Symposium on Ordinary Differential Equations: Minneapolis, Minnesota, May 29 - 30, 1972 (Lecture Notes in Mathematics, 312) 3540061460, 9783540061465

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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich

312 Symposium on Ordinary Differential Equations Minneapolis, Minnesota, May 29-30, 1972

Edited by William A. Harris, Jr. University of Southern California, Los Angeles, CA/USA and Yasutaka Sibuya University of Minnesota, Minneapolis, MN/USA

Springer-Verlag Berlin.Heidelberg New York 1973

AMS Subject Classifications(1970): 34-02,34A20, 34A25, 34A50, 34B10, 34B15, 34C05, 34C25, 34E05, 34E15, 34J05, 45M10, 39-02, 39A15, 4%02, 49A10

I S B N 3-540-06146-0 Springer-Verlag Berlin • Heidelberg • N e w Y o r k I S B N 0-387-06146-0 Springer-Verlag N e w Y o r k • Heidelberg • Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than privatc use, a fcc is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg 1973. Library of Congress Catalog Card Number 72-97022. Printed in Germany. Offsetdruck: Julius Beltz, HemsbachlBergstr.

This v o l u m e ,

as well as the S y m p o s i u m

to H u g h L. Turrittin for the m a n y

itself, is dedicated

contributions he has m a d e

the past years to the d e v e l o p m e n t of this subject. m o s t of us directly or indirectly encourgement

and guidance.

through

over

H e has influenced

his inspiration,

In life as well as in M a t h e m a t i c s

has been a true friend and a true c o m p a n i o n .

he

PREFACE

This v o l u m e is the proceedings of a S Y M P O S I U M DIFFERENTIAL

EQUATIONS

ON ORDINARY

that w a s held M a y Z9-30, 1972 at the

University of Minnesota, honoring

Professor H u g h L. Turrittin upon his

retirement, the t h e m e of the S Y M P O S I U M

was current researches arising

f r o m central problems in differential equations with special emphasis in the areas to which Professor Turrittin has m a d e contributions. The first paper of these proceeding entitled " M y Mathematical Expectation" by Professor Turrittin sets the tone and clearly establishes the scope and

breath

of the S Y M P O S I U M .

The editors wish to thank Professor

J . C . C . Nitsche, H e a d of the

School of Mathematics, the S y m p o s i u m C o m m i t t e e and m e m b e r s

of the

S chool of Mathematics for their support of the S y m p o s i u m and their w a r m hospitality and

September

197Z

generosity.

~ArllliamA. Harris, Jr. Yasutaka S ibuya

CONTENTS

My

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

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

A d m i s s i b i l i t y and the I n t e g r a l E q u a t i o n s of A s y r n p o t o t i c H, E, G O L L W I T Z E R

Theory...

Differential Inequalities and B o u n d a r y P r o b l e m s for Functional Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . L, J, G R I M M and L~ M , H A L L Singularly P e r t u r b e d B o u n d a r y V a l u e P r o b l e m s W , A , H A R R I S , Jr.

On Meromorphic

54

Type

65

Solutions of the Difference Equation

y(x+l) --y(x) + I + TOSIHUSA Branching

23

41

Revisited . . . . . .

Bounded Solutions of Nonlinear Equations at an Irregular Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P, F , HSIEH

1

y(x)

74

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

KIMURA

of P e r i o d i c S o l u t i o n s . . . . . . . . . . . . . . . . . . . . W, S, LOUD

87

Effective Solution for Meromorphic Second Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. J U R K A , D , L U T Z , a n d A. PEYERIMHOFF

100

O p t i m a l C o n t r o l of L i m i t C y c l e s or w h a t C o n t r o l T h e o r y c a n do to C u r e a H e a r t A t t a c k o r to C a u s e one . . . . . . . . . . . . . . . . . LAWRENCE MARKUS

108

T h e S t a b l e M a n i f o l d T h e o r e m Via a n I s o l a t i n g B l o c k . . . . . . . . . R I C H A R D M .cQ E H E E

135

S t a b i l i t y of a L u r i e T y p e E q u a t i o n . . . . . . . . . . . . . . . . . . . K, R. M E Y E R

145

A Nonlinear Integral Equation Relating Distillation E. R. RANG

151

Processes

....

