Singularly perturbed systems

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Singularly perturbed systemsPatrick J. Rabier a & Shiva Shankar ba Department of Mathematics, Southern Methodist University, Dallas, Texas,75275, USAb Tata Institute of Fundamental Research, Bangalore Centre, Indian Instituteof Science Campus, Bangalore, 560012, IndiaVersion of record first published: 02 May 2007.

To cite this article: Patrick J. Rabier & Shiva Shankar (1987): Singularly perturbed systems, ApplicableAnalysis: An International Journal, 24:1-2, 53-99

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Singularly Perturbed Systems

Communicated by R . P. Gilbert

PATRICK J. RABIER Southern Methodist University, Department of Mathematics, Dallas, Texas 75275, USA.

SHIVA SHANKAR Tata Institute of Fundamental Research, Bangalore Centre, Indian Institute of Science Campus, Bangalore-560012, India.

Abstract This paper discusses existence and nonexistence of C' quasi-steady-states to singularly perturbed problems near a singular point. In contrast to the existence and uniqueness result well known for the same problem near a regular point, the answer depends on generic conditions involving both the differential and the transcendental equations of the system. When existence is guaranteed, multiple solutions will typi- cally appear but their number cannot exceed an explicit upper bound. By comparison, infinitely many generalized solutions exist in the same situation. The verification of the generic conditions as well as the practical determination of the exact number of solutions reduces to algebraic calculations. The bulk of our approach combines results in bifurcation theory (generalizations of the Morse lemma to vector-valued functions) with standard methods in ODE'S. Arguments from algebraic geometry (generalized Bezoutls theorem) and homotopy theory (pa- rametrized Sard's theorem) are also involved, at a lesser degree however, to draw optimal conclusions.

(Received for Publication June 2, 1986)

1. INTRODUCTION. --

The theory of singular perturbations of differential equations con-

siders systems of the form

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P. J. RABIER AND S . SHANKAR

x ( t 0 ) = xc , y ( t O ) = Y, , (1 .2 )E

where x E lRn, y e IRp and f and g a r e f u n c t i o n s de f ined on some

open s u b s e t of IR x lRn x IRp wi th v a l u e s i n IRn and IRp r e spec t ive -

l y . The problem addressed i s t o determine t o what e x t e n t t h e so-

l u t i o n t o ( l . l ) , - (1 .2)E approaches a s o l u t i o n t o t h e s i n g u l a r

system

a s E tends t o 0. I n c o n t r a s t t o t h e i n i t i a l cond i t i ons ( 1 . 2 ) E , t h e

i n i t i a l d a t a xo and yo a s s o c i a t e d wi th (1 .3) a r e no t independent

s i n c e they must s a t i s f y t h e r e l a t i o n g ( t O , x O , y O ) = 0. So lu t ions t o

t h e system (1 .3) w i l l be r e f e r r e d t o a s quasi-steady-states of t h e

system (1.1) - ( l . 2 ) E , Assuming ( a g / a y ) ( t O , x O , y O ) i s nons ingu la r ,

the second equa t ion i n (1.3) can be solved through t h e i m p l i c i t

f u n c t i o n theorem i n t h e form y = y ( t , x ) w i th y ( t O , x o ) = yo. A

corresponding s o l u t i o n t o equa t ion (1.3) can then be ob ta ined by

so lv ing t h e w e l l posed d i f f e r e n t i a l equa t ion

As noted i n [ll], most of t he a v a i l a b l e theory i s r e s t r i c t e d t o

models (1 .3) i n which t h e n o n s i n g u l a r i t y cond i t i on on a g / a y ho lds

a t eve ry p o i n t of t h e ze ro s e t of g. Although some p rog res s has

s i n c e been made, a s r epo r t ed i n t h e r ecen t survey a r t i c l e by

Kokotovic [12], no gene ra l theory seems t o have been developed t o

cons ide r t h e ca se when g i s s i n g u l a r a t some p o i n t ( tO ,xO,yO) of

i t s zero s e t . I t i s t h e aim of t h i s paper t o c o n t r i b u t e towards

f i l l i n g t h i s gap by g iv ing s u f f i c i e n t c o n d i t i o n s f o r e x i s t e n c e o r

nonexis tence of s o l u t i o n s t o (1 .3) n e a r a p o i n t ( t O , x O , y O ) a t

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SINGULARLY PERTURBED SYSTEMS 55

which aglay is s i n g u l a r .

A n a t u r a l e x p e c t a t i o n of s i n g u l a r p e r t u r b a t i o n theory would be

t h a t t h e x - component of t h e s o l u t i o n t o (1.1) - (1 .2) approxi-

mates t h e x-component of a s o l u t i o n t o (1.3) on some i n t e r v a l

[ t O , ~ ] whi l e t h e y - component of t h e s o l u t i o n t o (1.1) - (1.2)

approximates t h e y - component of t h e corresponding s o l u t i o n t o

(1.3) on some s m a l l e r i n t e r v a l [ t l , ~ l only . As a r e s u l t , i f i t i s

s t i l l n a t u r a l t o seek s o l u t i o n s ( g ( t ) , x ( t ) ) of (1.3) such t h a t 8 1

is of c l a s s C on [ tO ,T] , only c o n t i n u i t y a t to should be r equ i r ed

of x. While ou r e x i s t e n c e and uniqueness r e s u l t s w i l l ensu re t h a t

bo th 8 and a r e of c l a s s c', a c o n d i t i o n f o r nonexis tence of

C' x CO s o l u t i o n s w i l l be g iven a s w e l l .

Af t e r performing an obvious change of v a r i a b l e s , t h e s i n g u l a r

p o i n t ( tO ,xO,yO) of t h e ze ro s e t of g may be supposed t o be t h e

o r i g i n w i thou t l o s s of g e n e r a l i t y . The problem thus i s t o f i n d t h e

s o l u t i o n s t o t h e system

v e r i f y i n g t h e i n i t i a l c o n d i t i o n

As i t i s i n s t r u c t i v e f o r f u t u r e comments, we f i r s t d e s c r i b e an

i d e a developed by Dolezal and Shankar [7 ] and Shankar [18], r e l y i n g

on methods having some of t h e s p i r i t of t hose i n t h e t heo ry of

measurable mul t i -va lued mappings: Suppose t h e f u n c t i o n g van i shes

on a s e t whose p r o j e c t i o n on to t h e J R x n n space c o n t a i n s a

neighbourhood of t h e o r i g i n i n IRx Rn. For each ( t , x ) i n t h i s

neighbourhood, l e t

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56 P. J . RABIER AND S. SHANKAR

I f i t i s p o s s i b l e t o s e l e c t a p o i n t y = y ( t , x ) from each of t h e

Y w i th y(0) = 0 , t h e reduced d i f f e r e n t i a l equa t ion ob ta ined by t , X

t hus s o l v i n g f o r y i n t h e f i r s t equa t ion (1.6) has t h e form (1.4) -

(1.5) (wi th xo = 0 ) . Hence, whenever t h e l a t t e r system admits a

s o l u t i o n $ ( t ) , then ( 2 ( t ) , y ( t , z ( t ) ) ) i s a s o l u t i o n t o (1.6) - (1.7).

Now, s o l u t i o n s t o (1 .4) - (1.5) need only be a b s o l u t e l y cont inuous

f u n c t i o n s of t i f one r e q u i r e s equa t ion (1 .4 ) t o be s a t i s f i e d

almost everywhere ( t h e s e s o l u t i o n s a r e cus tomar i ly r e f e r e d t o a s

s o l u t i o n s i n t h e s ense of Caratheodory; ano the r r e l a t e d no t ion

t h a t w i l l n o t be d i scussed h e r e i s t h a t of s o l u t i o n i n t h e s ense of

F i l l i p o v [8 ] ) . I f t h i s gene ra l i zed d e f i n i t i o n i s taken, a l l t h a t

i s needed f o r cons i s t ency i s t h a t y ( t , x ) be a Caratheodory f u n c t i o q

namely cont inuous i n t and measurable i n x ( s e e e .g . K r a s n o s e l ' s k i i

[ 3 ] ) . An impor tant consequence i s t h a t such gene ra l i zed s o l u t i o n s

need no t be unique and i n f a c t need no t be f i n i t e i n number: In-

deed, suppose t h a t y ( t , x ) and y 2 ( t , x ) a r e two d i s t i n c t admis s ib l e 1

s e l e c t i o n s . As only m e a s u r a b i l i t y i n x i s r e q u i r e d , i t i s p o s s i b l e

t o f i n d i n f i n i t e l y many o t h e r admis s ib l e s e l e c t i o n s by "a l t e rna -

t i v e l y " t ak ing y ( t , x ) t o be y l ( t , x ) and y 2 ( t , x ) . Each s o l u t i o n

provides a d i f f e r e n t reduced system (1.4) - (1 .5 ) , hence i n f i n i t e l y

many s o l u t i o n s t o (1 .3 ) . These comments a r e e s p e c i a l l y a p p r o p r i a t e

when (ag/ay) i s s i n g u l a r s i n c e such a s i t u a t i o n w i l l t y p i c a l l y

r e s u l t i n t h e e x i s t e n c e of m u l t i p l e p o s s i b l e s e l e c t i o n s , b u t we 1 s h a l l see how f i n i t e n e s s of t h e number of c l a s s i c a l ( C ) s o l u t i o n s

i s recovered under gene ra l cond i t i ons . Our approach i s based on

methods r e c e n t l y developed t o d e a l w i th multi-parameter b i f u r c a t i o n

problems, which then f i n d a somewhat unexpected f i e l d of a p p l i -

c a t i o n he re . We came t o cons ide r t h e i r u s e f u l n e s s a f t e r making

t h e fo l lowing two obse rva t ions : F i r s t , t h e v e c t o r f (0) = i ( 0 )

s p e c i f i e s t h e d i r e c t i o n of e v o l u t i o n of t h e t r a j e c t o r y ~ ( t ) a t

t = 0. ~ h e r e f o r ' e , i t s u f f i c e s f o r t he p r o j e c t i o n of t h e ze ro s e t

of g on t h e &x lltn space t o c o n t a i n p o i n t s about t h e d i r e c t i o n

( l , f ( O ) ) a lone b u t n o t n e c e s s a r i l y an e n t i r e neighbourhood of t h e

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o r i g i n . Of cou r se , t h i s ze ro s e t must admit a p p r o p r i a t e s e l ec t ions .

The second obse rva t ion is t h a t t h i s can be checked w i t h r e l a t i v e

ea se when g i s a homogeneous polynomial mapping, o r , more genera l ly ,

when i t s ze ro s e t nea r t h e o r i g i n ( l o c a l ze ro s e t ) co inc ides w i t h

t h e l o c a l zero s e t of a homogenous polynomial mapping. I t s o hap-

pens t h a t t h e g e n e r a l s i t u a t i o n can p r e c i s e l y be reduced t o t h i s

p a r t i c u l a r ca se under a s u i t a b l e nondegeneracy c o n d i t i o n invo lv ing

t h e f i r s t nonzero d e r i v a t i v e of g a t t h e o r i g i n , which we now in-

t roduce .

