Contributions to the Theory of Nonlinear Oscillations (AM-29), Volume II 9781400882700

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ANNALS OF MATHEMATICS STUDIES

Number 29

AN NALS O F M A TH EM A T IC S S T U D IE S

Edited by Emil Artin and Marston Morse 1.

Algebraic Theory of Numbers, by H e r m a n n W e y l

3.

Consistency of the Continuum Hypothesis, by K u r t G o d e l

6.

The Calculi of Lam bda-Conversion, by A l o n z o C h u r c h

7.

Finite Dimensional V ector Spaces, by P a u l R. H a l m o s

10.

Topics in Topology, by S o l o m o n L e f s c h e t z

11.

Introduction to Nonlinear M echanics, by N. K r y l o f f and N. B o g o l i u b o f f

14.

L ectu res on Differential Equations, by S o l o m o n

15.

Topological Methods in the Theory of Functions of a Complex V ariable,

L e fs c h e tz

by M a r s t o n M o r s e

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 Mouvem ent, by M. A. L i a p o u n o f f

19.

Fourier Transform s, by S. B o c h n e r

20.

and K .

C h a n d ra se k h a ra n

Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. L

efsch etz

21.

Functional Operators, Vol. I, by J o h n v o n N e u m a n n

22.

Functional O perators, Vol. II, by J o h n v o n N e u m a n n

23.

Existence Theorem s in Partial Differential Equations, by D o r o t h y

L.

B e r n s t e in

24.

Contributions to the Theory of Games, Vol. I, edited by H. W . K u h n and A. W . T u c k e r

25.

Contributions to Fourier Analysis, by A. Z y g m u n d , W . T r a n s u e , M . M o r s e , A. P. C a l d e r o n , and S. B o c h n e r

26.

A Theory of Cross-Spaces, by R o b e r t S c h a t t e n

27.

Isoperim etric Inequalities in M athem atical Physics, by G. P o l y a

and

G . Szeg o

28.

Contributions to the Theory of Games, Vol. II, edited by H . K u h n and A. W . T u ck er

29.

Contributions to the Theory of Nonlinear Oscillations, Vol. II, edited by S. L

30.

efsc h etz

Contributions to the Theory of Riemann Surfaces, edited bii L . A h l f o r s et al.

CONTRIBUTIONS TO THE THEORY OF NONLINEAR OSCILLATIONS V O L U ME I I

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

S. LEFSCHETZ

E. A. CODDINGTON

N. LEVINSON

h

. f . De BAGGIS

j

. M cC a r t h y

H. L. T U R R I T T I N

E D IT E D BY S. LE F SC H E T Z

PR IN CE TO N P R I N C E T O N U N IV E R S IT Y PRESS

1952

Copyright, 1952, by P r inceton U ni v e r s i t y Press London: Geoffrey Cumberlege Oxford Univ e r s i t y Press

The papers i n this volume by DeBaggis, Lefschetz, and Turrittin, were prepared under contract w i t h the Office of Naval Research, and equally sponsored by the Office of Ai r Research. The p aper by Codd i n g t o n and L e v i n s o n was prepared under contract w i t h the Office of Naval Research. Reproduction, translation, publication, use and disposal in whole or i n part by or for the United States Government w i l l be permitted

Printed i n the United States of A merica

iv

PREFACE

This monograph, a s e q u e l t o Annals o f Mathematics Study No. 20, o f f e r s another c o l l e c t i o n o f co n trib u tio n s to d i f f e r e n t i a l equ atio n s. l a s t d e a l more o r l e s s w it h o s c i l l a t o r y problem s.

A l l but the

I n Mary L. C a r t w r i g h t ’ s

paper t h e r e a re g i v e n new and more a c c u r a t e e s t i m a t e s o f th e p e r io d and a m p li­ tude o f th e o s c i l l a t i o n o f th e van d e r Pol e q u a t io n 2

(1)

x + p ( x - i ) x

+ x=

o

f o r fi l a r g e . E. A. Coddington and Norman L ev in s o n examine s u f f i c i e n t c o n d i t i o n s f o r th e e x i s t e n c e o f p e r i o d i c s o l u t i o n s o f (2)

x = Ax + p +

(x , t , ^ )

where x i s an n - v e c t o r , A a c o n s t a n t m a t r i x , ^ a s m a ll p aram eter and f p e r i o d i c in t . DeBaggis g i v e s n . a . s . c . f o r the s t r u c t u r a l s t a b i l i t y o f a system (5 )

x = f ( x , y ), y = g ( x , y ). The paper by L e f s c h e t z c o n s i s t s o f two p a r t s .

