Constrained equations with impasse points

Constrained Equations with Impasse Points

Ana M. Guzman-Gomez

_´

_´

Departamento de Matematicas, Facultad de Ciencias, Uni´ ¨ersidad Nacional Autonoma´

de Mexico, Mexico, D.F., 04510, Mexico´ ´ ´

Submitted by Hal L. Smith

Received May 13, 1996

We give local normal forms for generic constrained equations of the form Ž . Ž .

f z z˙sH z where f is a smooth real function defined on a manifold M and H is

a smooth vector field on M. Q1997 Academic Press

1. INTRODUCTION

In this paper we will study constrained equations of the form

f z z

Ž .

˙

sH z ,

Ž .

Ž

1.1

.

Ž `. m

where f is a smooth C real function onR and H is a smooth vector field onRm_{. Here all functions, fields, manifolds, maps, etc., will be C}`_.

w x Ž .

In 1 , Rabier posed the problem of studying equations of the form 1.1 , because they can be useful to understand explicit systems of differential equations. In the same paper he described qualitative properties of the

Ž .

solutions of 1.1 , in a neighborhood of certain points contained in the zero

Ž Ž . Ž . Ž . .

set of f those satisfying f z s0 and df z H z /0 .

Ž .

Equation 1.1 is equivalent to a particular case of an equation of the form

A z z

Ž .

˙

sH z ,

Ž .

Ž

1.2

.

Ž . m w x Ž .

where A z is a linear map ofR into itself. In 2 normal forms for 1.2 are given, in the case when ms2.

w x

In this context, our results extend those given in 1 in the sense that

Ž .

they contain generic cases of equations of the form 1.1 , giving normal forms for their germs around a zero of f.

292

Other kind of constrained equations are those of the form

F x , y

Ž

.

s0

1.3

Ž

.

xsG x , y ,

Ž

.

˙

where F :Rm_=Rn_ªRn _{and G :R}m_=Rn_ªRm_.

The relation between this kind of constrained equations and those of

Ž . m Ž .

the form 1.1 is that in the tangent space ofR , 1.1 is of the form f z p

Ž .

yH z

Ž .

s0

zsp.

˙

w x Ž . Ž .

In 3 Takens studies Eq. 1.3 assuming that Fs ­Ur­y , . . . ,1 ­Ur­yn and some properties on the potential U. He defines in an adequate way discontinuous solutions and gives a general existence theorem; he also gives local normal forms for some low-dimensional cases.

Nevertheless, these results cannot be applied to our problem because w x y1_{Ž . Ž} n_.

one of the basic hypotheses in 3 is that F 0 l K=R is compact if K is a compact subset ofRm_{. In our case this is satisfied, locally, just in}

Ž Ž . . Ž Ž . Ž . .

trivial cases f z₀ /0 or on not generic ones f z₀ s0 and H z₀ s0 . The present paper is organized in the following form. In Section 2 we give some examples and definitions. In particular we give criteria about

Ž .

when two equations of the form 1.1 are equivalent; for this purpose we follow two approaches, the first takes into account the solutions, and the second just considers their images and the zero set of f.

In Section 3 we state our main results, which are applied generically,

m _{Ž .}

taking the product fine C topology for pairs f, H defined on a m-manifold.

Theorems 2 and 3 give, for each equivalence relation, normal forms for

Ž .

the germs, at the zero set of f, of pairs f, H . In Proposition 1 we describe the invariant geometric properties of them.

Finally, in Section 4 we give the proofs, using Lemma 4 which is an

Ž .

analytic characterization of germs of pairs f, H that are equivalent to each normal form.

I thank Santiago Lopez de Medrano and Jesus Lopez Estrada for their

_´

_´

comments and fruitful discussions and the ‘‘Laboratorio de visualizacion

_´

matematica’’ for obtaining graphical representations of the dynamics of

_´

some equations.

2. DEFINITIONS AND EXAMPLES

Ž . `Ž m. Ž

DEFINITION1. A solution of 1.1 is a mapping zgC I,R where I

. Ž Ž .. Ž . Ž Ž ..

Ž

graph of a solution, and a phase cur¨e is the image of a solution as a m_.

subset ofR .