Def in i t ion 1.1: For m 5 1, p 5 1 i n t e g e r s , l e t g : xm+p + xp be k

o f c l a s s C w i th k 2 1. The k- form

will be said t o be regular on i t s zero s e t i f f o r every

w e B ~ + ~ - 101 such t h a t Q(w) = 0 , i t s d e r i v a t i v e

k LYJ (w) = kD g(0) . ( w ) ~ - ' € L(IR~+' , 7Rp) ,

- i s s u r j e c t i v e . 1 - 1

Note f o r k = l t h a t r e g u l a r i t y of Q = Dg(0) on i t s zero s e t

means t h a t DG(0) i s s u r j e c t i v e . With D e f i n i t i o n 1.1, one has

Theorem 1.1: (Bucher, Marsden and ~ c h e c t e r [ 3 , Theorem 1 .31 ) .

Assume i n the no ta t ion of De f in i t ion 1.1 t h a t nig(0) = 0 ,

0 5 i z k-1, and t h a t t h e k- form Q i s regular on i t s zero s e t .

Then there e x i s t neighbourhoods W and W* of the o r i g i n i n xm+' 1 and an origin-preserving C -diffeomorphism 0 : id + W* such t h a t

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58 P. J. RABIER AND S. SHANKAR

Remark 1.1: I n [ 3 , Theorem 1.31, t h e mapping g is assumed t o be a t

l e a s t of c l a s s ck+'. However, t h e r e s u l t remains v a l i d when g i s

of c l a s s ck ( c f , [ 3 , Remark 2, p. 4121). I f k = 2 and p = 1, Theorem -

1.1 is nothing b u t a weak form of t h e Morse Zema. 1 - 1

A; one might expec t , Theorem 1.1 w i l l be used t o c a r r y t h e

problem from one open neighbourhood of t h e o r i g i n t o t h e o t h e r . I n

t h i s p roces s , a c e r t a i n number of t e c h n i c a l i t i e s must be overcome,

which may obscure t h e e s s e n t i a l f e a t u r e s of our approach. We hope

t h a t t h e b r i e f summary given below w i l l he lp t h e r eade r t o l o c a t e

them among o t h e r d e t a i l s of l e s s e r importance.

We begin wi th a few comments r ega rd ing Theorem 1.1. A f i r s t

p o i n t worth mentioning i s t h a t an i n c r e a s e of r e g u l a r i t y of g w i l l

n o t p rov ide a diffeomorphism b e t t e r than c1 i n g e n e r a l ; a counter

example wi th g a polynomial i s g iven i n [ 3 , Remark 3 , p. 4121. I t 1

i s t hen remarkable t h a t C r e g u l a r i t y of 0 i s e x a c t l y what i s need-

ed h e r e . Next, t h e nondegeneracy cond i t i on of D e f i n i t i o n 1.1 is

known t o be gene r i c among forms of t h e same degree k , i . e . ho lds

f o r Q i n an open and dense s u b s e t of t h e space of k - forms ( [ 3 ,

Theorem 4.21). This w i l l a l low us t o s t a t e e x i s t e n c e a s w e l l a s

nonexis tence r e s u l t s bea r ing t h e same gene r i c c h a r a c t e r . I n p a r t i -

c u l a r , i t i s an a p o s t e r i o r i obse rva t ion t h a t a l l t h e s i t u a t i o n s

when a somewhat g e n e r a l theory f o r e x i s t e n c e of C' s o l u t i o n s can

be developed reduce e i t h e r t o t h e well-known c a s e when ( a g / a y ) ( 0 )

€ Isom (IRP) o r t o t h e c a s e when Dg(0) = 0 i n v e s t i g a t e d i n t h i s

paper. Indeed, l e t us assume t h a t (ag/ay) (0) 6 Isom (I?). Then,

r ep l ac ing g by i t s reduced mapping ob ta ined through Lyapunov-Schmidt

procedure ( s ee e . g. Vainberg and Trenogin [l9]) we can suppose

(ag/ay)(O) = 0 wi thou t l o s s of g e n e r a l i t y . Doing s o mod i f i e s p bu t

does n o t a f f e c t t h e cond i t i on p ~ l . I f Dg(0) # 0 , one h a s r = rank

Dg(0) - > 1 whi le t he nul l -space Ker Dg(0) h a s dimension n + p - r and

c o n t a i n s t h e subspace I01 x xp of IRx lRn x lRp s i n c e (ag/ay) (0) = 0.

Hence, t h e p r o j e c t i o n of Ker Dg(0) on to t h e f a c t o r ~ R X B ~ i s a

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proper subspace E w i th dimension n f l - r c n . I t i s then an imme-

d i a t e c o r o l l a r y t o Theorem 3 .1 t h a t t h e sys tem (1.6) - (1 .7) h a s no 1

C s o l u t i o n whenever (1 , f ( 0 ) ) ~4 E , namely f o r a l l bu t e x c e p t i o n a l 1 0

v a l u e s of f ( 0 ) (however, e x i s t e n c e of C x C s o l u t i o n s i s no t pro-

h i b i t e d i n t h e s e " in t e rmed ia t e " c a s e s ) . I f t h e f u n c t i o n g i s inde-

pendent of t , t h e r e s u l t i s t h e same and t h e arguments a r e s i m i l a r

upon r e p l a c i n g lR x IRn by lRn, (1 , f ( 0 ) ) by f (0) and Theorem 3 .1 by

Theorem 2.1. These obse rva t ions can be confirmed by many elemen- 2

t a r y examples. For i n s t a n c e , w i th n = p = l and g (x ,y ) = x - y , t he

system (1.6) - (1 .7) becomes

and c l e a r l y has no s o l u t i o n i f f (0) # 0.

The above c o n s i d e r a t i o n s j u s t i f y t h a t t h e c a s e Dg(0) = 0

should be considered a s complementing t h e one when (ag/ay)(O) is

nons ingu la r . I f Dg(0) = 0 , t h e s i g n i f i c a n t term becomes t h e f i r s t

nonzero d e r i v a t i v e of g a t t h e o r i g i n and t h e nondegeneracy condi-

t i o n of D e f i n i t i o n 1.1 g e n e r a l i z e s t h e assumption t h a t Dg(0) i s

s u r j e c t i v e , a s a l r e a d y observed. Ac tua l ly , t h e analogy between t h e

two c o n d i t i o n s can be cont inued a l l a long t h e s tudy . For i n s t a n c e ,

assuming Dg(0) s u r j e c t i v e does n o t imply t h a t (ag/ay)(O) i s non-

s i n g u l a r , and, s i m i l a r l y , t h e nondegeneracy c o n d i t i o n of D e f i n i t i o n

1.1 a lone w i l l n o t be s u f f i c i e n t t o p rov ide a s u b s t i t u t e f o r t h e

i m p l i c i t f u n c t i o n theorem. But observe t h a t s ay ing t h a t (ag/ay)(O)

i s nons ingu la r amounts t o s ay ing t h a t Dg(0) i s s u r j e c t i v e and t h a t

t he p r o j e c t i o n of t h e (n+ l ) -d imens iona l space Ker Dg(0) onto t h e

f a c t o r lR x IRn i s s u r j e c t i v e ( b i j e c t i v e ) . I f s o , ( 1 , f (0 ) ) be longs

t o t h i s p r o j e c t i o n i r r e s p e c t i v e of t h e va lue of f ( 0 ) . Due t o i t s

much more complicated s t r u c t u r e , t h e ze ro s e t of t h e k - f o r m Q

a s s o c i a t e d w i t h t h e f i r s t nonzero d e r i v a t i v e of g a t t h e o r i g i n

does n o t always p r o j e c t onto t h e whole space lR x IRn when k > 1. But,

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60 P. J. RABIER A N D S. SHANKAR

aga in , we s h a l l s e e t h a t t h e f i n a l r e s u l t e s s e n t i a l l y depends on

whether ( l , f ( O ) ) belongs t o t h e complement (no s o l u t i o n ) o r t h e

i n t e r i o r U of t h i s p r o j e c t i o n . Whenever U i s nonempty, t h e nonde-

generacy cond i t i on of D e f i n i t i o n 1.1 a long wi th t h e parametr ized

S a r d ' s theorem w i l l a l l ow u s t o recover t h e d e s i r e d s u b s t i t u t e f o r

t h e i m p l i c i t f u n c t i o n theorem, f o r a l l p o s s i b l e cho ices of ( l , f ( O ) )

i n an open and dense s u b s e t of U. Exi s t ence of s o l u t i o n s f o r t h e

corresponding va lues of f ( 0 ) ( a l s o i n an open s u b s e t of lRn) w i l l

fo l low. I n p r a c t i c e , t h e admis s ib l e va lues f o r f ( 0 ) can be found

by e l i m i n a t i n g t h e s o l u t i o n s of a system of a l g e b r a i c equa t ions .

Algebra a l s o h e l p s f i n d i n g upper bounds f o r t h e number of s o l u t i o n s

when f v e r i f i e s t h e s t anda rd L i p s c h i t z cond i t i on p rov id ing unique-

nes s i n t h e con tex t of o rd ina ry d i f f e r e n t i a l equa t ions ( [ l o ] ) .

Remark 1.2: I f Dg(0) = 0 i n (1 .6) ( p o s s i b l y a f t e r Lyapunov-Schmidt

r educ t ion ) we have made i t c l e a r t h a t t h e impor tant term i s t h e

f i r s t nonzero d e r i v a t i v e of g a t t h e o r i g i n . As a r e s u l t , excessive

s i m p l i f i c a t i o n of models l ead ing t o systems of t h e form (1.3) by

neg lec t ing too many h ighe r o r d e r terms w i l l have an e f f e c t i n com-

p l e t e c o n t r a d i c t i o n wi th t h e d e s i r e d s i m p l i f i c a t i o n , by making t h e -

nondegeneracy cond i t i on of D e f i n i t i o n 1.1 f a i l t o ho ld . 1 - 1

Of cour se , a d e t a i l e d r e l a t i o n s h i p t o t h e s i n g u l a r pe r tu rba t ion

(1.1) - ( 1 . 2 ) remains t o be e s t a b l i s h e d through a gene ra l theory

o r i n p a r t i c u l a r ca ses . Although we do no t add res s t h i s q u e s t i o n

h e r e , we n o t e t h a t a p rope r ty l i k e l y t o p l a y a r o l e i n i t i s t h a t

f o r every admis s ib l e va lue f (0) and f o r every s o l u t i o n ( ~ ( t ) , ~ ( t ) )

of (1.6) - ( 1 . 7 ) t h e d e r i v a t i v e ag/dy is nons ingu la r a t ( t , z ( t ) ,

~ ( t ) ) f o r eve ry t f 0 n e a r t h e o r i g i n ( s ee Remarks 2.4 and 3.2).