I n the f i r s t t h e r e i s

g i v e n a co mplete d e s c r i p t i o n o f the c r i t i c a l p o i n t s o f an a n a l y t i c a l system ( 3 )- In th e second the s o l u t i o n s o f the e q u a t io n o f van d e r Pol i n th e f u l l phase p lan e a re studied ' and d e s c r i b e d . The t i t l e o f the paper by John McCarthy t e l l s

i t s own s t o r y .

H. L. T u r r i t t i n d is p o s e s c o m p le t e ly o f th e fo rm a l problem o f the s o l u t i o n o f an e q u a t iio on (b )

. x = A( t , E )x

where £ i s a param e ter, x i s an n - v e c t o r and A an n m a tr ix whose terms a re power s e r i e s i n e . He a l s o shows t h a t th e fo rm a l s o l u t i o n s a r e , under c e r t a i n con­ d itio n s , a ctu al so lu tion s. S. L e f s c h e t z Princeton U n iv e rs ity January, 1952

v

CONTENTS Preface

v

B y So l o m o n Lefschetz I.

V a n d er P o l ’ s E q u a t i o n for Rela x a t i o n Oscillations

3

B y M a r y L. Cartwright II.

Perturbations of L i n e a r Systems w i t h Constant Coefficients Possessing Periodic Solutions

19

B y E. A. C o d d i n g t o n and N. L e v i n s o n III.

D y n a mical Systems w i t h Stable Structures

37

B y H. F. DeBaggis IV.

Notes o n D iffer e n t i a l Equations

61

B y So l o m o n Lefschetz V.

A Method for the C a l c u l a t i o n of Limit Cycles by Successive Approximation

75

By John McCarthy

VI.

Asymptotic Expansions of Solutions of Systems of Ordinary L i n e a r D i f f e rential Equations C o n taining a Parameter B y H. L. T u r r i t t i n

81

CONTRIBUTIONS TO T HE T H E O R Y OF NO N L I N E A R OSCILLATIONS VOL. II

I.

VAN DER P O L ’ S EQ U A T I O N F O R R E LAXATION OSCILLATIONS B y M. L. Cartwright §1 .

Introduction.

(1 )

The equat i o n

x - k ( l - x

with

k

2

) x + x =

'

0

large and positive has only one periodic solution, other t h a n

x =

0,

and this is of a type usually described as a r e l a x a t i o n osc i l l a t i o n (as

1

opposed to a sinusoidal oscillation). tained a graphical solution fcr

It was discussed by v a n der Pol w h o obp k = 10 and by le Corbeiller who, using

Lienard’ s method, showed that the period

2T = 2 k (3/2 - log 2) + 0 (k), and e * k -> «» . Other authors^ have also dis4 cussed the equation, in partic u l a r D o r o d n i t s i n has obtained a n asymptotic

the greatest height f ormula for

T

h = 2 + 0 (1)

as

w i t h smaller error terms but his analysis is difficult to

follow . This pap e r Is based o n the joint w o r k of Professor J. E. Littlewood and myself, largely on w o r k w h i c h was done before that contained in our other published papers o n n onlinear differential equations. k

We shall show that as

00

(2 )

T

1 0 ge 2

) +

+

h= 2 + ^ 5

(3) where

= k (5/ 2 -

&

and

p

0

( - ^ 3)

+ ° ( ^ 75) ’

are constants determined as follows:

The eq u a t i o n

> has one and only one solution

(1) (2 ) (3 )

(4 )

*( 5 )

such that

0

as £

B. v a n d er Pol, Phil, Mag. 2 (1926), 978-992. Ph. le Corbeiller, Journal Inst. Elec. Eng., 79 (1936), 361-378. D. A. Flanders and J. J. Stoker, "The Limit Case of R e l a xation O s c i l l a t i o n s ”, i n "Studies I n N o n l inear V i b r a t i o n Theory", e d . R. Courant (Inst, f or Maths, and Mech., New Y o r k University, 19^6, typescript), J. Haag, Ann. E c . Norm. Sup. 60 (19^3), 35-111, 61 (1944), 73-117, J. P. LaSalle, Quart. App. Maths., 7 ( 1 9 ^ 9 )* 1-20. A. A. Dorodnitsin, "Prik. Mat. i. Mech.", 11 ( 1 9 ^ 7 ).

