Bifurcations and global stability of families of gradients

M. Carneiro; J. Palis

doi:10.1007/BF02698875

Bifurcations and global stability of families of gradients

Carneiro, M.; Palis, J. 2007-08-30 00:00:00 BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS by M. J. DIAS CARNEIRO and J. PALIS Dedicated to Ren~ Thom on his sixty fifth anniversary. In 1967 it was proved that among gradient vector fields on compact boundaryless manifolds, the elements of an open and dense subset are structurally stable: under small perturbations they have their orbit structure unchanged up to orbit preserving homeo- morphisms [14], [16]. From this result it follows that the stability of a gradient flow is equivalent to the hyperbolicity of the singularities and transversality of their stable and unstable manifolds. At the end of that decade, Thorn was asking about the bifurcations and stability of families of gradients, specially about k-parameter families with k <~ 4. The question is very challenging and indeed it might ammount to a rather formidable program, since not even just locally near a singularity the question for k = 4 is solved (and this problem by itself is very interesting). A point to stress here is that the dynamic bifur- cations of a gradient family are in general considerably richer than those of the corres- ponding family of potentials; see [3], [4], [21], [22] for comments. Also from the global point of view this comparison is relevant to understand Thom's question: often near a bifurcating singularity there appear secondary bifurcations due to tangencies between invariant (stable and unstable) manifolds from far away singularities. Finer dynamic analysis is then needed to describe the bifurcation diagram and to prove stability of a generic family. In this line, in 1983, in a paper dedicated to Thom on the occasion of his sixtieth birthday, the question for k = 1 was settled [15]: among one-parameter families of gradients, the stable ones are dense. These stable families can be charac- terized up to high codimension degeneracies; see Section I. The purpose of the present paper is to provide a proof of a similar result for two- parameter families of gradients. New techniques, specially concerning singular invariant foliations, are introduced to study the bifurcation diagrams and to prove stability. Let us state our result in a precise way. Let M be a compact boundaryless C ~ 104 M. J. DIAS CARNEIRO AND J. PALIS manifold. Gradients of real functions can be considered either with respect to a fixed Riemannian metric or to all possible ones. Although our result is true in both cases, we will restrict ourselves to the last one. Let x~(M) denote the set of C ~~ two-parameter families of gradients endowed with the C ~ Whitney topology, the parameter being taken in the unit disc D in R ~, and denote by rc2:M � D-+D the natural projection. We say that X~, ~2~ e z~(M) are equivalent if there are homeomor- phisms H:M � D-+M X D and ~:D-->D such that ~2H=q0n, and, for each ~x e D, h~, is an equivalence between X~ and X~,~), where h~ is defined by H(x, ~) = (h~(x), ~([x)). That is, h~ sends orbits of X~, onto orbits of X,c~ for each ~ e D. The family X~ is called (structurally) stable if it is equivalent to all nearby elements in ;(g(M). Our main result can now be presented as follows. Theorem. -- There is an open and dense subset ~t in z~(M) whose elements are stable. Several comments are in order. First of all, as we observe in Section IV, the para- meter space in our theorem can be taken to be any compact surface. Second, while the result makes one hopeful of giving a similar positive answer about stability of k-parameter families for k = 3 and k = 4, it is known that this is not true for k/> 8 [18] ; actually, it is not true even locally near a singularity [19]. On the other hand, positive local results near a singularity were obtained for k = 3 and to some extent k = 4 in [22], [4]; however, the question for k = 4 is still essentially open and very interesting. We also point out that our result was obtained by Vegter [22], [23] for manifolds of dimension less than or equal to three. These papers and [15] were the starting point of our work. However, the analysis of codimension-two bifurcations in higher dimensions is considerably more elaborated and led us to introduce new kinds of singular invariant foliations (that might even be useful in other contexts) ; see, for instance, w 1 of Section III below. The paper is organized in the following way. The first two sections are prepara- tory ones, so the reader gets acquainted with some basic concepts and tools and the previous result for one-parameter families. To serve as references for other cases, already in Section II we use these tools to exhibit the bifurcation diagrams and to prove local (in the parameter space) stability of families with quadratic and higher order contact between invariant manifolds. In Section III we complete the definition of the subset of families ~ (up to a slight modification still to be performed in the last section) and prove the local stability of its elements, except for the ones already considered in the preceeding section. In Section IV we globalize the result to the whole parameter space. To be more specific, in Section I we recall the characterization of the stable one- parameter families and from it infer what shall be a corresponding characterization for two parameters. This leads to a list of cases that begins with codimension-one bifurcations, namely a saddle-node and a quasi-transversal tangency. We then have BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 105 combinations of these two cases, like the simultaneous occurence of two saddle-nodes. There are also the purely codimension-two cases: a codimension-two tangency (cubic contact or lack of dimensions) and a codimension-two singularity. Also, three cases arise from the degeneracy of one of the transversality conditions concerning center- stable and strong-stable, rep. unstable, manifolds that are required for one-parameter families. In Section II we present the basic concept of compatible systems of invariant foliations and provide a brief description of how they are constructed. This kind of foliations have been used in previous work like [6], [14], [15], [16] and [22]. Here, the concept has to be considerably extended to include several new types of singular foliations. Using these foliations, we treat in this section the initial cases of quadratic and cubic (or higher order) contact between invariant manifolds. Finally, in Section III we obtain ~ as the intersection of several open and dense subsets of x~(M), each one corresponding to families that present one of the bifurcations listed in Section I. We prove that every family in ~ is stable at every value of the parameter Ix 9 D; i.e., the family restricted to a small neighbourhood of ~ in D is globally stable on M. We then show in Section IV that we can piece together, in terms of the parameter space, our construction of the equivalence between two nearby families in ~/, thus proving the result. We are thankful to several colleagues, including Arnold, Takens and Thorn, for their interest and valuable comments. We are specially grateful to Khesin (see Theorem B, Section II) and noteworthy the referee of the present paper. Section I. -- Bifurcations of codimension two We first recall the bifurcations of codimension one and some generic conditions that are imposed in order to obtain stable one-parameter families as in [15]. Let X~, 9 R, be a family of gradients on M and dim M = n. a) Saddle-nodes. -- We say that X~ has a saddle-node singularity p if one of the eigenvalues of dX~(p) is zero and all the others nonzero. Moreover, restricted to a center manifold through p, X~ has the form Z~(x) = ax 20x -t- O(1 x [a) with a ~e 0 (about center manifolds see [10]). For each ~z near ~, there is a Ez-dependent center manifold restricted to which X~ has the form o ]8 X~(x) = (ax 2q-b([z-~))Ox +O([x + [x(~t--~)l + I(~ x-~)[~) with a + 0. The saddle-node unfolds generically if also b ~e 0. This condition is satisfied by the elements of an open and dense subset of families. b) Quasi-transversal orbits. -- Let p, q be hyperbolic singularities of Xg; that is, all eigenvalues of dX~ at p and q are nonzero. Let W"(p) and W'(q) be their unstable 14 106 M. J. DIAS CARNEIRO AND J. PALIS and stable manifolds. Suppose y is an orbit of tangency between them and assume that dim T r W"(p) + dim T r WS(q) = dim M -- 1 for r e y. In local coordinates near r, we have X~ -- Oxl" W"(p) -~ (xl, ..., x,, 0, ..., 0), u = dim W"(p), w'(q) = (xl, ..., 0, ..., 0, x,+l, ..-, g( 2, ..., where n = dim M and k = dim(T,W"(p) n TrWS(q)). We say that is quasi-trans- versal ifg is a Morse function. For each ~ near ~, we can write similar expressions for X~, W"(p~) and W"(q~), g being replaced by a ~-dependent function g~. We then say that y unfolds generically if ~- (r)[~=g ~e 0. c) Generic conditions. -- We now list a number of generic conditions concerning the stability of families of gradients. c. 1. Distinct eigenvalues. -- We assume that the eigenvalues of Xg at the singularities associated with an orbit of tangency have multiplicity one. Hence, there exists a smallest expansion (respectively contraction) and we can consider the strong unstable manifold W "" corresponding to all but the smallest positive eigenvalue (see [15]); similarly for the strong-stable manifold W". And, corresponding to the smallest positive eigenvalue and all negative ones, we have a C 1 center-stable manifold W c,, which is transversal to W "". Similarly for a center-unstable manifold W c". We observe that, in the presence of an orbit of tangency the assumption on the multiplicity one of the eigenvalues at singula- rities are generic (open and dense) for two-parameter families of gradients. In fact, a failure of these conditions gives rise to a subset of codirnension at least 3: an orbit of tangency corresponds to subsets of codimension at least one and a multiple eigenvalue of the linear part to subsets of codimension at least two (since it is a symmetric operator). c. 2. Noncriticality. -- We assume that the strong stable and strong unstable mani- folds of a saddle-node are transversal to the unstable, resp. stable, manifolds of all other singularities. Similarly for the singularities associated to a tangency. c. 3. Transversality of center-unstable and stable manifolds at a tangency. -- If ? is an orbit of tangency between W~(p) and W~(q), we require W~"(p) to be transversal to W~(q); similarly for W"(p) and W~"(q). c.4. Linearizability. -- For a family X~ with one of the bifurcations of type I through IX below, we assume that Xg is C" linearizable transversally to a center manifold of a saddle-node or near the singularities associated with an orbit of tangency. Actually, this linearization is also required for each ~ near the bifurcation value ~. The integer m is taken to be bigger than p + 2, where p is the maximum ratio of positive, resp. negative, eigenvalues of dX~ at the singularity (cf. c. 1). By [I7], [20], the linea- BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 107 rization may be taken to depend differentiably on the parameter and these conditions are generic for two-parameter families. We do not assume this hypothesis when we deal with strictly codimension-one cases: a quasi-transversal orbit of tangency, treated in Theorem A, Section II, or a saddle-node, treated in w 7.B, Section III. Given a family of gradients, a parameter value is called regular if it corresponds to a stable field, in this case a Morse-Smale gradient field (hyperbolic singularities and transversality between stable and unstable manifolds); otherwise, it is a bifurcation value. Now, for an open and dense subset of arcs each bifurcation value is isolated and it corresponds to a unique tangency or to a nonhyperbolic singularity, for which condi- tions a), b) and c) are satisfied. These arcs are stable [15]. What we prove in Sections II and III is the analogue of these results for two- parameter families of gradients. The subset of families we consider must now include codimension-two bifurcations; they are listed here and studied in detail in Section III except for case VIII (cubic contact) which is considered in Section II. We keep denoting a family of gradients by X~, but now ~ varies in the unit disc D in R ~. I. A quasi-transversal orbit of tangency with criticality. -- For some ~z = ~, there are singularities p, q such that W"(p) and WS(p) have a quasi-transversal tangency. However, unlike in c. 2 above, there is another singularity s such that W"(s) and WS'(p) are nontransverse along a unique orbit, which is quasi-transversal, or similarly there is such an orbit in W""(q). Except for that, all conditions in a), b) and c) above are satisfied. II. Two quasi-transversal orbits of tangency. ~ Two orbits of tangency may occur simultaneously, but they must satisfy the generic conditions b) and c) above. III. A saddle-node with criticality. -- The unstable manifold of some singularity is nontransverse to the strong stable manifold of a saddle-node along a unique orbit which is quasi-transversal, or similarly with respect to the strong unstable manifold of a saddle-node. All other generic conditions in a), b) and c) are satisfied. IV. Two saddle-nodes. -- Two saddle-nodes occur simultaneously and both satisfy the generic conditions a) and c) above. V. A saddle-node and a quasi-transversal tangency. ~ These two bifurcations may occur simultaneously; again we assume all generic conditions we have mentioned con- cerning hyperbolicity of the other singularities, linearizability and transversality in a), b) and c). VI. A quasi-transversal orbit of tangency along which the corresponding stable and a center-unstable manifolds are tangent. -- Since all center-unstable manifolds are tangent on each orbit of the unstable manifold, the condition does not depend on which center- unstable manifold we consider. We also require all singularities to be hyperbolic and the generic conditions in b), c. 1, c. 2 and c. 4 to be satisfied. We will show in w 5 of Section III that the orbit of tangency may be taken to have quadratic contact. 108 M. J. DIAS CARNEIRO AND J. PALIS VII. Codimension-two tangency originating from lack of dimensions. -- A tangency occurs between W"(p) and WS(q), for some singularities p and q, so that the sum of their dimensions is equal to (dim M)- 2. Several generic conditions are imposed including the ones already mentioned. VIII. Tangency corresponding to cubic contact. -- Similar to the previous case, but now W"(p) and W'(q) have cubic contact along a unique nontransversal orbit of intersection. IX. Codimension-two singularity. -- Xg has a unique nonhyperbolic singularity which has a single eigenvalue zero; restricted to the corresponding center manifold, X~ has the form Xg(x) = (x 8 + O(I x t 4) ~x" Actually, we will treat here the case of a codimension c singularity for all c >/ 1 under the hypothesis of a single eigenvalue zero. Section II. -- Invariant foliations and invariant manifolds A basic tool in the proof of stability of a family of gradients is to construct invariant foliations which should be globally compatible: they ought to be preserved so that we can fit together localized constructions of flow equivalences or conjugacies. It is also helpful to restrict the family to invariant submanifolds in order to " reduce dimensions ", for instance, to obtain the bifurcation diagram. The strategy has been successfully adopted in several previous papers [6], [11], [15] and we refer to them for more details. In this section we recall the notions of compatible system of (invariant) foliations and of center-unstable and center-stable foliations, applying them to prove local stability of families presenting either quadratic or cubic (or more generally simple) contact between stable and unstable manifolds of hyperbolic singularities as in b) and VIII of Section I. The first case has been proved in [15] since it corresponds to a typical codimension-one bifurcation. However, our treatment is different from the one in [15] and, in fact, it contains some of the main new and old arguments involved in the proof of several other bifurcations. For this reason it will be repeatedly quoted in subsequent cases. Definition H. 1. ~ Let ~ be a hyperbolic singularity for Xg, ~ e R 2, and U C R 2 a neighbourhood of ~ and V a neighbourhood of a in M such that for each ~ e U there is a unique singularity a(~t) of X~ in V, with ~(~) = ~. A (local) unstable foliation F~(a) for X~ is a continuous foliation such that a) The leaves are C" discs, m 1> 2, varying continuously in the C m topology, with distin- guished leaf F~(~(~x), ~x) = W"(~(~)) n V � { ~ }, b) Each leaf FU(x, ~) is contained in V � { ~x }, c) F~(a) is invariant: X~, t(F"(x, ~x)) 3 F~(X~. t(x), ~x), t/> 0, BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 109 d) For each ~t 9 U, the intersection of a leaf of F"(a(~t)) with WS(~r(~z)) is a point. A (global) unstable foliation F"(~) is just the positive saturation by the flow of X~, 9 U, of the local unstable foliation. Similarly we define a stable foliation Fs(cr). Let us suppose that the vector field X~ has only (finitely many) hyperbolic singu- larities with their unstable and stable manifolds intersecting transversally. We may order the singularities ~l(~z) <~ . .. ~< crt(~z ) for ~z 9 U, U a small neighbourhood of in 112, in such way that if WU(~(~)) c~ W~(%.