Ž . Ž .

DEFINITION 2. z is an impasse point of 1.1 if f z s0 and the

y1_{Ž .} impasse set is f 0 .

Remark. In a generic case, f and H do not take the value 0 at the same point, so the solutions cannot take the impasse points as values. Nevertheless, for each impasse point z , there exist solutions z, such that0

Ž . Ž .

lim z t sz . In fact the phase curves of 1.1 are the integral curves of H0

tªt0

excluding the impasse set.

It is also a generic property that in a neighborhood of an impasse point

Ž .

f is a submersion; so locally, the set of impasse points is a my1 -submanifold ofRm_.

It follows from these observations that, generically, in a neighborhood of an impasse point, the family of phase curves together with the impasse set

m _{Ž .}

is diffeomorphic to a family of parallel lines in R with a my1 -submanifold ofRm_.

EXAMPLE1.

x

˙

1

y s

_{ž /}

. x

ž /

˙

y

Using the diffeomorphism

1 1

2

C

Ž

x , y

.

s

ž

x , yy x

/

'

2 2

one gets that C sends the impasse set and the phase curves to the respective curves of the equation

x

˙

1

2

x qy s .

Ž

.

ž /

_y

ž /

0

˙

In both cases the closure of the phase curve that passes through 0 is tangent to the impasse set and the others do not touch it or are transversal

Ž .

to it. See Fig. 1 .

EXAMPLE2.

x x

˙

1

˙

1

x

ž /

s

_{ž /}

yx

ž /

s

_{ž /}

.

y 0 y 0

FIG. 1. The impasse sets and the phase curves of two equivalent constrained equations.

In these examples, the impasse set and the phase curves of each equation coincide, nevertheless their dynamics are different: in the first case the solutions move away from the impasse set and in the second they

Ž .

get closer to it Fig. 2 .

EXAMPLE3.

x

˙

1

3 _y

x qxyqz

˙

s .

Ž

.

0

ž /

0

0 z

˙

Here the closure of each phase curve is an horizontal line, the one that passes through 0 has a 3rd order-contact with the impasse set, S, while the

Ž 1

others have 2nd order-contact if they pass through a certain curve, S ,

.

contained in the impasse set ; almost every line is transversal to the

Ž .

impasse set Fig. 3 .

We will give two equivalence relations for germs of equations of the

Ž .

form 1.1 . The first takes into account the trajectories of the equations, while the second considers their phase curves.

FIG. 3. The phase curves can touch the impasse set with different contact orders.

ˆ ˆ

Ž . Ž .

DEFINITION 3. We will say that the two triples f, H, z and f, H, z

ˆ

ˆ ˆ

wŽ . Ž .x

are path-equi_¨alent f, H, z f f, H, z

_ˆ

if there exist a diffeomorphism

Ž m . Ž m .

C: R , z ª R , z such that its germ at z satisfies

_ˆ

ˆ

y1

_ˆ

f C#Hs

Ž

f(C

.

H ,

Ž

2.1

.

where C#H is the vector field ofRm _{induced from H by the differential} of C.

ˆ ˆ

Ž . Ž .

DEFINITION4. We will say that f, H, z and f, H, z are phase-equi

ˆ

¨

-ˆ -ˆ

wŽ . Ž .x

alent f, H, z ; f, H, z

ˆ

if their germs satisfy

ˆ

C#HslH

y1

_ˆ

f(C saf ,

Ž m . Ž m .

where C: R , z ª R , z is a diffeomorphism and

ˆ

l and a are two real smooth functions onRm _{that do not vanish at z.}

_ˆ

Ž .

Remark. With generic hypothesis H/0 at the impasse points ,

path-Ž .

equivalence means that, in a neighborhood of z , t0 ªz t is a solution of

ˆ

ˆ

Ž . Ž . Ž Ž .. Ž . Ž .

f z z

˙

sH z if and only if tªC z t is a solution of f z z

˙

sH z

Žnotice that in the complement of the impasse set, Eq. 2.1 can be writtenŽ .