3

M. L. CARTWRIGHT

b

y ^

1,

- log hi. 2

Let 0 at some point Z in HC, th e n since h < 0 and x is continuous there is a sub-arc X Z of HC such that x = 0 and y >

0

for

x > y ^ z.

6

M. L .

CARTWRIGHT

Hence k - iHpTiT + h W

Z ..

> 2 ■ x = J x y dt > 0 '

which gives a co n tra d ic tio n , and so x { 0 for a l l X in HC. Part

(ii)

follows at once from ( i )

and the o r ig in a l eq uatio n, and

also i f h > x > y > 1 .

xy which gives

*

=

h

- x-- > k

Ixl

)? ( x

- £ ) dx = kl

-

y2 ) - log x / y .

-’y

( i v ) and the p a r tic u la r result fo llo w s .

F in a l l y •• • 2 •• *2 x = - k (x -1 ) x - 2 kxx = k ( x 2 -i ) |x| i

- 2 k x i2 > k (x

which is

-x

- 2 kx x 2 + |x|

-1 )

2 k(x-i )

( 3 ). §V.

Lemma 2 consists of one-sided in e q u a lit ie s except i n ( i i ) ,

it is obvious that nothing b etter than the left-hand in e q u a lity i n ( i i )

and can

be obtained near H , but i n order to o b ta in a more accurate estim ate of t ^ c we need in e q u a lit ie s

of the opposite k in d .

For th is purpose we need to know that

h is not too near 1 and we div ide the arc HC by points Y , LEMMA 3 .

Suppose that h >

Z and E .

and that HY is a n arc

on which x -6

(1 )

k ( x 2 -i )

where 0 < kQ ( 4 ) .

Also y =

|x|

y

I H

( 1 ) is cer ta in ly negative fo r x

c e r ta in ly e x i s t s .' Since

= 1 and x < 0 in HC, y ^ 1 . ( x 2 -i ) - x < - cJ

x dt < - k ( 2 for k > k Q , and (ii)

L E MMA 5. Suppose that Y Z is the longest arc satisfying the hypo theses of Lemma b, and that E is a point i n ZC at w h i c h

e y_ 1 + A k

2^ ,

t h e n if X is

M. L. CARTWRIGHT any point in ZE M

Ix|
A . ze Ak ° At Z (i) holds in virtue of Lemma 2 (ii) and ( 4 ), and since x < t

(iii)

it w i l l continue to hold as long as x ) if x {

0

whi c h is the case at Z by ( 5 ).

0

But

0

M

k ( x 2 - 1 ) |x| £ 2kx x 2 - ix|
0 and x is increasing so that we can repeat the argument, and (i) holds throughout ZE. Part (ii) follows at once, and

t TO- f ze

•E

^

IxI

A

1

e2 )

6.

5

K -

+- f- f

log ^

+

I

If E satisfies the hypotheses of Lemma

+ Ak


A _ and k > k (S,A). O P — 1 /*^ The two largest error terms k(e - 1 ) and A k ' come f r o m the interval

EC whi c h

wil l need special consideration. §6. We now proceed to study the arc C A ’,f raming

our

lemmas so as

to cover all solutions starting d o w n fr o m C*. L E MMA 7. If a solution starts f r o m C, and if F is a point in CA' at w h i c h f max (A1/2 k"1/3 , |c| ) ,

and

(ii)

tcf < M A 1/2 k '1/5 .

B y the energy equa t i o n for CF w e have x 2 y_ c 2 + and so |f| > max (A 1 ^ 2 k ,

_

tcf

1

- x2

|c|) w h i c h gives (i).

n

Jp TxT

dx /

C

^ Jf

It also follows that

dx_______ «

1 !2

-p / m A

(1 . x 2 ji/2 - arc

003 f ^

LE M M A 8 . If a solution starts f r o m C and reaches F w here f = 1 - A k , A > 0, t h e n

(i)

tf a , < M/(A k1/3) ,

and (ii) Put x rj y k ^

for

=

0

1


2

y

1V1

Ak-

Part (ii) follows immediately fro m the integrated eq u a t i o n §2 (3) applied to CA' . §7.