T o t a l l y I m p l i c i t y M e t h o d s f o r N u m e r i c a l S o l u t i o n of S i n g u l a r I n i t i a l Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . E, R, BARNES and D. L. RUSSELL

164

VIII

D i c h o t o m i e s for Differential and Integral Equations . . . . . . . . . GEORGE Ro S E L L

188

A n Entire Solution of the Functional Equation f(~) + f(wl) f(0~-ll) = I, (~5 = i) . . . . . . . . . . . . . YASUTAKA

SIBUYA

and

ROBERT

H

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . .

194

CAMERON 203

M y M a t h e m a t i c a l Expectations H.L.

Turrittin

i. Introduction This expository presentation is intended to be a brief review of m y m a t h e m a t i c a l research, beginning with m y

first paper [i], in 1936.

Since I a m

about to retire, it is an appropriate occasion to look once again at m y publications in a critical fashion. This S y m p o s i u m

on O r d i n a r y Differential Equations is, I feel, indeed a

v e r y particular and gratifying expression of e s t e e m on the part of m y and students.

T h e papers presented here at the S y m p o s i u m

colleagues

have b e e n of special

interest and have provided n~.e with several m a t h e m a t i c a l inspirations which, it is hoped, will materialize later in the f o r m of specific results, express m y

i w i s h to

gratitude to all those w h o have taken part in the S y m p o s i u m ,

w h o have helped organize it, and especially I w i s h to thank Professor

to those

Yasutaka

Sibuya and Profes s o r W i l l i a m A. Harris, Jr. T h e title is intended to indicate that the e m p h a s i s will be placed on what I had expected to do as I initially attacked each n e w p r o b l e m , I actually accomplished.

rather than on w h a t

In m o s t cases there is quite a difference in these two

things, as I shall point out as w e proceed° A s a retiring professor there is no point at this time in trying to conceal m a t h e m a t i c a l ideas.

T h e y are likely k n o w n m a n y

of you anyway.

In short I a m

not trying to stake out claims to certain unsolved p r o b l e m s for future study. Precisely the contrary, since I did not succeed in solving certain problems, nothing would please m e

m o r e m a t h e m a t i c a l l y than to see others

solutions or extend or use s o m e of m y

results.

find the

2.

Equations involving a p a r a m e t e r In 1930 1 b e c a m e

a graduate student in m a t h e m a t i c s

Wisconsin and Professor Rudolph E. L a n g e r b e c a m e mentor.

Z a n g e r had already b e c o m e

my

graduate adviser and

interested in what he called turning point

p r o b l e m s in the theory of ordinary differential equations, problems may

at the University of

see [15-18].

Such

occur w h e n the differential equation contains a parameter.

Professor Langer's suggestion for m y

Ph.D.

thesis

At

I studied the solutions of

an equation of the type

(i)

dny ~x n

Pi(x,p)

where

and that,

n ~

+

=

IP I > R > 0 .

(x,p) i

i=l

Here

0

dn-iy

= 0,

dx n - i

~ Pij(x)/p j , j=O

if the n roots

¢n + Plo(x)

pirp

(i = 1 , . . . ,

is a l a r g e

n),

parameter.

is convergent if a < x

T 0 > 0 ,

then there exists a transformation o~

X = w h e r e the square m a t r i x region

~

j=0

T -j BjY~

B 0 is nonsingular and the series converges in s o m e

~T I > T 1 > T0~which will cut off the series in the equation (12) and convert

(12) into the Canonical f o r m

dY dT

S

=

T

q(~

j--O

T -J CjlY °

All this is true; but Birkhoff thought that he had also proved is it n e c e s s a r y to take

s greater than

q + i.

However

has p r o d u c e d a c o u n t e r - e x a m p l e showing Birkhoff's bound

F.R.

that in no case Gantmaeher

(q + I) is wrong.

15

Since

R.E.

Langer

student of L a n g e r ,

was

a

Ph.D.