(~z)) ~e ~, then ~([z) ~ cr~(~z), and if i 4: j, then i < j whenever ~,(tz) ~< crj(~t). For each singularity a~(~) which is not a sink, we consider an unstable foliation F"(ai(Ez)). Definition II.2. -- The foliations F"(al(~t)), ..., 1~(%_1([z)) form a compatible unstable system if whenever a leaf F of FU(a~(~z)) intersects a leaf F of FU(~(~z)), i<j, then F D F. A similar definition holds for a compatible stable system, F"(%+x(~z)), ..., F*(at(~)). The construction of such systems is detailed in [6], [14], [15]; in [6] a center-unstable foliation FCU(ak(~z)) is also obtained which is compatible with the system F"(%(~z)), ..., F"(ek_a(~)) in the above sense: a leaf of F"(a,(~z)) that intersects a leaf of F~"(%(~)) actually contains this leaf. Each leaf of the center-unstable foliation is a C 1 disc and is the union of leaves of an unstable foliation F~. For fixed ~ the foliation F~"(%(~x)) is tangent to the vector field X~. In order to construct Fr it is assumed that the linear part DX~(%(~x)) has a smallest contraction, that is a negative eigenvalue of smallest absolute value, and hence we may take a center-unstable manifold Wr as the distinguished leaf of Fr Actually, since in the bifurcations of type I to VII we assume the linearizability condi- tion c.4 for X~ near the singularity ~r,(~x), there is a natural choice for Wr in this special coordinates, namely, W~"(%(~t)) is linear. Another important tool that we have often been using in bifurcation theory, as in the present work, is the following parametrized version of the well-known Isotopy Extension Theorem (see [12]). Let N be a C' compact manifold, r >t 1, and A an open subset of IIL ". Let M be a C ~ manifold with dimM>dimN. We indicate by C~(N � A,M � A) the set of C * mappings f: N � A ---> M � A such that = = rdf, endowed with the C * topo- logy, 1 ~ k ~< r. Here, r~ and ~' denote the natural projections re:N � A-+ A, r~':M � A--->A. Let Diff[(M � A) be the set of C ~ diffeomorphisms q0 of M � A such that r~'= r~' % again with the C ~ topology. Isotopy Extension Theorem. -- Let i 9 C~(N x A, M � A) be an embedding and A' a compact subset of A. Given neighbourhoods U of i(N x A) in M x A and V of the identity in Diff~(M � A), there exists a neighbourhood W of i in C~(N x A, M � A) such that for each j 9 W there exists ~ 9 V satisfying ~i = j restricted to N � A' and ~(x) = x for all x r U. This theorem is used to extend homeomorphisms h which are defined on top dimension submanifolds with boundary N C M whose restrictions to the boundary 110 M. J. DIAS CARNEIRO AND J. PALIS are C 1 diffeomorphisms, C x close to the identity. Hence, by applying the above theorem to h [ ON we obtain an extension H to all of M and defining H : M ~ M such that H ] N = h and H ] M\N = H we get the desired extension. One needs this parametrized version in order to obtain such extensions on each leaf of an invariantfoliation. We refer to [15] and [11] for some applications of these ideas in very similar situations. The use of the above invariant foliations is illustrated in Theorem A below. Before that, we recall the definition of local stability. Definition H. 3 (Local Stability). -- A family X~ 9 )~g(M) is stable at ~ 9 R 2 if there is a neighbourhood U of ~ in R 3 and a neighbourhood ok' of X~ in x~(M) such that for each family X~ 9 q/there is a value ~ e U and a homeomorphism H:U x M-+R 3 x M of the form H(~x, x) ----= (q~(~), h(~, x)), with ? : (U, ~) -+ (112, ~) also a homeomorphism onto its image, such that h~ : M ~ M is a topological equivalence between X~ and X,I~I, where h~(x) = h(~, x) for x 9 M. Theorem A. -- Let X~, ~ 9 113, be a family of gradients and -~ a bifurcation value suck that Xg presents exactly one orbit of quasi-transversal intersection between stable and unstable manifolds, which unfolds generically as described in b) of Section I. Suppose that all singularities of Xg are hyperbolic and the conditions described in c. 1, c. 2 and c. 3 of Section I are satisfied. Then, X~ is stable at ~x. Proof. -- First we describe the bifurcation set for X~, ~z close to ~. For ~ in a neighbourhood U of ~ in 112, we order the singularities of X~, ~x(~) ~< --- ~< at(~t), as above, and assume that the orbit of tangency y belongs to the intersection of WU(~k(~)) and W~(~k+~(~)). Let us assume that dim W"(~k(~) ) § dimW~(%+~(~))>/ n § 1 for, otherwise, similar arguments apply. Let Z(~) be a small cross-section intersecting y and Z = U~ 9 1J Z(~z). From the assumptions ofquasi-transversality and generic unfolding, there are C oo coordinates (~z, xl, ..., x,,yl, ...,y,_,, zl, ..., z,_,, Wl) in N centered at p=ynZ(~) such that n = 0, 0), WS(ek+,(~.)) n Y, = (~, x, O, z, F(~., x)) with r = dim[T, W~(ak@)) n T r W'((~/c+l(~.)) ] -- 1 and x ~ F@, x) [ dV(r,,o) is a Coo M~ functi~ such that rank td [0F ~_(t -,0]l ~ ) = r + 1" Theref~ the tangency between W~(a,(~)) and W'(%+~(tx)), for [x near ~, is characterized by the equa- OF tions F(~t, x)= 0, Ox (~x, x)= 0. By the implicit function theorem, there is a Coo curve P in 113, containing ~, such that ~t 9 F if and only if W~(%(~)) is quasi-transversal BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 111 to W'(ak + 1 (V t))- Moreover, it is easy to see that the noncriticality and transversality condi- tions (c. 2, c. 3 of Section I) imply that these are the only bifurcations near ~. Let X~, be a nearby family so that all conditions described above are also satisfied. We get a curve P near F which represents the bifurcations of the family X~. Let Z~-- - (Vt, x, 0, 0, wl) be a normal section in Y.. From Morse's Lemma with para- meters (see [13] or [5]), there is a diffeomorphism M: Z~--+Z ~ of the form M(W, x, wx) = (~(~t), h~(~, x, wl) ) which sends WU(~k(~)) n ~c to Wu(~k(~?(~))) n Z ~, W*(c~+l(~)) n Z ~ to W'(~+~(~0(Vt)) ) n X ~. Here ~ is a diffeomorphism defined in a neighbourhood of ~ such that ~(F) ---- r and which is close to the identity. Let us assume thatf~, the potential function of X~, has distinct critical values for near ~. Iff~ is the potential function of )~, then, since f~ and J~ are C ~ close Morse functions, there are C ~~ families of diffeomorphisms H~ : M --+ M, ),~ : I --+ I such that f~oH;~= X~of~, and so grad~;,o~,f~ is equivalent to grada-f ~ ----X~. (Here, g~ and ~ are the respective metrics and I an interval.) Hence, there is no loss of generality if we assume that X~ and X,~) have the same potential. The equivalence between X. a and X,c~ wilt be a conjugacy outside a neighbourhood V of the closure of the orbit of tangency y in M x R 2. Inside V it will preserve the level sets off~. Let us now describe this distinguished neighbourhood V. Let ck(bt ) =f,(a,(b0 ) andf~(,k+l(bt)) = ck+l(b0. If, > 0 is small and V,(b~) is an open neighbourhood of ,,(b~) in M, we consider A,(tz ) =f~-l(c,(~) -- ,) n %(~) and B,(tz ) ----f~-x(c,(~) + ,) n %(~) fori-----k,k+ 1. Let V,(b0-----{x e'~,(bt); X~,,,(x) n B,(bt ) 4 = 13 for t> 0 or X~,(x) tnA,($) 4:0 for t< 0}u{a,(~)}, i:k,k + 1, be neighbourhoods of a,(~) and ~,+~(~), respectively. We connect V,($) to V,+~(~) along y in the following way. We consider D C B,(~), a small closed disc centered at y n B,(~) such that [.J,>~0X~.,(D) does not intersect the boundary of the closure of A,+~(~) in M, and define D(tz ) :{xe M;X~.,(x)eD for some t~< 0 and Xv, t(x ) e A~+l(bt ) for some t > 0 }. Let V(~) = V,(bt) u D(~,) u V~+I([,) and V -~ tJ~e~ V(bt). Observe that, in order to glue continuously a conjugacy in the com- plement of V(bt ) with a level preserving equivalence, we adjust the metric in a neigh- bourhood of the part of aV(bt ) which is a union of trajectories, in such way that 1[ X~, [[o~ = 1. Moreover, since the critical levels in V will be preserved, this ajustment is such that the time it takes to go fromf~-~(c,(~)) to A~(~,) and fromf~-~(c,(bt)) to Bi(b0 is constant. We now briefly describe the construction of a center-unstable foliation F~*(a,(W)) ; we refer to [15] for more details. Let A~(tz) denote the sphere A~(tz) n W'(a,([,)) which is transversal to Xu, and intersects every nonsingular orbit in W*(e,(~,)), i.e. it is a fundamental domain for W'(e~(t*)). It contains A~'(~) = A,(t, ) n W"(**(~)) as a codi- mension-one (equatorial) sphere. Recall that W"(a,(~)) is foliated by a unique codi- mension-one C ~ foliation F"(a,(t*)) -- the strong stable foliation. We can write 112 M. J. DIAS CARNEIRO AND J. PALIS A~(V) = D+(V)u C,(~)u D~-(~). Here Cs(~) is a small tubular neighbourhood of A~'(~) in A~(~) ; D+(~) and DT(~t ) are closed discs whose respective boundaries aD+(~) and a DT(~) are the intersection of leaves F + and F- of FS'(%(~)) with A~(~). The subset C~(Vt) is taken in such way that if W"(~+(~)) n C,(Vt ) 4: O, then W"(~+(~)) n A~(pt) 4= O and W"(a,(B)) n C,(B) is transversal to the induced foliation FSS(%(~)) n C,(B). This is possible because of the noncriticality assumption (c.2) of Section I. The condition also allows one to construct a one-dimensional C 1 foliation F~(%(~)) on G,(~) which is compatible with the system F"(al(~)), ..., F"(~k_l(~) ) and is transversal to F+"(%(~)) n C,(~) and to a D~-(~) u a D+(~). Let F~(~) be a u-dimensional continuous foliation with C 1 leaves on Ak(~) which is compatible with the system F~(~I(~)), ..., F"(%_1(~)) and is transversal to A~(~). We now point out the following key fact: if P~ : Ak(~)\A~(~) ---> Bk(~) is the Poincar6 map between the non-critical levels Ak(~) , Bk(~) and b~ : Bk(~) +-) is a homeomorphism preserving leaves of P~(F"(Gk(~))), then the induced map (P~)-I o b~ o P~ extends conti- nuously to a full homeomorphism of Ak(~). We observe that preserving leaves means that the map sends a leaf of the foliation into another one. This motivates the definition of a center-unstable foliation as F~"(%(~)) ----- U+/>0 X~.,(F~(~)), a distinguished leaf being a center-unstable manifold W~"(%(~)). We distinguish two parts in F ~" with different types of leaves. One, denoted by F~"(%(~)), has (u + 1)-dimensional leaves, with u = dim(W~(%(~))), each leaf corresponding to a point of D+(~) w D~-(~). The other part of the foliation, denoted by F~(%(~)), has a typical leaf of the form U~eF~ F~(%(~))~, where F~ is a leaf of F~(~(~)) and Fr is the leaf of F~"(%(~)) through the point x. Notice that the ]eaves in F~"(~(~)) have dimension u + 2. The fact pointed out above concerning extensions of homeomorphisms together with the existence of a weakest contraction (see condition (c. 1)) imply the following stronger property. A homeomorphism b~ : B~(~) +-~ that preserves F~"(%(~)) and the portion of F~"(%(~)) inside a conic region which contains the center-unstable manifold induces, as before, a homeomorphism on all of A~(~). Our conic region corresponds to a bundle of solid cones, with constant width over the sphere B~(~) n W~(%(~)). The construction of a center- stable foliation F~'(% + ~(~)) is dual to this one. We proceed in the same way to construct a center-unstable foliation F~"(~(q~(~))) and a center-stable foliation Fe'(~+,(q~(~))) for the vector field X~(~). Let us now construct an equivalence between X~ and :~(~1" We start by sending sources to sources and sinks to sinks. Then, proceeding by induction on j and using the Isotopy Extension Theorem as in [15], pages 413 and 414, we obtain a continuous family of homeomorphisms and k~,: U~>~+~ W"(a~(~)) + U~;~++~ W"(+~.(9(~))). On each step, say from i -- 1 to i, we use the unstable system in order to go from a fundamental domain of W*(a,_I(~)) to a fundamental domain of W*(a,(~)), and use BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 113 also the fact that Ws(,i(~)) and WS(~i(~(~))) are ~1 close on compact parts. By imposing that the equivalence preserves the unstable system F"(%(~)), ..., F"(%_1(~)), we see that h~ induces a homeomorphism in part of the sphere A~'(~). By using induction on the indices j, starting withj = k -- 1, and using again the Isotopy Extension Theorem, we extend this homeomorphism to the whole of A~*(~) (space of leaves of F~U(ak(~)). By preserving the central foliation FC(%(~)), we define a homeomorphism on 0 D+(~) ~J 0D~-(~). Using once more the Isotopy Extension Theorem and induction we extend this homeomorphism to D+(~) w D~-(~). In this way, we obtain a homeo- morphism on the space of leaves of the center-unstable foliation FCU(,k(~) ). We proceed dually to get a homeomorphism in the space of leaves of the center-stable foliation F~ ). Our next task is to construct a homeomorphism in the cross sections Z(~) (which we assume to be contained in a level set off~) preserving the center-unstable and the center-stable foliations. We first construct this homeomorphism on the section X~(~) = WC~(~+l(~))nZ(~). Let F~C Zcs(~) be a C 1 foliation compatible with W'(ak+i(~)) n Z(~) whose leaves have complementary dimension and are transversal to Z~(~), where as above Z~(~) is a smooth cross section which is tangent to WCS((Tk..bl(~s ('~ Weu((~k(~) ) I~ Z(~). Let (Vl, vi, w,,) be coordinates for Zr such that W"(Gk_I(~) ) n E"~(~) = {Vl = vi = 0} and W~"(%_i(~t)) nZ~'(~) ={v I=0} where v I= (v2, ...,V~_,);WT,= (Wl,..,W,) and s = dim W~(%+l(~)). FIG. I 15 114 M. J. DIAS CARNEIRO AND J. PALIS In Z"'(~), we consider two conic regions C(ai) = { v~ -- 8~(~) II ~ It ~ ~> 0 ) and C(a~) = { v~ -- ~(~)[[ v~ ]1~< 0 }, where 0 < aa(~)< a~(~) are chosen so that 00(aa) and 0C(a~) are transversal to F~ The intersection of F~"(a~(~)) with Z"'(~) gives rise to a foliation in C(a~) which is singular along W"(a~(~)) n y o,(~). This foliation is extended continuously on each leaf, say F~, of F~"(~(~)) n Z"'(~) in such way that OC(a~) n F~ is a leaf and it is non-singular in the interior of C(a~) n F~. We denote by F~"(a~(~)) this new foliation. By construction Fg"(.~(~)) is topologically transversal to F~2, so the projection Eo'(~) ~ E"(~) along the leaves of F~', restricted to each leaf of Fg~(a~(~)), is a homeomorphism. Hence, by performing the same cons- truction for X~,), since we already have a homeomorphism h~:Z~ -+ Z~ we can define h~' by sending F~2 to F~(~ and F0"(bt ) to ~'~"(~0(~)) preserving leaves of the center-unstable foliations. The main property of h~' is that it preserves the leaves of type F~"(~(~)) inside the conic region C(aa). Therefore, as we pointed out above, the induced homeomorphism ~(~t~' ~-~ o h~C~ o P~* automatically defines an extension to the fundamental domain A~(~). By proceeding analogously in the section = n we obtain a homeomorphism h~" which preserves a foliation F~ u compatible with WC~(,k(~) ) and the center-stable foliation F~ ) n Zc"(~). Finally, we match h~' and h~, ~ to obtain a homeomorphism on the whole section X(~). We do this by first considering a C a foliation FC"(~) transversal to Z*(~) and of complementary dimension, such that Fs"(~)n E~(~)= F~ ' and F'"(~)n Z~"(~)= F~ ~. We then require the homeomorphism to preserve this foliation, as well as the center-unstable and center- stable foliations. The homeomorphism extends to V(~)n (UtE, Xt,~(Z(~))) just by preserving the level sets off~ and by sending orbits of X~ to orbits of )(,(~. In particular, it defines a homeomorphism on a closed disc D,(~) contained in the level set Bk(~) and on a closed disc Dk+a(~) contained in the level set Ak+~(~). By construction, near the boundary of these discs the homeomorphisms are actually C 1 diffeomorphisms close to the identity. Hence, since all stable manifolds W'(~(~)), k + 1 ~< i ~< t are transversal to Wu(~k(~)) outside D,(~), we can proceed by induction on i and apply again the Isotopy Extension Theorem to get a homeomorphism on all of Bk(~) which preserves the intersections of the stable system F"'(ak+l(~)), F'(ak+2(~)), ..., F~(~t(~)) with B,(~). Similarly, we obtain a homeomorphism on the level Ak+~(~) which preserves the unstable system. We complete the definition of the equivalence inside V(~) by preserving the level sets off~ and of course sending orbits of X~ to orbits of X,c~. Thus, we have defined two families of homeomorphisms which depend conti- nuously on ~, and : BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 115 Let F~ be the level set off~ that contains the cross-section Z(~) and suppose that f~ is nondecreasing with respect to the ordering of the singularities. Observe that if xr [Uj~<k+iWS(a~([z))] u [U~>kW"(cr~(~z))], then the orbit of x intersects F,. Therefore, to extend the equivalence to all of M, we just construct a continuous family of homeomorphisms on F~ which are compatible with h~ and h~, and which preserves the stable and unstable systems. This is done by using once more the Isotopy Extension Theorem and induction on the dimension of the stable manifolds that intersect Fc, exactly as in [15]. 