j b e c a u s e t h e n i> s and (B1 *Bs )=0 by

c o n stru ctio n . To p r o v e t h a t i f

|a, . |^(0 ) =

X

. fc > 1 '' X >„'i t 6 0 , n 2 (o )

= 1, and applying

I.

ORDINARY DIFFERENTIAL EQUATIONS

9

T w i l l g iv e the s o lu tio n s o f the v a r i a t i o n a l eq u atio n s i n ( 4 ). C o ro lla ry 1. 2.

The v a r i a t i o n a l e q u a t i o n s o f a

H a m ilto n ia n s y s te m w i t h two d e g r e e s o f fre ed o m a r e i n t e g r a b l e by q u a d r a t u r e s . Proof o f c o r o ll a r y 1 . 2 :

dqz _ aH_ dt 3Pr Then i t

C o n s id e r t h e s yste m

^ dp^ ^ _ _0_H dt a qr

(I^

2)

i s known [ W .]p .3 l^ t h a t by means o f th e e n e r g y

i n t e g r a l H (p1 , p 2 , q 1 , q 2 ) = c o n s t . , and i f

j H 4= 0, t h i s ^ 1

s ystem c a n be redu ced t o a syste m o f one l e s s d e g r e e o f freed om : ^ 2 = 1 ^ , ^£2 dq1 3 P2

=

C o n s e q u e n t ly c o r o l l a r y eq u atio n s o f t h i s

- JJ£ dq1

1.1,

9 q2

a p p lie s to th e v a r i a t i o n a l

l a t t e r s y s te m .

r e d u c t i o n was b a se d on

Since th e i n i t i a l

3 H/ap1 +

th e c o n c lu s io n w i l l ,

i n g e n e r a l , be v a l i d o n l y l o c a l l y . C o ro lla ry 1 . 3.

Let x = x ( t , c ^ , . . . , c ^ ) be f o r each

c = ( c 1 , . . . , c^ ) i n some domain and a l l t a s o l u t i o n , a n a ly tic

( 5 )

i n t h e c^, o f t h e e q u a t i o n

a

r

1

-

j

v

v

-

.

-

.

’ V

,

1

1

"

1

’ *

.........................n )

where t h e X^ a r e h o lo m o rp h ic i n t h e i r a r g u m e n ts , and t h e

S . P. DILIBERTO so lu tio n is

such t h a t th e rank o f th e J a c o b i a n m a t r i x

(c f i x e d )

3 ( X t .............. Xn) 3 (t,c 1, . . . , i s k+i .

cn )

Then t h e r e e x i s t s an o r t h o g o n a l t ran s fo rm a t i o n

U = (u j j ) w i t h u^ . c o n t i n u o u s l y d i f f e r e n t i a b l e , U g i v e n e x p l i c i t l y I n term s o f x ( t , c 1 , . . . , c ^ ), such t h a t t h e v a r i a t i o n a l eq u atio n s

(6 )

i f -

Vt

;

V -

(v

dt

) ,

v

. J

J

j

x=x( t

, C,

. . . ,

o f t h e o r i g i n a l system t r a n s f o r m u n d er j (7 )

=W,J

and A k+i

/ A B\ w=^C D J ,

f

= U r\ t o

C= ( o)

sq u a re and t r i a n g u l a r .

Proof o f c o r o ll a r y 1 . 3 :

C o n s t r u c t t h e f i r s t k+1

columns U1 o f U by u s i n g t h e Gram-Schmidt p r o c e s s on th e v e c to r s

, |~

. . . , n ) be any n -k -1

( i= l , 2 , . . . , k ).

L e t U^ ( i= k + 2 ,

d i f f e r e n t i a b l e normal o r t h o g o n a l

v e c t o r s i n t h e o r t h o g o n a l complement o f t h e s p ac e o f t h e f i r s t k+i

Ui f s (su c h a r e e a s i l y c o n s t r u c t e d from

th e

, and t h e g e n e r a l i z e d norm als t o t h e

jnr ,

curve u ( t ) ) .

U s in g t h e f a c t t h a t

0

s o lu tio n s o f th e v a r i a t i o n a l eq uatio n s

v

are

[S.L.],p.52,

t h e p r o o f now p a r a l l e l s t h a t o f theorem 1 . One r e s u l t i m p l i c i t l y c o n t a i n e d I n t h e c o r o l l a r i e s 1.1

and 1 .3 and t h e i r p r o o f s i s :

s y s te m o f n l i n e a r f i r s t

"The s o l u t i o n o f a

o rd er d i f f e r e n t i a l eq u atio n s

f o r w hich k s o l u t i o n s a r e known c a n be red u ced t o a s y s te m o f n -k e q u a t i o n s p l u s a q u a d r a t u r e .