I felt it m y

student of

GoD.

c l a i m or at least c o r r e c t his u p p e r b o u n d

prove was

that,

PhoD.

on

if q = -i~ the c o r r e c t u p p e r b o u n d

so

All I w a s

able to

is not (q + I), but (q + 2);

[11]. The rescue

on

a

duty a n d great expectation that I could r e s c u e

Birkhoff's

see

Birkhoff a n d I w a s

s

job w a s s u p p o s e d to be d o n e by p r o v i n g t h a t

would be correct

if we w o u l d o n l y a d m i t

some more

Birkhoff's general

bound

transformation,

s a y o n e of the f o r m

P(~ ~-J/qBj) Y, ¢0

X=,

j=O for a p p r o p r i a t e an expectation

constants on my

have already made

p

part.

some

and

q

with

B0

Nevertheless

progress

nonsingular.

W.B.

T h i s result is only

Jurkat and D.A.

in this direction.

A l s o see

Lutz,

D.A.

[41],

Lutz's

lecture in these P r o c e e d i n g s .

ii.

Extensions We

now

presented me

a n d generalizations

come

to the r e s e a r c h

on this o c c a s i o n

p r e s e n t the p r o b l e m

of the L e t t e n m e y e r problem

that I w o u l d

liked to h a v e solved a n d

rather than giving this e x p o s i t o r y

in a greatly simplified f o r m .

the subject of a s y m p t o t i c

series,

which

seem

at first glance m a y

theorem

lecture.

In introducing

o n e m i g h t b e g i n with E . L .

to be v e r y e l e m e n t a r y ;

namely

First let students to

Ince's e x a m p l e , [42], c o n s i d e r the

equation dw ''

=W+

dz

where

temporarily

we

a solution a p p r o a c h i n g

a r e interested zero

as

Wl(Z) =

in

z -~ o~ C

.l,

z

+

1 z

'

w

and

--

z

as real variables

of the f o r m

c2 '" + zZ

.o. +

c

n n z

+

...

and seek

16

Formally

one finds that

=

wl(z )

~

(_i)j-l(j_l)! /z j ;

j=l but unfortunately this series diverges for all finite values of

z.

However

the

solution w e w a n t does take the f o r m

wZ(z) and~ after integrating

=

f

-

n

e z-(~ d~ ~

z>O,

,

z

t i m e s by parts,

one finds

n

w2(z) = ~ ( - 1 ) J ' l ( j - 1 ) ! z-J + Rn(Z), j=l where

the r e m a i n d e r R

term

n

A n e a s y estimate

co

(z) = (-I)n n! /

e

z shows

(7

w

Wl(Z ) is an a s y m p t o t i c

and

z

to b e c o m e

the c o m p l e x Iz I > 0

z-plane,

and

complex

Indeed the a s y m p t o t i c

if

expansion

z>0.

representing

our solution.

If w e p e r m i t

variables a n d extend our solution analytically into

an estimate by

I arg z I < --

do n+l

that

IRn(z)l < ~!n+l ' Thus

z-(y

I

0

X

f

x0

T h e n (CG, C G)

Proof.

We

]G_l(X)K(x,t)G(t) Idt_< ff < i, x_> x 0.

is admissible for

(3.1).

m u s t first s h o w that a solution

w h e n e v e r the continuous function u

replaced by

u

satisfies

P(x)u(x) = u(x) on

I

f satisfies the s a m e algebraic condition with

f° Suppose that it has been s h o w n that

P(x)k(x,t)G(t) = k(x,t)G(t) holds w h e n

x

and

t satisfy 0 < t < x.