9 Remark. -- Let 9 : (U, -~) ---> (q~(U), ~(~)) be the reparametrization obtained in the proof of Theorem A. If D i C D2 C U are closed discs centered at ~ and p : R 2 -+ R is a C = function with supp p C U, O I D1, 0 ~< p ~< 1 and p ---- 0 outside U, then by defining F~ ---- (1 -- O(~))f~ + p(~)f~, g~ ---- [1 -- p(~)] g~ 4- p(~) ~ and X~ ---- gradgt, f~ , we obtain a two-parameter family such that X~,----X~ for ~ r U, :X~ = X~ for [~ E D1. Since 9 is C 1 close to the identity, there is another reparametrization + : (U, ~) --+ (+(U), +(~)) such that + restricted to Di is equal to 9 and is equal to the identity outside a neighbourhood U2 of the disc D2. Observe that the system of foliations constructed in the proof of Theorem A can be taken to be the same for X~ and X~ when r U 2. Hence, having the identity as the homeomorphism on the space of leaves of these foliations and repeating the proof of Theorem A, we obtain an equivalence h~ between X~ and X~(~ such that h~ ---- ida, for ~ r U2. This fact # very relevant in order to prove global stability, see Section IV, and it applies to all bifurcation cases treated in this and the next sections. We now prove local stability of bifurcations of type VIII, an orbit of tangency with cubic contact. Actually, as M. Khesin pointed out to us, using the theory of V- equivalence (or contact equivalence) as in [1], [8], [9] and the arguments in the proof of Theorem A, one can show the local stability of a much wider class of families of gradients. Recall that two germs fl and f~ : (R', 0) --+ (R, 0) are V-equivalent if there is a germ of diffeomorphism h : (R', 0) ---> (R', 0) and a smcoth germ M : (R', 0) --+ R such that fi(x) ---- M(x) .f,(h(x)) (so, h sends the "variety "fl-~(0) to f2-~(0)). Let ~ be a bifurcation value for a family X~ such that ~, C W"(a(~)) tn WS(a'(~)) is an orbit of tangency along which the manifolds have simple contact of type Ak, Dk, E6, E~, E 8 as in Arnold's list [9]. By that we mean that for p ~ ~, dim[T~ W~(a(~)) + T, W'(a'(~))] ----- dim M -- 1 and if 2] is a smooth cross section at p, there are ~-dependent coordinates (x,y, z, wl) centered at p such that W"(a(~)) nY.={z-----0, wi----0},W'(o'(~)) nE={y----0, Wl----F(~, x)} with f(x)= F(0, x) being equivalent to one of the following normal forms: A, ---- ~+~ + Q, k 1> 1, Q.(x,, ..., x,) a non-degenerate quadratic form; + + 0_, E~ + 0_; + + U_; + + O_, ~D(x~, ..., x,) a non-degenerate quadratic form. We require F(~, x) to be a V-versal 116 M. J. DIAS CARNEIRO AND J. PALLS unfolding off(x) (~ = 0) [9]. So, for a nearby family X~ with corresponding unfolding ~'(tx, x) we have F(F , x) = M(F , x) F(q~(F), h(F, x)) with M(0, 0) 4: 0, ~ and h~ being local diffeomorphisms. Therefore, there is a local diffeomorphism Z * -+ ~,~ of the form (q~(~), h(~z, x), M(F, x) -1 wl) which sends W~(~(tx)) n Z ~ to W"(~(q0(F))) n ~,~ and W~(&(F)) n Z ~ to W"(~'(q0(tx))) n ~c (as in the proof of Theorem A, X ~ denotes a smooth cross-section tangent to WC~(~(~)) n W"(~'(~)) n Z). Hence, from the non-criticality condition c.2 and the transversality between W""(~(~)) and W'(~'(~)) and between W""(&(~)) and W"(~(~)) (condition c.3), we construct compatible unstable and stable systems and proceed exactly as in Theorem A to get an equivalence between X~ and X~. 9 Thus, we have the following Theorem B. -- Let ~ be a bifurcation value for a family X~ such that X~ presents exactly one orbit of tangency with cubic contact, or more generally simple contact, which unfolds generically. Suppose that all singularities are hyperbolic and conditions c. 1, c. 2, c. 3 of Section I are satisfied. Then, X~ is stable at ~. We will see in w 6 of Section III that if dim[T~ W"(,) + T~ W"(,')] <~ dim M -- 2, the family may present other tangencies (secondary bifurcations) besides the tangencies between W"(,(tx)) and W"(*'(~)). This will impose several delicate adjustments in order to extend a local equivalence (in a neighbourhood of the orbit of tangency) to an equivalence on all of M. Section III. -- Local stability In this section we continue to prove local stability of the bifurcations mentioned in Section I; in the previous one we have already studied the families that exhibit one orbit of simple contact between stable and unstable manifolds. As mentioned before, by local we mean that we only consider the parameter varying in some small neigh- bourhood in R ~ of an initial bifurcation value ~. In each case, we start by requiring several additional generic assumptions for the family X~, then we obtain the bifurcation set near ~ and finally we prove stability. w 1. Bifurcations of type I: one orbit of tangency with criticality (1 .A) Generic conditions describing the bifurcation. -- A family X~ in z~(M) has a bifurcation value ~ of type I if the following holds: BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 117 (1.1) The vector field Xg presents a unique orbit y of quasi-transversal intersection between say, the unstable manifold WU(p(~)) and the unstable manifold W'(q(~)) of hyperbolic singularities p(~), q(~), (1.2) Linearizability of the family near p(~) and q(~) with the respective linear part with distinct eigenvalues (conditions c. 1 and c.4 of Section I), (1.3) There is a unique orbit y' of quasi-transversal intersection between the unstable manifold Wu(a@)) of a hyperbo]ic singularity ~(~) and the strong stable manifold ofp(~), W'8(p(~)), (1.4) Let W"8(p(~)) C W"'(p(~)) be the codimension two submanifold of W"(p(~)), invariant by X~, which corresponds (in linearized coordinates near p(~)) to the negative eigenvalues of dXg(p(~)) except for the two ones of smallest norm a!(~), a~(~). Then W"(a(~)) is transversal to WW(p(~)); in particular, the orbit of tangency y' does not belong to Wv*(p(~)), (1.5) The orbit of tangency 5" unfolds generically as in b) of Section I, so there is a C 1 curve F in the parameter space such that [X s F if and only if W~(p([X)) is not transversal to WS(q([X)), (1.6) The orbit of tangency 5"' also unfolds generically so that there is a C 1 curve F' in the parameter space such that [X e F' if and only if W"(a([X)) is tangent to W"'(p([X)). Moreover, the curves I' and F' intersect transversally at K, (1.7) The vector field Xg satisfies the linearizability conditions near ~(~) and the eigenvalues of dXg(a(~)) have multiplicity one so that if we take a C 1 center-unstable manifold WC"(a(~)) for a(~), then it is transversal to W"~(p(~)). We also assume the non-criticality conditions c. 2 of Section I, and that every unstable manifold is trans- versal to WS'(a(~)) and every stable manifold is transversal to W"U(q(~)) and the hypothesis c.3 which says that W~(p(~)) is transversal to W"(q@)) and WC"(q@)) is transversal to W~(p(~)), (1.8) Let n~_~ = dim[T, W"(a(~)) n T, W'(p(~))], for r e 5"', and consider the invariant manifold VsC W~(p(~)) of dimension nk_ 1 corresponding to the nk_ 1 nega- tive eigenvalues of dX~(p(~)) of biggest norm. There is a subspace E(r) C T, M such that lira dXg ,(r).E(r) = Toc~, W~"(a(~)) and lim dXg ,(r).E(r) = T~, V ~, t ---~-- oo ' ~ --~ ~- 0o , (1.9) According to (1.2) there are C m ix-dependent coordinates (xl,...,x~, Yl, -..,Y~) in a neighbourhood of p(~) in M such that X~ = - ~i([x) x~ + --, ,-, - oy, where 0 < ~([X) < ... < a,(~) and 0 < ~([X) < ... < [3,([X). 118 M. J. DIAS CARNEIRO AND J. PALIS The manifold W'(q(~)) is transversal to the plane (0, x,,0, ...,0,y~, ...,y~). This can be formulated intrinsically by saying that W'(q(~)) is transversal to any C" invariant manifold that contains the orbit of tangency y' and the unstable manifold W~(p(~)). Similarly, the unstable manifold W"(a(~)) is transversal to the plane (0, x2, ..., x~,yl, O, ..., 0). (1.B) The bifurcation diagram. -- Let zx(~t) ~< ... ~< %(~t) ~< %+1(~t) ~< ... ~< zt(~t) be the ordering of the singularities of the vector field X~ for ~t near ~ as in the previous section. We assume p(~) = %(~t), q(~t) = %+a(~t) and without loss of generality we may also assume that z(~t) = z~_l(~t). First observe that since W~"(%(~)) is transversal to Wg(gk+l(~) ) and W~*(%+1(~)) is transversal to W~(%(~)) the only possibility for non-stability of the vector field X~ comes from either the tangency between W*(a,(~)) and W'(a,+l(~)) or between W"(~,_l(~Z)) and W~(%+1(~)). This follows from the fact that if W"(%(~)) is transversal to W"(a,(~)), then W"(%([z)) is transversal to W~(%+~(~z)) for ~z near ~. Proposition. -- Let X~ be a family presenting a bifurcation value ~. of type I as described above. Then, there is a neighbourhood U of -~ in R ~" such that the bifurcation set for X~ in U is the union of two C ~ curves F u P0, such that ~t e P if and only if X~ presents a unique orbit of quasi-transversality y~C W~(~k(~t)) c3 W'(e~+l(~t)) and bt ~ F0 if and only if there is a unique orbit of quasi-transversality ~,~C W"(%_~(~t)) ~ W~(%+l(bt)). The relative position of F and F 0 is illustrated in Fig. III. Proof. -- Using a ~t-dependent C ~ (m/> 3) linearization for X~ near %(~t) and the transversality between WU(%(~)) and W~(%+1(~)), we may construct a C a sub- C$ C8 -- manifold W e' C M � R 2, Wk+ 1 = [J~wr � { ix} such that W (r k+l contains the closure of y. Moreover, W~+I and WC'(zk+l(tz)) admit smoothing C ~ struc- tures, r/> 3 (see Chapter II. 1 of [15]). In the sequel, we shall take r = m 1> 3. Analo- gously, from (1.7) the center-unstable manifold W~u(%_~(~)) can be extended to a C 1 manifold W~_I that contains the closure of the orbit y' which also admits a C r smoothing structure. Let W e-- W~k-1 n W~+I and consider the restriction of Xa to W~(~t)----W~"(zk_~(tx))nWr In this setting both W"(%_,(tz)) and W~(z~+I(Ez)) have codimension one and we may also assume that X~, ] Wc(Ez) is of class C m-1 having a C ~ tz-dependent linearization near %+1(~z) for ~ near ~: '*k-1 0 '~k 0 ,=1 Oy, with 0 < ~x(~t) < ... < ~,k_~(bt), 0 < ~l(0t) < ... < ~,k(bt) and n k_~ = dim[T,, W"(z) c~ T,, W*(p(~))] for z' ~ y', n k -= dim[T, W~(p(~)) c~ T, W'(q(~))] for z e y. BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 119 If Z * C { x~ ----- I } and S * C {Yl = 1 } are two cross-sections with coordinates (ix, xl, xi,yX,yL) and (a, vl, v,, vi, wry), respectively, such that y' n E*(~) = (~, 0, a~, 0, 0), 7nSC(~) = (~,0,0,0) where x I= (x3, ...,x,k_l ) and Yr.= (Y,, ...,ynk) then w'(~+,(~)) n s(~) = { vl = F(~, v~, ~i, w~)} and W~(ak_l(~)) ('~ E~(~t) --{ x I = C(~t, xi,yl,yr. ) } where F and G are C m functions, m >/ 3. The quasi-transversality assumptions mean that the functions wr. ~ F(~, 0, 0, wr.) and x I v-~ G(~, xi, 0, 0) have non-degenerate (Morse) critical points at WL = 0 and x I ---- ai, respectively. Hence, W"(%(~t)) is tangent 9 9 D to W*(%+a(~t)) if and only if F(~t, 0, 0, ws) = 0 and ~ (~, 0, 0, wr) = 0 forj = 2, n k 9 By the generic unfolding of the orbit of tangency Y and the implicit function theorem, we get a C m-1 curve P in the parameter space such that the corresponding vector field X~, ~ e F, presents an orbit of quasi-transversality between W"(,k(~x)) and W~(%+l(~t)). Therefore, F belongs to the bifurcation diagram. Analogously, solving 0G the equations G(~, xi, 0, 0) = 0, ~ (~z, xi, 0, 0) = 0, we obtain a curve F' containing such that, for ~t e F', the vector field X~ has an orbit of quasi-transversal intersection between W"(a,_~(~z)) and W"(%(~x)). The condition (I.6) says that F and P' are transversal at ~ (the tangencies have independent unfolding). We may suppose that ~=0, r={~=0}and r'={~x=0 }. To study the tangencies between W~(%_1(~)) and W"(%+~(tz)), we write in the above coordinates WU(~_~(~)) n S~ = {(e -~l'~,t G(~, XI, g--~l(g) t, e--~L(g) t /I)L), e--a2(~)t, e--at(g)t(Xi ~_ ai), WL)} with e- ~)t = diag(e- ~(~)t, .--, e- ~,~(~)~) and e- ~(~)~ = diag(e- %(~)~, . .., e- ~'~-,(~)~). Hence, the tangency between W~(%_I(~x)) and W'(g~+~(~z)) in S~(~) is expressed by the system of equations: e -=~tG(~) -- F(~) = 0 OG OF ~(~) -~(~).e",-~" =o ~= a, ...,n~_~ (El) e- =t t 0G OF Oy-~ (~)'e-~i* -- Ow~ (~) = 0 j = 2, ..., n~ -- al G(~) -- [3~(iz ) e-h'~" OG ~, e_~/~, , o.r--~ (~) - ~7~ 9 . (~). ~ 120 M. J. DIAS CARNEIRO AND J. PALLS -= (bt, xi, e -~ll~'~, e -~L~g~ wL), ~ -= (Vt, e -~c~, e-~t(xi -k ai), wz). Making t -+ q- oo OG OF we obtain F(~,O,O, wL) =- O, ~x~ (~'x~' 0,0) = O, Fw~ (~'O'O'wL) = 0 and G(~t, xi, O, O) = 0 which is non-singular due to the generic and independent unfolding of the orbits of tangency. Let ~ ---- min (~x(0), 0c1(0 ), 0~2(0) -- 0~1(0)} > 0 (SO ~j([s ---- %(Vt)/= /> 1, "~j(~t) = ~(tt)/0t >/ 1, for Vt near 0). By setting e-=~= z, we may extend the system to a neighbourhood of the origin in a C x fashion to apply the implicit function theorem and get a C ~ curve F o in the parameter space, tangent to the curve I' at 0, such that ~t ~ F 0 if and only if X~ presents a tangency between W"(~k_x(~t)) and W'(~k+l(Vt)). It also follows from the above equations that along P0 the manifolds W"(~_l(Vt)) and W'(~+x(Vt)) present an orbit of quasi-transversal intersection. For the singularities ~(~) with i < k -- 1, the transversality between W=(~,(0)) and W'(~(0)), and between WC~(~(0)) and W'(~+x(0)), guarantee the transver- sality between W"(~(~z)) and W'(%+1(~)) for tz near 0. 9 V 1 VI Lc(pL) FIG. II We now proceed to describe the above equations (El) in terms of tangencies between foliations: this geometric interpretation will be useful in the proof of the stability of the bifurcation of type L Let F~+l(t~) be the C" foliation in SO(or) which is compatible with W~(~k +l([z)) defined by ~l(tz, vl, v~, VI, WL) = (V 1 -- F([L, v2, /JI, WL) q- F(~L, V2, 0, WL) , 02, WL). It follows BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 121 from the quasi-transversality between W"(%_~(O)) and W'(ak(O)) that the restriction of n~_~ to W"(ak_x(~)) n S~(tz) is a submersion with fold along the set { e-"' G(~, xi( ~, e-", wL), e -~'', e -~'t, wr.), e-"*, e -~`t Xl([L , e-", w,), WL} where xi(~x , z, w~) is a C 1 function which is of class C m- ~ for z > 0. Hence, the subma- nifold L*(I~)C Sr defined by (v a, v2, v~X/~" xi/(~. , ~,,'/", w~), w~) v~ >>. 0 is transversal to F~x(tx ) and, to W"(*~-I(t~)) ~ S"(~) and it contains the locus of tangency between W~(%_x(~)) c~ S*(~) and F~_~(~); see Fig. II. Let F"(tx) be a codimension-two C x foliation in L*(vt) which is compatible with W"(%_~(~))n L~(t~) and with W"(.~(~)) ~ L~ Then, from the quasi-transversality between W"(%(0)) and W'(.~+x(0)), we get that W*(.~+I(~) ) c~L*(~) is tangent to F"(~) along a two- dimensional C ~ manifold Sg(~) which is the graph of a C 1 map w~, = f~L(~, V~), v~ >t 0. In short, the second set of equations in (El) above defines L*(~) and the second and third ones define S~(vt). In this way the bifurcation set is described as follows: a) The point W~(%(Vt)) n Sg(~x) belongs to the curve Wu(o'k+l(~L)) ('~ Sg(~L) if and only if there is a quasi-transversal orbit of tangency between W"(.~(tx)) and W*(.~+ a(~)), b) The curves T~(~) = W~(~+~(Vt)) c~ S~(~) and T"(~) = W"(~_I(~)) n S~(tx) are tangent in Sg(~) if and only if there is a quasi-transversal orbit of tangency between W~(%_l(~X)) and W*(%+x(~)). It corresponds to the curve I' 0 in the parameter space while condition a) corresponds to 17. I" o FIo. III We now further analyze the bifurcation diagram, specially the second condition above, in order to initiate the proof of the local stability in this case. First, notice that the curve T~(~) is a leaf of a foliation pg defined by a C 1 one-form + O(4+') dv2' > 0, in the region R~ = { v~ >/ ~ I Vl l ~'/'}" Notice that if ~1 q- el/> ~ then 0 is a hyper- bolic singularity of sink type with linear part equals to ~ v2 dvl -- ~1 vl dv2. If e2 > ~1 -/- ~1, then by setting w = v~ ~' + ,,>/-, we obtain, after dividing by ~2 w- 1 + ,,/c,~ + ~,~, the expres- sion of a C 1 one-form with a hyperbolic sink at the origin with linear part equals to ( 0o) (~x + ~1) w dw -- ~1 vl + ~1 W~y 1 (0) dv2. Hence, in both cases, the foliation p~ = 0 ~lv~l+~l'm'OG P~=~2v~dvl--~lVldV2--[ 122 M. J. DIAS CARNEIlZO AND J. PALIS has a singularity of sink type at 0 and is such that all leaves, except one, are tangent to the axis v2 = 0. Therefore, a tangency between p~ and T*(~z) occurs along a C I curve l~ aF v~, + ~,)/~, aG defined by ~2 V2~v 2 -- ~17)1 -- ~1 " Z 0Y + 0(V[ +') = 0, V 2 >/ 0. Using again the OF 0G hypothesis (1.9), which means that 0v-~ (0) 4:0 and Oy---- 1 (0) 4: 0, we obtain that/~ is transversal to T*(~) and to { 9~ ---- 0 } and the contact along t~ is quadratic. Thus, the bifurcation set is characterized by the position of the points Pk+I(V) ----- T~(~) n t~, pk_l(Ez)---= T~(~)r~t~ and pk(~z)= W~(a~(~))raS~(~z). By taking ~z in small neigh- bourhood V of R ~ and shrinking S(~z), we can modify p~ in a neighbourhood of the boundary of R~ in order to include the curve v~ = ~ I vl I (~'/~(~) as a leaf of p~ and to extend it linearly by setting p~ --= o~ 2 v 2 dv x -- ~1 vl dv2. The inverse image of p~ by the Poincar6 map between level sets P~ : Ak(~) -+ B~(tx ) gives a continuous one-dimensional foliation with C 1 leaves which are topologically transversal to A~(~z): this remark is easy to check by using the linearization of X~ and it will be important in the proof of local stability at the end of the paragraph. Suppose now that X~ is a nearby family. Let q0 : V -+ R ~ be a local homeomorphism which sends the bifurcation set of X~ to the bifurcation set of X~. Then, we define a homeomorphism h~: S~([z) -+ S~(q0(~z)) which sends p~ to ~(g) and the curve T*(tx) to T*(q~(~z)) as follows. We first define a homeomorphism between the lines of tangency tg, "[~,~) such that p~_l(tZ) is sent to/~_x(q~(~)) and P~+I(&) goes to/~+~(~?