I n o t h e r words, no s o l u t i o n t o (1.6) - (1.7) goes through a s ingular

p o i n t of ag/ay d i s t i n c t from and nea r t h e o r i g i n .

The case when g i s independent of t i s cons idered i n Sec t ion 2.

I n S e c t i o n 3 , t h e g e n e r a l c a s e when g depends on t i s reduced t o

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t h e one considered i n Sec t ion 2. Sec t ion 4 i s devoted t o t h e s tudy

of s imple b u t appa ren t ly non-standard examples showing how t h e a lge -

b r a i c c o n d i t i o n s involved can e s s e n t i a l l y be i n t e r p r e t e d i n geomet-

r i c a l terms.

The i d e n t i t y o p e r a t o r on lRm, mil, w i l l be denoted by I . m

2. THE CASE WHEN g IS INDEPENDENT OF t .

This s e c t i o n cons ide r s t h e ca se when t h e f u n c t i o n g i n t h e s i n g u l a r

system ( 1 . 6 ) - (1 .7 ) i s independent of t . Then g i s a map from a

domain c o n t a i n i n g t h e o r i g i n i n Rn x IRp w i th v a l u e s i n Ep, s o t h a t

t h e problem now i s t o de termine f u n c t i o n s and s a t i s f y i n g

wi th t h e i n i t i a l c o n d i t i o n s

k I t w i l l be assumed i n t h i s s e c t i o n t h a t g i s of c l a s s C ,

k > - 2 , w i th

Let Q be t h e k - form

Denote by

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t h e ze ro s e t of Q , and by

i t s p r o j e c t i o n on to t h e space TRnalong iRp, a n o t a t i o n t h a t i s con-

s i s t e n t w i th t h e d e f i n i t i o n of t h e o p e r a t o r n a s t h e p r o j e c t i o n

from I R ~ x IRP on to IF.".

A necessary cond i t i on f o r t h e e x i s t e n c e of C' s o l u t i o n s t o t h e

system (2 .1) - (2.2) i s e s t a b l i s h e d v i a t h e fo l lowing nonexis tence

r e s u l t .

Theorem 2.1: If

1 then there is no C solution ( c , ~ ) to the sgstem (2.1) - (2 .2 ) . ~f

in addition

Proof: Because of ( 2 . 3 ) , t he Taylor expansion of g about t h e

o r i g i n y i e l d s

Assume now t h a t (,E,x) i s a s o l u t i o n of c l a s s C' x CO t o (2.1) - (2 .2 ) . Then

(')As Q(0, .) i s homogeneous of degree k , t h e c o n d i t i o n Q(0,O) = 0 is always f u l f i l l e d .

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By assumpt ion ( 2 . & ) , f (0) i s a nonzero v e c t o r . Then, a s

z ( t ) / t and t h e r e f o r e ~ ( t ) a r e b o t h nonzero f o r t > O s m a l l enough.

Then, d i v i d i n g (2 .6 ) by 1 1 ( ~ ( t ) , ~ ( t ) ) ~ f o r such t > 0 y i e l d s by t h e

homogeneity of Q

The re fo r e ,

The u n i t s p h e r e i n lRn x IRp be ing compact, t h e r e i s an e lement w i n

x lRP, I / w ! = 1, and a sequence ( t ) w i t h 0

such t h a t

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64 P. J . RABIER AND S . SHANKAR

Iloreover (2.8) imp l i e s t h a t

Now a s t h e sequence ( ~ ( t ~ ) / t i ) i s bounded (cf . ( 2 . 7 ) ) , ou r

assumptions ensu re t h a t t h e sequence ( x ( t p ) / t ) i s bounded too . a. Indeed, t h i s i s c e r t a i n l y s o i f i s cl. I f is CO on ly and

( x ( t L ) / t 2 ) i s no t bounded, t hen t h e r e is a subsequence ( t ) such

t h a t

and from (2 .9 ) , w would be of t h e form (O,v) , v 6 R', llvll = 1, i n

c o n t r a d i c t i o n t o assumption (2 .5 ) . Thus, by e x t r a c t i n g a subse-

quence i f neces sa ry ( a l s o s u b s c r i p t e d by R f o r s i m p l i c i t y of no-

t a t i o n ) , i t can be assumed t h a t

f o r some v i n np. Together w i th (2 .7 ) , t h i s imp l i e s t h a t t h e

l i m i t w i n (2.9) is of t h e form

Equation (2.10) and t h e homogeneity of Q t oge the r show t h a t

i n c o n t r a d i c t i o n t o assumption (2.4) and t h e proof i s complete. 111

I n l i g h t of t h e above theorem, a neces sa ry c o n d i t i o n f o r t h e 1

e x i s t e n c e of C s o l u t i o n s t o (2 .1)- (2 .2) i s t h a t t h e v e c t o r f ( 0 ) be

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con ta ined i n n Z ( Q ) . The r e s t of t h i s s e c t i o n is devoted t o proving

a converse of t h i s s t a t emen t .

Assume t h a t t h e mapping Q i s r e g u l a r on i t s zero s e t ( c f . Defi-

n i t i o n 1.1 wi th m = n ) . Then, by Theorem 1.1, t h e r e e x i s t neigh-

bourhoods W and W* of t h e o r i g i n i n IR"X IRp and an o r ig in -p re se rv -

i ng c1 diffeomorphism @ : W + W * s a t i s f y i n g DO(0) = I such t h a t n+P

Iw* f W* ; g(w*) = 03 <=> Iw* = @(w) ; Q(w) = 0 ) . (2.11)

Without l o s s of g e n e r a l i t y , W can be taken t o be an open b a l l cen-

t e r e d a t t h e o r i g i n , an assumption t h a t s h a l l be i n f o r c e through-

o u t t h i s s e c t i o n .

As a f i r s t s t e p , t h e t r ans fo rma t ion of t h e sys tem ( 2 . 1 ) - (2.2)

under t h e a c t i o n of t h e diffeomorphism 0 needs t o be examined. The

problem i s t o f i n d f u n c t i o n s ~ ( t ) and ~ ( t ) , de f ined and of c l a s s

c1 i n some i n t e r v a l about t h e o r i g i n s a t i s f y i n g (2 .1) - (2 .2) . By

c o n t i n u i t y , t h e p o i n t ( ~ ( t ) , x ( t ) ) b e l o n g s t o W* f o r sma l l enough t ,

so t h a t

d e f i n e s a c1 curve s a t i s f y i n g t h e i n i t i a l cond i t i on

For (x ,y ) E W*, s e t

where u and v a r e c1 f u n c t i o n s of (x ,y) w i th v a l u e s i n IRn and dp

r e s p e c t i v e l y , van i sh ing a t t h e o r i g i n . S i m i l a r l y f o r (u ,v ) 6 W ,

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66 P. J. RABIER AND S. SHANKAR n where x and y a r e C' f u n c t i o n s of (u ,v) w i th v a l u e s i n IR and lRp

r e s p e c t i v e l y t h a t a l s o van i sh a t t h e o r i g i n .

I n t h e n o t a t i o n of (2.14), t h e curve (2.12) becomes

~ ( t ) = ( u ( z ( t ) , ~ ( t ) ) , v ( z ( t ) , ~ ( t ) ) ) = ( g ( t ) , y ( t ) ) . F u r t h e r , a s ( z ( t ) , x ( t ) ) = O(x( t ) ) by d e f i n i t i o n , and hence t h e

1 f u n c t i o n s z and x a r e of c l a s s C i f and on ly i f t h e f u n c t i o n s t and x a r e , t h e cha in r u l e and equa t ions (2.11) t r a n s l a t e t h e sys tem

(2.1) - (2.2) t o t h e fo l lowing one: Find f u n c t i o n s ~ ( t ) , ~ ( t ) of

c l a s s c1 i n some i n t e r v a l con ta in ing t h e o r i g i n such t h a t

w i th t h e i n i t i a l c o n d i t i o n s

g (0 ) = 0 , $0) = 0 . But, from (2 .11) , g o 0 ( x ( t ) ) = 0 i f and only i f Q ( x ( t ) ) = 0. I n

1 o t h e r words, t h e problem now i s t o f i n d C s o l u t i o n s ( g , ~ ) i n some

i n t e r v a l about t h e o r i g i n t o t h e system

s a t i s f y i n g t h e i n i t i a l cond i t i ons

The above t r ans fo rma t ion thus e s t a b l i s h e s t h e equ iva l ence of

systems (2.1) - (2.2) and (2.16) - (2.17) i n t h e s ense t h a t

( ~ ( t ) , ~ ( t ) ) is a C' s o l u t i o n of t h e former i f and only i f - 1

( ~ ( t ) , ~ ( t ) ) = 0 ( c ( t ) , x ( t ) ) is one of t h e l a t t e r . Furthermore,

t h e o r i g i n a l problem has been reduced t o a more t r a c t a b l e one, a s

now t h e s tudy of t h e zero s e t of t h e f u n c t i o n g i s r ep l aced by

t h a t of t h e s imp le r s t r u c t u r e of t h e ze ro s e t of t h e k-form Q . 1 For e a s e of e x p o s i t i o n , e x i s t e n c e of C s o l u t i o n s t o (2.16) -

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(2.17) is established for nonnegative t. The homogeneity of Q al-

lows exactly the same considerations for nonpositive t. Finally,

solutions in some open interval containing 0 are obtained by piec-

ing together appropriate solutions for t,O and t2O. 1

Suppose now that (:(t),x(t)) is a C solution of (2.16) -

(2 .l7). Denote by (~(t) ,x(t)) the corresponding solution to (2.1)

- (2.2). By definition

from which it follows by the k-homogeneity of Q, that for t > 0

In the limit as t tends to 0, this becomes

As u and v are C' functions of (x,y), it follows from the chain

rule that

-1 Now the relation D@ (0) = I can be rewritten as

n+P

and therefore

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Equation (2.18) then shows t h a t i ( 0 ) i s c h a r a c t e r i z e d by t h e

cond i t i on

To sum up, t h e p o i n t ( # t ) / t , x ( t ) / t ) must tend t o some p o i n t +

of t h e form ( f ( O ) , v ) i n t h e ze ro s e t of Q a s t tends t o 0 . I f , i n s t e a d of c h a r a c t e r i z i n g ( x ( t ) , ~ ( t ) ) a s a s o l u t i o n t o

t h e system (2.16) - (2 .17) , one d e f i n e s ~ ( t ) = ( e ( t ) , x ( t ) ) through

a s o l u t i o n ( z ( t ) , x ( t ) ) o f t h e system (2.1) - (2.2) a s i n d i c a t e d by

(2 .12 ) , . t hen t h e above p rope r ty remains t r u e under t h e much weaker

assumption t h a t i s continuous and has a r i g h t der iva t i ve a t t h e

o r i g i n . Indeed, c h a r a c t e r i z a t i o n through t h e system (2.16) i s n o t

needed whi le t h e cha in r u l e can e a s i l y be extended t o cover t h i s 1 0

case . I n p r a c t i c e , t h i s a l lows f o r ( z ( t ) , x ( t ) ) t o b e a C x C

s o l u t i o n t o t h e system (1.3) having a r i g h t d e r i v a t i v e a t t h e o r i -

g in . This obse rva t ion i s impor tant f o r g e t t i n g opt imal uniqueness

r e s u l t s .