The

resolvent formula X

u(x) = f(x) +

6k(x,t)f(t)dt

J0

together with the k n o w n relationships P(x)f(x) = f(x) = G(x)G_l(x)f(x)

and P(x)k(x, t)G(t) = k(x, t)G(t) clearly imply that P(x)u(x) : u(x) holds for all x the identity displayed in the last f o r m u l a line.

in I. It r e m a i n s to establish

The resolvent

co

series

k(x,t) =

~iJ Ki(x ,t), w h e r e

Kl(X ,t) = K(x,t) and

1 X

K.(x,t)1 =

The hypotheses on

K

f

t

K(x's)Ki-l(S't)ds'

i__> 2.

imply that

K(x,t)G(t)

=

G(x) G l(X)K(x,t)G(t

k is given by the

31

and h e n c e x

P(x)KZ{X,t)G(t)

f lK(x, s)G(s)G_l(S)K(s,t)G(t)ds t

= P(x) X

= ft K ( x ' s ) G ( s ) G - l ( S ) K ( s ' t ) G ( t ) d s An induction a r g u m e n t

shows

= Kz(x't)G(t)

that

P(x)Ki(x,t)G(t) = Ki(x,t)G(t), and h e n c e the previous identity holds with We

now

proceed

to show

that

u

K.

1

is in

i>2

replaced by

C G

whenever

k. f is in

C G.

The

identity P(x)u(x) = u(x) permits us to write x

v(x) = G_l(x)f(x ) +

G_l(X)K(x , t)G(t)v(t)dt

/

0 where

v(x) = G_I (x)u(x) is m e a s u r a b l e and uniformly bounded on c o m p a c t

subsets of I. If

M

denotes the bound

for v

on

[0,x0] , then the previous

f o r m u l a line leads to the estimate x0

(3.3)

when

x _> x 0.

T h e hypotheses of the t h e o r e m i m p l y that the first two t e r m s

in the right m e m b e r Let

X

IG l(X)K(x ,t)G(t)liv(t) Idt LG_I(X)K(x , t)G(t) Idt + f x0

Iv(x) i _< if IG + M 0 f0

w(x)

and let

of this inequality are b o unded by s o m e constant

denote the finite least upper bound of

s vary between

x 0 and

x°

We

iv(s) i on

N

[x0,x ]. Fix

on

I.

x >x 0

conclude f r o m (3.3) that

S

Iv(s)I! N + w(x)

f

IG_l(S)K(s,t)G(t) Idt
O, a n d t h u s

@n(t) = 0,

where

An(t) >

w(t) , v(t), w(t), v(t)

From

the L e m m a ,

O, n = 1, 2 , . . .

en(t) = Vn(t ) - Wn(t ).

.

We now show

The function

that

On(t) is a

~

solution of

L@

=

n

--~f[Vn]

satisfying h o m o g e n e o u s

-

f"

~rwnl + "Bn

- Yn

- ZA

boundary conditions.

n

,

W e can write, using the m e a n -

value theorem, L @ n = ~2@n(t) + ~3@n(g(t)) - Z( IBI] + BI) @n(t)

- Z( IB z i + B 2) @n(g(t)) +fin - ~/n

= [Fz - z( iBlt + B1) ] On(t) + [f% - Z( ;B z t + BZ)] On (g(t)) n

Set

llenll = sup tel

n

len(t) I . 1

i@n(t) I--< f0 i~z- 2( LBII + BI)i I G(t'm) I IT @nil dT

1

+ fo l~3- Z(IB z I + BZ)[ IG(t,~)

ire n 11 dT

1

+ [ "0
0, and

fk(x) -~ x 0 as

T h e local stable manifold t h e o r e m states that, for sma]l embedded

We

U, W +

submanifold of U, with the e m b e d d i n g as s m o o t h as

is a n

f.

T h e local stable manifold t h e o r e m has a long history dating back .

t

to Poincare.

(See H a r t m a n ' s notes

[2,p. 271].)

The standard proof uses

p o w e r series techniques in the analytic case and the contraction m a p p i n g priniciple in the

Cr

case.

t h e o r e m in a B a n a c h space. block as defined by

More modern

proofs use the implicit function

In this paper w e exploit the concept of an isolating

C o n l e y and Easton[l]

to give a proof using only e l e m e n t a r y

topology of Euclidean spaces and e l e m e n t a r y linear algebra.

Techniques

similar to those presented in this paper have b e e n used in certain case of the three-body p r o b l e m to prove that the set of parabolic orbits is a s m o o t h submanifold

[3].

In the next section w e give a precise statement of the local stable manifold t h e o r e m in Euclidean space.

In section 3 w e develop properties of

an isolating block w h i c h w e use in section 4 to prove the t h e o r e m in the Lipschiiz case.