W e now jump from A ’to the arc AB w h i c h is the refl e x i o n of

A ' B 1. Whereas at C the magnitude of c made little difference to the subsequent be h a viour of the solution, after A the behaviour depends very much o n the magni tude of a. If a is very small, the p o ssibility of the solution turning do w n be fore it reaches B, or taking a very long time to reach B, cannot be ruled out, but it is of some interest to see that e v e n if a - 0 (k 1 ) the subsequent b e ­

M. L. CARTWRIGHT

10

2

ha v i o u r may be similar to that of the periodic solution for w h i c h a ru -k 1 /2 except that h ~ 3 1 instead of 2 and that it takes a fairly long time to get away fro m A. LEMMA

9.

A solution starting f r o m A wit h a

3 /k

reaches B with (1)

|b - a - ffcl < t ab < M i provided that k > k .

^

If further a >

fcab < F T ¥ + "Tc- for k > Let G be the point x = ^ .

T h e n in A G

increases fro m A as long as x £ ^ ka.

Since x £

G is reached wi t h t&g < ^-/a ^ x ^

>

g- k.

3

On 1

AG 2

?■ + kx(l - - x ) ^

, £k,

where £>

0 ,t h e n

ko ‘ x

^ kx

- x,

and

^ in A G and ^ ka

so x ]> ^ ,

by the integrated equation ? | A

xdt

k + T ^ l o c - ^ kx = k + l k x '

Hence g ) ^ + ^ k , and t = tag

C — fdx_ ✓ 2 _____ dx_____ . M log k for k \ k J. I T ' I + 1 kx k 0 ‘ A -’ o k + 4

Next x > i- k near G; If eve r x =

1- k

on GB, let X be the first occasion.

The n

but also x > s - SG for k

x dt > k + - p - - 1 = - ^ r - - k > E k

> 2 w h ich gives a contradiction.

So x ) ^ k o n GB, B is reached wi t h

t b K.

and (1 ) follows from the integrated eq u a t i o n §2 (3) for AB. If a >£k, since x increases in A G w e have t M while the left hand side is positive.

Hence p < M and so t^

have |b + k ( p - p 5 /3 for k > k 0 (£).

2 / 3)1

< M/(£k). p j'

= tk +

U s i n g (2) again, we

P - e L

-

|
1

1.

in PQ, we have

x = - k x(x2 - 1 ) - x

- kx

,

and so divi d i n g by x and integrating fro m P to Q we obta i n l o g ( q / p ) £ - k tpq . Hence k tpq £

2

log(£k)

f r o m w h i c h the result for t

follows. The restrictions o n 1 . L E M M A 12.

If the hypotheses of Lemmas 10 and 11 are

satisfied, t h e n t ^

< M/(ek) and

|b + k (h ’ for £ < £q , x < - x

S k Q (e, 5 ) .

and as above

- Tk -

x < M.

Since h = 0

I * dt < - M S h

f r o m w h i c h the result for t ^ follows. B y the integrated e q u a t i o n for B H we have

M. L. CARTWRIGHT

12

H J - x dt { M tb h {

|b - k (h - j h 3 - |)I =

B for

«5 < < 50 , £ < §9 .

£0

and k > k Q (t, 5 ).

We have now covered the equivalent of a half wav e H C A ’ B’ H 1, and

w e m a y review the results as follows:

in Lemma 1 we showed that a solution

starting at a m a x i m u m H not too far above x = 1 arrives at C w i t h |c| bounded b y a constant d e pending on h. I n Lemma 2 we showed that a l t hough x { 0 in HC, x remains small until x approaches 1, and consequently the s o l ution takes a long time to reach C unless h is nea r 1. Lemmas and 5 show that the one-sided estimates for x and for the time in L emma 2 do in fact give a good approximation, provided that h > and x is not too near 1. Lemma 6 deals w i t h the arc just before C, and L emma 7 wit h that just after C a s suming that the solution starts from C, and together they show that the time t a k e n to cross the strip |x - 1 | £ A k

2/ 3

is at most M A 2 k

t i o n starting fro m C reaches A ’in time M A

L emma

1/ 2

8

shows that a s o l u ­

k ~ 1^ 3 w i t h ja'I > |k + |c| - M

L e m m a 9 shows that a solution starting fro m A w i t h a y 3 /k rises to B in time . p 0 (log k/k) w i t h b > - 0 (k) and gives a b e tter result if a is large. Lemmas

10,

11, and 12 deal w i t h a solution rising f r o m B w i t h b large.

show that tfeh =

(log k/k), and give a formula for h d e p e nding o n t>.