The above c o n s i d e r a t i o n s sugges t t h a t i t i s bo th necessary and

s u f f i c i e n t t o c h a r a c t e r i z e t h e s t r u c t u r e of t h e ze ro s e t of Q about

p o i n t s ( f (O) ,v ) . As we s h a l l now s e e , t h i s can be s u c c e s s f u l l y

accomplished when 0 i s a r e g u l a r v a l u e of t h e p a r t i a l mapping

Q ( f ( O ) , . ) , namely when t h e d e r i v a t i v e D V Q ( f ( 0 ) , v ) is anisomorphism

of IRp f o r every s u l u r i o n v e IRP of che equa t ion Q(f ( 0 ) , v ) = 0.

Remark 2.1: Because of t h e s t a n d i n g assumption k ~ 2 , one has

Q(0,O) = 0 and DvQ(O,O) = 0. S ince p z l , i t i s i m p l i c i t l y r equ i r ed - t h a t f ( 0 ) # 0 f o r 0 t o be a r e g u l a r va lue of Q( f (O) , . ) . 1 - 1

Remark 2.2: Suppose t h a t 0 is a r e g u l a r v a l u e of Q( f (O) , - ) and v

i n lRp is such t h a t Q(f(O),v) = 0. I f v is considered a s an e l e -

ment of tP, then DvQ ( f (0) ,v) E L (up, up) i s c l e a r l y and isomorphism

a s w e l l . This obse rva t ion w i l l be used l a t e r on f o r ob ta in ing an

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e s t i m a t e on t h e nwnber of s o l u t i o n s t o t h e equa t ion Q(f(O) ,v) = 0. -

1 - 1

The assumption t h a t 0 i s a r e g u l a r va lue of Q ( f ( O ) , - ) i s re-

l a t e d t o t h e c o n d i t i o n t h a t t h e k-form Q i s r e g u l a r on i t s ze ro

s e t , a s shown i n P r o p o s i t i o n 2 .1 below.

Proposition 2.1: Assuming t h a t Q i s regular on i t s zero s e t , there

i s an open and dense subset A of xn such t h a t , for f ( 0 ) e A, 0 i s

a regular value o f Q(f (O) , . ) ( t h i s conclusion aZZows t h e possi-

b i l i t y t h a t t h e zero s e t o f ~ ( f ( o ) , - ) be empty for some or any

choice o f f (0) i n A).

Proof: Saying t h a t Q i s r e g u l a r on i t s zero s e t amounts t o s ay ing

t h a t 0 i s a r e g u l a r va lue of t h e r e s t r i c t i o n of Q t o (E?xmP) - { O } .

From t h e hypo thes i s , i t fo l lows i n p a r t i c u l a r t h a t 0 i s a

r e g u l a r va lue of t h e r e s t r i c t i o n of Q t o (IRn - {O}) x IRP. With

the pa rame t r i zed Sa rd ' s theorem ( s e e Abraham and Robbin [I], a

ske t ch of t h e proof i s a l s o g iven i n Chow, Mal le t -Pare t and Yorke

[5]) , 0 i s a r e g u l a r va lue of t h e mapping Q(a , . ) f o r almost a l l

a f nn - 101, hence f o r a lmost a l l a € xn. On t h e o t h e r hand, t h e

polynomial n a t u r e of Q ensu res t h a t 0 i s no t a r e g u l a r va lue of

Q(a , . ) i f and only i f t h e components of a v e r i f y a ( f i n i t e ) sys tem

of a l g e b r a i c r e l a t i o n s ( c f . Seidenberg [17]) namely belong t o an

a l g e b r a i c s u b v a r i e t y of B". From t h e above, t h i s s u b v a r i e t y has

measure ze ro i n IRn, hence i s p rope r and t h e r e f o r e i t s complement

is open and dense i n lRn. 111

I f 0 is a r e g u l a r va lue of Q(f (0) , - ) , t h e s o l u t i o n s i n xP of

t h e equa t ion Q(f(O) ,v) = 0 a r e i s o l a t e d . By Remark 2.2, t h e s e

s o l u t i o n s a r e i s o l a t e d i n cP a s w e l l . Then by t h e gene ra l i zed

Bezout ' s theorem ( see Mumford [15]) t he i rnumber cannot exceed kP

( f o r p = l , t h i s i s j u s t t h e fundamental theorem of a l g e b r a ) . Bet-

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t e r upper bounds can be ob ta ined f o r s p e c i a l c l a s s e s of polynomials

by e l i m i n a t i n g t h e "zeroes a t i n f i n i t y " ( s e e L i , Sauer and Yorke

[ 1 4 ] ) . Let t h i s f i n i t e number of s o l u t i o n s be denoted by v and s e t

w i th t h e obvious abuse t h a t i v . 1 . = $ i f v = O . 1 J 1 = l , v

E x i s t e n c e o f C s o l u t i o n s t o (2 .16 ) - (2 .17 ) i s guaranteed by t h e

fo l lowing.

Theorem 2.2: Asswne tha t the function f i s continuous and tha t the

k- fomn Q i s regular on i t s zero s e t . L e t

(2) If 0 i s a reguZar value of Q(f (O) , . ) , then i n the notat ion of

(2 .20) , there i s Ro > 0 such tha t for any given 1 5 j 5 v , there 1 is a C solut ion (LI,~) t o the system (2.16) - (2.17) i n [ o , R ~ )

sa t i s fy ing

Proof: Reca l l t h a t t h e assumption of nondegeneracy ensu res t h a t

f (0) # 0 ( c f . Remark 2.1). Also, a s by hypo thes i s f ( 0 ) is con-

t a i n e d i n nZ(Q), t h e s e t ivj}j=l ,v, (2 .20 ) , i s nonempry. F i x then

l ( j l v .

Because DVQ(f (O), v ) is an isomorphism of IR' and by t h e i m - j

p l i c i t f u n c t i o n theorem, t h e ze ro s e t of Q around t h e p o i n t ( f (O),v,)

co inc ides w i th t h e graph of a C~ f u n c t i o n 8 = @ . ( u ) de f ined on J

j J some neighbourhood of f (0) i n I!Xn and s a t i s f y i n g

( 2 ) ~ t w i l l f o l low from t h e proof of Theorem 2.2 t h a t t h i s nondegeneracy assumption ensu res t h a t f ( 0 ) a c t u a l l y belongs t o t h e i n t e r i o r of nZ(Q).

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SINGULARLY PERTURBED SYSTEMS

C l e a r l y , t h i s neighbourhood can be t a k e n t o be a n open b a l l B, in -

dependent of t h e i n d e x j and c e n t e r e d a t f ( 0 ) . Def ine t h e cm mani-

f o l d

By s h r i n k i n g B i f n e c e s s a r y , i t can b e assumed t h a t 0 6 B, t h e s e t s

B. a r e d i s j o i n t f o r 1 5 j 5 v , and t h a t t h e p a r t i a l d e r i v a t i v e s of J

e , a r e bounded on B. I f A > 0 i s such t h a t bo th u and Xu be long t o

B, t hen t h e homogeneity of Q i m p l i e s t h a t

Def ine t h e open s e t

and ex t end e t o A on C by s e t t i n g j j

where u be longs t o B. Although a p o i n t i n C does n o t have a unique

r e p r e s e n t a t i o n a s a s c a l a r m u l t i p l e of a p o i n t i n B, n e v e r t h e l e s s

i t f o l l o w s from (2.21) t h a t 2. is w e l l d e f i n e d . Indeed l e t ul and J

u2 be long t o B and suppose t h a t X1 > 0,X2 > 0 a r e two s c a l a r s such

t h a t Xlul = A2u2. Then

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P. J. RABIER AND S. SHANKAR

By (2.21) a s bo th ul and u2 belong t o B , i t fo l lows t h a t

1 e . (U ) = - e j (ul) , J 2 x2

so t h a t by t h e d e f i n i t i o n of j

Observe a l s o t h a t A a s def ined s a t i s f i e s j

f o r every u i n C and A > 0. Moreover i t i s c l e a r t h a t t h e s u b s e t of

Z(Q)

coincides w i t h the g r q h of a; and t h a t t h e d i s j o i n t n e s s of t h e J

B . ' S imply t h a t t h e C . ' s a r e d i s j o i n t a s we l l . F i n a l l y , a s C co- J J j

i n c i d e s l o c a l l y w i th a p o s i t i v e s c a l a r m u l t i p l e of B;, C; i s a cm J J

manifold and t h e mapping g i s then cm on C. I n words, C . i s a j J

conic s l i c e of Z(Q) (wi thout IOI).

From t h e d e f i n i t i o n of 8 i t fo l lows t h a t f o r every u i n C , j

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where A > 0 i s any r e a l number such t h a t u /A€B. By d i f f e r e n t i a t i n g ,

we g e t

which imp l i e s by t h e boundedness of DB. on B t h a t t h e d e r i v a t i v e of - J 0 . i s bounded on C. J

I t was observed e a r l i e r t h a t g iven any c1 s o l u t i o n ( ~ ( t ) , ~ ( t ) )

of t h e system (2.16) - (2 .17 ) , t h e p o i n t ( , u ( t ) / t , $ t ) / t ) must tend

t o some p o i p t of t h e form ( f ( O ) , v ) i n t h e zero s e t of Q , t hus of

t h e form ( f ( O ) , v . ) f o r some index j . By c o n t i n u i t y and a s t h e B . ' s J J

a r e d i s j o i n t , ( : ( t ) / t , x ( t ) / t ) must t hen belong t o some B f o r j

sma l l t > 0. Then , u ( t ) / t i s i n B and t h e r e f o r e

t

As every p o s i t i v e s c a l a r m u l t i p l e of a p o i n t i n B be longs t o C by

d e f i n i t i o n , i t fo l lows t h a t f o r t > 0 m a l l enough, $ t ) E C w i th

Since such a conclus ion r e l i e s on ly on t h e f a c t t h a t ( , u ( t ) / t ,

$ t ) / t ) t ends t o ( f ( O ) , v . ) , i t remains v a l i d i f , i n s t e a d of de f in - J

i ng ,u and a s a s o l u t i o n t o t h e sys tem (2.16) - (2 .17) , one de-

f i n e s = (2,:) through r e l a t i o n (2.12) from a x CO s o l u t i o n

( z , ~ ) of t h e system (2.1) - (2.2) having a r i g h t d e r i v a t i v e a t t h e

o r i g i n ( r e c a l l t h a t r e l a t e d comments have been made e a r l i e r ) . I n

t h i s con tex t , r e l a t i o n (2.23) w i l l be used aga in i n Theorem 2.3.

