Vector fields near the boundary of a 3-manifold

. The simplest patterns of qualitative changes -bifurcations- located around a compact two dimensional submanifold, that occur on smooth one-parameter families of vector fields on a three dimensional manifold, are studied here.

The points in S at which Xf ¥ 0 (resp. Xf = 0) are called S-regular (resp. S-singular) points of X The points of S where b 2 is satisfied are called 6old ��nguf�t{eh; they form smooth curves in S along which X has qua dratic contact with S . The set where b 3 is satisfied is the union of isolated points of cubic contact between X and S , located at the extremes of the curves of fold singularities, called ��P ��ngul�eh.
In fact, by projecting S along the orbits of X into a surface N transverse to the orbit through a singularity, we get a Whitney singularity of fold (case b 2 ) or cuspidal type (case b 3 ) [ W] .  iii) k 1 is open in X� , in the topology induced from X r iv) For a residual set of smooth curves y : R � X r , y meets k 1 transversally The background on the study of stability and bifurcation properties of vector fields on manifolds with boundary can be found in [A , P-P, Pe, s This paper is organized as follows. In Sections 2 and 3 the singularities of vector fields, of critical and tangencial types are studied. Section 4 is devoted to the study of local structural stability of families of vector fields. In Section 5 some considerations concerning Transient Vector Fields are made. These sections prepare the way for the proof of Theorems 1 and 2 given in section 6.  ii) If and then is C r -equivalent to 00 2. 4. REMARK. In the C class , the equivalence obtained follows directly from Chincaro's work [C ] and it is clear that they have to respect the stratification as given in § 3.

2,5. NORMAL FORMS FOR A SINGULARITY OF TYPE � l (b)
.For simplicity's sake we give the normal forms in terms of a "straightened" vector field and a "twisted" boundary.
Let p in S be an S-hyperbolic critical point of X . In what follows we will study the local behavior of trajectories near such a point and their relation- If a # 0 then the order of tangency of the trajectory of X through PROOF. We have to consider the following cases: a) All eigenvalues of DX (p) are real .
In this case there is, at least, a pair of invariant two dimensional manifolds of X in general position which are transverse to L X at p . Moreover all trajec tories through any point of L X -{p} tend asymptotically to directions transverse to L X . These facts permit us to conclude the proof in this case.
b) The eigenvalues of DX(p) are A E JR and A 1 ; a ib with a # 0 and b I 0.
Assume first that the vector field is linear. We may choose canonical coordinates y; (y 1 ,y 2 ,y 3 ) around p such that the orbits of X are given by ¢ t (y) ; Assume z 0 f 0 ; then all trajectories of the vector field lie in the set � given by the equation if a > 0 then the trajectories of X lie in branches of a "hyperbole" (see Observe that Det( a y � � Y z (O)J is non zero provided that the eigenspaces of DX P are transverse to S at p . This implies that L X projects regularly on the The conclusion of the lemma in this particular case is now immediate.
We are going to prove that this situation persists in the general nonlinear case.
We can introduce a c 1 change of coordinates in a full neighborhood of p in such a way that X is represented by the equations: (3. 2) way).
(u,z) -+(0,0) J Assume for instance that a < A < 0 (the other cases are treated in a similar The solutions of (3.2. 1 ) through (u ,z ) 0 0 satisfy Arguments similar to those used in the linear case can be used to finish the proof. D 3.3. REMARK. a) A proof of the above lemma can also be obtained by performing a sphe� ical blowing up of the critical point; b) using the above notation, we observe that if a � 1 then there exists a trajectory tending to p (in positive or negative Let N be a one dimensional section transverse to S at p and p : M , p � N , p be a projection "parallel" to S • The required function is defined by We can pick a neighborhood B of X in X r r-1 . and a C -m app�ng h : The conclusion follows by continuity. In what follows we define a stratification of a neighborhood of a critical point p E S which is essential for the proof of this proposition.
First of all, we shall distinguish some subsets in V • We have to elaborate the following list:

)
2) T(Y) and Py must be distinguished as well as the trajectories of Y passing through them.
Consider the following possibilities: 3) Py is a node; this means that the corresponding eigenvalues A i are real; 3a) the invariant one dimensional manifold of Y tangent to T 1 (resp. W�) .
3b) the invariant two dimensional manifold of Y tangent to the linear space generated by T 1 and T 2 (resp. W� 2 ) .