ˆ ˆ

ˆ

Ž . Ž ..

as C# Hrf s Hrf , while if 0 is a regular value of f and f, phase-equivalence means that, locally, C sends the integral curves of H to the

ˆ

ˆ

EXAMPLE 4. The two equations mentioned on Example 1 are

phase-equivalent, and in spite of the fact that C does not satisfy Definition 3,

˜

y1r3 y2r3 2

Ž Ž . Ž

they are also path-equivalent. Take C x, y s 2 x,y2 x q 1r3 _{. .}

2 y . The equations on Example 2 are phase-equivalent but they are not path-equivalent.

3. RESULTS

Our main results are Theorems 2 and 3 given below; we give normal

Ž .

forms for the germs at impasse points of generic pairs f, H .

Ž .

In the case of phase equivalence, the normal forms are f ,_k ­r­z , 0 ,₁ where kFm and

f z , . . . , z szk_qzky2_{z qz}ky1_{z q???qz z qz .}

Ž

.

k 1 m 1 1 2 1 3 1 ky1 k

Ž .

DEFINITION 5. We will say that z is a k-regular impasse point of f, H

Ž . Ž .

if f, H, z ; f ,_k ­r­z , 0 , and that z is a regular impasse point if it is a₁ k-regular impasse point for some kFm.

The following proposition gives a description of the geometry of the

Ž .

pairs f, H in a neighborhood of a k-regular impasse point. Here the sets Sl_{, defined below, play an important role}

S0sfy1

Ž .

0

Sl_{s zgS}ly1_{NH z gT S}ly1 _.

Ž .

z

4

1 ₄ l

EXAMPLE5. In Example 1, S s 0 and for l)1, S sf; in Example

2, Slsf for l)0 and in Example 3, S1is the curve where the closures of 2

4

the phase curves have contact of order 2 with the impasse set, S s 0 and Slsf for l)2.

PROPOSITION1. If z is a k-regular impasse point, then in a neighborhood

of z:

Ž .a For l-k, S is a submanifold ofl Rm of codimension lq1 and S0_>S1_{> ??? >S}ky1_.

Ž .b For lGk, Slsf.

Ž .c E¨ery impasse point is regular and the set of l-regular impasse points is Sly1_{_S}l_.

Ž .

The next theorem guarantees that the pairs f, H satisfying that all of its impasse points are regular are generic. This is true if we take the

Ž .

THEOREM 2. Let M be a paracompact manifold and L be the subset of

`<sub>Ž .</sub> `_{Ž .}1 _{Ž .}

C M,R =X M made of the pairs f, H that satisfy that e¨ery impasse point is a regular impasse point.

L is an open and dense set with Whitney’s Cm _{fine topology.}

In general, phase-equivalence does not imply path-equivalence, but in the case of regular impasse points these two properties are almost the same, except that one has to consider the directions in which the solutions of the equations run through its phase curves. More precisely we have the next theorem

Ž .

THEOREM3. Let z be a k-regular impasse point of f, H .

If k is odd, then

­ ­

f , H , z f f , , 0 or f , H , z f yf , , 0

Ž

.

ž

k

/

Ž

.

ž

k

/

­z1 ­z1

and if k is e_¨en, then

­

f , H , z f f , , 0 .

Ž

.

ž

k

/

­z1

4. PROOFS

Before starting, we will define m operators which will be useful to

Ž .

characterize the k-regular impasse points of a pair f, H . This characteri-zation will be stated on Lemma 4.

DEFINITION 6. For each kFm, let S be the operatork

S : Ck `

Ž

Rm,R

.

=X`

Ž

M

.

ªC`

Ž

Rm,Rk

.

S f , H s f , D f , . . . , Dky1_{f ,}

Ž

.

Ž

.

k H H

where Di _{f denotes the ith derivative of f in the direction of H, i.e.,} H

Ž . Ž . Ž . i Ž . Ž iy1 .Ž . Ž .

D f zH sdf z H z and D f zH sd DH f z H z .

Remark. The operators S are independent of the coordinate system ink the sense that for any diffeomorphism w,

S f , H (wy1_{sS f(w}y1_,w#H

Ž

.

Ž

.

k k

so these operators are well defined if instead ofRm _{we take any manifold.}

1 _`

Ž .

Ž .