0

be observed that the tran s i t i o n interval

PQ

is comparable in some ways to the

tr a n s i t i o n interval Y Z in partic u l a r the time f or each is §1 0 .

They It may

0

(log k/k).

The periodic solution traces the reflec t i o n of the hal f wave

ABHCA' b e low x =

0,

and we shall now proceed to estimate T and h b y applying

the previous lemmas w i t h this in mind. THEOREM

1.

M \

M Ak

F o r the periodic solution ,

I / m A 1 /g k 1/ 5 ’

1/ 5

(ii)

la

- | k| ^k f or k > k Q , we

w h i c h with (ii) gives (iii).

L emma 12, w e have |h - l - h 3 + || ( M i 1 / 2

,

and so if we write h = 2 + £ we have it, (3 + 2 s + j i ; 2 )i { m a

which gives (iv) provided that A

y

A 0 and k

,/2

y

k ' 1*/5 ,

kQ (A).

U s i n g this in

V AN DER P O L ’ S EQUATION

13

Finally T = tah + thc + t o a l B y Lemmas 9 ,

1 0 , 11

and

12

t -i = t ah ag / M ^ k

M £k

+

4*

t -i ~f~ "t-t + "t + t -i gb bp pq qh

M log; k k

M / M log; k £k ^ k

f o r every £ > 0 , provided that k > kQ (e). and using (iv) in Lemma 2 we have

B y Lemmas

7

and

putting e =

1

+ Ak

8

t

, (M A

1/ 2

k

tvhe

and similarly from Lemmas

3 * b,

5 and

6,

_o /*

1 D we have, as

in §5,

thc = V s M

+ tyz + tze +

M log k

^ T k

+ —

k

+ ki

.

t ec

. M A

&

3 2

21

log

■ ^173

+ “ ^T 7 3

A Q , k > k Q (A, S) = k Q (A). §11.

It remains to remove the large constant A and replace it by

constants w h i c h can be determined more precisely.

It is obvious that the

errors depending o n A originate in the arc EF, i.e. i n the strip |x - 1 | ^ A k

with x
1 as

h y p erbola

2

^ rj Q + 1 =

0,

£ ->

w e see that

+ 00

d rjQ /d^ > 0

below and to the right of the upper b r anch and d t ] Q /d£
±00 unless 2 £ »7 0 +

1

-»

.

0

.

0

above

Hence one

set of solutions descends fro m + o© as £ increases fr o m until it crosses the upper branch of the h yperbola and t h e n ascends a g a i n to + , a nother set

The curves *)0 (4 ) are shown In black w i t h the separating curve rj* o n w h i c h i]0 (£ ) — 1►

0

as

I — ►-»

thicker t h a n the o t h e r s . crosses rjo = + oo .if |

0 at

§= %Q , say, wit h d r i o /d§ = 0 0 at % + 00 i n such a w a y that lim rjQ > 0 , the n

and ascends to

and so *|0 (|) ^ 2 as 5 ± 93 • B o t h sets vary continuously wit h value of and fo r m ope n sets. There is therefore a solution, or closed set of solutions, separating the two sets, and it must lie b e t w e e n the hy p e r b o l a and r[o = 0 for negative § . It follows that o n such a s o l u t i o n ^ * d r | * / d £ > 0 , and since for fixed ^ d r ^ / d ^ increases as rf decreases, the lower solutions increase more rapidly t h a n the upper w h i c h is impossible if there is more tha n Y|Q ( 0 )

V A N DER POL'S EQUATION

15

one solution o n wh i c h % —> - ^ b e t w e e n the h y p e r b o l a and rj = 0. Hence there is a unique solution separating the classes. LE M M A 1 4 . Suppose that T) (£) satisfies (2) and that a is defined

as in Lemma 13(or

w he r e if r|(0)

> o, we have y) 1 for some £, such that - A £ % 0 for all I and 90 (i) 00 as £ for ^- A , provided that A > A q (1 d l ~

2

+

cl

S

and let

= ol + t,

.