([x)). This is only necessary in the region tXl i> 0 or ~xz/> 0 since, otherwise, the curves T*(tz ) and T"(~z) do not intersect ~. This gives a homeo- OF morphism on part of the space of leaves of a foliation -~ defined by dv~ -- ~v z (~z, v~) dr. z = 0 and which has T*(~) as a distinguished leaf (T*(~z) is defined by v~ = F(~z, v~)). We extend such a homeomorphism to the axis v~ =: 0, v~ ~< 0, also preserving its intersection with T~(~z), to complete the definition of a homeomorphism on the space of leaves of ~. This yields a homeomorphism also on the space of leaves of ~. By the reparametrization above, when p~_l(~X) coincides with p~+x(~z), the same occurs with /~e_l(q~(~)) and ~+l(~?(t~)). Therefore, since ~ and ~ have quadratic contact along IF, we define h~ by sending leaves of ~ to leaves of 9"~(~) and leaves of -~ to leaves of $,(~). In the sequel we will fully develop the proof of local stability. (1.s The local stability of the bifurcation of type L -- Let )~, be a family near X~ so that all conditions described in (I. A) are satisfied with respect to a bifurcation value near ~ where )~ presents one orbit of quasi-transversality with a criticality. The equivalence between X~ and X,(~) (after choosing an appropriate repara- metrization q~) will be a conjugacy outside a neighbourhood of the closure of y ~3 y' in M. This neighbourhood U is the union of two distinguished neighbourdhoods U,_I and U, +a of the orbits of tangency y' and y, respectively, which were constructed in the previous section (Theorem A). Inside U the equivalence will preserve the level sets BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 123 of the potentials of X~ and X,(~, which can be assumed to be the same for both families. We suppose that a compatible unstable system F"(%([Z)), ..., F"(ak_2([z)) together with a center-unstable foliation F~U(ak_l([z)) have been already constructed and also a compatible stable system FS(~k+2(~)), ..., F'(~t([Z)) with a center-stable foliation F9 We consider similar compatible systems for the nearby family { X~} and assume that we already have homeomorphisms defined on the space of leaves of these foliations. We first construct a center-unstable foliation FCU(ak([z)) which is compatible with the unstable system and with FC"(ak_l([z)). Besides the presence of criticality, this last compatibility condition with a singular foliation is the novelty here. As in the proof of Theorem A, we begin by describing the central foliation Fc(%(Ez)) in a neighbourhood of the point of tangency rk(0 ) = y' c~ A~(0). Recall that A~([Z) is the sphere Ak([Z ) c~ W'(%([Z)), where Ak(~z ) is the part of the boundary of Uk_l([z) c~ Uk+a([Z) which is contained in a non-critical level of the potentialf~. The central foliation is constructed in C~([Z), a tubular neighbourhood of W"(%([Z))c~ A],(~) in A~([Z) which is bounded by two spheres 0D+([Z) and 0 D[([z). We take [z-dependent coordinates (x~, xx, xa) for the cylinder C],([Z) centered at rk(0 ) such that = { = 0 }, Wc"(%_~([Z)) c~ C~([Z) ----{(xl, xx, xz ---- G*"([Z, x~, xi) ) } and WU(6k_I([Z)) (~ C~([Z) = { X 1 = G([Z, xi) , x]. = G~(~, x,, xx) }. The leaves of F~(%(~t)) inside W'"(ak_I([Z)) c~ C~([z) are the integral curves of the vector field Z~ defined by A1 = [xl -- G(~, xi)]* + Z 0G ([Z, xi) , 0G = ([z, Hence, F~(ok(V)) has a "saddle-node " singularity at rk([z), the point of tangency between W"(ok_~(~z)) c~ C~(~z) and FS'(%([z)) n CZ([Z ). f~ Fro. IV 124 M. J. DIAS CARNEIRO AND J. PALIS Now, since F~"(%_I(~)) n C~(~t) is transversal to F8~(%(~)) c~ C~(~t) we are going to lift this foliation to each leaf of F""(%_1(~)). Recall that there are two types of leaves of F~(ak_l(~)) which are denoted by F~"(%_l(~X)) and F~"(%_I(EX)) with dimensions equal to (dim W"(%_x(~)) -k- 1) and (dimW"(%_a(~)) -~ 2), respectively. Let 0 < 81 < 8~ be small numbers and consider the following two conic regions in a neigh- bourhood of r~(~) in C~(~t): C(Sx) = (I xt -- G(~, xx)t2/> 8~ II x~ -- G~(pt, xl, xx)ll2 ) and C(~2) ={1 x~ -- G(V, x~)[2~< ~211 Xa -- G""(V, x~, x~)l]2 }. The numbers 8~, 8, are taken so that the boundaries of these regions are transversal to F~"(ek_l([x)). Inside C(~1) , we just lift the foliation F~(ek(V)) C W~ to each leaf of F~"(%_I(V) ) via the projection (Xl, xi, xj) ~-* (xl, xj). Hence, all leaves of type F~"(%_I(V)) are subfoliated by a one-dimensional foliation diffeomorphic to the one described above. In the region C(~2) we take the codimension-one foliation defined by the non-positive level sets of the map (x~, x2, xa) ~-* " ] xa -- G(~, x~)] 2 -- ~, [] x a -- G~"(~, x~, x~)]] ~ and intersect it with F~"(%_~(~x)) to obtain a codimension-one foliation transversal to F*~(%(~)). The central foliation, in the intersection of F~"(a,_~(~)) with C(82), is given by the one-parameter unfolding of Z~ lifted to intersections of levels of ~h with leaves of F~"(ak_a(V) ). That is, ifF denotes a leaf, then in F r~ n-l(0) we just lift the vector field Z~ and in F n r~i-a(-- a) we lift the perturbed field Zi. ~ = Z i + a Oxt The region between C(~1) and C(~2) is used to match continuously the above foliations. To do that, we need to modify the intersections of F~"(%_~(~)) with the complement of C(8~) in order to include the boundary of C(82) as a new leaf. In doing so we can glue a singular central foliation near the tangency point r,(0) with a non- singular foliation F~(a,(~)) in C~(~t) which is compatible with the unstable system F"(~(~)), ..., F"(%_~(~)), F~ see Theorem A. Before concluding the construction of Fo~(%(~)), we indicate how to extend a homeomorphism hl, defined on a neighbourhood of the tangency point r,(0) in Wo"(%_l(~))r~ C~(~t) and which preserves the central foliation F~(a,(~)), to a full neighbourhood of r~(0) in C~(~). Inside the conic region C(8~), we just use the homeomorphism in the space of leaves of (~_~(~)) and the projection (xa, x~, xa)w-~ (x~,x~) to lift h i to each leaf of F cu F cu (~ (~_~(~)) c~ C~(~). Then, since F~(a~(~)) is non-singular outside C(8~) we extend it to a neighbourhood of C(~2) t~ 0 D+(~) in the sphere O D+(~). This neighbourhood is taken to be bounded by non-singular levels of n~ and in this boundary the homeo- morphism is actually a diffeomorphism C 1 close to the identity. Therefore, using the Isotopy Extension Theorem we can proceed as in Theorem A to extend it to all of 0 D+([x) in a way that it is compatible with the homeomorphism defined on the space of leaves of the unstable system. In the following lemma, we show how to obtain this homeomorphism h~. Lemma 1. -- Let F~(~) be a central foliation for the family X~(~) defined by a vector field Zi which has a saddle-node at ~(~) as above. Then, there is a homeomorphism h i defined BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 125 on a neighbourhood of r,(O) and depending continuously on ~, which sends leaves of F~(B) to leaves of and to Proof.- Using the above coordinates to describe WC"(ek_a(F))nWCS(~,(~)), we observe that the projection (Xl, xi) ~ xl, along the leaves of FS"(ak(~)), gives a Liapunov function for Z~ and that Z~ restricted to W"(ak_l(F) ) n C~([x) has a hyper- bolic singularity at rk(~). Therefore, the proof consists in showing that there exists an equivalence between Z~ and Z,(~) near saddle-nodes which preserves the level sets of a Liapunov function. To simplify the notation, we drop the parameter ~ in the following arguments and denote by W~(Z~), j = u, s, cu, cs, the invariant submanifolds of Z c. For a > 0, we let D- = { X 1 = -- ~; } and D + = { x 1 = * } be non-critical levels of such that for small tz the singularities rk(F) and ;k(q0(tz)) are contained in Ix, 1< e. We first obtain an equivalence on W*"(Z~), starting by taking a C 1 diffeomorphism close to the identity from the closed disc D-(Z c) = W"(Z c) n D- to W"(Z ~) n D-. Then, we take a tubular neighbourhood of D-(Z ~) in W*8(Z ~) n D- with fibers forming a radial foliation ~8. Each fiber of ~" is a C 1 curve transversal and exterior to the boun- dary 0 D-(ZC). Positive saturation of ~ by the flow of Z r intersected with D +, gives rise to a one-dimensional foliation in W~"(Z *) n D + which is singular at the point Wc(Z c) n D +. Hence, performing the same construction for 7,~, we define a homeo- morphism from We'(Z ~) n D + to W""(Z e) n D + which preserves this foliation and it is a diffeomorphism outside the point We(7, r n D +. By preserving the level sets of and the trajectories of Z ~ inside WeS(Ze), we obtain an equivalence between Ze 1 Wc"(Z c) and ~c j W~,(~c). Proceeding dually, starting now at the level D +, we get an equiva- lence between Z~[ Wc~(Z ~) and 2~ w~-(2o). The corresponding radial foliation in Wc"(Z c) n D + is denoted by b". We are now going to match these two equivalences. In D-, we raise over each point of the disc D-(Z c) a continuous foliation g" with C 1 leaves, transversal to Wc~(Z~), which is compatible with W"(ok_I(F)) n D-. Each leaf of g" has dimension equals to d, = dim W""(Z"), and Wc"(Z c) n D- is taken as a distinguished leaf. In the complement of the component of D-\W*(ak_l(bt)) which contains Wc"(ZC), we take a (d, -t- 1)-dimensional continuous foliation g~, with C 1 leaves, which is trans- versal to W~"(Z ~ n D- and such that the boundary of each leaf of g~ is a leaf of g" in W~(ak_~(F)) n D- and g~ n Wc"(Z ~) is the radial foliation ~". Dually, we construct in D + the foliations g" and g[ with dimension d 8 = dim W"~(Z e) and (d, + 1), respec- tively. Still denoting by g~ the intersection of the positive saturation of g~ by the flow of Z e with D +, we observe that for each leafg~, ~ ofg~ the intersection g~ n g[, ~ is a one- dimensional foliation which is singular at the point b = g[,~ n 0D+(Z *) C W"(,,_I(B)) n D +. Hence, as in the proof of Theorem A, we take two families of closed conic regions E C g~, ~ with vertices at b, such that We*(Z c) n gl, ~ is contained in the interior of E~. We then modify g"n g~, ~ to get a new one-dimensional foliation oct ~ such 126 M. J. DIAS CARNEIRO AND J. PALIS that ~bn E~ = gun E~ and outside E b it is non-singular and transversal to g~ ----- g~, b n W~(%_l([z)). Clearly, this can be done depending continuously on b and Ix. Once the same construction for Z~ is performed, we are prepared to define a homeo- morphism h + on D + and conclude the proof of Lemma 1. The basic property of h + is that by preserving 9ff b it induces (via projection along the trajectories of the respective vector fields) a homeomorphism on D- which is a continuous extension of the homeo- morphisms already defined on D- n W~(Z ~) and D- n W~8(Z~). The definition of k + goes as follows. Let A+ be the closure of the component of D+\W"(%_I(~Z)) that contains the disc D+(Z ~) (i.e., the set x 1 = ~; G(~, xi) t> ~). In A +, we take a continuous foliation by C 1 closed discs compatible with g~ (the positive saturation ofg~), which is transversal to Wc*(Z ~) n D + with complementary dimension, such that D+(Z ") is a special leaf. The boundary of each of these discs is a sphere in W"(%_x(&)) n D + which is trans- versal to the foliation g~ n W*(ak_~([z)) , with complementary dimension. Thus, by preserving this family of discs and the foliation g8 and using the homeomorphisms already defined on W~'(Z ~) n D + and on W""(Z ~ n D + = D+(Z ") (space of leaves of these foliations) we obtain a homeomorphism on A+. This gives a homeomorphism on the space of leaves of 9ff b which are outside the conic region E b. Since in the space of the leaves of3r ~b which are inside E b is the disc D-(Z ~) (where we also have defined a homeomorphism), we have obtained a homeomorphism in the total space of leaves of 3r ~b. To complete the definition of h +, it is enough to preserve ~b and a codi- mension-one foliation whose space of leaves is W~'(Z ") n g~. b, which can be defined by G(~z, xi) = a, a ~< r This concludes the construction of h +. As observed above, the equivalence between Z ~ and Z~ is obtained by preserving levels of n, using h + and preserving trajectories of the fields. 9 A center-unstable foliation F""(ak(~)) can now be defined as in the proof of Theorem A except in a neighbourhood of rk(0 ). In this neighbourhood, we want to distinguish a leaf that contains the tangency point rk(~z ) and contains the possible tan- gencies between W'(%+1(~)) and W"(ak_a([Z)). So, we let Z~ be a neighbourhood ofrk(0 ) inAk([z ) n W"~(%+x(~)) n W""(%_~(~s and consider coordinates (xl, xi,yl,y~), as in (1 .B), to define a continuous family of vector fields Y~ in ~"(~) fory x >1 0 by Xl = [Xl -- G(Ez, XDyl,Y~/~'YT.)] 2 -? y" OG ,=8 ~ (~' xI'yl'Y~J~'YT') ~'' OG OF (y~,./~,,y~/~, x~,yT.) Y~-- ""/~' 4 = ~ ([z, xi,yl,y~J~'_yL) -- ~ j~=O, )t=O. Y~ is tangent to W"(%_I(~Z)) n Y~([z) and its restriction to A~(~z) is equal to Z~, the vector field which defines the central foliation F"(%(~t)). For fixed (~.,y~,y,), Y~ also has a singularity of saddle-node type and its singular set is a C ~ manifold contained in W"(%_a(~)) n Y~"(~). Moreover, the image of this singular set by the Poincar6 BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 127 map P~,: Z*([z)--~S*(~t) coincides with the set of points of tangency between W~(%_I([z)) n S*(~t) and the foliation F~+l(~t) defined in (1.B). In particular, this image contains the tangencies between W~(%_l(~t)) and WS(%+l(~t)) in the level set Bk(~x ). Therefore, over each leaf of F*(%(~z)) we can raise a (u + 1)-dimensional sin- gular foliation in Ak(~t ) ca W~ such that its intersection with Z*(~t) is tangent to Y~. The positive saturation of this foliation gives part of the leaves of F*~(%(~)) which are contained in WCU(ak_l(~x)). The process to define FC~((rk(~)) inside the other leaves of F*u(%_~(~t)) is analogous to the one described above to obtain the central-foliation F*(%(~t)), i.e. one uses conic regions and projections onto W*~(%(~t)) ca A~(~t). Since outside a neighbourhood of the tangency point rk(0), F*"(%(~t)) is exactly as in Theorem A, we have completed the definition of F*~(%(~t)). Now comes the main step in proving the stability of the bifurcation of type I: to define a homeomorphism in the level Bk(~t ) which preserves the intersections of leaves of the stable system and the center-unstable foliation FC~(ak(~)). Since the stable system is transversal to the singular set of F*~(%(~t)), it is transversal to all leaves of F*~(%(~t)) outside a neighbourhood of the tangency point r~+x(lz). Hence, it is enough to obtain a homeomorphism in a small neighbourhood S([x) of rk+l(~Z ) in Bk(~t ) and proceed with a cone-like construction as in Theorem A outside S(~t). The same is valid in the section S*~(~t) = S(~t) ca W~ since WC"(%(~z)) is transversal to W"(%+~(~t)). The novelty here is to obtain a homeomorphism on the dual section S*S(~t) = S(~) ca W~*(%+~(~)). Fol- lowing the methods of the non-critical case (Theorem A, Section II), we want to preserve the intersections of FCU(ak(~)) with S~"(~z) and a C '~ foliation F~*, which is compatible with W'(%+x(~)) and transversal to W*~(%(~)) ca S*"(~) with complementary dimen- sion, m >t 3. However, due to criticality, this process must be modified in a neighbourhood of L*(~t). Let us recall the notation used at the end of (1 .B). First, ~u 0" v8 S~(~) ---- W (k_~(~t)) ca S~s(~t) and F~+~(~t) is a C" foliation in S~(~) which is compa- tible with W"(~e+a(~)) and has codimension equals to dim [W~(%(~t)) ~ S"(~t)] q- 2. Further, L~(~t) is a submanifold of S~ which contains the set of tangencies between F~_l(a) and W"(v~_~(~z)). In L'(~) there is a C 1 codimension-two foliation F~, which is compatible with W"(%_x(a)) ca L~(~) and with W"(%(~t)) ca L~(a). Finally, S~(a) is a two-dimensional manifold of L~(a) which is transversal to W"(%(~t)) and contains the set of tangencies between W'(%+l(~z)) and F~,. The curve T"(~x) = W"(%_a(~Z)) ca Sg(~t) is a leaf of a singular foliation defined by the one-form ~ and the curve T'(~t) = W'(%+1(~)) ca Sg(~t) is a leaf of a foliation %. These foliations are defined in (1. B). We now start constructing a homeomorphism h~ s : S~"(~t) ~ S~"(q0(~)). Using the reparametrization q~(~x), also obtained in (1. B), we define a homeomorphism from Sg(~) to Sg(~(~t)) that preserves the singular foliation p~ and the curves T'(~t) and T~(~t). Next, we extend the homeomorphism to L'(~t). We take a two-dimensional foliation (SN)~ in L~(~x) which is compatible with W'(%+I([z)) and is singular along the curve T~(~t). 