LEMMA 4. z is a k-regular impasse point of f, H if and only if the

following three conditions hold:

Sk

Ž

f , H

. Ž .

z s0

Dk_{f z /0 4.1}

Ž .

Ž

.

H

Sk

Ž

f , H

.

is a submersion at z.

To prove Lemma 4, we will use the two following lemmas.

m _{Ž .}

LEMMA5. Let f, g, and a be real functions onR , with a 0 /0, and

Ž m . Ž m . Ž . Ž .

C: R , 0 ª R , 0 is a diffeomorphism such thatC# ­r­x sl ­r­x ,

Ž .

wherel0 /0.

Ž .a If gsaf thenk

Ž .i g 0Ž .s Ž­gr­x 0.Ž . s ??? s Ž­ky1gr­xky1.Ž .0 s 0 and

Ž­kgr­xk.Ž .0 /0

Ž .ii S g,_kŽ ­r­x is a submersion at 0..

Ž .b If fsg(C then f satisfies properties i and ii if and only if gŽ . Ž .

satisfies them.

Ž . Ž .

Proof. a Let gsaf ; clearly, i is valid. The Jacobian matrix ofk

Ž .

S g,k ­r­x at 0 is of the form

0 ??? ..0 0!a

Ž .

0 0 ??? 0

. . .

. ? ) . ??? .

. . .

0

Ž

ky2 !

.

a

Ž .

0 0.. ) 0 ??? 0

k!a

Ž .

0 ) ??? ) 0 ??? 0

0

Ž .

so S g,_k ­r­x is a submersion at 0.

Ž .b Let fsg(C and l such that C#Ž­r­x.sl ­Ž r­x . We have.

that

l l ly1 i

­ f ­ l ­ g

l

s

_Ý

B_i l, . . . , (C, l

ž

ly1

/

i

­x _is1 ­x ­x

l l_{Ž .} l

where each B is an homogenous polynomial and B_{i l} l₁, . . . ,l_l 'l₁.

Ž .

Using this identity and the fact that l0 /0 one gets the first part of the statement.

Ž .

One also gets that Sk f,­r­x has the form

­ ­

Sk

ž

f ,

/

sSk

ž

g ,

/

(C M ,

where M is the triangular matrix

1 0 0 ??? 0

1 2 ky1

0 B1 B1 ??? B1

2 ky1

0 0 B2 ??? B2

Ms .

. . .

. . .

. . .

_{0 0 0 ??? B}ky1

0

ky1

Ž .Ž .

Differentiating at zero and using that S g,_k ­r­x 0 s0 we get

­ t ­

D S

ž

k

ž

f ,

/

/

Ž .

0 sM 0 D S

Ž .

ž

k

ž

g ,

/

(C

/

Ž .

0 ,

­x ­x

Ž .

where D? is the Jacobian matrix of the respective map.

Ž . Ž .

M 0 is not singular, since l0 /0 and C is a diffeomorphism, so

Ž . Ž .

S g,_k ­r­x is a submersion at 0, if and only if S_k f,­r­x is also a submersion.

Ž .

LEMMA 6. Let w: R, 0 ªR be a function of order k at 0 and f a

deformation ofw.

Ž .

f is r-¨ersal if and only if Sk f,­r­x is a submersion at 0.

Proof. A deformation is versal if and only if it is infinitesimally versal.

ŽSee 4 . In the case of right-equivalences, this means that each germw x. a, can be represented as

˙

asw

˙

hqSc f ,i i

˙

where h is a function germ and each c is a real number, and f_{i i}s

Ž­fr­m ?_i.Ž, 0 ..

In a neighborhood of 0 we can write

˙

ky1

f xi

Ž

.

sP xi

Ž

.

qx r xi

Ž

.

a x sP x qxky1_{r x}

Ž

.

Ž

.

Ž

.

w

˙

Ž

x

.

sxky1w1

Ž

x ,

.

where P and P are polynomials of degree not greater than k_i y2. Using this notation, we have that f is r-versal if and only if for each P and r there exist real numbers, c , and a function h such that in a_i neighborhood of 0,

4

For every collection c , the second equation is always true because_i

Ž . 4

w₁0 /0, so the r-versality of f is equivalent to the fact that P_i generates the space of polynomials of degree not greater than ky2, which means that the matrix

­f ­f

0 ??? 0

Ž .