The f u n c t i o n 8 extends cont inuously a t 0 by A. (0) = 0, s o t h a t t h e j J

r e l a t i o n (2.23) i s t r u e f o r t = O a s w e l l .

Because t h e f u n c t i o n 8 is cm on C , t h e chain r u l e y i e l d s f o r j

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Hence, f o r t > 0 sma l l enough and i n t h e n o t a t i o n of (2 .12) , t h e

system (2.16) - (2.17) becomes

and where

Thus t h e problem now i s t o determine a f u n c t i o n ~ ( t ) of c l a s s C'

i n [ o , R ~ ) f o r some Ro > 0 , s a t i s f y i n g ~ ( 0 ) = 0 and such t h a t

- - [z ( u , ~ . ( u ) ) + ax ( , F . ( u ) ) ~ e . ( : ) l i = f ( t , e ( : , e . ( c ) ) , a u Q 3 % a~ 2 J Q J J

(2.24)

f o r t € (O,Ro) . The term I: (ax/ au) CU,B. ~ u ) ) + (axla") (u,B. (u))DU (u) I e L ( n n , n 7

J J . j (which f o r ~ = ~ ( t ) becomes t h e c o e f f i c i e n t of ~ ( t ) i n (2 .24)) ,

denoted by M(u) i n t h e s eque l , i s de f ined and cont inuous only on

C fl W and not on a neighbourhood of t h e o r i g i n i n gene ra l . Now t h e

r e l a t i o n D0(O) = I can be r e w r i t t e n a s ( 3 ) n+P

- 1 ( 3 ) ~ h i s i s t h e analogue of formula (2.19) w i th 0 r e p l a c i n g 0 .

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The key point then is that DZ being bounded on C, the function j

u + (ax/ av) (u,s. (u)) D T ~ (u) (resp. the function M) can be extended J

continuously on (C fl W)U {O} by setting its value at the origin to

0 (resp. In). In particular by the continuity of at 0, equation

(2.24) is satisfied at the origin as well.

Existence of a solution to (2.24) can now be exhibited as fol-

lows: First, it can be assumed that the function M, already ex-

tended to (C f l W ) U { O } as indicated above, admits a continuous ex-

tension to the closure of C fI W. Indeed, it suffices to replace ? by a smaller closed cone with its apex at the origin and W by a

smaller closed ball centered at the origin. Next, by Tietze's .-. theorem, PI can be extended to a continuous function M on all of lRn.

Similarly, the continuous function h. (t,u) = f (t, ~(u,;. (u))) can be J

J n continuously extended to a continuous function h. on IRxlR .

J It follows from (2.25) that c(0) = M(0) = In. Thus h(u) is an

invertible linear mapping for u in some neighbourhood of the origin

in R". Clearly, if for small enough tz0, : is any C' solution of

equation (2.24), then

with, of course, the initial condition

As [h(u)]-' ij(r,u) is a continuous function of (t,u), a standard

result in the theory of differential equations (see e.g. [lo, Theo-

rem 1.11) ensures the existence of a c1 solution ~ ( t ) to (2.26) - (2.27) for t in some interval containing the origin.

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The r e l a t i o n $(O) = f ( 0 ) exp res ses t h a t t h e curve % is t angen t

t o t h e l i n e I R f ( 0 ) a t t h e o r i g i n . A s f ( 0 ) be longs t o B c C , i t i s

e a s i l y s een t h a t t h e r e e x i s t s Ro > 0 such t h a t ~ ( t ) i s i n C fl W f o r

O < t < R o ( c f . F igu re 2.2).

But a f t e r s h r i n k i n g R i f necessary and f o r 0 2 t i R o and u E C fl W ,

t h e mappings fi and fi c o i n c i d e wi th M and h r e s p e c t i v e l y . Thus, j

from t h e d e f i n i t i o n s , x ( t ) , 0 5 t < R o , j l

i s a l s o a C s o l u t i o n of

equa t ion (2 .24) , i n which case ( ~ ( t ) ,;. ( $ t ) ) ) i s a s o l u t i o n of 1 J

(2.16) - (2.17) t h a t is C away from t h e o r i g i n . I t is a l s o c l e a r

t h a t Ro can be chosen independent of t h e i ndex 1 2 j ( v .

To complete t h e p roo f , i t needs t o be v e r i f i e d t h a t t h e second 1

component of t h e s o l u t i o n , namely x = Fj 0 is C a t t h e o r i g i n .

Towards t h a t end, observe t h a t r e l a t i o n (2.22) w i th X = l / t imp l i e s

Hence

(2.29)

which shows t h a t i s d i f f e r e n t i a b l e a t t h e o r i g i n . F i n a l l y , d i f -

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f e r e n t i a t i n g (2.28) a t t > 0 r e s u l t s i n

I n t h e l i m i t a s t tends t o 0 and s i n c e 6 i s cffi around f (0) E C , j

t h i s y i e l d s

dx - l i m - ( t ) = B.(f (O)) = @ . ( f ( O ) ) = v

d t j ' (2.30) t+O+ J J

From (2.29) and (2.30) t h e f u n c t i o n = 6.0: i s of c l a s s c1 a t t h e J

o r i g i n and t h e r e f o r e (IJ, 5.0:) is a s o l u t i o n t o (2.16) - (2.17) I

s a t i s f y i n g t h e r equ i r ed p r o p e r t i e s . 111

It may be con jec tu red t h a t uniqueness , o r more p r e c i s e l y e s t i -

mates on t h e number of s o l u t i o n s t o (2.1) - (2.2) a s guaranteed by

Theorem 2.2, would fo l low from L i p s c h i t z c o n t i n u i t y of f . While

t h i s is t r u e , a s e s t a b l i s h e d i n Theorem 2.3 below, i t cannot how-

eve r be deduced from an examination of t h e t ransformed system

(2 .24) . This i s because t h e mapping M is on ly cont inuous a t t h e

o r i g i n . Ne i the r can i t be made L i p s c h i t z by i n c r e a s i n g t h e regu-

l a r i t y of g , f o r a s r e c a l l e d i n Sec t ion 1, t h e b e s t t h a t can be

assumed of t h e diffeomorphism 0 i n g e n e r a l is t h a t i t is C' a t t h e ffi

o r i g i n , even when g i s C ( i f k = 2 and p = l however and g i s of

c l a s s cm, then t h e Morse lemma ensu res t h a t @ e ern-'), The r e l a t i o n (au/ay) (0) = 0 i n (2 .19 ) , t o g e t h e r w i th t h e

c o n t i n u i t y of DQ imply t h a t given any E > 0 t h e r e e x i s t neighbour-

hoods of t h e o r i g i n , which can be wi thou t any l o s s of g e n e r a l i t y * *

taken a s t h e W and W of Theorem 2.2, such t h a t f o r a l l (x ,y) i n W ,

Also, because (au/ax) = I ( c f . (2 .19)) t h e r e is a cons t an t k > O *

such t h a t f o r a l l (x ,y) i n W ,

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S i m i l a r l y , i t fo l lows from (2.25) t h a t f o r a l l (u ,v) i n W ,

and

Remark 2.3: The neighbourhood W i s , a s be fo re , a b a l l c en te red a t * t h e o r i g i n . Moreover, by sh r ink ing W and thus W = O ( W ) i f neces-

s a r y , i t can be assumed t h a t W* is conta ined i n a convex s e t ( b a l l )

on which i n e q u a l i t i e s (2.31) and (2.32) con t inue t o hold . This * -

obse rva t ion w i l l permi t t h e u s e of t h e mean value theorem i n W . 1 - 1

Theorem 2.3: In addi t ion t o the asswrrptiols of Theorem 2.2, sup-

pose tha t the function f i n (2.1) i s Lipschi tz continuous with re-

spect t o (x,~? i n W*. Then there i s Ro > 0 such tha t for any given j , 1 15 j 5 v , there i s an unique C xCO solut ion ( z , ~ ) to the system

(2.1) - (2.2) i n [o ,Ro) , r igh t -d i f f eren t iab le a t the origin, s a t i s -

fy ing

and no such solut ion e x i s t s tha t does not s a t i s f y (2.35) for some

index 15 j 5 v . In addi t ion , aZZ these solut ions u1.e automutical- 1

LY c . 1

Proof: Let ( 2 , ~ ) be a C x c O s o l u t i o n t o t h e system (2.1) - (2.2)

r i g h t - d i f f e r e n t i a b l e a t t h e o r i g i n . As po in t ed o u t i n t h e proof - 1

of Theorem 2.2, t h e curve (LI,~) = 0 ( z , ~ ) i s w e l l de f ined and l i e s

on C . f l W f o r t 2 0 sma l l enoughand some index j p r e c i s e l y charac- J

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t e r i z e d by t h e c o n d i t i o n (2 .35) . Hence, no such s o l u t i o n can f a i l

t o s a t i s f y a l l t h e r e l a t i o n s (2.35) t o g e t h e r . On t h e o t h e r hand, j

be ing now f i x e d , Theorem 2.2 gua ran tees t h e e x i s t e n c e of a t l e a s t

one C' s o l u t i o n , denoted by (zllf l) , i n some i n t e r v a l 1 0 , ~ ~ ) and

such t h a t t h e curve (El,xl) = 0 ( ~ ~ , x ~ ) l i e s on C . fl W. Hence J

Let ( z2 ,x2 ) be ano the r c1 x CO s o l u t i o n on [ O , R ) r i g h t - d i f f e r e n -

t i a b l e a t t h e o r i g i n and s a t i s f y i n g (2.35) too . From t h e above,

t h e curve (x2,x2) = 0-1(z2,x2) l i e s on C . I7 W f o r 0 2 t 5 RA, J

RA 5 Ro sma l l enough and ,

I n p a r t i c u l a r , f o r 0 2 t 5 RA, t h e images

a = 1, 2 , (2.39)

By Remark 2.3, t h e mean va lue theorem a long w i t h t h e r e l a t i o n s

(2 .31) , (2.32) and (2.39) y i e l d s

(2.40)

Denote by 1 . I R t h e supremum norm of cont inuous mappings de f ined on

some i n t e r v a l [o,R]. Then f o r 0 < R < RA i n e q u a l i t y (2.40), which

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i s v a l i d f o r a l l t i n [o,R;), y i e l d s on t a k i n g t h e supremum of b o t h

s i d e s ove r [o ,R] ,

Note t h a t E i n t h e above e x p r e s s i o n i s independent of R f o r

0 < R < RA.