128 M. J. DIAS CARNEIRO AND J. PALLS The foliation (SN)~ is tangent to a family of C a vector fields Y~ with singularity of saddle-node type defined by OF ~ = [v~ - F(~, v~, O, w~,)] ~ + ~ ~w~ (~' v~, O, wd w~, OF OG ,,1/~ .... ~ "~"~" @'~'wL) v~'+~Y% ~v~ Ow~ (~' v~, o, w~) ~ (~, x~(~, P2 ~ 0. S~(tt) is a distinguished leaf of (SN)~, and F~ is transversal to (SN)~ except along the curve T~(~). Hence, the intersection of (SN)~ with (~)-1 (v~) is a continuous one- dimensional foliation transversal to WC"(a,(~)) ca SC'(~), where ~ is the projection into S~(~) along the leaves of F~. We can apply Lemma 1 to obtain a homeomorphism WC~(ak(~)) n SC"(~t) to WC~(~k(~(~t))n S~'(9(~t)) which sends trajectories of Y~ to trajectories of ~',(~) and also preserves W~(ak(~)) n S~S(~). This gives a homeomorphism in the space of leaves of (SN)~. We now have homeomorphisms defined on the space of leaves of two complementary foliations: S~,(~) (whose leaf space is S~(~z)) and (SN)~. Thus, we have a homeomorphism from L*([z) to [,*(~?(~z)) which preserves W"(%([z)), W~(%_x([z)) and W'(,k+a(~z)). We extend this homeomorphism to all of S*'(~z). We recall that there are two types of leaves of F*"(,,([z)) which are denoted by F~"(,k(~) ) and F~"(a,(~)) such that dim F~"(%(~z)) = dim W"(%(Ez)) + 1 and dim F~'(%(~z)) = dim W"(%(~)) + 2. The foliation F~"(,k(~) ) has a saddle-node type singularity along a u-dimensional sub- manifold, which contains the tangencies between W"(,k_x(~)) and W'(%+l(~z)). Outside this submanifold, F~"(ak(~z)) is transversal to WS(,k+l(~Z)). Let us construct a foliation F~ * in SC*([z) which is compatible with W~(%+x([z)) and transversal to W*"(%(~z)) ca S**([z) with complementary dimension. We first take a C m foliation F~, ~, m/> 3, which is compatible with W~(,,(~z)) and transversal to S*(~z) such that F~ *' ca S*(~) = F,+i([z )*~ ; the foliation F~,*s is defined by a submersion r:~ *~. We then define F~, ~ by taking the pull back via ~" of the one-dimensional foliation in L*(~z) defined by (~,)-~ ('~) ~ (SN)~. By construction, L~(bt) is a typical leaf of F~"(%(~t)) ca S~(~), i.e. the restriction of r~' to each leaf of type F~"(a,(~)) ca SC"(~z) near L~(~) is a homeo- morphism onto L~(~t). In particular, this is valid for all singular leaves of F~"(~,(~t)) . Therefore, we can define a homeomorphism on a neighbourhood of L~(~t) using the two complementary foliations: F~"(a,(~t)) ca SC"(~t) and F~ s. Such a homeomorphism has one important property: it induces a homeomorphism in the level set A,(~t)\A]~(~t) which extends continuously to the sphere A~.(~t) = A,(~t) ca W"(a,(bt)). Let us explain this point. Denote by P~" : S~"(~t) ~ A,(~z)\A~(tx) the restriction of the Poincar6-rnap. Since F~ ~ is of class C", m/> 3, and the singular foliation p~ is preserved on its quotient space, the image by P~" of the singular foliation induced on each leaf of F~"(a,(~)) by z:~" is a continuous foliation with C ~ leaves which are topologically transversal to A~(~t). Therefore, a homeomorphism which preserves F~"(a,(~t)) and F~," extends automatically to the sphere A~(~t). As we pointed out above, the idea is to construct h~,' by preserving F~ ~ BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 129 and FC"(,k(V)) outside a neighbourhood A,(V) of Lc(V). Inside A2(~) it preserves F~ 8 and F*~(ek(~) ). We detail this construction. Let AI(V)C A~(~) be two wedge-shaped regions in SC*(~) that contain LC(~) in their interior and which are bounded by non- singular leaves of type F~(ak(~)). We also require that all singular leaves of FC~(%(~)) are contained in the interior of AI(~). Each Ai(~) , i = l, 2, is the image by the Poin- card-map Ak(~)-+S(~) of a solid cylinder transversal to A~(~). Inside the subset VS\-- I ~ ) (R~) n AI(~) , R~ as in (1. B), we preserve the two complementary foliation F~ ~ and F~(ak([z)) n S~8([z). In the complement of this set in AI([~ ) we preserve the comple- mentary foliations F~(~k(~)) c~ SC*(~) and F'*([~). Since the intersection of the boundary of / w\--I (rcv) (R~) with each leaf of F~"(%([z)) n S~"(~x) is a leaf of type F~"(%([x)) and F~ ~ is a codimension-one foliation in F~", we obtain a homeomorphism on Al([x). In the complement of A2([x ) in S"'(~t) we proceed with the cone-like construction of Theorem A to define a homeomorphism preserving F~ ~ and F~"(ak(~)). The region A2([x)\AI(~t ) is now used to match these homeomorphisms. Notice that each non-singular leaf of type F~"(ak([z)) is parametrized by a point in the sphere A~*(a) = Ak(~x ) n W"8(%(a)). We assume that the boundaries of AI([x ) and A,(~t) correspond to codimension-one spheres S~ and S~ in A~"(~x) centered at the point of tangency r,_~(0). Hence, the matching is done as we move radially from S~ to S~ preserving F~* in a subset R, of R,. This subset is bounded by the pre-image of two leaves of p~ whose distance gets smaller as we approximate the outer sphere S~. Finally, when we reach a point in S~, this region collapses into the unique leaf of a~ which is transversal to the axis v2 = 0. The picture illustrates this process in a section complementary to W"(%([x)) in S~*(~x). It shows how the region R~, foliated by leaves of p,, shrinks to a curve. leof of P,u. - R x x inA1(~) y inA2(zU)\ A|(/U,) neor ~2(~) ze a~(y) Fro. V 17 130 M. J. DIAS CARNEIRO AND J. PALIS In this way we have obtained a continuous family of homeomorphisms h2 : -+ As pointed out above, this is enough to get a homeomorphism on the neighbourhood S(~) of the tangent point rk+l(~X ) in the fence Bk(~). We can now, using the methods in Theorem A, obtain an equivalence between X~ and IK,~) on the neighbourhood U(~) = Uk_l([x) w U~+I(Ex ) of the orbits of tangency 7' and 7 preserving level sets of the potential f~: this is possible because we have preserved the center-unstable foliation F~"(%_t([z)) throughout the process. The extension of this equivalence to all of M is done as in Theorem A: outside the distinguished neighbourhood of the orbits of tangency we obtain a conjugacy between X~ and X,~. w 2. Bifurcations of type II: two orbits of quasi-transversality (2. A) Description of the bifurcation. -- This is a codimension-two bifurcation presented by families { X~ } in ?(~(M) such that, for ~ ~ 112, the vector field Xg presents exactly two orbits Yx and Yz of quasi-transversal intersection between stable and unstable mani- folds of hyperbolic singularities: Y1C W"(pl(~) ) n W'(ql(~)) and y, C W~(p~(~)) n W'.(q~(~)). In addition, we assume the following conditions: (2.1) C" linearizability of X~ near each of these singularities with the eigenvalues of dXg at these points having multiplicity one, m being sufficiently large as specified in Section I, (2.2) Non-criticality of any other singularity with respect to the strong-stable or the strong-unstable manifolds: if p e M is a singularity of Xg different from Pl(~), P2(~), ql(~) and q~(~), then W"(p) is transversal to WS"(pl(~)) and to WS"(p,(~)) and W"(p) is transversal to W""(ql(~) ) and to W~"(q2(~)), (2.3) W~"(pi(~)) is transversal to WS(q~(~)) and W~'(q~(~)) is transversal to W"(p,(~)) for i= 1, 2, (2.4) Generic and independent unfolding of the orbits of tangency of the family X~, so that there exist two C 1 curves F 1 and P 2 in the parameter space crossing each other transversally at the point ~ such that ~ e F ~ if and only if W"(p~(v)) is not transversal to W"(q~(~)), for i --= 1, 2. We distinguish two possibilities, (II. a) and (II. b), that will be treated separately: a) two of the above singularities coincide, namely ql(~) = Pz(~) or qz(~) = Pl(~) (which are dual) or the easier case Pl(~) = P~-(~), b) all singularities above are distinct. BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 131 (2.B) The bifurcation diagram of type (II.a). -- Let us first assume that Pl(~) =~k--l(~), ql(~) =P,(~)=%(~) and q,(~) =~k+t(~) in the ordering al(~) ~< ... ~< %(~) of the singularities of X~ described in section I. We begin by ana- lyzing the restriction X~ of X~ to the center manifold W cuff w~ = (k-l(~)) n w0,(~k+i(~)) near %(t~). Let (xl, xi,yl,yT.) be C" linearizing coordinates for X~ near ~k(Vt): 0 0 0 0 ~ = Z ~,(~) x, + ~l(~)yl + y- ~(~)yj Oy~ with 0 < ~l(t~) < ... < ~,(~), 0 < ~1(~) < ... < ~,(~), x1 = (Xl,.-., xs), Y*. = (Yx,.--,Y~), u = dim[W"(ek(~)) n W*'(.~+I(~))] and s = dim[W'(ek(~) ) n WC"(%_l(Vt))]. By the quasi-transversality assumption in a cross section l~c(Ex)C{x 1 ----1 }, we have W~(%_l(~))r3Xc(W)={(xI, G(~,xI,yL),yL)} , with 13 being a C '~ function such that x I --~ O(~, xi, 0) has a non-degenerate critical point at 0. Hence, we get from the generic unfolding of the orbit Y1 that the map Vt ~ G(~, xt(~) , 0) is a submersion at ~, where xi(vt ) is the solution of 0G Ox~ (~' xD O) = O. Also, by taking coordinates (vx, vi, wL) in a cross-section S*(t~)C {Yl = 1 } such that W"(~k(~) ) n S~(tz) = {(0, O, wr.)}, we have ws(~+~(~)) n s0(~) = {(F(~, v~, w~), ~i, w~)}, where F is a C '~ function such that w L ~* F(~, 0, WL) has a non-degenerate critical point at 0. Hence, the conditions of generic and independent unfolding imply that the map (~, xi, w,~ / ~ G(~, ,~, 0), ~ (~, xi, 0), F(~, x. 0/, ~ (~, O, ~,./ is a local diffeomorphism at the point (~, O, 0). Therefore, if ~x ~ WL(~X ) is the solution OF of (~x, 0, w~) = O, then the curves Ow L r~ = { G(tz, xi(~) , 0) ---- 0 } and Fz ---- { F(~, 0, WT.(~)) -- 0 } belong to the bifurcation diagram near ~. Also, Vt e Fx if and only if W~(%_~(~)) is not transversal to W'(%(~)) and ~ e Fz if and only if W~(%(tz)) is not transversal to W*(%+~(iz)). Furthermore, the intersection of W"(%_l(iZ)) with S*(~) is dcscribed by the equations v~ ---- e -~''l')t, vx = e -~'(g)t xi, e -~"~)~ -- G([s XI, e -~(~)~ Wr,) ---- O. Using that the bifurcation unfolds generically and the implicit function theorem, we obtain a third C 1 curve I'3 in the parameter space tending to ~ (but disjoint from F 1 M. J. DIAS CARNEIRO AND J. PALIS and F~ outside this point), such that ~ ~ Pn--{~ } if and only if W"(%_~([~)) is not transversal to W"(a,+a(~)). Along the curve F a the family X~ presents one orbit of quasi-transversality between W~(%_x({x)) and WS(a,+x(~)). 1"1 1"~ 1-, 2 FIG. V[ (2.C) The stability of the bifurcation of type (II.a). -- As in Theorem A, we focus our attention to a neighbourhood U(~) of the closure of the orbits of tangency Ya and 7~ in M which is constructed by glueing together distinguished neighbourhoods of these orbits. We construct in U(~) flow equivalences that preserve compatible systems of foliations, so that they can be extended to flow equivalences on all of M. Suppose we have already constructed a compatible unstable system F"(el(~), ..., F"(~k_2(~)), F*"(ak_l(~)) and a compatible stable system Fr , ..., FS(%+2(~x)), ..., FS(%(~x)), together with a homeomorphism in the space of leaves of these foliations. We start by constructing a center-unstable foliations F*"(%(~)) com- patible with the unstable system whose main leaf, W*"(%(~)), is a C 1 invariant manifold contained in W*"(ak_l(~)) and transversal to W"(%_a(~X)) and which contains all possible tangencies between W"(~,_ 1 (~t)) and Ws(a, + 1 (~x)). This construction resembles very much the one done in w 1 for the orbit of tangency with criticality. In the cross section Z~(~) C { x I = 1 } consider coordinates (xi,yl,yT.) centered at rk(0 ) ----- 71 n Z~(0), as in (2.B) above. Let the vector field Z~, tangent to W"(ak_l(~t)) , be defined by 0G Jl = (Y~ -- G(~t, x~, y~IL/~yL)) 2 + Z ~, (~x, xI,y~L/~'y~) ~, 9, = ~ (~, x I, y~l~/'yL) -- OF yl,,_,,,/0, "~v, ' j~ = 0, 1 -- ~ (Y~'/~' "DYT.) eI X /~=0, i---- 1,...,s, fory~>0. Since [~j(~)>~(~) and ~,(~)>e~(~) for i/> 2 and j1> 2, this extends to a (I ~ vector field in N~(~) which has for each (~,y,.) a singularity of BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 133 saddle-node type. The singular set of Z~, Sing(Z~), is a submanifold of dimension u of W"(ak_x(F) ) c~ E~(~) which is topologically transversal to Ws(a,(~))nZ~([x). Its image by the Poincarf-map P~:Y,'(~)~S~(~) contains the tangencies between W~(ak_~(F)) and W'(a,+~([x)) in S'(~), a cross-section in {Yx = 1}. We consider a foliation F~([x) in Z~(~t) which is tangent to the vector field Z~ and singular along Sing(Z~) having C 1 leaves of dimension (u + 1). We distinguish a leaf M~([x) which is transversal to W"(ak_~(~) ) and such that M~(F)c~W"(ak_l([X))= Sing(Z~). Let F~(~t) be a uk-dimensional (u k --dimW"(ak(~) ) foliation in W'"(a~_~(~))nZ(~) which is compatible with W"(ak_a(~) ) n Z(~) and such that F~([x) n Z'(~x) = F~(~). Positive saturation of F~(~x) gives part of the center-unstable foliation F~"(~(~)) inside W'"(~,_I(F)), which has a distinguished leaf denoted by W~"(~). In the next figure we see these leaves in a slice complementary to W"(~,(~)). The construction of the other leaves of F'"(~(F) ) corresponding to points near the singular set Sing(Z,) follows as in w 1 of the present section. Dually, we obtain an s,-dimensional singular foliation F~(~), (s~ = dim W'(~(~t))) in the level set B~(~) = [f~l[f~(ak([X)) -F ~]] which is compatible with the stable-system F~'(~+~([x)), F"(~+~(~)), ..., F"(~t(~x)). We denote by M~(~) the distinguished leaf of F~(F) that contains the point p~+l(~X) = yz(~x) n B~(~). ()a~ n s c" ()LI) " (oK_l(F) )n s Cs (p) w" (ok ()a))ns 9 I)~) FIG. VII Let Xc(~t) be the C 1 curve defined by X'(~) := W~([z) n M~(F) n S~(~) and consider the points pe_l([Z) = W"(~e_l(~z)) n V(~t), W~(ak(F)) n V(~) =Pk(F) (= ( 0 }) and Pk+ I(B) = W"(ak + l(~t)) c~ V(B). Notice that p~_ 1([z) is only defined for B on a connected 134 M. J. DIAS CARNEIRO AND J. PALIS component of V -- Pl, where V is a neighbourhood of ~ in R ~. The curves P~ and F 3 of the bifurcation set correspond to {p~+l(t~)=pk(tz)} and {pk+l(tz)=p~_l(~z)} respectively. We start the construction of an equivalence between X~ and a nearby family X~. Consider a reparametrization ~0: (V,~)-+ (R 2, ~), such that q~(Fi)= Fi for i = 1, 2, 3 and which sends the regions A,~ between the curves onto corresponding regions 2~ as in the picture (Fig. VIII). To obtain an equivalence between X~ and X~(~, we first want to define a homeo- morphism on the level set B~(W) which preserves the foliations F~(W) and F~(~) ~ B~(tz). The main step is the construction of a homeomorphism on S(tz), a neighbourhood of p~+~([~) in B~(lz). Let M~"(~t) = W~"(tz) r~ W~(%+~(~z)) c~ S([z) and let (v~, w~) be a system of coordinates for M*"(~t) such that ( vl - ~(~, w~) = o } = w~(~+~(~)) r~ M0~(~), { v~ = 0 ) = W"(%(tz)) c~ M~"(tz) and WU(6k_l([s n M""(t~) = { v~ ,/~', -- G(t~, v~"/~".w,,) = O, v~ > 0 }, where F, G are of class C 2. Hence, by construction the foliation F~(B) r Mr is tangent to the vector field Y~ defined by eg ~ = (vl - Y(~, w~))~ + Ow~. (v., wd .~,~, - ~14"" + Oy~ (~' v~'~'" w~). ~,~, w for v 1 > 0. We extend it to v I x< 0 by setting ~ 0F /?1 = (Vl -- W(~, wL)) 2 -Jr- ~w (~' wE) /i/L' w~ = ~w~ (~' w~). Let L"(~, vl, w~) = [v~ ''~''(~' -- G(~, v~ L/~''~' WL) + G(~, 0)] ~'("'/~'~' be a C 1 sub- mersion defined for vl~> 0; observe that (L~)-I(G([x, 0) ~'/~'(~l) = W~(%_l(tz)) and (L~)-x(0) = W~(%(~t)). For vl < 0, we extend it as L*(~t, Vl, //JL) = Vl" It is easy to check that -- L"(~t, vl, wr) is a Liapunov function for the vector field Y~. We apply Lemma 1 of w 1 to get a homeomorphism M~"(~x) -+ l~IC"(q~(~x)) which is a topological equivalence between Y~ and ~r(~) preserving the level sets of the respective functions L~ and I,~. The same procedure is used in order to get a homeomorphism on the cross section M"*(~)= W~*(~)n W""(%_1(~))r~ Y~(~), where E(~)is a neighbourhood of %(~) in Ak(~). Now, to complete the definition of the homeomorphism on the cross-section BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 135 r ~ r 3 A12 / J Az3 r 2 A21 A12 FIG. VIII S(~) n W"(ak(~t)), which corresponds to part of the space of leaves of the foliation F~([z), we take a C 1 foliation on W*(%([z)) n S(~t) transversal to Wes(O'k+l(~)) ('~ WU(qk(~)) ('~ S([~) and of complementary dimension. We make the same construction for W"(~(~(~))) n S(~(~)) and obtain the extension of the homeomorphism by requiring that this foliation be preserved and also the intersections of the center-stable foliation F"(~k+l(~) ) n W"(%(~)) n S(~). This homeomorphism is further extended to the whole sphere (fundamental domain) W"(~(~t)) n Bk(~t ) by preserving the stable system FS(%+2(Ez)), ..., FS(~rt(~t)) in a compatible way with the homeomorphisms on the space of leaves of these foliations. Similarly we obtain a homeomorphism on the space of leaves of the center-unstable foliation F'"(~k([z)), which is compatible with the unstable system F"(~x(~t)), ..., The homeomorphism on the level set B~(~t) is then well-defined since we want it to preserve the complementary foliations F~(~t) and F~"(~t) n Bk([z ). As we saw in Theorem A, Section II, this is enough to obtain the equivalence in the neighbourhood of the singularity %(~z). The arguments to define the equivalence in B~_a(~) and, afterwards, its extension to all of M are now very similar to those in Theorem A. (The corresponding facts in case (II. b) are somewhat more delicate and will be treated in more details in the sequel.) 