Ž .

­m1 ­mm

. .

. .

. .

ky1 ky1

­ f ­ f

_ky2

Ž .

0 ??? _ky2

Ž .

0

0

­m ­1 x ­m ­m x

must have rank ky1.

Ž .

This means that Sk f,­r­x is a submersion at 0 because w is of order k at 0.

Proof of Lemma 4. Without loss of generality zs0 and Hs­r­x. This is because the operator S and the concept of a k-regular impassek point do not depend on the coordinate system.

If 0 is a k-regular impasse point, in a neighborhood of it, f(Cy1saf ,k fora andC satisfying the hypothesis of Lemma 5, using this fact, we have

Ž . Ž .

that the triple f,­r­x, 0 satisfies 4.1 .

Ž .

To see that 4.1 are sufficient conditions for 0 being a k-regular impasse point, we will use the fact that two r-versal deformations with the

ˆ

same number of parameters, say mq1, f, and f, of two r-equivalent germs satisfy

ˆ

fsf(C for a diffeomorphismC having the form

C

Ž

x ,m

.

s

Ž

j

Ž

x ,m

.

,w m

Ž

.

.

,

Ž my1 . Ž my1 . Ž m . Ž

where j: R=R , 0 ªRand w: R , 0 ª R , 0 are smooth. In w x4 is given a proof in the case of V-versatility, in our case the proof is

.

analogous.

Ž . Ž . Ž .

If f,­r­x, 0 satisfies 4.1 then f ?, 0 has a zero of order k at 0, so is r-equivalent to xk _{or yx}k_{. Also, by looking at the germ of f as a} deformation, Lemma 6 guarantees that f is r-versal. Using the result given

Ž . Ž .Ž .

above, we have that fs"fk(C where C# ­r­x s ­jr­x ­r­x , so 0 is k-regular impasse point.

Ž .

Proof of Proposition 1. Take a neighborhood of z such that Sk f, H is

Ž .

In this situation, we can show inductively that

y1 l

S s Slq1

Ž

f , H

.

Ž .

0 .

Ž

4.2

.

Ž .

From this follows a .

Intersecting with another neighborhood such that Dk_{f has no zeros on} H

it, we get Sk_{sf and every impasse point is l-regular for some lFk, so}

Ž . Ž .

using 4.2 we get c .

To prove Theorem 2, we will use the following

`_Ž my1

.

LEMMA7. Let V be the set of functions ggC R=R ,R such that

Ž . y1Ž .

the germ of g at any point x ,₀ m₀ gg 0 is a r-_¨ersal deformation of

Ž .

g ?,m₀ .

`_Ž m _. m

V is an open and dense subset of C R ,R with the Whitney fine C topology.

Proof. For kFm let V_k be the set of functions g such that 0 is a

Ž .

regular value of S g,_k ­r­x , and let V_mq1 be the set whose elements are

Ž .

the functions g such that S_mq1 g,­r­x is always different from 0. Using Lemma 6, one gets that

mq1

Vs

F

V ._k

ks1

Each V is an open set; we do not prove it because it is a particular case_k of the next theorem.

To show that V is dense we will use the Thom Transversality Theorem_k

Žsee 4 .w x.

m_Ž m _. m _{Ž .} Take coordinates of the space of m jets J R ,R such that j f z is represented by

z , . . . , z , p0_{, . . . , p}l _{, . . . , p}m _,

Ž

1 m Ži , . . . ,i1 m. Ž0 , . . . ,0 , m.

.

l _Ž l i1 im_{.Ž .}

where p_{Ži , . . . ,i .}s ­ fr­z₁ ??? ­z_m z and lsi₁q???qi ._m 1 m

4 mŽ m .

For each kg 1, . . . , m let C be the submanifold of J_k R ,R given by equations

p0_{s0, p}1 _{s0, . . . , p}ky1 _s0

Ž1 , 0 , . . . ,0. Žky1 , 0 , . . . ,0.

and Cmq1 given by

p1 _{s0, . . . , p}m _s0.