4)
Py is a nodal-focus: this means that A 1 is real, A 2 a ± ib with We list the following distinguished sets: w s 1 : the invariant one dimensional manifold tangent to T 1 (resp. W u ) 1 5) P y is a focal node: this means that A 1 is real, A = 2 a ± ib with a < A 1 < 0 (resp. 0 < A 1 < 2) We list w s 2 : the invariant two dimensional manifold tangent to T 2 (resp. W u ) 2 6) P y is a nodal-saddle: this means that A. are real, i = 1 ,2,3 and sets are distinguished: w s 3 ' u and w u w 12 1

7)
is a focal saddle: this means that (resp.

8)
The boundary of V •

We list
J.
Using the above notations the following (resp. w s s and W u ) 1 ' w 1 2 3 is real and A 2 = a ± ib with and (resp. and 9) We include in our list the intersections between each two distinguished sets listed above.

8. REMARK. If T(Y) � P y then:
a) the trajectory of Y passing through T(Y) never meets the distinguished sets given in 3, 4, 5, 6 and 7 (see above). This can be seen immediately by considering the restriction of X to some suitable two dimensional invariant set and using the results of [ T] ; b) in a similar way we can check that no trajectory of Y contained in is a dis tinguished one dimensional invariant manifold of Y listed in 3, 4, 5, 6 and 7 of (3. 7).

STRATIFICATION OF A NEIGHBORHOOD OF P y •
Denote by E 0 (Y) the union of all 0-dimensional distinguished sets defined above. If we observe that E c S • If 0 then P y E E 0 (Y) and P y � S • In this way we define E 1 (Y) , E 2 (Y) and E 3 (Y) = V. We have to consider the following cases: a) P y is not a saddle. In this case we may consider X transverse to oV b) P y is a saddle. In this case there is a one dimensional submanifold of ()V which is the set of external tangencies between Y and ()V • This submanifold must be included in the list of distinguished sets and of course it is far away from L Y We see that E 0 C E 1 C E 2 c E 3 define a stratification on V in such a way that Whitney's Conditions are naturally satisfied [Th] .
Let us now indicate how Proposition 3. 6 can be obtained from the above stratification.
Let pE S be a singularity of X of type k 1 (a) • This proof will be done in a geometrical way. We will use results and techniques contained in [ S . l] and ( T) • 13 PROOF OF PART i). To prove the sufficiency condition we proceed as follows: a) we have to study each of the cases listed in 3, 4, 5, 6 and 7 of (3.7); b) we analyze the bahavior of I X n 3V with respect to the intersection between 3V and the distinguished invariant manifolds of X . , c) because of the definition of a singularity of type L1 (a) , this behavior persists for small perturbations of X in r xl ; d) the equivalence is first defined on av and then extended to the full neighborhood as a stratified mapping. From Lemma 3.2 it follows that any trajectory passing through a point q E I X n av encounters L X just once. and U Consider the following cases: I) The eigenvalues of DX(p) are real with A l < A z < A 3 < 0 (Node).
Assuming the above notations, call v 1 I X n 3V = u 1 U u 2 (see Figure 3. 4) We see that, both v 1 and U have two connected components consisting of isolated points and semi-intervals respectively. Moreover, Cl(U) and v 2 meet transversally.
For Y E B we have the analogous objects v 1 We define the equivalence h : JV � av by imposing q i � q i , and U. � U. , l l v 2 � v 2 and then extending to V (for example by preserving the rate of arc length) respecting the stratifications.
II) The eigenvalues of DX(p) are real with \ 3 < 0 < \ 1 < \ 2 (Nodal-Saddle). If Y is a small C rperturbation of X in X r , we have similar objects The required homeomorphism on (JV must preserve the above distinguished sets. Now the conclusion of the proof in this ease is straightforward. III) The eigenvalues of DX(p) are /.. 1 E lR and /.. 2 a ± ib with >.
Let v l We have to start the construction of the equivalence by imposing that r and U must be sent to r and U respectively. Notice that: a) U (resp. U) is transverse to S n 8V b) any trajectory of X (resp. Y) meets 8V exactly once; c) we can use the rate of arc length with respect to S and 8V to extend the homeomorphism.
) > 0 . We will proceed to construct the equivalence between Y 1 and Y 2 as in Part i) .
Notice that any Y E B with G(Y) I 0 has also the following sets to be dis- are real, nonzero and distinct.
2. X(x 1 ,x 2 ,x 3 ) = (ax 1 + x 2 , x 1 -ax 2 , Ax 3 ) and f(x 1 ,x 2 ,x 3 ) = x 1 + x 2 + x 3 with a f 0 , A # 0 and a # A .  converging to X such that each X is never n co -s equivalent to X (at p) If X(p) = 0 and one of the conditions i), ii) , iii) and iv) of Definition 1. 2 is violated then X f/:. r 1 (p) . We will further discuss more completely the condi-tions ii) and iii).
Assume that X, at p , satisfies the above conditions i) and iv).
If the Condition ii) is dropped, this means that two eigenvalues of DX(p) are equal then we can pick a suitable invariant 2-dimensional manifold W and restrict our study to X IW . In this way we select a sequence (X n ) in X�(p) such that W is still an invariant 2-dimensional manifold of each X n and the eigenvalues of D (X I ) (p) are distinct. Now by using the same techniques and arguments as [T] we n W prove directly that X � � l (p) .
To illustrate the last situation, suppose that the eigenvalues of DX(p) are A l = A 2 < A 3 < 0 . We choose W as the invariant 2-dimensional manifold of X asso ciated to A l . In addition we take local coordinates around p such that X(x,y,z) = (A 1 x + h 1 (x,y,z), ax + A l Y + h 2 (x, y,z), A 3 z + h 3 (x, y,z)) for some nonzero scalar a • We can choose (X n ) , given by such that H(A) is a C0-S-equivalence (at p) between (Â 1 (A) and (Ã 2 (h(A)) .
With this concept we get naturally the structural stability definition in ¢ r . Let us denote by A r (p) the collection of the elements ( E ( r such that: 3) i;(-s) and ((€) are in (p) . We have that i; is structurally stable in § 5. TRANSIENT VECTOR FIELDS.
We clarify the intrinsic features of the proofs of Theorem l and Theorem 2 by giving in this section some constructions and preparatory results concerning transient vector fields in suitable neighborhoods of S in M . In [P ] the structural stability of transient vector fields on a manifold has been studied.