In section 5

w e complete the proof of the theorem.

This r e s e a r c h w a s ~u]~j)orte~ b y N~q~ Ora~t G P 27275

136

2.

Preliminaries We

first

we

shall must

state introduce

Fix integers on

Rm

the

R n"

and

m

stable

some

notation.

and

We

R m x R n,

(x,y) ~

local

n

and fix n o r m s

[[ I[

further use

We

use these n o r m s

ll = m a x

We

shall say A

space,

but

(not necessarily Euclidean) by

]I IF. F o r

(tlxll,

Ilylf)°

to denote the linear operator n o r m

subordinate to

M

IF.

to define the unit discs:

= {x ~ R m :

E

in Euclidean

let

We

A

theorem

denote both these n o r m s

I(x,y)

Let

manifold

Hx)I

%=

IY ~ R n :

I =

I (x, y) ~

GL(R m

0.

pc is s m o o t h

if pc

is either

cr-small,

137

Let manifolds

that,

Note

Rm

f :

×Rn ~ Rm

x R n.

Define the local stable and unstable

as:

w+(f)

:

{7, 6 5:

fk(z) E

I

for all k > 0 },

w-(f)

=

{z 6

fk(z)

I

for all k < 0 } .

if A

is c a n o n i c a l l y

x:

~

hyperbolic,

By suitably choosing a coordinate manifold theorem

f = A + pc

: ~ -~I Z. Lipr-small, %o

is

Let

Furthermore, r > O, then

Lipr;

i__f p

I.e., cr-small,

f-l, one also concludes that W-(f)

W + and

W + and

W-

T h e 5s olating T h e unit disc

W-

if

pc is

r > i, then

are the stable and unstable manifolds

defined b y Conley and E a s t o n

[i].

is an isolating block

xR n :

Let

Wl(X ,y) = x, wz(x ,y) = y.

7/= { (x,y) 2.

6

Rm

Le___t A

C O- small, and let f = A + pc.

for f in the sense

F o r our purposes, the important properties

are those listed in the proposition below.

Proposition

to that fixed point.

Block ×R n

on IRm

is a s m o o t h

intersect at exactly one point, a fixed point for f.

5 c R m

maps

be smooth,

is the graph of a function

pc is

pc

C r.

submanifold.

3.

¢ , W+(f)

q0 is as s m o o t h as ~ is

W - ( A ) = {0} × 1z.

to t h e f o l l o w i n g :

be canonically hyperbolic, let

Then, for small

B y considering

Thus

A

and

patch we can reduce the local stable

a s s t a t e d in t h e i n t r o d u c t i o n

Then~-er~ ]~ and let

W+(A) : ~ x { O}

w I and

w 2 be the projection

Also let

x R n : llY]r >

l[xH } •

be canonically hyperbolic, le____tPs T h e n for small

¢,

b__~e

138

(3-1)

=if(I) c

(3-2)

f: 11 x01Z

Furthermore, z I ,z 2

if

-~ I 1 x (Rn-12)

p

6. I, with

11 ,

is

Lip0-small,

zI - z2

~

f(zl)-f(z2) ~

(3-4)

II=z(f(z1) -f(~z))il

Since

Let

_>

Since

Ol Z

v> 0

s u c h that, if

v II

~Z(Zl- zz) ii "AI" - ~l O

}.

4 can be applied to the m a p

is the graph of a Lipschitz function

~ : I 1 -~ 13 .

f to conclude that However,

a stronger

result is true: If_ Ps

Proposition 6.

is cl(Lipl)-small,

continuous (Lipschitz) function _Proof. Proposition Z,

Let

V=

4:

W + is the graph of a

I 1 "~ 13 •

{ ~0} ×I3 " x0 E I 1 } .

rrI ~ (1) c ~ and

then

~ : ~ ×813

B y (3-i) and (3-Z) of

-~ ~ x(L(R m , R n) - 13) is a h o m o t o p y

143

equivalence.