9 (9.. D) The case of two orbits of quasi-transversality corresponding to disjoint pairs of singu- larities (type II. b). -- This case goes much in parallel with the previous one: the main difference consists in a more careful construction of an equivalence for the two nearby 136 M. J. DIAS CARNEIRO AND J. PALIS families. This is due to the existence of intermediate singularities between the ones corresponding to the orbits of tangency. Let X~ be one of these families and let us order the singularities at the bifurcation point ~, so that Y1 is an orbit of quasi-transversality between W~(a,(~)) and W"(a, + 1(~)) and Y2 is a similar orbit between W~(a,+,(~)) and W"(a,+,+~(~)) for some k/> 2. We first observe that the construction for ~x near ~ of compatible unstable and stable systems, which are now denoted by F~(~z), ..., F~"(~), Fg~(~z), F~'+~(~z),..., Fg~_,(~z) and F~ ,+~k~z), , , F~+~(~), ~ .. ., F ~+,~), ~s , , F~+~+~(~z), ~ . .., F}(~z), respectively, is very similar to the previous case. The difference is that now we have to construct F~+~(~z) for 2 ~<j~< k -- 1. To do this, we just note that although like before F~_l(~Z ) is a singular foliation, its singular set, Sing(F~_ t(~x)), is the union of two manifolds which are trans- versal to W~(a~+~(~)) for j~> 2. Moreover, since each leaf of F~_~(~) accumulates in a C a fashion on Smg(F,+l(~x)), any foliation on this singular set can be extend to the leaves of F~_~(~t) in a continuous way. Thus, when constructing F~+~(~), in a compa- tible way with Fr it is enough to do so in Sing(F~_~(~z)), and then extend it to each leaf of Fg~_~(~). The same reasoning applies to F~'+~(~), 2 ~<j~< i -5 k -- 1 and to V;~_~(~). In the construction of the above foliations, we can also require W~(~(~x)) to be foliated by leaves of F~(~). In particular, since W'(e~+~+,(~)) is transversal to We"(~i+~(~)), we conclude that W'(~h+~+l(~)) is transversal to W~(~(~)), ~x near ~. Thus, the bifurcation set of X~ near ~ consists exactly of two C 1 curves r 1 and Pz that intersect transversally at ~: ~ e F~ if and only if W~(e~(~x)) is quasi-transversal to W"(%+~(~x)) and ~ e Fg if and only if W"(~;~+~(~)) is quasi-transversal to W'(%+~+~(~)). (2.E) Local stability. -- Let X~ be a family of type (II.b) and let X~ be a nearby family with main bifurcation value ~ near ~. Let (F, G) : (V, ~) ~ (R 2, 0) be a C 1 map defined in a neighbourhood U of ~ in R * such that F-l(0) = { ~z ~ V; W"(a~+k(~z)) is quasi-transversal to W"(a~+k+l(~)} and G -1 ={ ~x E V; WU(a~(~)) is quasi-transversal to WS(a~ + l(~x)}. By the hypothesis of independent unfolding, (F, G) is a local diffeomor- phism. Therefore, if (~', G) is the corresponding map associated with the family { ~2~ }, we can define the reparametrization q~ = (~', G)-lo (F, G). To prove that X~ is equivalent to X,(~) we take two distinguished neighbourhoods U~(?) and U~ +k(~t) of the closure of the two orbits of tangency, ~'1 and 72, as constructed in Theorem A. Inside these neighbourhoods the equivalence h~ will preserve the level sets of the potential function f~ and outside them it will be a conjugacy. The idea of the proof is to first define a continuous family of homeomorphisms on the space of leaves of the unstable F ~" ' ' The important point here is to preserve the leaves system from F~(~z) up to ,+kk~z). of the stable system which are contained in the stable manifolds. Dually we define a family of homeomorphisms on the space of leaves of the stable system from F}(Ez) to F~_ ~ (Ez). We then obtain a homeomorphism on the fence Bi + k(~) C f~- ~ (f~(~ + k(~) ) -+- s), preserving F~k(~z ) n B~+,(Ex) and the stable system. At this point we obtain an equi- BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 137 valence on a full neighbourhood of the singularity a~ + k: we use the cone-like construction in Theorem A, Section II, and preserve the level sets of the potential f~. The equi- valence is extended to the distinguished neighbourhood U,+ k by preserving level sets of the potential and repeating the cone-like construction near a~+k+ 1. With this we define homeomorphisms on the space of leaves of F ~+lk~z) c8 " ' and complete the definition on the fence B~+I(~)Cf~-l(f~(a~+l(~)) § ~) also preserving Fg~_l(t~ ). Since the folia- tion Fg"(~) is preserved in this process, the equivalence can be extended to the second distinguished neighbourhood U,(~) again by the methods explained in Theorem A. Let us give more detail on this construction. We assume that we already have homeomorphisms on the space of leaves of F[(Vt), ..., F~'_I(~) and F~"(~) as well as on the space of leaves of F~k+~(~), ..., F~(~). The homeomorphism on the space of leaves of the foliation F~_~(t~) is obtained as in (II.a) using Lemma 1. Next, we obtain homeomorphisms h~+ ~, ~ on the space of leaves of the foliation F~'+ j(t~) forj = 2, ..., h -- 1, and, also, of F~_k(~). We will perform the construction for j = 2, since the general case can be done by induction in a similar way. Construction of h~+2,~. -- Let us suppose that W"(~i+l(~) ) nWS(~+2(~))4: O. We denote by F~_2(a~+~(~) ) the set of leaves of F"'(a~+I(Vt)) which are contained in W*(a,+2(~)). We recall that A,+I(~) and B,+I(~) are two small fences contained in the non-critical levels f~-~(f~(a~+a(~))+ ~) for 9 > 0 small, respectively. We are going F ~s (~ to define a homeomorphism on B~+I(t~ ) n W"(~,+2(~) ) which preserves ,+~(~+1(~)). So, we first construct a homeomorphism on the space of leaves of this foliation: this is done leaf by leaf using the Isotopy Extension Theorem, as in the previous cases. Since we already have a homeomorphism on the space of leaves of the foliation F~_I(t~), we obtain a homeomorphism on A~+~(~)t~W"(a~+~(~)) which preserves Fr ,+2ka,+~(~)), / / Fg~ ~(t~) and a complementary foliation F~: this is exactly like in the proof of Theorem A when we restrict ourselves to W*(a,+z(~)). Therefore, through the Poincar6 map P,+a,~ : A,+~(~)\W"(a~+~(~)) ~ B~+a(~)\W"(a~+~(t~)) , we get the required homeo- morphism on W~(a~+~(B)) c~ B~+~(~). Let D'(i + 2, t~) be a fundamental domain for WS(a,+2(~)) which is contained in the non-critical level set ffa(f~(a~+l(Vt)) + ~). Using the Isotopy Extension Theorem and the compatibility of the homeomorphisms on the space of leaves of the foliations F~(~), ..., F~"(B), we obtain the extension of the homeomorphism W*(~+2(~) ) ~ B,+I(~) -+ W~(~+~(q~(~))) ~ B,+~(q~(~)) to D'(i + 2, ~), finishing the construction of h~ + 2, ~. 9 As mentioned before the construction of the other homeomorphisms hf+~,~, for 3 ~< j ~< k, is analogous to the one described above: we proceed by induction, using the $ $ leaves of the stable system Ff~_~(~), F~+~(t~), ..., F~+~_a(~) which are contained in We are now prepared to define an equivalence on the distinguished neighbourhood U~+,(~) of the orbit of tangency u To do that we again apply Lemma 1 to obtain a homeomorphism on the space of leaves of F~_,(t~ ). The construction is dual to the one 18 138 M. J. DIAS CARNEIRO AND J. PALIS used to obtain a homeomorphism on the space of leaves of F~_l([Z ). This homeomor- phism, together with the homeomorphism h~+k, ~ constructed above on the space of leaves of F~_k(~t), yields the definition of an equivalence on the neighbourdhood U~ + k(~t) according to the methods in Theorem A. We conclude our arguments with the cons- truction of an equivalence in the distinguished neighbourhood Ui(~z ). We have already defined a homeomorphism on the set [J2~<j~<k+xW*(~r~+i(~t))('~]~i-1-1(~) which pre- serves the foliation F~_t(~z). We can then extend this homeomorphism to the remaining part of the space of leaves of F~_a(~z) corresponding to the leaves contained in W~+~+2(~t), ..., WI(~z). This extension, which is by now standard, is compatible with the homeomorphisms already defined on the space of leaves of the corresponding stable foliation. With this, since we also have preserved the foliation F~"(Ez) throughout the process, we can define the equivalence on the neighbourhood U~(~t) again by the methods in Theorem A. To obtain the globalization of the equivalence to all of M, we just choose the non-critical level F, =ffX(c) where fg(a,+,(~)) < c (A(6i+/r and proceed as it was done at the end of Theorem A. 9 w 3. Bifurcations of type 111: saddle-node with criticality In this paragraph, which is similar to w l, we treat the case of a saddle-node with criticality. Let X~ be a family in z~(M) such that for a value ~ ~ R *, the vector field Xg presents a unique nonhyperbolic singularity p(~) which is a saddle-node unfolding generically, as defined in Section I. Suppose that there is one hyperbolic singularity q(~) such that the unstable manifold of q(~) is transversal to the stable manifold of p(~), but there is one orbit y of quasi-transversal intersection between W~(q(~)) and W"(p(~)), the strong stable manifold of p(~). In addition we assume the following conditions to hold for the family X~. (3. A) Other generic conditions. (3.1) The pair (p(~), -((~t)) unfolds generically at ~t =~. This means that, provided that the saddle-node unfolds generically, there is a C I curve Fs~ in the para- meter space such that ~ e Ps~ r if and only if the vector field X~ exhibits a saddle-node singularity p(~z), and an orbit of tangency between W"(q(~z)) and W~*(p(~)) occurs only for the isolated value ~ in rs~. This is equivalent to say that, if ~t e I~sN ~ a"(~) and [z ~Pst~-*a~([z) are two C 1 curves in M such that a"([z)~W"(q(~z)), a~([x) ~ W~(p(~t)) and a~(~) = &*(~.) = r e y, then the projection of ~r~(~) -- ~8~(~) onto T, M/T, W"(q(~)) § T, W""(p(~)) is not zero, (3.2) X~ is C" linearizable near q(~) and partially linearizable near the saddle- node p(~) as described in Section I (c.4), its linear part having distinct eigenvalues at these points and m 1> 3, (3.3) W~ is transversal to W"~(p(~)), BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 139 (3.4) Let W'*(p(~)) be the invariant manifold of codimension one in W*'(p(~)) whose tangent space at p(~) is complementary to the eigenspace corresponding to the weakest contraction for XgIW"*(p(~)). Then, W"(q(~)) is transversal to W"*(p(~)), (3.5) There are no other criticalities: for any singularity a different from p(~) and q(~), W"(a) is transversal to W"(q(~)) and to W*"(p(~)) and W'(z) is trans- versal to W"(q(~)) and to W""(p(~)). All other invariant manifolds intersect trans- versally. (3.B) Tke bifurcation set. -- The hypothesis of generic unfolding of the saddle- node implies that there exists a C 1 curve I's~ r near ~ in the parameter space, such that along rs~ the family X~ presents a saddle-node bifurcation. Fs~ is the image of the singular set of the restriction of the projection (x, V-) ~ ~ to the manifold ~ (x) = 0 , where (x, V-) are C ~ coordinates in a neighbourhood of (p(~.), ~.) in M � I! ~" and f~ is the potential function associated to the family X~. Let W~*(p(~t)) be a C 8 center-stable manifold. From the linearizing assumptions, we can write 0 0 X 2 = X~ ] W~*(p(~t)) = B(~t, x) Ox + A,~(~t, x)y,- j= a 021 in a neighbourhood of p(~t) in M, where s = dim W"(p(0)) and (x, yl,...,y,) are ~t-dependent C" coordinates, m >/ 2, such that the eigenvalues of the matrix A(~t, x) ~- (A~j(~t, x)), � are distinct and negative. The ordering (Yl, ...,Y,) corres- ponds to the ordering al(~t) < ... < %(~t) of the absolute values of the eigenvalues of 0B A(~t, x). The generic unfolding of the saddle-node implies that B(0, 0) = ~x (0, 0) = 0, 0 3 B 0B Ox ~ (0, 0) ~e 0 (say positive) and ~ (0, 0) 4 0. Therefore, there is a diffeomorphism r x) = (r ~2(~t, x)) such that B o ~(~t, x) -~ x ~ + ~t x. Using the change of coor- dinates x ~- ~2(~t, ~), y~ =.~j, ~ = ~l(~t), we have [ . = [ + Multiplying by the nonvanishing function 0~2 0~ (~' ~)' we obtain a family X~ equivalent 0 0 to X 2 near p(~.) such that X~ = (~ -t- ~-1) ~-~ + I~N~(~, ~).~ ~--~. From now on we drop the bars to simplify the notation. Let :~2 C {y~ = 1 } be a cross section such that 140 M. J. DIAS CARNEIRO AND J. PALIS W*'(p(0, [x~)) n Z~' -= {(0, 1,y~, ...,y,)}. Since W*"(q(0)) is transversal to W'?(p(0)), we may write W*"(q(v)) c~ Z~' = { (x, yL,y,= =Yx(~, x,y,.))} and W"(q(~)) n Z~ ~ = { (F(~,y~.),y~.,yK(~, F(Ex,y~.),y~)) }, with YL = (Y2, ...,Y,q),Yx = (Y,q+a, ...,Y,), 1 + s, + dim W""(p(0)) = dim W"(q(0)), F(0, 0) = 0, ~ (0, 0) = 0 and \Oy~ Oy~ (0, O)/2<<.j.~,q nondegenerate (we assume dim W~(q(0)) + dim W"~(p(0))/> n § 1). For ix x < 0, we have two distinguished hyperplanes in E~{, namely x ----- ~ ~, which correspond to WS"(pl([/,)) ~ Z~' and to W"(p2(~x)) c~ Z~', where pl([X) and p2(~x) are the two hyperbolic singularities that collapse to form the saddle-node. There- OF fore, W"(q([x)) is nontransversal to W'(p2(~) ) if and only if Oy---~ (~'Y~): 0 and ~v/~ ~x~ = F([x,y~.). From the hypothesis of quasi-transversality and the implicit function theorem, we obtain a C a curve P in the parameter space defined by ~ ~1 = F(~, D~(~x)), OF OF where Yr. = flL(~) is a C 1 solution of ~ (~, y~) = 0. Since --8~t2 (0, 0) 4:0 (by the inde- pendent unfolding hypothesis), we obtain that F is a C x curve tangent to Fs~ at 0. There are no other criticalities and W"(q(~)) is transversal to W~(p(~)), and, thus, the bifurcation diagram for the family X~ for ~x near 0 is exactly I' ~ Fs~. Remark. -- Along I' the field X~ presents one orbit of quasi-transversality between W"(q(~x)) and W'(p~(~)). If dim W"(q(0)) -t- dim W"(p(0)) -= n, then the above equa- tions simplify to x = F(~) and P is given by ~ ~x~ = F(~). (3. C) Stability. -- Let X~ be in z~(M) such that X~ presents a saddle-node with criticality and the family satisfies all the conditions described in (3. A). If X~ is close to X~ so that it also has a bifurcation of type III for ~ near ~, then we will show that { X~ }~ e v is equivalent to { ~2~ }~ e v', where U and U' are open neighbourhoods of and ~ in IR ~. We may assume in the usual ordering of the singularities of X~, al(0t) ~< a~(~x)~< ... ~< ~t(~), that %(~) : fl'k_bl([s :p([-~)for ~ ~ Fs~ and q([z) : ~k_l([.L). We will see at the end of this paragraph that there is no loss of generality in doing so. We consider a distinguished neighbourhood Uk_l(~) of %_1([x) as constructed in Theorem A and connect it along the orbit of tangency y to a neighbourhood V(~) of the saddle-node. As in previous cases, we construct an equivalence k~ that preserves the level sets off~ inside Uk_ ~ (~x). In V(~) it preserves two continuous invariant foliations with C 1 leaves and depending continuously on ~; these foliations, denoted by F~" and F~', have complementary dimensions. The leaves of F~" have dimension equal to dim W""(%(~)) and its space of leaves is the center-stable manifold W~'(%([~)). BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 141 We start constructing the equivalence between X~ and X~ on the neighbour- hood V(iz) by obtaining an equivalence between X~S=X~[W~ ) and )~8 ___ )~ ]W0,(~(~)). To do that, let us consider a compatible unstable system FI(&), ..., F~_~(~z), F~U_l(~) as before. We take a continuous family of C 2 cylinders C(~) in WCS(ak(t~)) and a continuous family of C 1 closed discs D([z) contained in some leaf F of the strong-stable foliation F~ s so that, for [z e Fs~, C(~) is transversal to WS~(%([z)) and C(~z) ~3 D(~z) contains a fundamental domain for W~(ak(~)). On C(Ez), we construct a C 1 foliation F~(V) of dimension one, which is compatible with the induced system F[(~z) n C(~z), ..., F~_2([z ) c~ C(~z), F~"_l([Z ) c~ C(~z). Also, F~(~z) is compatible with W"(%_t(~)) c~ C(~z) and has a unique singularity of saddle-node type which is the point of tangency between W"(a,_x(~z)) r~ C(~z) and F~ ~ c~ C([z); outside this point, F~(~z) is transversal to F~" c~ C(~z). The construction of F~ is exactly like in the pre- vious paragraph. Let M~ be a distinguished leaf of F~ namely the curve in W~ n C(~z) defined by yr. = ~r.