Ž1 , 0 , . . . ,0. Žm , 0 , . . . ,0.

Proof of Theorem 2. Openness. We claim that L is a finite intersection

m `<sub>Ž .</sub> `_{Ž .}

of open sets with the C fine topology of C M,R =X M . Using Lemma 4 we get that

mq1

Ls

F

L_k,

ks1

Ž . Ž .

where Lmq1 is the set of pairs f, H such that Smq1 f, H has no zeros,

4 Ž .

and for kg 1, . . . , m the elements ofL_k are the pairs f, H such that 0

Ž .

is a regular value of S_k f, H .

Using that M is paracompact, one gets that M is the locally finite union

4

of a family of compact sets Q_a such that each Q_a is contained in a coordinate system.

Ž .

L_k is open if and only if the restriction of each pair f, H gL_k to Q_a

m _{Ž .}

forms an open set with the C topology for pairs f, H defined on Q ._a For kFm, this set is open because it is the inverse image of the set of functions greater than zero under the continuous operator

O

Ok: C`

Ž

Q ,a R

.

=X`

Ž

Qa

.

ªC`

Ž

Q ,a R

.

2

ky1 _­

_Ž

_s

_Ž

_{f , H}

_.

_.

2 k

i

f , H ¬ D f q ,

Ž

.

Ý

Ž

H

.

Ý

ž

/

­

Ž

z , . . . , zi i

.

is0 1 k

where

­

Ž

G , . . . , G1 k

.

­Gl

sdet

ž

/

.

­

Ž

z , . . . , zi₁ i_k

.

­zi_{j lg1 , . . . ,k , j4 g1 , . . . ,k4} For ksmq1, the proof is analogous.

Density. First we will prove that if M is a manifold that admits vector fields with no zeroes and H is one of those vector fields, then the set

BM_s

_fgC`

_Ž

_{M ,R}

_.

_N

_Ž

_{f , H}

_.

_gL

₄

H

`_{Ž .} is dense in C M,R.

For each zgM, take a chart w of M at z defined on a neighborhood V such that w#Hs­r­x.

`_{Ž .} V y1

Notice that fgC V,R is in B_H if and only if the germ of f(w at

Ž . Ž y1.y1Ž .

each point x ,₀ m₀ of f(w 0 is a r-versal deformation of

y1_{Ž .} f(w ?,m₀ .

Consider the operator

O

O: C`

Ž

V ,R

.

ªC`

Ž

w

Ž

V ,

.

R

.

Ž

This operator is an homeomorphism with the Whitney fine topology see w x5 , so it preserves density. It follows from Lemma 7 that B. V_H is a dense

`_{Ž .} subset of C V,R.

With the usual techniques and using the fact that BM _{is open, we can} H

extend this result to M. To do this, first check that for every compact subset K of M, the functions whose restrictions to K are in BK _{form a}

H

`_{Ž .} M

dense subset of C M,R. This implies that BH is dense.

Now we will prove that for any manifold M,L is dense. Take NN and MM

`<sub>Ž .</sub> `_{Ž .}

open sets of C M,R and X M , respectively. Let HgMM be a vector

4

field on M with isolated zeros given by z , and f a function in_i NN such

Ž .

that f z_i /0.

4

Let MsM_ z . Since H has no zeros on M, we can take g_i g

`_{Ž .} M <

C M,R lBH arbitrarily close to f M, we can suppose also that z is noti

y1_{Ž .} an accumulation point of g 0 .

M

Notice that the condition that ggB_H just depends on the germs of g

y1_{Ž . 4}

at the points in g 0 , so we can perturb g in a family V of disjoint_i

y1_{Ž .}

neighborhoods of each z , that does not intersect g_i 0 . This perturbation

˜

can be chosen in such a way that it can be extended to a function f on M which is in NN and has no zeroes in DV . Clearly, in this situationi

˜

M

fgBH lNN.

The technique to carry this out is to choose, for each i, a function `_{Ž w x.}

ligC M, 0, 1 with support contained in V , which is equal to 1 in ai neighborhood of z , and takei

˜

fsSlifq

Ž

1yli

.

g .