As b o t h (z l ,x l ) and ( z 2 , x 2 ) a r e s o l u t i o n s t o t h e sys tem (2.1)

o , and x s a t i s f y t h e r e l a t i o n - (2 .2) f o r 0 5 t < R' %2

By h y p o t h e s i s , f i s L i p s c h i t z con t i nuous i n ( x , y ) w i t h c o n s t a n t

Lf > 0 s a y , s o t h a t t h e above e q u a l i t y y i e l d s

Theref o r e

f o r a l l R i n [o,R;). Le t now R > 0 be s m a l l enough s o a s t o s a t i s -

f y 0 < LfR < 1. Then t h e u s e of t h e e s t i m a t e (2.42) i n (2.41)

r e s u l t s i n

The mean v a l u e theorem, a long w i t h (2 .33) , (2.34) and (2.38) im-

p l i e s t h a t

Now i n t h e proof of Theorem 2.2, i t was observed t h a t t h e f u n c t i o n

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SINGULARLY PERTURBED SYSTEMS 8 1 - 8. i s L i p s c h i t z cont inuous on C. The re fo re , from (2.36) and (2.37), l

and by modifying t h e cons t an t k i f neces sa ry , t h e above i n e q u a l i t y

p rov ides

This t r ans fo rms (2.43) t o

C l e a r l y , a s E can be decreased by s h r i n k i n g t h e neighbourhoods W

and w*, which does no t a f f e c t t h e v a l u e of k, i t may be assumed

t h a t E (k + E ) < 1. Then t h e c o e f f i c i e n t of It1 - x2 1 i n t h e r i gh t -

h a n d s i d e of t h e above i n e q u a l i t y can be made s t r i c t l y l e s s t han 1

by choosing R > O s u f f i c i e n t l y sma l l . For t h i s va lue of R ,

so t h a t t h i s t o g e t h e r w i th (2.36) and (2.37) g ives

on [o,R]. 1 0 Summing up, what t h e above e x h i b i t s i s t h a t any o t h e r C x C

s o l u t i o n ( z2 ,x2 ) on [ O , R ) r i g h t - d i f f e r e n t i a b l e a t t h e o r i g i n w i th

&(0) = V . co inc ides w i t h (cl ,xl) on some i n t e r v a l [o,R] w i t h J

0 < R < R o e This i s n o t enough t o deduce uniqueness because K may

a p r i o r i depend on t h e s o l u t i o n (,x2,x2). I n o t h e r words, what has

been e s t a b l i s h e d s o f a r i s t h a t any two s o l u t i o n s must c o i n c i d e on

some i n t e rva l . However, uniqueness on t h e e n t i r e i n t e r v a l [ O , R )

can now be e a s i l y ob ta ined by r e p e a t i n g t h e same arguments: De-

f i n i n g X a s t h e upper bound of 0 5 R < R f o r which (zl ,xl) and

( Z E ~ , ~ ~ ) co inc ide on [o ,R] , t h e above proof shows t h a t R > 0. Sup-

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pose R < Ro. Then, by c o n t i n u i t y , (zl ,xl) and (e2 ,x2) co inc ide on -

[o,:] and t h e p o i n t (x1(R) ,x l (Z)) = (x2(R) ,x2(K)) belongs t o C , f l W 3

( s i n c e (xl,xl) t a k e s i t s v a l u e s i n C . f l W f o r 0 5 t < Ro). By a 3 c o n t i n u i t y argument, ( x 2 ( t ) , x 2 ( t ) ) s t i l l l i e s on C f o r t around 2

j and t h e same proof a s above wi th R r ep l ac ing 0 imp l i e s t h a t (:l,xl)

and ( z2 ,x2 ) now co inc ide on [R,R+6] f o r some 6 > 0 (no te t h a t t h e

r e l a t i o n s (2.31) - (2.34) remain a v a i l a b l e ) . Thus, t h e two solu-

t i o n s co inc ide on t h e l a r g e r i n t e r v a l [0,R+61 which c o n t r a d i c t s t h e -

d e f i n i t i o n of R. As a r e s u l t , R = R and t h e proof i s complete. /I/

As was observed e a r l i e r , t h e homogeneity of t h e k-form Q a l -

lows e x a c t l y t h e same conc lus ions t o be drawn f o r nonpos i t i ve t a s

were f o r nonnegat ive t . Thus, t h e assumptions of Theorem 2.3 a l s o

imply a f t e r a p o s s i b l e mod i f i ca t ion of Ro, t h a t f o r any given j ,

1 2 j 5 v , t h e r e i s an unique s o l u t i o n ( z , ~ ) t o t h e system (2.1)

- (2.2) i n (-Ro,O] s a t i s f y i n g (&o) , i ( 0 ) ) = ( f (0) , v . ) . The only way J

t o p i e c e t o g e t h e r t h e s e s o l u t i o n s f o r t 2 0 and t 5 0 i n o r d e r t o

o b t a i n s o l u t i o n s t o t h e system (2.1) - (2.2) de f ined on (-R , R ) 0 0

is t o match t h e branches corresponding t o t h e same index 1 5 j 5 v. 1 Hence, t h e system (2.1) - (2.2) has exactly v C solutions defined

1 on (-Ro,Ro) which a r e t h e only C x CO s o l u t i o n s d i f -

f e r e n t i a b l e a t t h e o r i g i n . The e s t i m a t e v 2 kP ob ta ined e a r l i e r

from Bezout ' s theorem then p rov ides an upper bound f o r t h e number

of c1 s o l u t i o n s t o t h e system (2.1) - (2 .2) .

Remark 2.4: Let Q be r e g u l a r on i t s zero s e t and suppose t h a t 0 i s

a r e g u l a r va lue of Q(f(O) , . ) . Let ( e ( t ) , x ( t ) ) be t h e unique C 1

s o l u t i o n e x h i b i t e d i n Theorem 2.3. Then, a f t e r s h r i n k i n g R i f

neces sa ry , t h e p a r t i a l d e r i v a t i v e ag/ay i s nons ingu la r a t

( ~ ( t ) , ~ ( t ) ) f o r 0 < / t i < Ro. Indeed, assume by c o n t r a d i c t i o n t h a t

t h e r e is a sequence ( t ) tending t o 0 , t E # 0 , such t h a t t h e r e is L

i, e RP, /15,lj = 1 and ( a g l a y ) ( z ( t , ) , ) l ( t E ) ) 5, = 0 . A f t e r ex-

t r a c t i n g a subsequence, one may assume t h a t (it) t ends t o < E Rp

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'0

k - l + Dividing by t

R 0 , we f i n d

I n t h e l i m i t a s k tends t o i n f i n i t y , t h e above r e l a t i o n y i e l d s

where v i s de f ined a s i n Theorem 2.3. I n o t h e r words, j

DQ(f(0) ,v j ) . ( 0 , ~ ) = 0, wh i l e Q ( f ( O ) , v . ) = 0, c o n t r a d i c t i n g t h e J -

assumption t h a t 0 i s a r e g u l a r va lue of Q ( f ( O ) , . ) . 1 - 1

1 I f i t were n o t f o r t h e e x t r a assumption t h a t t h e C x CO so lu-

t i o n s de f ined f o r t , O sma l l enough must be r i g h t - d i f f e r e n t i a b l e a t

t h e o r i g i n , Theorem 2.3 would y i e l d t h e p l e a s a n t conc lus ion t h a t , 1

under t h e s t a t e d assumpt ions , eve ry C X C O s o l u t i o n t o t h e system

(2.1) - (2.2) de f ined f o r tzO smal l enough is a u t o m a t i c a l l y c1 and

t h a t t h e s e s o l u t i o n s a r e f i n i t e i n number (and extend a s C' func-

t i o n s f o r It sma l l enough). Such a r e s u l t i s now e s t a b l i s h e d

under an a d d i t i o n a l assumption a l r e a d y encountered i n Theorem 2.1 1 0

(on nonexis tence of C x C s o l u t i o n s ) .

Theorem 2 .4 : I n a d d i t i o n t o t h e nsswnptions of Theorem 2.3, sup-

pose t h a t

1 Then, every C x C0 s o l u t i o n t o t h e system (2.1) - (2.2) def ined

f o r t 20 ( re sp . - < 0 ) smaLl enough i s r i g h t ( r e sp . Left) d i f feren-

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t i a b l e a t the or iq in ( s o t h a t t h i s condi t ion i s no t r e s t r i c t i v e i n the uniqueness statement o f Theorem 2.31.

1 0 Proof: Let ( 2 , ~ ) denote a C x C s o l u t i o n t o t h e system (2.1) - (2.2) de f ined f o r t , O sma l l enough. We only need t o prove t h a t

l i m x ( t ) / t e x i s t s . Equ iva l en t ly , we may show f o r every sequence t +o+ ( t z ) w i th

t h a t l i m x ( t k ) / t z e x i s t s and i s independent of ( t9 . ) . Arguing a s 9.-

i n Theorem 2.1 , boundedness of t h e sequence x ( t z ) / t z and charac-

t e r i z a t i o n of i t s accumulation p o i n t s a s p o i n t s v ~ I R P such t h a t

Q ( f ( O ) , v ) = 0 fo l low from cond i t i on (2 .44) . I n p a r t i c u l a r , any

accumulation p o i n t of x ( t ) / t z is n e c e s s a r i l y one of t h e v p o i n t s 9.

v i . Boundedness of t h e sequence x ( t z ) / t L ensu res t h a t i t s conver-

gence i s e q u i v a l e n t t o uniqueness of such a v w h i l e e x i s t e n c e of i'

l i m + x ( t ) / t i s equ iva l en t t o independence of v on t h e sequence t + O j

( t i ) . It is e a s i l y checked t h a t e i t h e r p rope r ty can be proved by

r e a c h i n g a c o n t r a d i c t i o n under t h e assumption t h a t t h e r e a r e

sequences ( 7 ) and (5 ) j u s t a s ( t z ) above such t h a t 9. Q

A c o n t r a d i c t i o n is indeed e a s i l y found because of t h e c o n t i n u i t y of

x ( t ) / t : C l e a r l y , i t i s no t r e s t r i c t i v e t o assume t h a t r k and B Q

belong t o two d i s j o i n t neighbourhoods V and V . of v and v . re- j 1 j

s p e c t i v e l y . The o s c i l l a t i o n of x ( t ) / t between t h e s e two neigh-

bourhoods shows t h a t x ( t ) / t must t hen have i n f i n i t e l y many accumu-

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

t h e s e p o i n t s a r e f i n i t e i n number. Rigorously , t h i s can be seen a s

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fo l lows : Denote by 2d > O t h e d i s t a n c e between v and v . and i 3

choose V . and V . such t h a t d is t (Vi ,Vj) 2 d. Suppose @ < -iQ (equal- J R

i t y between B R and T~ i s obvious ly imposs ib l e ) . Con t inu i ty of t h e

func t ion

on [ s ~ , T ~ ] and t h e i n t e r m e d i a t e va lue theorem ensu re f o r every

0 < E < 1 t h a t t h e r e i s uQE € ( B P , ~ P ) such t h a t

A s i m i l a r d e f i n i t i o n f o r o can be made i f T L & < e Q ,

As both T L

and B Q t end t o 0, one has l i m o k = 0 and every accumulation p o i n t

of t h e sequence x ( o L ) /u: i s n e c e s s a r i l y a v f o r k = k ( ~ ) and k

l z k z v . As e x i s t e n c e of an accumulation p o i n t i s guaranteed be-

cause (ueE) i s a sequence j u s t a s ( t ) a t t h e beginning of t h e P

proof , we f i n d , a s a r e s u l t of (2.46)

But t h i s i s c l e a r l y imposs ib le f o r every 0 < E < 1 s i n c e f i n i t e l y

many v k l s only a r e a v a i l a b l e . 111

3 . THE CASE WHEN g I S DEPENDENT ON t.