([z), where ~z ~ U ~ ~r.(~z) is the C m-~ solu- OF tion of ~-~v ~ (~,y~) = 0 and, as above, (x,y~,y,r) are C ~ coordinates for C([z) near the point of tangency y(~t) c~ C([z), m t> 3. For ~h ~< 0 in this curve, there are three distin- guished points P~-I(~-) = Wu(~k-l([z)) ~ M~(~z), p,(~z) = W~(a~(~z)) n M~ and p,+~([z) = W~(%+~(~z)) n M~(~t), so that the curve p,_~(~z) =p~+~(~z) represents the values of the parameter such that W"(a,_~(~z)) is quasi-transversal to W~(%+~(~z)). Therefore, in the three-dimensional manifold M~ Uv~vM"([z), we have two C m-~ surfaces intersecting transversally at 0 defined by M~ = { x = F(~z, ~([z))} and M~ = {B([z, x)= 0}. So, let ~:M~ M ~ be a diffeomorphism of the form ~(~L, X) = (q?l(~s ho(~L, X)) such that ~(M~) = { x -- tx~ = 0 }, B o ~(~, x) = x ~ + ~. Then, it is clear that X~ * is topologically equivalent to ~ 0 ~ O X~' = (x ~ + ~1) Ox + ZA,~(~, x)y, Oy~ and the manifold i~I~(~x) = 2~~ ~ W"(,r is represented by { x -- tz2 = 0 }. If we repeat the construction for the nearby family X~, we obtain X~* equivalent to a family with the same normal form along the central manifold (still denoted W~ and with the same expression for the manifold ~I~(~t). Hence, X~" [ W~ is conjugate to X~ ] W~ and the conjugacy preserves the distinguished point x~_x(~), which is the projection via the strong-stable foliation F~ * of the point p~_ x(~x) = W~(o~_ ~(tx)) ~ M~ Thus, X~,s] WO(tx) is equivalent to X,,~, l W~ with ~: (U, 0) -+ (It ~, 0) being a homeomorphism that sends the region A i onto fi~, as in the picture (Figure IX). This gives a homeomorphism in the space of leaves of the strong-stable foliation F~ ~. We now define a homeomorphism h~":WO*(,~(tx))--~WeS(~(q)(tz))). Let us consider, as in previous paragraphs, a continuous family of compatible homeomorphisms h~, for i = 1, ..., k -- 1, defined on the space of leaves of the foliations F~(tz), . .., F~_~(tz), F~a([~). We define h~* in the same way as in Theorem A, Chapter III of [15], the only M. J. DIAS CARNEIRO AND J. PALIS difference arising from the singularity of the central foliation F~,. Hence, we begin by applying Lemma 1 to get a homeomorphism between W cu (k_X(~t))C~ (y C(~t) and W""(~_~(?(~t))) r~ (~(?(~t)) preserving the central foliation. We then proceed as in w 1 to extend this to a homeorr.orphism on C(~t) which is compatible with h~, i = 1, ..., k -- 1 and sends F"(~) to F"(~0(bt)). This induces a homeomorphism on the boundary of the disc D(~t) which is extended to its interior, the extension being compatible with the homeo~orphisms h~. Finally, we define h~ 8 by sending F~," to F~(~). A z / A! ~"SN Fxa. IX Now, over each point of C (~t) u D (~t) we raise a u-dimensional (u = dim W "" (o k (~t))) continuous foliation F~, ", with C 1 leaves compatible with the unstable system F~(~t), ..., F~_~(~t), F~_x(~t) and with W~((~_l(bt)). Positively saturating it by the flow Xr, ~ and adding the strong-unstable foliation restricted to W~(ak(~t)) for ~1~< 0, we obtain a strong-unstable foliation F~" whose space of leaves is W~8(ok(~) ). We then construct a complementary foliation denoted by F~,' compatible with a stable system Fr for i = k q-2, ...,L We start by constructing a compatible stable system F~+2(~t), ..., Fl(~t), together with a homeomorphism in the space of leaves of each of these foliations. Let L+(~) be a leaf of F~, 8 in W"'(~t) such that F ~8 n WC(~t) consists [J., of a point x, with coordinate r > 0 small. Over each point x of L+(~t) we take F.~." the part of the leaf of the strong-unstable foliation that contains x and is contained m the neighbourhood V(~t). If we let Dr, ~ = [-J~eT,§ then Dr, , is a C ~ disc of codi- mension one which is C" outside L+(tz). We also take a continuous family of C ~ cylin- ders CX(tz) in We~(~t) transversal to W~"(bt), so that CX(bt) t3 D c" contains a fundamental It, tg domain for W"(bt), where D*~r,, = Dr,, c3 WC"(ak(bt)), and the vector field X r is tangent to Cl(bt) n W c" In C~(bt), we let Fd(bt) be a one-dimensional central foliation compa- r,~ ~ tible with the stable system F~+~(tz), ..., F~(tz). Over each leaf of Fa(bt) we raise an (s + 1)-dimensional foliation compatible with the stable system. Over each point of Dr,. we raise an s-dimensional continuous foliation compatible with the induced system F'(a,(bt)) t3 Dr,., k + 2 ~< i~< t. The center-stable foliation F~ 8 is the (s + 1)-foliation obtained by saturating negatively the foliation and adding to it the center-stable mani- fold W*8(iz) for [z x < 0. We repeat the same constructions for }~,r~" BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 143 We can now get a homeomorphism h+ . +,+-+ D+<~+,+ compatible with the homeomorphism on the space of leaves of the stable system, by first constructing it on 0 D e" and then extending it to the interior of the disc. The equivalence between X~ and X+I+) in the neighbourhood V(~t) is finally obtained by preserving the complemen- tary foliations F~ ~ and F~,". Since we are preserving the center-unstable foliation F~"_t(~t), we may extend it to a neighbourhood Uk_t(bt ) of ++_a(~t) by preserving the level sets of the function f+. The globalization of the equivalence to all M follows exactly like in w 1 of this section or in Theorem A, Section II. Finally, if in the ordering of the singularities of X+, ~1(~) ~ 9 9 9 ~ (~k--l(~) ~ r ~ (Yk+l(~) ~ " " " ~ (Yl(~L), with cr,(~z) : ak+a(~t ) :p(~t) along the curve of saddle-nodes I~sz~, the orbit of quasi- transversality occurs in the unstable manifold of a singularity ~r~(~t), with j<~ k- 2, we proceed as in w 2, case II. b. That is, we construct a compatible system of unstable foliations F~(ax(~)), ..., F~"(~(~t)), F"(a~+~(~t)), ..., F~(%_~(~t)) and follow the same steps as above. Again, we connect the distinguished neighbourhood U~(~t) of aj(~t) to the neighbourhood V(~z) along the orbit of tangency. w 4. Bifurcations of type V: saddle-node with an orbit of tangency So far we have treated the cases which present at most one secondary bifurcation: in a neighbourhood of the bifurcation value ~, the family X~ presents for ~t 4: ~ at most one new bifurcation. Contrary to this, the bifurcations corresponding to types V, VI and VII of the list in Section I may present several secondary bifurcations. This lead us to analyze orbits of tangency between several invariant manifolds and a certain invariant foliation. For this reason, to prove stability, a globalization of Lemma 1 in w 1 (Lemma 2 below) will be necessary. In this paragraph we study the case where Xg presents a saddle-node p(~) and an orbit y of quasi-transversality. We assume that T belongs to the unstable manifold W"(q(~)) of a hyperbolic singularity and the stable manifold W+(p(~)) of the saddle-node. The case where the quasi-transversal orbit occurs between invariant manifolds of hyper- bolic singularities, will be discussed at the end of this paragraph. Besides the assumptions that we have already used in previous cases, like linearizability and partial linearizability for X~ near q(~) and p(~), generic and independent unfolding of the saddle-node and the orbit of quasi-transversality, and transversality between W+"(q(~)) and W+(p(~)), several others are required here. They are satisfied by generic families X~ + z~(M) which present a bifurcation of type V. (4. A) Other generic assumptions. (4.1) Let W+"(p(~)) be the codimension-one invariant submanifold of W""(p(~)) such that T+<g> W+"(p(~)) is complementary to the eigenspace corresponding to the 144 M. J. DIAS CARNEIRO AND J. PALIS smallest nonzero eigenvalue of dXg(p(~)) (weakest expansion). Then, for r e y, there exists a linear subspace E, C T, W"(q(~)) with dim E,=dimW'"(p(~)) such that lim~_~oo dXg.~(r).E, = T~(g)W""(p(~)). Moreover, if ~ is a singularity of Xg different from p(~) and q(~), then W'(~) is transversal to W"(p(~)), W""(p(~)) and W""(p(g)), and W"(~) is transversal to W""(q(~)), W~(p(~)) and W"(p(~)). (4.2) Let F~ " be the unique codimension-two invariant foliation in W"(p(~)) which has W*"(p(~)) as a distinguished leaf. F~- " is compatible with F u" each leaf L iz ~. ' of F"-- " is subfoliated by leaves of F~ ". Suppose L + W""(p(~)) and that n*" : L -+ R Ix is a submersion that defines F~ " in L. Then, the restriction of n~ to each stable manifold W'(a(~)) n L is a Morse function with distinct critical values. For any stable manifold such that W~(~(~)) c~ L is tangent to F~ ~, the eigenvalues of dXix(a(~)) are distinct. In this case the center-stable manifold W~*(a(~)) n L is transversal to F ~" g. Comments. -- Clearly, these conditions do not depend on the leaf L. Also, if W'(a(~)) c~ L is compact, it is easy to perturb Xg so that n "~ [ W~(a(~)) n L is a Morse function with distinct critical values and W~"(~(~)) c~ L is transversal to F~- " To get Ix " the genericity of these hypotheses, we use the ordering ,~(~) .< ....< ,,(~) .< ,,+o(~) .< ....< ,~(~) of the singularities of Xg such that p(~) ---= %(~), assuming that W""(a,(~)) is transversal to W'(%(~)) for k + 2 ~< i ~< l -- 1 ; i 4- 1 ~< j ~< t and proceed by induction using trans- versality arguments, in particular, transversality between W~"(%(~)) and F~'. (4.B) The bifurcation set. -- Assume that in the ordering of the singularities ~1(~) -< ..- -< ~k-l(~) -< ~k(~) -< ~+1(~) -< ... -< ~t(~) of x~, we have %(fx)----~k+l(tx)=POx) for ~ e I'sy , the curve of saddle-nodes, and that %_a(~) = q(~) ; also assume ~ = 0. Using the transversality between W~(%_a(~)) and W~(ak(~)) and the partial linearizability of X~ near ak(lx), we extend W~"(%_l(~)) to a neighbourhood of the closure of the orbit of tangency y so that it contains the saddle- node. We may suppose that we have a normal form for Xix I W~"(%_1(~)) near ~k(~) and, as in w 3, we can write 2 0 O X~ ~ = X~ I W~(%-1(~)) = (4- x 1 § ~a) ~ § Z~,~(xa, ~)Y' 0-~j + Z~(x~, ~) z~ Oz~ in a neighbourhood of p(0), with all eigenvalues of Ax(Xl, ~) = (ei~(xa, ~)) being nega- tive and of B~.(xl, ~) = (~(xl, ~.)) being positive. In these coordinates we assume that the zl-axis corresponds to the direction of the weakest expansion, u = dim W""(p(~)); we choose the positive sign in the above expression. Let Z~([x) be a cross-section inter- secting the orbit of tangency T. Then, the intersection of W*(ek_i([x)) with Z~_"(tx) is BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 145 {(YI, Zl, ZL) ] Zl = F([z,YI, Zr,)} with YI ~ F(pt, YI, 0) being a deformation of a Morse function. Moreover, the generic and independent unfolding of the orbit of quasi- transversality implies that the map (~h, ~t2)~-* (~Zl, F(~t, Yr(~t), 0)) is a local diffeo- OF morphism, where ~ ~ Yi(~t) is the solution of Oy--- I (~,yr, 0) = 0. Therefore, the curve Fk_x, k of quasi-transversality between W~(%_l(~)) and W"(ak([z)) is locally defined by { ~t 1 < 0 } n { F(~, YI(~), 0) = 0 }. By changing coordinates, we get Fk- 1, k = { ~2 = 0, ~t 1 < 0 }. Since there are no criticalities, other bifurcations may occur only in the region ~t 1 > 0, where the corresponding vector field does not present singularities near the point p(0). To analyze these possibilities, we let Y~.u(~) be a small closed disc contained in the section { xl = ~1 } such that Z~"(~) c~ W"(p(~)) is contained in a leaf of the strong unstable foliation. The positive number ~ is taken so that if W"(%(~)) n OZ~_"([z) + O, then W"(%(0))c~ W""(p(0))~e O. Hence, if there is an orbit of tangency between W"(ak_x([x)) and WS(%.(~)), then it necessarily intersects the interior of Z~u(~x). Moreover, from (4.1) and (4.2), these tangencies may occur only near the points of tangency between W8(%(0))c~ W~(p(0))n Z~u(0) and the foliation Fg ". For each j t> k -t- 2 we denote by Pj,1, ...,Pj, ,(j) these points. Let (vI, Wl, wL) be C m coordinates in Y~?(~t) such that W"(ak([x))n Y~_"(~)= (0, wl, wL), m>~ 3. We may assume that WS(%(0)) has codimension one in Z~?(0); if not, we just restrict ourselves to W~(%(0)). Then, from (4.2), near each point p~ we may write WS(6j(~)) ~ ~_u(~) = { Wl = Gji(~, VI, WL) } with Gj~(0, vi, 0) having a nondegenerate critical point at vt(pj,). Let us extend Fg", previously only defined on W*(p(0)), see (4.2). Let Zr t) = {(YI, Zl, zL) e ZL~(~ t) [zL = 0} and r~": Zc_u(~t) -+ ZL(~t) n~,"(Yx, z~, z~,) = (yx, Zl -- F(~,yI, zr,) q- V(~,yx, 0), 0) be a submersion that defines a C" foliation F~" compatible with W"(%_1(~)) n Z'_"(~t). For latter purpose, the flow saturation of this foliation will still be denoted by F~". Using the normal form for X~," near p(0) to get a linear expression for the Poincar~ map P~ . Y.'__~(~) -+ Z~?(~t) for ~ > 0, we obtain that the restriction of ~ to is singular along disjoint C "-1 manifolds M~i(~z) for i = 1,..., n(j), with dimension equal to I I l= dimW'(p(0))c~ W~"(q(0)) and which depend differentiably on a. As [z -+ 0, all these manifolds become C 1 close to they,-plane and for a~ = 0 they collapse into this set. Since the points {p~ } belong to distinct leaves of the foliation Fg ", the images Mj,(~t) = ~"(~I~,(~)) are disjoint submanifolds of codimension one in E~_(~). If W"(%(~t)) has minimal dimension (equal to dim W'(p(0))), then 19 146 M. J. DIAS CARNEIRO AND J. PALIS From this construction we conclude that W"(ak_l(~t)) is tangent to W'(%(~t)) in ZL"(~t) if and only if M~(~t) is tangent to W~(%_I(~)) n ZL(~) for some i = 1, ..., n(j). Hence, for each (j, i), we consider possible tangencies between the manifold Mt~(~) and the foliation defined by (YI, Zl) ~ Zx -- F(~t,yx, 0) + F(tz, 0, 0). Using now the hypothesis of quasi-transversality between W'(p(0)) and W*(q(0)), we obtain for each ~t in a neighbourhood of 0 in { ~tx ~ 0 } a unique point of tangency q~,(~) ~ M~(~). The map ~t ~ q~(~) is of class C 1 in a neighbourhood of 0 in { [z x t> 0 } and qi~(0, ~t,) = 0. Therefore, X~ presents a quasi-transversal orbit of tangency between W"(%_l(~)) and W'(%(fz)) if and only if q~(~t) belongs to W=(%_~(~t)) n ZL(~t). These values of ~t correspond to a finite number of disjoint C a curves I'~._ a, ~. tangent to the ~q-axis at 0. The bifurcation diagram is as in the figure. ~SN k-l,j T~k- ! ~r Fro. X (4. C) Stability. -- Let X~ be a family in ~(~(M) which presents a bifurcation of type V at ~ and satisfies all the assumptions described in (4. A). If X~ is a nearby family, with ~ as the corresponding bifurcation value, then we show that there are neighbour- hoods U and lJ of ~ and ~ in R ~ such that { X~ }~ E ~ is equivalent to { )(~ }~, ~ O. We assume that ~ = ~ = O. We start by taking a compatible unstable system F~(~), ...,Fk_~(~),Fk_l(~) and neighbourhoods Uk_I(V) of ak-x(~) and V(~) ofp(~) in M which are connected along the orbit of tangency y. From the description of the bifurcation set, each point of tangency between F~" and W*(%(0)) yields a quasi-transversal orbit between W~(%_~(~)) and W*(%(~)). So, we consider distinguished neighbourhoods U~(V) of each such singularity and connect them to V(~) with tubes along each orbit of tangency Ts~" The equivalence will preserve the level sets of f~ inside the neighbourhood Ui(~). Using the transversality between W~(~h(0)) and W~(%(0)) for i>j >/k + 2, and pro- ceeding as in w 2 of the present section, we construct a compatible center-stable system ~+~(~), F,+3(~) , .., F~'(~). It may happen that for some i i> k + 2, the stable manifold W'(a~(0)) is transversal to F~" (for instance, when a((0) is a sink). In this case we take, as in w 2, the stable foliation F~(~). BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 147 To obtain the equivalence between X~ and X~, on a neighbourhood V of the saddle-node p(~) in M � R *, we construct a center-unstable foliation FCU(~) compatible with the unstable system and the unstable manifold W"(~k_l([Z)). The method to cons- truct F~"(~) is similar to the one already used in w 1 and w 2. The main difference here is that we want the singular set of FZ"([z) to contain the points of tangency between W~(~_I(~)) and the manifolds WS(~(~)) for j = k + 2, ...,L Let E~_(Ez) be the leaf space of the foliation F~" constructed in (4.B). We recall that to describe the bifur- cation set we have obtained codimension-one submanifolds M~,(~z)C Y.