I f d esired

the_ r e d u c t i o n c a n be a c c o m p lis h e d by a n o r t h o g o n a l tran sfo rm atio n .

I.

ORDINARY DIFFERENTIAL EQUATIONS

§tk. F i r s t D i a g o n a l i z a t i o n Theorem . The method o f p r o v i n g theorem 1 y i e l d s d e v i c e s w h ic h e s t a b l i s h Theorem 2 : (8)

&

where t h e

Let

=

A(t) y

. a r e r e a l and

c o n tin u o u s f o r a l l t .

Th ere

e x i s t s a non- s i n g u l a r B =(bi j ) w i t h b ^ j c o n t i n u o u s l y d i f f e r e n t i a b l e and u n i f o r m l y b o u n d e d , such t h a t i f x=B 1y t h e n (9).

jjf

=Cx

where C is_ d i a g o n a l ( c^ j = o , i+ j

).

F u rth e rm ore t h e c^ ..

a r e bounded i f t h e a ^ . a r e b ou n d ed . T h is im proves t h e r e d u c t i o n o f theorem l a t t h e e x p e n se o f a d m i t t i n g t h e p o s s i b i l i t y t h a t t h e e le m e n ts o f B 1 become unbounded, and r a i s e s t h e q u e s t i o n . a s t o when B 1 o f th eorem 2 w i l l have bounded e le m e n ts | B | > 5 >o).

(e.g.

I f one dropped th e r e q u ire m e n t t h a t B,

its e lf,

be bounded t h e e x i s t e n c e o f t h e r e d u c t i o n s would —1 dR be t r i v i a l . Namely, s i n c e C=B(AB--rr-) we would m e r e ly dB h ave t o ch oose B such t h a t g ^ A B and C would be d i a g o n a l - - In f a c t ,

id e n t ic a lly zero.

P r o o f o f theorem 2 :

L et y 1 , . . . , y n be any b a s e

o f s o l u t i o n s o f (8) and d e f i n e B^* = y^/l ly^'l I •

Pro­

ceed in g as b e fo r e

0-5 = B ' 1 (ABJ- jprJ) 3-1

+itloe">J| 1

= B-’ — V M

= B _1B^

y

*3

I I

lo g

d t

| |yJ||

I

l y

11

d

S . P. DILIBERTO Thus,

° i j = ( 5 t l og

11 yX| 1 5 ^ i 1 ‘ bJ' )= 5i j i t

l og

11 ^

1

= 0 fo r i + j P a r a l l e l i n g c o r o l l a r y 1 . 2 o f theorem i we have u s i n g th eorem 2: C o ro lla ry 2 . 1 :

I f i n c o r o l l a r y i .3 o f th eorem 1

we remove t h e re q u ire m e n t t h a t U 1 be_ bounded t h e n A can be made d i a g o n a l . P r o o f o f c o r o l l a r y 2 .1 k+1 columns o f U from th e l a s t n -k -1

C o n stru ct th e f i r s t

^

a s i n theorem 2 and

as in c o r o l l a r y i . 3 .

The v e r i f i c a t i o n

t h a t t h i s t r a n s f o r m a t i o n i s o f t h e d e s i r e d k in d i s im m e d ia te . §5. N a t u r a l B a s e . We d e f i n e t h e n a t u r a l b a s e

{y 1 { ( i = i , 2 , . . . , n) o f

s o l u t i o n s o f a syste m o f l i n e a r d i f f e r e n t i a l e q u a t i o n s t o be a b a s e s u c h t h a t y i ( o ) = I i', i . e . ,

the *i-th u n it

vecto r. We s h a l l make r e p e a t e d u se o f t h e f o l l o w i n g o b v io u s lemmas. Lemma 1 .

Let

jx.^i and

Ju^i be t h e r e s p e c t i v e

n a t u r a l b a s e s f o r t h e ad .joint system s

CX ;

a m - -UC ;

C - (Clj);

T h en , x^ B 0 Lemma 2 .

i> j Let

;

uv1 s Q

i< j

j y 1 ! be any b a s e o f ^

= Ay and

t h e n a t u r a l b a s e o f th e redu ced s y ste m (v i a theorem -1 ) ix- = Cx where c . . = 0 i f

i> j .

e x i s t c o n s t a n t s d^p and d^p such t h a t

Then t h e r e

I . Bx1 -

ORDINARY DIFFERENTIAL EQUATIONS

£ r=l

d.p yr

;

13

B ’ V 1- 5 I d * r x r r^=1 x

.