Consider now t h e ca se when t h e f u n c t i o n g i n (1.3) depends on

t . Then g i s a map from a domain con ta in ing t h e o r i g i n i n

IR x lRn x nP wi th v a l u e s i n lRp, so t h a t t h e problem i s t o de termine

f u n c t i o n s and s a t i s f y i n g

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86 P. J . RABIER A N D S. SHANKAR

with t h e i n i t i a l cond i t i ons

k A s i n S e c t i o n 2 , i t w i l l be assumed t h a t g i s of c l a s s C , k 1 2 ,

where now D denotes d i f f e r e n t i a t i o n wi th r e s p e c t t o t h e t h ree var-

i a b l e s ( t , x , y ) . Let Q be t h e k- form

Denote by

t h e zero s e t of Q and by

n i t s p r o j e c t i o n onto t h e space IR x W along TRP.

F i r s t , observe t h a t an e q u i v a l e n t form of t h e problem (3 .1 ) - (3 .2) i s t o f i n d f u n c t i o n s k ( p ) , ~ ( o ) , x(Q), de f ined i n some i n t e r -

v a l about t h e o r i g i n such t h a t

w i th t h e i n i t i a l cond i t i on Dow

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SINGULARLY PERTURBED SYSTEMS 8 7

Now note that ~ ( c J ) = p describes the solution to the differential

equation with initial condition

so that the problem (3.3) - (3.4) transforms to one of determining functions & ( p ) , %(p), ~(p), in some interval about the origin such

that

with the initial conditions

Define the function h by (so that h is independent of t)

Let z denote the vector (t,x) in 1 ~ x 1 ~ ~ . Then the system (3.5) - (3.6) can be rewritten as

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P. J. RABIER A N D S. SHANKAR

z(0) = 0 , ~ ( 0 ) = 0 , % (3.9)

where t ( p ) is the function (tJp), ~ ( p ) ) . The system (3.8) - (3.9)

i s now precisely of the kind considered i n Sect ion 2, the results

of which therefore, merely need to be rewritten in the present con-

text.

Since h(0) = (l,f(O)), Theorem 2.1 translates to

Theorem 3.1 : I f

1 then there i s no C s o Z u t i o ~ (~,x) t o the system (3.1) - (3.2). If i n addi t ion

1 then there i s no C XCO solut ion (~,x) t o the system (3.1) - (3.2). As in Section 2, the assumption that Q be regular on its zero

set (cf. Definition 1.1 with m = n+l) permits the use of Theorem

1.1, which transforms system (3.1) - (3.2) to the problem of find-

ing c1 sblutions (X,~,Y) in some interval about the origin to the system

and satisfying the initial conditions

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~ ( 0 ) = 0 , ~ ( 0 ) = 0 , ~ ( 0 ) = 0 . (3.3.1)

A s h(0) = (1 , f (0 ) ) r e p l a c e s f (0) of t h e p rev ious s e c t i o n , we

s h a l l r e q u i r e 0 € lRP t o be a r e g u l a r va lue of Q ( l , f (0) ,.).

Remark 3.1: I n c o n t r a s t t o t h e problem t r e a t e d i n Sec t ion 2 , t h e

above cond i t i on a l l ows t h e ca se when f ( 0 ) = 0 , a s now ( l , f ( O ) ) i s a

nonzero v e c t o r i r r e s p e c t i v e of t h e va lue of f (0 ) . 11

The analog of P r o p o s i t i o n 2.1 i s

Proposition 3.1: Assuming t h a t Q i s regular on i t s zero s e t , ihere

i s an open and dense subset A of lRn such t h a t for f (0) e A, 0 i s a

regular value o f Q ( l , f ( O ) , - ) .

Proof: Replacing IRn by IRx lRn i n t h e proof of P r o p o s i t i o n 2.1, we

f i r s t deduce t h a t t h e s e t of p o i n t s ( t , a ) € lF.xIRn such t h a t 0 i s a

r e g u l a r va lue of Q ( t , a , . ) i s an open and dense s u b s e t of IRxlRn.

The same conclus ion thus a p p l i e s t o t h e s e t of p o i n t s ( t , a ) w i th

t # O . Due t o t h e homogeneity of Q , i t i s immediate t h a t 0 i s a

r e g u l a r va lue of Q ( t , a , - ) w i th t # O i f and only i f 0 i s a r e g u l a r

va lue of ~( l , : , . ) . I n p a r t i c u l a r , i t fo l lows t h a t t h e s u b s e t A of

p o i n t s of lRn such t h a t 0 i s a r e g u l a r va lue of Q ( l , a , . ) i s n o t

empty. Arguing a s i n P r o p o s i t i o n 2.1, t h e complement of A i s an

a l g e b r a i c s u b v a r i e t y of lRn which is n o t t h e whole space IRn from

t h e above r e s u l t . Thus, A i s open and dense and t h e proof is com- -

p l e t e . I - /

A s b e f o r e , choosing f ( 0 ) such t h a t 0 i s a r e g u l a r v a l u e of

Q ( l , f ( O ) , - ) imp l i e s t h a t t h e s o l u t i o n s of Q ( l , f ( O ) , v ) = 0 a r e i so -

l a t e d and f i n i t e , n o t exceeding kP i n number. Let t h i s f i n i t e num-

b e r of s o l u t i o n s be denoted by v and s e t Dow

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{vj }j=l,v = Iv E TRP; Q(l,f(O),v) = 01 . (3.12)

Note that the function h defined in (3.7) is continuous if and

only if the function f is continuous. Theorem 2.2 then translates

to

Theorem 3.2: Assume t h a t the function f i s continuous and t h a t the

k-form Q i s regular on i t s zero s e t . Let

I f 0 i s a regular value of Q(l,f(O),.), then i n the notat ion of

(3.12), there i s Ro > 0 such t h a t for any given l l j ~ v , there i s 1 a C so lu t ion (z,~,:) t o the system (3.10) - (3.11) i n [O,R ) .

Finally h is Lipschitz continuous w.r.t.(z,y) if and only if f

is Lipschitz continuous w.r.t.(x,y). Then from Theorem 2.3 follows

Theorem 3.3: I n addi t ion t o the asswnptions o f Theorem 3.2, sup-

pose t h a t the funetion f i n (3.1) i s Lipschi tz continuous wi th re-

spect t o (x,y) near the or ig in . Then there i s Ro > 0 such t h a t for 1 any given j , 1 2 j z v , there i s an unique C x CO soZution ( z , ~ ) t o

the system (3.1) - (3.2) in [o,R~), r igh t -d i f f e ren t iabZe a t the

o r i g i n , s a t i s f y i n g

mid no such so lu t ion e x i s t s t h a t does not s a t i s f y (3.13) for some

index 1 2 j 5 v. I n addi t ion, aZZ these so lu t ions ure automat-icalZy

cl.

Proof: Applying Theorem 2.3 to the system (3.5) - (3.6) yields a 1 0 C' solution defined on some interval [0,n0) unique among C x C

solutions right-differentiable at the origin satisfying

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(i(0) ,&o) ,y:(0)) = (1,f (0) ,vj) . Here, "c'" relates to the component 5 = (&,p) of the solution.

But since & ( p ) = p inany case, equivalence of the systems (3.1) - 1

(3.2) and (3.5) - (3.6) shows that is a C x CO solution to

(3.5) - (3.6) if and only if (%,y,) is a C x CO solution to (3.1) -

(3.2) and the proof is complete. IS/

Remark 3.2: Just as in Section 2, the hypotheses of Theorem 3.3

prove the existence of exactly v solutions to the system (3.1) - (3.2) defined on some interval (-Ro,Ro). Also, Remark 2.4 remains

valid with obvious modifications. 111

Finally, a simple transcription of Theorem 2.4 yields

Theorem 3.4: In add i t ion t o t h e asswnptions o f Theorem 3.3, sup-

pose t h a t

1 Then, every C x C' so lu t ion t o t h e system (3.1) - (3.2) def ined for

t - > 0 ( resp . t 5 0 ) small enough i s r i g h t ( resp . l e f t ) d i f f e r e n t i a b l e

a t t h e o r i g i n ( s o t h a t t h i s cond i t ion i s not r e s t r i c t i v e i n t h e

miqueness statement o f Theorem 3.3).

4. SOME SIMPLE EXAMPLES.

To illuminate the geometric content of the preceding theorems,

consider now the simple cases when p = 1, k = 2 and

(i) n = 1 and g is independent of t.

(ii) n = 1 and g is dependent of t.

(iii) n = 2 and g is independent of t.

Then g is a real valued C' mapping defined on a domain con-

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92 P. J. RABIER A N D S. SHANKAR

t a i n i n g t h e o r i g i n i n lR2 o r I R ~ and such t h a t Dg(0) = 0. As p = 1,

i t i s e a s i l y s een t h a t r e g u l a r i t y of t h e q u a d r a t i c form 2 Q(w) = D g(O).(w)' on i t s zero s e t i s equ iva l en t t o s ay ing t h a t 0

i s a nondegenera te c r i t i c a l p o i n t ( s i n c e t h e determinant of t h e

q u a d r a t i c form Q is t h e Hess ian of g a t 0 ) . The ze ro s e t - 1

Z ( Q ) = Q (0) t hus e s s e n t i a l l y depends on t h e e igenva lues o f Q a s

s t a t e d i n t h e Morse lemma.

( i ) The case when n = 1 and g i s independent of t .

If s o , Z(Q) reduces t o I O } o r is t h e union of two d i s t i n c t l i n e s

i n t e r s e c t i n g a t t h e o r i g i n . As f ( 0 ) # 0 i s i m p l i c i t i n t h i s s tudy

( c f . Remark 2 .1) , only t h e ca se when Z(Q) # I01 w i l l be considered .