Therefore, the a r g u m e n t used in the proof of Proposition 3 gives

US

r~y

Now

choose

¢

z ~

I and

~i'

~2

~ I3"

4 can

exactly

point for any

one

I 1 "> 13 -

Thus The

~Z ~ for

I[Df(z))* ~i - (Df(z))*~2 IT >

Proposition

4:

~_ ~ y .

so small that Dr(z) ~

(5-3)

for

~ £_(r) n

T~1

be applied

Note

that

restricted

proof

to

Theorem

can

now

1 when

W +

y

used

to conclude

Therefore

=

T h e n by (5-2) w e have

v I] ~i - ~2

the arguments

f and

F ( V.

6

show

Pe

to

W +

of Proposition

If we

Thus

z ~ I.

W +

O f_ -k (~) n I

and

is a homeomorphism

II

in the

that

r

proof D W+ _

on

Proof.

For

7.

of a function

is therefore

compact.

and

hence

t~ is continuous.

is complete.

that

= D ~, then w e have c o m p l e t e d the proof of

is C I or

Lip l-small.

Let (Xo,~o) ~ W+. R m

-~R n

graph (~) = For

~

E

Thus w e have only left to prove

4 we

(5-4) where

Then

~ 0 = D~? (x0).

let

{ (x,y) ~

Rm

xRn

L ( R m , Rn), let

graph(N)= In section

proved

that

~p

is

U {graph(~)~ Lipschitz,

~, } .

i.e.

W + = g r a p h (~) c (x,~?)) + g raph (13), " + "

contains

is the graph

the following: P~opo~J

of

indicates vector space addition.

y = g (x) } .

144

N o w let z 0 = (x o,~(x0))o neighborhood

2~ of

~0 ' there exists a neighborhood

W + By

(5-1) and

(5-3),

It is sufficient to show that, given any

N U c

there

z0 +

exists

Let

integer

(13)))

~0 = Dq~(x0) We

can now

prove

Theorem

Lip I or

C 1 _ small.

Suppose cr-l-small. function

i.

Propositions

Pc

Lemma 6 and

W e proceed is Lip r or

By

proof of T h e o r e m

c

z 0 + graph

c

(%~).

z 0 + graph (%{). 7 is complete°

1 by induction 4

r.

establishes the theorem w h e n

7 establish the theorem w h e n

p~

pc

is

r _> 2.

cr-small,

Proposition 7,

W +

~= D ~ .

Then

Pc is

Lip r-I

is the graph of a Lip r-I or Hence

~

or

C r-I

is Lip r or

C r and the

and Isolating

Blocks,"

1 is complete.

References I.

C.

Conley

Trans. 2.

and

Amer.

!m. Hartman,

R.

Easton,

Math.

Soco,

"Isolated

Invariant

Vol.

No.l

158,

Ordinary Differential

Sets (1971),

Equations,

35-61.

John Wiley and Sons,

N e w York, 1964. 3.

R. M c G e h e e ,

is

by induction.

B y inductive hypothesis,

~.

that 13 c (Dfk(z0))*(%{).

such

and the proof of Proposition

Proof of T h e o r e m Zip0-small.

k

W + c fk(z0) + graph (%), and hence

f-k( w + N U I) = W + N U Hence

z 0 suchthat

U I of fk(z0) such that

N(fk(z 0) + graph

U = fk(u'). B y (5-4),

of

('~).

a positive

Thus there exists a neighborhood f-k(u'

graph

U

"A Stable Manifold T h e o r e m

with Applications to Celestial Mechanics,"

for Degenerate Fixed Points (to appear).

Stability of a Lurie Type Equation

ti.

R.

Meyer

In their study of nonlinear electrical circuits Brayton and M o s e r [I] investigated the asymptotic behavior of a system of nonlinear differential equations that describe the state of an electrical network. give conditions that insure nonoscillating solutions.

The a i m w a s to

The criterion obtained in

[I] was v e r y restrictive and M o s e r in [Z] obtained m o r e g e n e r a l criteria by using the m e t h o d of P o p o v of automatic control theory.

The m e t h o d of P o p o v

has been very successful in the study of the stability properties of the Lurie equations (see [3] for a detailed discussion). At first glance the equations of Brayton and M o s e r bear no r e s e m b l a n c e to the usual Lurie equations but this note will s h o w that by a change of variables the equations take a f o r m similar to the Lurie equations.