T h e n by

~ 00 and so *)0 (£) 2 ^ £( 0. Substituting

■2 ^ 1 + ^273 ^ 2(,7o + 7i } + 5I = 0 »

w h ere ^ ( 0 ) = 0 , and since *)0 (£) 2. £ > 0 , we have

dth

. xtA)!^! + _ j _ ( ^

E-h, I

2

k 2/5 p

m +

£-

In,

where "X (A) denotes a number d epending o n A but not o n k or % . Since rj ^ (0 ) = 0 by a w e l l k n o w n me t h o d ^ it follows that for every A > 0 (6 )

h , I £ -X(A) k " 2 /J
0 and every A > 0 , provided that - A { ^ i A and k > k ( A > A 0 and k > k Q ( S , A ) . If Y)o ( 0 ) = o l - 8/2 5

=

§0

>

where

0


A b (6 ).

we also have

+

1)


S ^ *)

+ 1.

This means that the solution of (2 ) increases more rapidly t h a n the solution of (3) through the same point 2 %

+

1

low the creases.

>

0-

in the region defined by £


0,

certainly lies b e ­

just defined ne a r €> = 0 , and continues to lie be l o w it as £ d e ­ It therefore reaches y = 0 for | w h i c h gives the second part. LEM M A

15*

Suppose that at and p are defined by §1 ( 5 )

and t is any positive number. T h e n if 9 ( £ ) solution of (2) for w hich |y(0) - 0 and every A > o we have I ^ | < £' for -A {

5

{ A provided that h

0, provided that A > A 0 (£), £' < £q(£,A), k > k0 ( £ , £ ' A j ) . As ^ -» - 00 , >7

k ( k Q (£,A).

T h e n (iii) follows f r o m Lemma 9.

/ 2- £^

X

R e flecting in x = 0

Putting (iii) in Lem m a

12

we

have

h - j h 5 + |+ and

pu t t i n g h As

( « + (3) k -4/ 5 = 0(k~4/ 5 ) ,

= 2 + every

£

k (| - log

2

)

) 0,A ) 0 and

(3 )

+

( oc+ (3 ) k "

k>

k

t, he = t, h y i

every £

+ tyz

s)

)2

3>

5

t l U LtfJ.

>2

> 0 ,provided that k > k Q (e). dx

?

ec

1

+ tze

, S U J S J . , k( l _ ^

f

- k(e -

(£ , A ) . B y Lemmas

*

for

1/ 5

•''c

But _1

IxI

r o d 4.

k 1/ 5 •'-A P

and so b y L emma 15 and (i ) above we have

|tec - k(e - 1

( 0 , provided that A > A 0 (£) and k > k Q (£,$,A). Putting (2) and (3) t o g ether w i t h ( 4 ), we have ’ t^c

as k

f

00

.

= k(

I

-

log

2 ) +

\

( CX. +

p

) k ' 1/ 3

+

rj ) ~ p ( 0 ) - t) = 0 .

(0 . 3 ) For

p- =

0,

the system (0 .3 ) has

rj -

0

as a s o l u t i o n . . If the J a cobian

J = det (x*j (T,fj.,rj) - E) does not v a nish at continuous solution Here det

= determinant, and E I n case

form

I] = *](p)

f(x,

for

p. =

0,

t h e n ( 0 .3 ) has a unique

| p. | sufficiently small w i t h

r^(o) =

represents the n-dimensional unit

t, (jl )does not contain t

explicitly, but

f ( x , p ) , the system ( 0 .2 ) has £ = p' as a s o l ution of period

the above hypothesis is n ever fulfilled.

0.

matrix. is of the T

so that

I n this case the t h e orem is modified,

and if (0 . 4 ) *

x' = f(x, p.)

This pa p e r was w r i t t e n in the course of w o r k sponsored by the Office of Naval Research.

19

20

C0DDINGT0N AND LEVINSON

has a periodic solution

x = p(t)

of period

T Q f or p-=

0,

and if the first

v a r i a t i o n (0 .2 ) has no more t h a n one independent solution of period T Q , ( 0 .4 ) has a periodic solution x = q(t,p.) for small 1^-1 , continuous in (t, p O . q(t,

0

The period,

T(^),

) = p(t), and

of

q(t,p)

T( 0 ) = T Q .

is continuous in ^ .