00
For any compact connected C 3-manifold P with a non-empty 2-dimensional boundary we say that a vector field is transient in P if each integral curve of it leaves the manifold in finite positive and negative time.
Let p E S be either a S-stable or a S-quasigeneric singularity of X E X r , B a small neighborhood of X in X r and F € be a fundamental neighborhood system of in s given either in Proposition 3. 5 or in Lemma 4.4) for each Y E B. PROOF. Consider I s x I s x I € an s-flow bo x around p E S with respect to X such 23 that I=[-£,£] , p = (0,0,0) and X(x,y, z) = (1,0,0) . The surface can be given locally by the grafic of z = g(x,y) with (x,y)E I £ xI £ . This proves the asser tions of the lemma for the original vector field X . Now the proof follows by continuity.

D
The following lemma is immediate.

5.2.
LEMMA. If X satisfies the hypotheses of Theorem 2 and X(q) � 0 for every q E S then there exists a neighborhood B of X in X r such that for each Y E B there is a fundamental system of neighborhoods w £ of S where is transient.

REMARK. We recall from [P] that a vector field is transient if and only if it
is a gradient field (for some metric) with no critical points. Assume for instance that p E S is a singularity of X of type (a) .
We are considering V the neighborhood of p as in 3.9.
As before, call L X (p) the connected component of L X containing p .
If q E L X (p) n av then consider V E (q) as in Lemma 5.2 and define the sets v -(q) n v} and If p is a saddle point of X then E must be choosen such that 0 < s < d(L X ,T X ) where T X is the set of external tangencies between X and av .
The proof of the next lemma is straightforward. Part ii) and iii) are immediate consequence of Proposition 4.2.