O n c e the equations

are written in this n e w f o r m it is then clear h o w to use the m e t h o d s developed in control theory to study their stability properties.

In particular it is clear

that Popov's m e t h o d would yield a stability criterion. construct a Liapunov function for these equations.

It is also clear h o w to

W e choose the latter to

reprove Moser's t h e o r e m in a straightforward way. The s y s t e m considered in [l,Z] is of the f o r m

(1)

~= =

-Ax cx

+ By -

f(y)

This research w a s supported by N00014-67-A-0113-0019

ONR

contract n u m b e r

146

where

x

matrices

is a n n-vector,

y

an

m

of a p p r o p r i a t e d i m e n s i o n s ,

v a l u e d function of the

m

vector

vector, A

a finite n u m b e r

behavior. form

and

assumption

f(y) = V G ( y ) - c A - i g y

G

where

It is also a s s u m e d

h a s a finite n u m b e r Moser

and

nonsingular,

C

and

A,

]3, C a n d

are constant f is an

m

vector

f so that all solutions of

of e q u i l i b r i u m states and h e n c e

The fundamental

for gradient.

B

y.

O n e w i s h e s to find conditions on approach

A,

O

that

on

(i)

rule out oscillatory

f is that it c a n be written in the

is a scalar function and G tends to infinity as

of critical points

y

V stands tends to

(Yl .... ' Yk )"

then obtains conditions on the coefficients

A,

B

and C

such that all

solutions of (I) tend to x = 0, y = yj, j = 1 ..... k. if o n e m a k e s K =-(A+BCA-I),

the c h a n g e of variables

D=-CA

-I

{~ = K u

u=x, v = - y - c A - i x

then the E q u a t i o n s

and

lets

(i) b e c o m e

- By

~ =vG(y)

(Z)

y=Du-v

if

y

is a scalar and

G(y) = In

Y

definite for ReX _> O,

ii)

ReX > O.

Z

h o l d s and

is c a l l e d Z(~-) T + Z(X)

O.

T h e m a i n t h e o r e m is then Theorem

i.

_If T(X) = I + D { XI -K}-I B

is positive real then all solutions

of_ (4) are b o u n d e d and if it is strictly positive real all solutions of (4) a p p r o a c h one of the equilibrium points ~TG(o i) = 0.

(0, c~i) w h e r e

cri

is such that

148

We

can state T h e o r e m

1 for the original s y s t e m of Equations (I) by

tracing b a c k the coordinate changes. T(I)

: I + (-CA-I)(XI+A

:I- C{XA+A

(7)

In t e r m s of the original matrices

: I - G{XA

+ BCA-I)-IB

z +BC}-IB

+ A Z} -IB {I + G ( X A + A Z ) - I B } -i

: { I +C(XA + AZ)-IB} -l thu s Gorollary i.

If T(>$-I = {I + C ( X A

+ AZ)-IB}

is strictly positive real and (3)

holds t h e m all solutions of (i) a p p r o a c h one of the equilibrium points (0, Yi ) where

Yi is a critical points of G.

Remark.

Moser

does not a s s u m e that (3) hold explicitly but one can easily

s h o w that (3) is equivalent to the condition that the residue at T(X) + T(~) ~:
0

IEk+l(X) i _< B,

Pk+l(X)

at the points

Pk+l(Xk+~ )

n , k = 0,1, Z,3,...,

is the polynomial of degree Xk+l,

Pk+l(X) = Evaluating

x k

First consider the case lira h-~O

]]k all satisfy

in the discussion w h i c h led to the proof

W h a t w e need, then, is s o m e control over the k

Ilk can all be found

r > i.

g r o w t h of

k 0.