The fact that ( 0 .4 ) at p - =

only linearly independent solution of period v a n ishing of a c e r tain

TQ

the n

Moreover

0

has p ’as its

is equivalent to the n o n ­

(n - l )-dimensional Ja c o b i a n

J1.

In ma n y important cases the hypothesis o n the equations of the first variation, or, what is the same, the non - v a n i s h i n g requirement o n the Jacobians J and

J^,

Is not met. F o r example in case

(0.5)

w

Is a

scalar the eq u a t i o n

w" + w = p-g(w, w ’, t, ^ )

where

g

is periodic of period

2 rrin

t,

,

if it contains

t,

has f o r its

first v a r i a t i o n w" + w =

0 }

w h i c h has sin t and cos t as independent solutions the hypothesis for neither result stated above c an

of period 2 rr. Thus be met for (0.5). The cases

of equa t i o n (0.5) relevant here have b e e n treated in detail by F r i edrichs and Stoker [2]. f(x,

0

It is our purpose to consider the case whe r e f(x, t, 0 ) of (0.1 ), or ) of (0 .4 ), are of the form' Ax, wh e r e A is a constant matrix. We

show that it is possible to give sufficient conditions for the existence of p eriodic solutions of small

x

1

= A x + p-f(x, t,p.) (and

Ip-! e v e n w h e n the Jacobians

J and

J1

x* = A x + p.f(x, p.)) for

vanish.

These conditions only

involve knowledge of the solutions of the degenerate linear system As is to be expected, if case

f = f(x, p.))

for which

J

(or

f

can be solved fo r recursively. J1 )

x* = A x .

is analytic, t h e n the solutions (and period in I n the following, systems

vanish w i l l be referred to simply as systems w i t h a

vani s h i n g Jacobian. 1.

PERTURBATION OF A SYSTEM W I T H A V A N I S H I N G JACOBIAN

F o r the linear system ( 1 .1 ) where

x A

s o lution

1

= Ax,

is a constant real matrix, assume that there exists a real periodic p = p(t)

w i t h period

2 tt.

This is equivalent to the fact that

there exists at least one characteristic root wh e r e

N

is a n integer (which m a y be zero).

X

of

A

of the f o r m

N = IN

W e shall be interested in the

perturbed system (1.2)

x'=Ax+yu.f(x,

t,ju),

wh e r e it is assumed that A is the constant matr i x g i v e n in (1 .1 ), fjL is a real parameter, and f is real, periodic of period 2 TT in t. (Since 2 tt need not be the least period of e x c l u d e d ).

f

in

t

the case of subharmonic oscillations is not

PERTURBATIONS OP LINEAR SYSTEMS It is clear that if

c and

d

21

are any real constants, t h e n

is also a real periodic solution of ( 1 .1 ) w i t h period f or what values of

c

and

d,

periodic s o lution t e n ding to

,

2 tt

c p(t + d)

It is not obvious

if any, the perturbed system ( 1 .2 ) m a y have a c p(t + d)

as p--»0 .

W e can not a pply the p r o ­

cedure m entioned in the Introd u c t i o n f or in the case ( 1 .2 ) the relevant Ja c o b i a n

J

vanishes.

However, it is possible to give sufficient conditions

fo r the existence of periodic solutions of ( 1 .2 ) fo r

y- ^ 0

provided

A

is

i n canonical form, and it is always possible to arrange this. Setting

x = Py

wh e r e

P

is a real n o n - s ingular constant matrix,

the system ( 1 .2 ) c a n be replaced by a system for matrix B = P 1A P,

when

p- =

0,

y

w here the coefficient

is i n real canonical form.

n e w system satisfies the same assumptions as ( 1 .2 ). assumed that

(1

.3 )

A

al r e a d y has the fo l l o w i n g real canonical f o r m

A =

w h e r e the elements not s h own are zeros. ma t r i x of

Moreover, this

It w i l l therefore be

a . (

a . even)

Eac h

Ay

j =

1,

..., k,

Is a

rows and columns of the f o r m sj Sj

E2 (1 . * 0

2

w h e r e all elements are zero except

sj = n j

Nj be i n g a positive integer, (In k - d i m e nsional unit matrix 0 < k < only two rows and columns in w h i c h rows and columns, 1,

sj

and

and

0

1

the foll o w i n g E^. w ill always denote the n, and E = E n ). A m a trix Aj m a y have case it is Each ma t r i x Bj has p S3 and Is of the for m

CODDINGTON AND LEVINSON w here

B. m a y have only one row and column, in w h i c h case B. J k J m the single element 0 . The matrix C has Y = n X

(O)

0

)

■ f ( P ( 0 ) . ») n v i=1

A p( J*-1 ) + b j A p ( o )

af

(P

dx.

