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

Table of contents :
CONTENTS
Preface
I. Van der Pol’s Equation for Relaxation Oscillations
II. Perturbations of Linear Systems with Constant Coefficients Possessing Periodic Solutions
III. Dynamical Systems with Stable Structures
IV. Notes on Differential Equations
V. A Method for the Calculation of Limit Cycles by Successive Approximation
VI. Asymptotic Expansions of Solutions of Systems of Ordinary Linear Differential Equations Containing a Parameter

Citation preview

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 =