Regardless of t h e l o c a t i o n of t h e two l i n e s t h a t Z(Q) is made o f ,

i t i s c l e a r t h a t nZ(Q) = IR ( s ee F igu re 4 .1 ; r e c a l l t h a t 7 denotes

t h e p r o j e c t i o n onto t h e u-axis i n t h e p r e s e n t s i t u a t i o n ) . Let

f (0) € IR- { O } be given. For v € IR,

and hence

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SINGULARLY PERTURBED SYSTEMS 93 2 2

Assume first that (a g/ay )(O) # 0. Since 0 is a nondegenerate

critical point, the equation Q(f(O),v) = 0 has two distinct solu-

tions. Both these solutions are such that the derivative 2 2

DvQ(f(0),v) is nonzero. On the other hand, if ( a g/ay )(O) = 0,

the condition that 0 is a nondegenerate critical point ensures that

(a2g/axay) (0) # 0. Then the equation Q(f (O),v) = 0 has exactly one

solution for which DvQ(f(0),v) # 0. It follows that 0 is a regular

value of Q(f(O),.) in both cases (recall p=l) and the singular

system (2.1) - (2.2) possesses either two distinct c1 solutions or exactly one by Theorems 2.2 and 2.3 (assuming that f is Lipschitz

continuous w.r.t. (x,y)). Note that the case when there is exactly

one solution is when one of the two lines in Z(Q) coincides with

the vertical axis. In any case, since n = 1, the condition

g(x,y) = 0 shows that each solution (g(t),x(t)) of the system (1.3)

must coincide with one of the two curves in the zero set of g. But

the parametrization depends of course on f. An interpretation of

such solutions regarding the singularly perturbed system (1.1) -

(1.2)E, using arguments of nonstandard analysis can be found in 2 2

Diener [6 1. Theorem 2.4 applies if and only if (a g/ ay ) (0) # 0.

Remark 4.1: More generally, a similar study can be made for arbi-

trary k12. Again, the zero set Z(Q) reduces to iOi or is the

union of a finite number (zk) of lines through the origin. Pro-

vided none of these lines is chosen as the "line at infinity",

assuming that Q is regular on its zero set amounts to assuming that

all the real roots of the corresponding dehomogeneization of Q are

simple. If so, 0 is a regular value of Q(f(O),.) as soon as

f(0) # 0. The verification of these statements is asimpleexercise.

Note if k is odd - so there is at least one line in Z(Q) - that nonexistence may nevertheless occur, namely when Z(Q) reduces to

this one line which in addition coincides with the v-axis. 111

(ii) The case when n = 1 and g is dependent on t.

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Examination of this case will bring to prominence some striking

differences with the previous situation when g was independent of 3

t. Here Q is a function of the variable w = (T,u,v) € IR . The

zero set Z(Q) reduces to the origin or is a cone with its apex at

the origin (see Figure 4.2). Of course, the axis of the cone which

determines its location in space can be any line through the origin 3 1 of IR . If Z(Q) = { O ) , nonexistence of C solutions is ensured by

Theorem 3.1. Now consider the case when Z(Q) is a cone. Clearly,

Z(Q) separates the space into two open regions. The region that is

the union of two convex components will be referred to as the 'in-

side" of Z(Q) while the other open region will be called the 'out-

side' of Z (Q) .

According to the analysis of Section 3, a necessary condition for

existence of c1 solutions is that (1,f (0)) € nZ(Q), where n denotes

the projection onto the (T,u)-plane (Theorem 3.1). If in addition

0 is a regular value of Q(l,f(O),-), then existence of c1 solutions is ensured by Theorem 3.2. As Theorem 3.2 is just a transcription

of Theorem 2.2, a necessary condition for this is that (l,f(O)) be-

long to the interior of nZ(Q). In the simple context under con-

sideration, this is also a sufficient condition. A geometrical

proof of this statement is immediate by observing that when (l,f(O))

lies in the interior of TZ(Q), the tangent space to Z(Q) at each

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p o i n t of i n t e r s e c t i o n wi th t h e l i n e p a r a l l e l t o t h e v-axis and

pas s ing through ( l , f ( O ) ) i s t r a n s v e r s a l t o t h e v-axis . It i s l e f t

t o t h e r e a d e r t o g i v e an a n a l y t i c proof of t h i s a s s e r t i o n (based on

t h e hypo thes i s t h a t Q i s r e g u l a r on i t s ze ro s e t ) . Thus, t h e con-

d i t i o n t h a t ( l , f ( O ) ) be longs t o t h e i n t e r i o r of nZ(Q) l e a d s t o an

examination of t h e p r o j e c t i o n nZ(Q). I t i s immediately s een t h a t

t h e fo l lowing t h r e e ca ses ( r ep re sen ted i n F igu re 4.3 below) must be

d i s t i n g u i s h e d :

(a) The v-axis l i e s i n s i d e Z(Q) s o t h a t t h e p r o j e c t i o n nZ(Q)

i s t h e whole ( T , u ) - p l a n e . I n t h i s ca se e x a c t l y two p o i n t s of Z(Q)

a r e mapped t o ( l , f ( O ) ) by T .

( b ) The v-axis l i e s on Z(Q), so t h a t t h e p r o j e c t i o n nZ(Q) is

the whole ( T , u ) - p l a n e minus A - 10) where A i s t h e i n t e r s e c t i o n of

t h e ( T , u ) - p l a n e wi th t h e p l a n e t angen t t o Z(Q) a long t h e v-axis .

Thus t h e i n t e r i o r of nZ(Q) i s nZ(Q)- iO}, and t h e cond i t i on t h a t

( l , f ( O ) ) be longs t o t h e i n t e r i o r of nZ(Q) amounts t o s ay ing t h a t

( l , f ( O ) ) 6 A. Besides t h e r e i s e x a c t l y one p o i n t of Z(Q) t h a t i s

mapped t o ( l , f ( O ) ) by n.

( c ) The v-axis l i e s o u t s i d e Z(Q) s o t h a t t h e p r o j e c t i o n nZ(Q)

i s t h e p a r t of t h e ( T , u ) - p l a n e conta ined between two d i s t i n c t

l i n e s i n t e r s e c t i n g a t t h e o r i g i n . For ( l , f ( O ) ) i n t h e i n t e r i o r of

nZ(Q), t h e r e a r e e x a c t l y two p o i n t s of Z(Q) t h a t a r e mapped t o i t

by n.

I f f i s L i p s c h i t z cont inuous w . r . t . ( x , y ) , i t t hen fo l lows 1

from Theorem 3.3 t h a t t h e number of C s o l u t i o n s t o t h e problem i s :

- Exact ly two f o r every f ( 0 ) € W i n t h e s i t u a t i o n of

Figure 4.3 ( a ) .

- Exact ly one f o r eve ry f ( O ) ' € W - {a?, where a i s t h e compo-

nen t a long t h e u-axis of t h e i n t e r s e c t i o n of t h e l i n e A w i th t h e

l i n e T = 1 i n t h e ( T , u ) - p l a n e . This i s i n t h e s i t u a t i o n of Figure

4 .3(b) (no conc lus ion can be drawn from t h e p r e s e n t s tudy i f

f ( 0 ) = a ) .

- Exact ly two when ( l , f ( O ) ) l i e s i n t h e i n t e r i o r of nZ(Q) and

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no solution if f(0) 6 nZ(Q) in the situation of Figure 4.3(c)

(again no conclusion can be drawn from the present study if (l,f(O))

lies on the boundary of nZ(Q)). Theorem 3.4 applies if and only if 2 2 ( a g/ay )(O)#O, i.e. in the situation ofFigure4.3(a) and (c) only.

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Note t h a t t h e cond i t i on t h a t ( l , f ( O ) ) b e i n t h e i n t e r i o r o f .irZ(Q) c a n b e

t r a n s l a t e d i n t o a c o n d i t i o n on f ( 0 ) a lone , depending on t h e loca-

t i o n of t h e u-axis w i th r e s p e c t t o nZ(Q) ( s e e F igu re 4.4; a and 6

denote t h e components a long t h e u-axis of t h e i n t e r s e c t i o n of t h e

l i n e T = 1 wi th t h e boundary of nZ(Q)). I n t h e ca se of F igu re

4 .4(a) t h e cond i t i on t h a t ( l , f ( O ) ) be longs t o t h e i n t e r i o r of nZ(Q)

( r e s p . ( 1 , f ( 0 ) ) 6 nZ(Q)) i s e q u i v a l e n t t o s ay ing t h a t

f (0) f (-m,n)U(B,fm) ( r e s p . f (0) f ( a ,B) ) . I t i s e x a c t l y t h e oppo-

s i t e i n t h e ca se of F igu re 4 .4 (b ) . .

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( i i i ) The case when n = 2 and g i s independent of t . 3

A s Q i s a q u a d r a t i c form on IR , t h e d e s c r i p t i o n of t h e s t r u c t u r e

of t h e zero s e t Z(Q) i s a s i n ca se ( i i ) , bu t now u i s a two dimen-

s i o n a l v a r i a b l e and t h e u-plane p l a y s t h e r o l e of t h e ( r , u ) - p l a n e ,

and n then deno te s t h e p r o j e c t i o n on to t h e u - p l a n e a long t h e

v-axis . The case when Z(Q) = IO} being obvious (no s o l u t i o n i f

f ( 0 ) # 0, t h i s cond i t i on is i m p l i c i t l y r equ i r ed h e r e ) , only t h e

ca se when Z(Q) is a cone needs t o be examined. Again, t h e t h r e e

p o s s i b l e l o c a t i o n s of t h e v-axis w i th r e s p e c t t o Z(Q) a s repre-

s en t ed i n F igu re 4.3 must be d i s t i n g u i s h e d . Nondegeneracy i n t h e

s ense of D e f i n i t i o n 2 . 1 is ensured when f ( 0 ) i s i n t h e i n t e r i o r of

nZ(Q). On assuming t h a t t h e mapping f i s L i p s c h i t z cont inuous

w . r . t . ( x ,y ) t h e conclus ions a r e a s fo l lows : 2 - Exis tence of e x a c t l y two C' s o l u t i o n s f o r f ( 0 ) e IR - 101 i n

t h e ca se ( a ) of Figure 4.3.

- Exact ly one C' s o l u t i o n f o r f (0) A i n t h e ca se of F igu re

4 .3(b) .

- Exact ly two C' s o l u t i o n s when f (0) i s i n t h e i n t e r i o r of

nZ(Q) and no c1 s o l u t i o n when f (0) 6 n Z ( Q ) i n t h e ca se of Figure

4 .3 (c ) . 2 2

Again, Theorem 2 .4 a p p l i e s i f and only i f (a g /ay )(O) # 0 (F igu re

4 .3(a) and ( c ) ) .

Acknowledgements.

We would l i k e t o thank D r . Adimurthy whose p e r c e p t i v e comments l e d

t o s i g n i f i c a n t improvements i n t h e p r e s e n t a t i o n of t h i s paper .

D r . R. Haberman is g r a t e f u l l y acknowledged f o r providing u s w i th

va luab le i n fo rma t ion on t h e c u r r e n t developments i n s i n g u l a r l y per-

turbed problems. F i n a l l y , we wish t o thank an anonymous and know-

l edgeab le r e f e r e e who po in t ed o u t t o us both Diene r ' s work i n non-

s t anda rd a n a l y s i s and L i and a l . ' s r e s u l t s sharpening Bezout ' s

theorem.

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