1 1 ]]]k+l - Yk+l I

for

In this case w e have

hXk-'~T1 =

T h e n since the m a t r i x

1

- "-i- A G0(Xk+l, Yk+l' Wk+l)

as close as w e w i s h to the identity m a t r i x by taking

can be

Xk+l' Yk+l' Wk+l

small, w e have -i -r 1 [D 1 + khXk+ 1 V D 2 C D 3 (k+l)(- -~ g G 0(xk+l, Yk+l' Wk+l) ) ]

r

(6.9)

Xk+ 1

~h

-

where

[(VDzCD3(k+I))

r

-i

Xk+l

+ E(Xk+l'Yk+l'Wk+l) + O( ~

) ],

E(Xk+l' Yk+l' Wk+l)

small and the t e r m applying

can be m a d e small by choosing Xk+l' Yk+l' Wk+l r x k+l O( - - ~ ) can be m a d e small by taking h small, both

uniformly for k < k 0.

Xk+l' Yk+l' Wk+l'

and

h

It follows therefore that by taking

all sufficiently small w e can guarantee that the n o r m

of the m a t r i x occurring on the left hand side of (6.9) is _< i.

T h e n it is an easy

matter to see that (6.5) also applies for k < k 0. If

r < i

we

have

lim

h-~0 uniformly for k < k 0. < 1 for h

1

0

It is then quite easy to s h o w that the ratio

(6.4) m u s t be

sufficiently small and for k < k 0 just by observing that the first

185

D 1 is unity and the first diagonal entry of the matrix

diagonal entry of

X V D 2 CD3(k+I ) ( - ~

can be confined to a c o m p a c t

AG0(Xk+l, Yk+l, Wk+l))

subinterval of (0, =)

sufficiently small,

for Xk+l, Yk+l, Wk+l

Thus again (6.5) holds for k < k 0 as well as To s u m m a r i z e ,

when

r /i, r > 0,

k < k 0.

k >k0o

h is sufficiently small and

Xk+ I ¢ [0, a], a sufficiently small,

(6.10)

1 1 i T]k+l -Yk+l

for k > 0

so that w e have,

1~1When

i 1 1 hn+l 7]k- Yk I + P0 B

k _
k0,

[0,a], provided

1 . These numbers Ilk

are sufficiently small, by n u m b e r s via

Theor~ml

i requires that b be

of the proof of T h e o r e m

b, ] i ~ + 211 _< b,

W e have, therefore, (a)

suitably restricted, (6.12)

(6.8) holds also for k + i, and the above reasoning is repeated

w e are dealing with points Theorem

that

15 and the inequalities (6.2), (6.3) all hold

a s s u m i n g this to be true, the portion

to get

~

The proof of T h e o r e m

In addition, these quantities m a y

guarantee that L e m m a

above shows that

a and

assume

With appropriate choices of 6, a and h,

TI~k+llf
0o

For

h

and

a

m a y be generated k < k0, the

r / 1 but special m e t h o d s m a y be

required if r = i. (b) This numerical m e t h o d is accurate to n-th inequality (6.8) of T h e o r e m

order, as expressed by

2.

Reforences i.

J.C.

Butcher: '~/_mplicit Runge-I/ulta processes",

Math. C o m p .

18(1964),

50 - 64. Z.

C. Lanczos: "Trigonometric functions,"

J. Math.

interpolation of empirical and analytical

Phys. 17 (1938), 123-199.

187

3.

C. Lanczos:

"Tables of Chebyshev Polynomials,"

(Introduction),

Nat. Bur. Stand. Appl. Math. Ser. 9 (1952). 4.

D. L. Russell: SIAM

5.

J. N u m .

"Numerical solution of singular initial value problems," Anal.

7 (1970), 399-417.

W. R. Wason: Asymptotic Expansions for Ordinary Differential Equations, Interscience Pub., N e w York 1965.

6.

K. Wright:

" S o m e relationships between implicit Runge-Kutta,

collocation and Lanczos B%T

i0 (1970), 217-227.

T methods,

and their stability properties"

D i c h o t o m i e s for Differential and Integral Equations G e o r g e R. Sell

i.

Introduction T he theory of exponential dichotomies for linear differential equations plays

an important role in the study of the qualitative properties of such equations. Consider the following situation, for example. T h e differential equation x: = ax, w h e r e

x e Rn

and

P2

on

Rn

and positive constants

P1 + P2 = I, leatPl

E