§6. L i a p o u n o f f ' s C h a r a c t e r i s t i c E x p o n e n t s . These e x p o n e n ts a r e a s s o c i a t e d w i t h e q u a t i o n s ( 1 0)

rjX- = A (t.)y

where i t

i s assumed t h a t t h e a^ . ( t ) a r e c o n tin u o u s and

u n i f o r m l y bounded.

We s h a l l r e c a l l a l l d e f i n i t i o n s and

two e l e m e n t a r y p r o p o s i t i o n s .

We s h a l l , h o w e v e r, u se

P e r r o n 's d e f i n i t i o n o f c h a r a c t e r i s t i c exp onent sin ce p r o p o s itio n I I is tic

[0 . P . 2 ]

( b e lo w ) i s not t r u e f o r c h a r a c t e r ­

e x p o n e n ts a s f i r s t d e f i n e d b y L i a p o u n o f f - - a s

L ia p o u n o f f h i m s e l f shows b y an example The c h a r a c t e r i s t i c exponent so lu tio n of

(10),

[ M. L . ] , p . 236.

X ( o r X( y ) o f y , a

i s d e f i n e d by

\ = lira 1 lo g I l y ( t ) M t —*OO Let X ^

\ 2< ...< ( X^

•

be a l l t h e d i f f e r e n t v a l u e s

that

X may assume f o r d i f f e r e n t y ( a l l s o l u t i o n s o f

(1),

of course).

Let e

be t h e number o f l i n e a r l y

1

s __

in d ep en d en t y f o r w h ic h X ( y ) = ^ ;

le t

e^ be th e

number o f l i n e a r l y in d e p e n d e n t s o l u t i o n y f o r w hich X ( y ) = e g.

Then e\ i s c a l l e d t h e m u l t i p l i c i t y o f X

Two im p o r ta n t r e s u l t s a r e P ro p o sitio n I :

[0 . P . 2 ] :

I f |a^ .(t)

___ tlim -

r-1

" t-*+ 00Jo >o i=l

i f

rz n jra;, n (r ) dr

11

We s h a l l f i r s t 'prove two lemmas. Lemma 3 .

I f u, - By where B and B 1 a r e b o u n d e d ,

th en X (u) = X ( y ) P roof: n X (u ) = \ (By )oo L t-a o ^ o

dt = Xi

where X^(i=1,2,...,n) are the distinct characteristic roots of A. ]JjTot only do the mean values (1 7 ) exist -they even exist uniformly, tnat is for each I

(i.e. is bounded). When the elements of A are periodic with common period, the \ iof (1 7 ) always exist; and if there are n solutions for which they are different then a reduction by means of theorem 2 can be effected so that B~1 is bounded. In the case of periodic co­ efficients, also, condition (1 8 ) is automatically satis­ fied . Theorem 5 asserts that conditions (1 7 ) and (1 8 ) are sufficient for the desired reduction. They are in a reasonable sense necessary, as shall be pointed out at the end of this section. Theorem 3 * Let (19) where the a. • are real, continuous, and bounded

I.

ORDINARY DIFFERENTIAL EQUATIONS

21

Assume: (H1 ) The system has n different characteristic exponents X , Xn(H2 ) For a maximal base Jy1 ! (2 0 )

lim

L

f dTt 1^

t

JG

II

dt

exists.

||yi M

(H^) For each y1 of the maximal base and its characteristic exponent \ ^

(21)

f / ;L m y 1 11

\

I ly I I

/

Jnf

--

----- r --------- - X . J dt = 0(1).

Then there exists a (non-singular) matrix B defined for all t , B and B 1' having bounded elements, such that if x=B 1y then ( 22)

*

-

00 L Jo 1 J

t

)

dr = u. .. , and 1 J

Second, this corollary is directed towards a generalization of the classical representations for solutions of systems of linear differential equations with either constant or periodic coefficients which may be 'Stated: the s-th component, x , of any solution x of the equation = Ax;

A = (a. .)

: a., constant (periodic)

may be written when all the characteristic roots are distinct n e ; constant (periodic ) xs

S . P. DILIBERTO

26

T h is theorem i s not t r u e f o r a lm o s t p e r i o d i c c o e f f i c i e n t s , as t h i s example due t o C am e ro n 'sh o w s: = *(t)y

;

=

y(t)

=

=

X( y )

=