(o)

0) Pi

(j-i)

+ P ( j) (1 ) P(J) depends o n p ( 0 ), where That the system (2.12) has a solution f o r p (i)

d (J’ 2),

and

and

b.

v

b2,

bM

is clear.

T H E O R E M 6. U n d e r the assumptions of T h e o r e m 5 the analytic solution q of (2.1 ) c an be obtained b y solving equations (2.12) i n s u c cession for the periodic coefficients p ^ ,

of period

2ir,

of the power series (2.9) for

p(s, jjl ) = q (s (1 + T / 2 TT),p.), e x p a n s i o n (2.10) f or

T / 2 tt.

and the constants The p.(i) vx; and b^

determined in (2.12) by the requirements that p ^ ( s

+ 2 tt) = p ^ ( s ) ,

PROOF.

in the are uniquely p ( o ) (s) = e sAa,

2 TTb,

Suppose there are two functions

fying the second e q u a t i o n i n (2.12), and two constants

( 1 ), b2,

-(1 ) ~2 ,

satis-

such that

(2 ) £(2) b 2 ) there correspond to the pairs (p (1 ) respectively satisfying the third e q u a t i o n of (2.12). S ubtracting the third e q u a t i o n for one case f r o m that fo r the other case, and d e noting p (1)

b 2 = t>2 - b g ,

we have

g f 2) = A p < 2 > + b, A p l,) + b £ A p < °>

n v i=l

af dx.

(P

;o)

0) p. (1)

F r o m the second e q u a t i o n for each case follows ds Let p (1 >(o) = a (1 > each is zero. Since

=

(0) == 0.

and

A p* v0 )

and p ^2 ^( 0 ) = * ( 2 ) The second (ot1 ) p ^ 1 ^ is periodic it follows that p (1 }(s) = e aAa < ' K

component of

-

CODDINGTON AND LEVINSON

3k

w h e r e only the exceptional components of

a/

1^

c an be different f r o m zero.

Thus a ^1 ^ has at most 2k + m - 1 components that are not k n o w n to be zero. P r o m the differential equa t i o n for p ^ 2 ^, and the fact that p ^ ( o ) = p ^ 2 ^(2 tt), it follows that (E - e 2™ )

-(2) = b

2tr (* e ( 2Tr^ ) A p ( 1 ) (cr)d **r\ fo

a

'

+ b2 A J - V ^ V ^ ^ d c r +

£

•’o

- |£ _

1=1

( p ( o ) ( 0,

b p,

cr (x 1 , x 2 ) -

ap0 + " a ^ ) + °*

I n the proof of the theo r e m w e shall assume that a) and b) are not satisfied and derive a contradiction. w e m a y assume

(x°, x°)

Without loss of generality, in the proof

at the origin.

Moreover, since, by L emma 3, each crit

ical point of (A) is a n isolated point we m ay encircle the o r i g i n b y a circle of radius

r > £,

and choose

6(e)

so small that there is exactly one c r i t i ­

cal point of the perturbed system w i t h i n this neighborhood. Suppose first that e qua t i o n

\

2

+ ° < a 0,

A.(o, 0) > 0,

0 ],

When

no definite statement ca n be made as to the stability or unstability We shall show that if (A) is structurally stable, it cannot possess, a limit-cycle y such that h(y) = 0 . But first we shall prove the

0

following lemma w h i c h is a slight ex t e n s i o n of the continuity theo r e m wit h respect to the initial conditions. Let us wri t e the system (A) in the v e c t o r f o r m

D Y N A M I C A L SYSTEMS W I T H STABLE STRUCT U R E S (A)

^5

x = P(x)

where

x, P. denote the co l u m n vectors

> (p^-) 9 resPe c '*:^v e ^ y •

this

n o t a t i o n the perturbed system (B) becomes (B)

y = P ( y ) + p(y).

The solutions of (A), (B) are giv e n b y the equations x =