˜

y1_{Ž .} y1_{Ž .} The fact that g is sufficiently close to f guarantees that f 0 sg 0 ,

˜

Ž . Ž .

so f, H g NN=MM lL.

Proof of Theorem 3. Before starting the proof, we will give a

characteri-m _{Ž . Ž}

_˜

_.

zation of the fact that two triples onR , f,­r­x, 0 and f,­r­x, 0 , are path-equivalent.

Let

x

tf

Ž

x , y

.

s

_H

f s, y ds,

Ž

.

0

where xgRand ygRmy1

Ž .

t can be interpreted as the time that takes the point 0, y to go to the_f

Ž . Ž .Ž . Ž

point x, y by the flow of the vector field 1rf ­r­x at least if f does

.

not change sign .

˜

Ž . Ž .

It is obtained immediately that f,­r­x, 0 f f,­r­x, 0 if and only if, in a neighborhood of 0,

Ž m . Ž m . Ž .

where C: R , 0 ª R , 0 is a diffeomorphism such that C# ­r­x s

Ž . my1

l ­r­x and h is a function defined onR . With this observation, we will begin the proof.

Ž .

Let z be a k-regular impasse point of f, H . Without loss of generality, we can suppose that MsRm_{, zs0, Hs­r­x and fsaf for a}

k

Ž .

function a with a 0 /0.

Let us denote by T the function ofRmq1 _{defined by} T x , y, s

Ž

.

st_af_k

Ž

x , y

.

qs.

Ž . Ž .

T satisfies that T ?, 0, 0 has a zero of order k at 0 and Skq1T,­r­x is a submersion at 0. To prove the last fact, observe that

Dt_afk

Ž .

0 1

­

­

D S

ž

kq1

ž

T , _­

/

/

Ž .

0 s

x DSk

ž

af ,k

/

Ž .

0 0

­x

Ž .

and we know, by Lemma 5, that Sk af ,k ­r­x is a submersion at 0, so applying Lemma 4 we have that the germ of T is a r-versal deformation of

Ž . kq1 kq1Ž

T ?, 0, 0 which is r-equivalent to x oryx depending on the sign of

Ž ..

a 0 .

The germ at 0 of the function

Tk

Ž

x , y, s

.

stf_k

Ž

x , y

.

qs kq1 _{Ž .}

is also a r-versal deformation of x r kq1 , so there exists a

diffeomor-Ž mq1 . Ž mq1 . Ž . Ž .

phism C: R , 0 ª R , 0 such that C# ­r­x sl ­r­x , and locally

˜

˜

TksT(C or yTksT(C

Ž

4.3

.

Ž .

depending on the sign of a 0 . Now, let C be defined by

˜

˜

˜

C

Ž

x , y

.

s

Ž

C1

Ž

x , y, 0 ,

.

C2

Ž

y, 0 , . . . ,

.

Cm

Ž

y, 0

.

.

,

˜

˜

where C_i is the ith coordinate function of C.

˜

Ž

˜

.Ž .

C is a diffeomorphism at 0 because C is one and ­ C_mq1r­y 0_i s0

˜

Ž .Ž . Ž Ž ..

and ­ C_mq1r­x 0 s0 this can be obtained by differentiating 4.3 . Also, the germ of C at 0 satisfies that

˜

t_"fk

Ž

x , y

.

stafk(C

Ž

x , y

.

qCmq1

Ž

y, 0

.

REFERENCES

1. P. J. Rabier, Implicit differential equations near a singular point, J. Math. Anal. Appl. 144 Ž1989 , 425. ]449.

2. M. Zhitomirskii, Local normal forms for constrined systems on 2-manifolds, Bol. Soc. Ž .

Brasil. Mat. 24 1993 , 211_]232.

3. F. Takens, Constrained equations; a study of implicit differential equations and their discontinuous solutions, in ‘‘Structural Stability, the Theory of Catastrophes, and Applica-tions in the Sciences,’’ Lecture Notes in Mathematics, Vol. 525, Springer-Verlag, New York, 1976.

4. V. I. Arnold, S. M. Gusein-Zade, and A. N. Varachenko, ‘‘Singularities of Differentiable Maps,’’ Vol. 1, Birkhauser, Boston, 1985.¨