~_(~z) such that W~(a,_l(~Z)) is tangent to WS(aj(~z)) if and only if W"(ak_l(~) ) n Z~_(~) is tangent to Mj~(~) for some i = 1, ..., n(j). For each pair (j, i) and ~ > 0, we let OF 0A~ Yj,(~,YI) = ~ ([z,YI, 0) -- 0YI (~,YI), where graph (A~)= M~(~z). Since lim Yj~(V, YI) OF ~,~,0 ---- ~ (~,YI, 0), we may extend OF this family to Yj~(~z,y~)=~v (~,y~, 0) for [Zl~< 0. For 0 < r < ~2 small, we let A~(~I)CAj~(r be open neighbourhoods of M~(~z) nW~(%_l(~)) such that Ai~(r ~ At,~,(~) = O for (j', i') # (j, i). We define a family of vector fields Y(~,y~) OF such that Y I Ai~(r ---- Yi~ and Y~ in the complement of U A~(~) is equal to Oy-- I (~,y~). As in w 1 and w 2, the central foliation which gives rise to the leaves of F~"(~z) inside W~(%_l(~)) is tangent to a vector field Z~ with a saddle-node singularity such that Z~ restricted to each Ai~ is equal to Y~. Associated to a central n=anifold of Z~ we have a special leaf denoted by W~"(~). This invariant manifold is completed for ~z~ ~< 0 by adding part of a center-unstable manifold which is linear in the above normal form coordinates. By construction, W~"(~) contains all tangencies between W"(~_a(~)) and W~(g~(~)), j >1 k + 2. The other leaves of F~"(V) are obtained exactly as in w 1. Complementary to F~"(~), we define a strong-stable foliation F~'. Since all stable manifolds Ws(~(0)) are transversal to W"(g~(0)), the method described in w 3 can also be applied here. However, since F~"(~) is a singular foliation, in order to have transversality between F~' and F~(~) outside W~"(~z) we modify F~' for ~q 7> 0 near the points of tan- gency psi(0). Let Z be two cross-sections such that Y.+(~z) c~ W~"(a,_~([z)) ---- ~([z) and suppose that F ~* is a strong-stable foliation in W*(a~(~z)) n Z+(~), as constructed in w 3, which is transversal to W~"(V). We can also assume that F". is transversal to F~"(~z) outside a neighbourhood of each point pi~(0). Let P~ : Z (V) -+ Z+(~z) be the Poincard map for ~ > 0. We modify F**. in a neighbourhood of pi~(0) in such way that each leaf of the induced foliation P~ -1 (F~,~) ~ ~ Y~e"([z) projects by n~, V~ onto a level set of a Liapounov function of the vector field Z~ in Z~_([z). Proceeding in this way for all stable manifolds W*(a~(~z)) and extending this modifyied foliation to each leaf of F~'(~) as in w 1, we get the required strong stable foliation F~*. By preserving F~/ and F~ ", we M. J. DIAS CARNEIRO AND J. PALIS can obtain an equivalence between the two families X~ and X~ on a neighbourhood of the saddle-node singularity similarly to w 3. Hence, to prove local stability of X~ we have now to obtain homeomorphisms on the space of leaves of these foliations. Let us frst construct a suitable reparametrization q~. Consider X~ restricted to W~(~) (the space of leaves of F~,8). Since W~(~) depends differentiably on ~, it is transversal to W8"(%(~)) for ~t e Fsz ~ and admits a C" smoothing structure, r>~ 3 (see [15]), we conclude that X~[W~(~) has a ~-dependent normal form near the saddle-node as in (4.B). In W~,"(~) we consider a codimension-two invariant foliation compatible with F~ ~ such that for ~ E FsN it has as special leaf W*"(%(~)), the codi- mension-two strong unstable manifold (see (4.1)). For ~1 > 0, F~, s is obtained by satu- rating the foliation used at the end of (4. B) and intersecting with the leaves of F~". This foliation is extended to a neighbourhood of p(0) for ~x 1 ~< 0 by adding to it a codi- mension-two linear foliation. In particular for ~ = 0 this gives the foliation defined in (4.A). For each ~ the leaf space of F~" is an invariant surface W(~) that contains a center manifold, and it is defined in the above coordinates by z L = 0. In W(~t) we take a fundamental domain C(~)w E ~, where E ~ ={Xl = ~, ]zx] ~< ~} and = c+ u c = {I I = [ I -< Let C ---- [J~ev C(~) and definc I x : U\Fk_~, k --> C t3 E c, the map that associates to each ~ thc point of intcrsection of W"(%_x(~) ) with C([~) u E~. If c#,(~) e E" reprcsents the leaf of F~," which contains the tangency point p#,(~), then the curve F~_~,j obtained at the end of (4.B) is defined by Ixl(Cj,(~)). Moreover, from the hypothesis of generic unfolding of the orbit of tan- gency y (0) ~e 0 we obtain that Iff~(E ") is a wedged shape region A C { [z 1 >/ 0 } with vertice at 0, which is bounded by two curves Ix1(4 - 8). We also have in A a singular foliation r defined by Ixl(x) for x ~ E ~ with special leaves Pik_l. ~. We define a repara- metrization q~ : (A, 0) -+ (A, 0) of the form (q~1(~1), q~,(~zl, ~z,)) which sends F to F. Since a conjugacy on a center manifold induces via the strong unstable foliation a homeo- morphism h ~ :C ~ C, we choose the reparametrization on U\A in such way that I~ o ~? = h ~ o I x. This gives a reparametrization on a full neighbourhood of 0. We now prove that X~ and 1~,(~) are equivalent. We begin by taking a continuous family of diffeomorphisms +~ :E~(~) ~ F2(q~(~z)) sending c~(V) to ?~i(q0(~)). Using a conjugacy we define a homeomorphism on the space of leaves of F~,". To define an equivalence between X~ ] W~"(~z) and X,(~)[W~"(~(~x)) we use a conjugacy which preserves F~" inside each leaf of F~,". Therefore, for [z~ > 0, it is enough to obtain a continuous family of homeomorphisms on a leaf Z~?([x), preserving F~" and the center- stable system, in order to get an equivalence on a neighbourhood of the saddle-node p(0) in W~,"(~). Contrary to this, for [xx <~ 0, the negative flow saturation of Y.~_"([z) just fills a conic region A(~) with vertex at the singularity a,+~(~t). Therefore, to get an equi- valence on a full neighbourhood of p(0) in W~"(~), we construct a two-dimensional foliation F~(~) in the complement of A(~z) which is compatible with the center-stable system and transversal to F~,". Thus, the equivalence is defined by preserving F~," and BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 149 F~(V). Let us construct a foliation F~(~z): this construction resembles very much the one of a central foliation in [15]; the main difference here is that we want it to be transversal to the codimension-two foliation F~ ". Let K(~t) = U,~e(~D~"(~t), where D~"(~t) is a closed disc centered at x and contained in the leaf of F~" over x. As above, C(~t) is the intersection of the fundamental domain C with the plane ~t = constant. Let E""(~z) be a closed solid cylinder in the leaf Z~"(~t) which is bounded by two closed discs K~'"(~t) ---- K(~t) c~ Z~"(~t), i = 1, 2, and by a cylinder S(~t). Over each disc K~'"(~t) we raise a one-dimensional continuous foliation Zc(~t) in K(~t) which is compatible with the center-stable system. We can assume that C(~t) is a leaf of Zc(~t). We construct Z'(~t) in such way that the union of the leaves of Z'(~t) which are over the spheres 0Kx(~) and 0K~"(~t) is the closed cylinder U,~c~0D~"(~t). Since the tangencies between W'(~rj(~t)) and F~" occur in the interior of E""(~t), the cylinder S([z) C 0E""(~t) can be foliated by a one-dimensional foliation 8~(~), which is compatible with the center-stable system, and whose leaves are C ~ and transversal to F~ ~ n S(Ez). Over each leaf of 8~(~t) we raise a two-dimensional foliation Z~(~z) also compatible with the center-stable system, with each leaf of ~(~t) being bounded by two leaves of X~(~). Thus, F~(~) is obtained by taking the negative saturate of Xe(~t) and of k~(~t) by the flow of X~. This finishes the construction of Y~(Vt ) which has as space of leaves the boundary of E""(bt ). Since we already have defined a homeomorphism on the space of leaves ofF~ ", in order to conclude the construction ~ of the equivalence between X~ ] W~"(Vt) and W,(,)I W~"(q~(~t)) it is enough to obtain a continuous family of homeomorphisms h~": E~([z)-+E~(~(~z)) which preserves F~ ~ and the center-stable system. The idea to obtain h~," is to " project " E""(~z) onto E""(0) along the leaves of F~," and to construct a homeomorphism from E""(0) to E""(0) which satisfies the above requirements. We then pull back this homeomorphism to E""(~) to get h~ ". This process is achieved by constructing a continuous foliation 3~ ~ on E"~= O~vE""([z), with C ~ leaves of dimension two, which is transversal to E""(0) and compatible with both the center-stable system and with the foliation F~ ". The construction of ~ is easy except at neighbourhoods of the tangency points p~,(0). Near each point p~(0), ~ restricted to WC"(a~([z)) is defined by intersecting F~ " with a three- dimensional foliation given by a continuous family of vector fields, parametrized by ~, which has a saddle-node type singularity at p~,(tz). Therefore the surfaces of tangency (~, p~,(~)) are special leaves of ~. The extension of 3f ~ to the leaves of the system F~ ~ near p~,(0) is done as in w 1. The foliation j~o was conceived so that it may be used to tri- vialize the foliation F~" along the center-stable system. Suppose that h~" : E~"(0) -+ E""(0) is a homeomorphism preserving Fg" and the center-stable system. Then, we define h~," : E""(~z) ---> E""(q~(tz)) by sending ~ n E""([z) to ~,~ ~ E""(~(~)). The reparame- trization ~ obtained above guarantees that the point p~(~) is sent to the corresponding one ~.~(q~(tt)). Thus, to finish the construction of an equivalence between X~ [ W,""(~z) and X,(~) [ W,~"(~(~z)) it remains to prove the existence of h~". This is the content of the following key lemma. 150 M. J. DIAS CARNEIRO AND J. PALIS Lemma 2. -- There is a homeomorphism hg": E""(0) -+ F,""(0) that preserves the folia- tion Fo ~, the center-stable system F~'(%(0)) ~ E""(0) and the stable manifolds W'(%(0)) c~ E""(0) for j = k + 2, . ..,t. Proof. -- By using a diffeomorphism which preserves Fo" we may assume that E~"(0) = ~,""(0) and F;"= ~'g". We may also assume that ~*~l W'(%(0)) and ~"]W*(~j(0)) have the same critical values for j=-k-t-2, .,t. Let us assume that W~+2 = W'(ak+2(0)) c~ E""(0) is compact and disjoint from the boundary of E""(0). If ~;"(wl, w~.) = wl is the projection along the leaves of Fg" then n~-2 = ng"[ W~+2 is a Morse function with distinct critical values. Analogously, for rck+ ~^~" = %^*" [ Wk+ ^' ~. Let ~k + 2 : WZ + 2 -+ "vV~ + 2 be a diffeomorphism C ~ close to the inclusion map. Then, ~k ^~" + 2 o % + 2 and r~k+ ~" 2 are C ~ close Morse functions with the same critical values. Therefore, there exists a C 2 diffeomorphism h2: W i,+2-+'v~r~+2, close to the identity, t~Z~ At~t~ t*U $ such that ~k+2~176 We define the restriction of ho to W k+2 by ho"" = ~Pk+2 o h-Xk+2. The same is done, in a continuous way, for all leaves of FZ+ ~ =F"(a~+~(0))c~ E""(0) which are contained in the center-stable manifold W~'+2 = W"'(%+~(0))n E""(0). Since W~' is transversal to F~?(0), we let F~_ 2 be a C a foliation in a neighbourhood of W~+ 2 which is transversal to W"'k+2 and compatible c$ with Fg" such that dim F~,~_~ = codim~o ~ W~+ 2. The foliation F~_~ is defined on a tube % + 2 along the stable manifold W~ + 3. The intersection of % + 2 with each leaf F of F~" is a closed box B~'k+2 bounded by a cylinder transversal to F~_~ together with two closed discs ~[ u ~ contained in leaves of F~_ 2, such that any leaf of the center- stable foliation F~*+~, whose dimension is equal to the dimension of W ~'~+~, intersects transversally ~ ~ ~. To obtain v~ + 2, we first define local tubes % + ~,~ in a neigh- bourhood of each critical point p~+ 2,~ between two non-critical levels 0~-+ ~,~ and 0++ 2,~. Let Z~+2, ~ be a C 2 vector field (as constructed in w 1) tangent to each leaf of F~"+ 2 = F"'(a~+ 2(0)) n E""(0), whose restriction to W~"+ ~ has a saddle-node singularity atpe+~,~ and whose restriction to W'~+2 is the gradient of r~;~_.. In the leaf 0;-+2,~ we take a closed box B~-+ 2,~ as above and positive saturate it by the flow of Z~ + 2,~ in the strip between ~-+2,~ and 0~+2,~. We add to this set the stable manifold W"(Z~+2,i) in order to obtain the local tube %+2,i- The local tubes %+2,~ for i = 1, ..., n(j) are then connected along W]~+ ~ by using the integral curves of Z~+ 2, a C ~ extension of Z~+2,i along the leaves of F~+~ in a neighbourhood of W~- such that Zi restricted to W~ is a Morse-Smale vector field. Therefore, by preserving the two complementary foliations F ~'~+~ and F~_~, we obtain a homeomorphism h~ ~ on the tube %+~. Observe also that E""(0) is bounded by a cylinder S(0) transversal to Fg" and two closed discs, each one contained in a leaf of Fg ". Since W"(%) is transversal to Fg ~ at the boundary of E""(0), the cylinder S O can be foliated by one-dimensional leaves 8~(0) compatible with the stable system and transversal to Fg ~. Hence, if W],+2 c~ 0E""(0) 4 = O, we take the diffeomorphism described above also preserving the foliation 8~ + ~ (leaves of 8~ which are contained in W~+ 2 c~ 0E"~(0)). Next, suppose that W~+~ = W'(%+~(0))c~ E~"(0) BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 151 is nonempty and does not intersect the boundary of E""(0). If W~+ 3 is compact, we repeat the above argument. If not, then WU(%+~(0))n W'(a~+8(0))4:0 and we consider the foliation F~+ 2, k + 8, consisting of the leaves of the center-stable foliation F~+ 2 which are contained in W'(,k+3(0)). We also take a diffeomorphism on the space of leaves of this foliation. This is possible because the intersection of WS(,k+3(0)) with a fundamental domain of W"(ak+2(0)) is compact. By preserving F~+o-,k+3 and F~_o-, we get a homeomorphism from WT,+ 8 n %+.2 to X3(~ + 8 n % + 2. Using the Isotopy $ A $ Extension Theorem, we get a homeomorphism %+s: Wk+8-+Wk+8, which is a C ~ diffeomorphism on Wk+s\~e+ 2.' The functions nk+ ~*u and ne+8^*~ o ~k+8 have the same critical values and coincide in W'e+8 c~ %+2- Let n~+8 be the homotopy (1 -- t) zck~"+3+ tnk^~+8o ~k+8 for te [0, 1]. Then ~+8 [ %+~ n W ~e+8= z~k+8- ~ ~ By defining a family of vector fields ~k -t + 8 on Wk+ ~ with supp ~+8C W],+8\%+~ , such that ~t ~'~+8.',t , we obtain that ~**+3 is topologically trivial. That is, there exists a continuous family of homeomorphisms h~ + 3 : W~ + 8 ~ W'k+8 such that n~+3 oh~+~ = n0+~ = r~_~. Hence we define h~" restricted to W~+~ by h~" = q%+~oh~+~. We do the same on each leaf of F~ + ~ contained in W e + ~, to extend h 0 to a neighbour- hood of W~+n in W ~s~+~. Again, since W~+~ is transversal to F o , we take a C ~ folia- tion F~_~ in a neighbourhood of W~+o- which is transversal to W and compatible with F 0 and with F~+~ such that dimF~+ a = codzm~.~0)We+ a. This foliation is constructed in a tube ~ + a along W~ + a exactly as in the previous step of this induction. If W~+an0E"~(0)4: O, we take the homeomorphism %+a also preserving ~r in W~ + a ~ 0E~"(0) 9 Proceeding by induction on the ordering of the singularities, we obtain the homeomorphism h~ ~ as wished. 9 Thus, we have obtained a homeomorphism on W~(~), the space of leaves of F~'. By applying Lemma 1 and the methods described in w l, wc obtain a homeomorphism on the space of leaves of F~(~). These homeomorphisms define an equivalence on a neighbourhood V(~) of the saddle-node in M as in w 3, by imposing that the two comple- mentary foliations F~' and F~u(~) must be preserved. To extend this equivalence to a distinguished neighbourhood U~_~(~) of ~e-z(~), we connect it to V(~z) with an invariant tube Wk_l(~) along the orbit of tangency. In the fence B~_~(~)C OUk_l([.s we let D~_x(ix) be the intersection of [.J,<0X~.~(V~) with B~_ a ([z). We can assume, after a reparametrization of time, that D~_ ~ (~) is contained in X~,_v(Z (~t)) for some T. Hence, we have defined a homeomorphism on D~_a(~) which preserves the center-unstable foliation F~"(%_a(~)). The same arguments as for Theorem A are now applied to extend this homeomorphism to the fence Be_l(~) pre- serving F~"(%_a(~)) and the center-stable system. We define a homeomorphism on U~_z(~) by preserving level sets and trajectories. Inside the tube T~_~(~) the equi- valence is a conjugacy. Analogously, we get an equivalence between X~ and ~2~,(~) on a distinguished neighbourhood Ue+ ~([z) of %+ ~(~). Proceeding inductively and using the compatibility of the center-stable system, we construct equivalences on distinguished ~"~+O- 152 M. J. DIAS CARNEIRO AND J. PALIS neighbourhood Ui(~) of ~([x), j/> k -t- 3. Finally, as in w 2, we extend the equivalence to all of M as a conjugacy outside these neighbourhoods. 9 It remains to deal with the case where the vector field Xg presents a saddle-nodep(~) and one orbit 7 of quasi-transversality between W"(q(~)) and W'(q'(~)), q(~) and q'(~) being hyperbolic singularities. We assume the linearizability conditions and the non- criticality condition with respect to the strong-stable and strong-unstable manifolds ofp(~), q(~) and q'(~) and also the generic and independent unfoldings of the saddle- node and the orbit of quasi-transversality. Similarly to the case (II. b) of w 2, since there are no criticalities, we conclude that the bifurcation set near ~ is the union of two C ~ curves I~T t3 I'sN intersecting transversally at ~, such that for & ~ I'Q~ the field X~ presents one orbit of quasi-transversality between W~(q([x)) and W'(q'(~)) and for ~ I'Q~ a saddle-node p([x). The equivalence between X~ and a nearby family >2~ is obtained without much difficulty using a combination of the methods developed in (II. b) of w 2 and w 3. 9 w 5. Quasi-transversal orbit with tengency between center-unstable and stable msnlfolds In this paragraph we consider a family X~ ~ )(~(M) such that for a value ~ ~ R ~ the vector field Xg presents a bifurcation of type VI: there is an orbit of quasi-trans- versality between W"(p(~)) and W'(q(~)), p(~) and q(~) hyperbolic singularities, satis- fying all the generic conditions described in Section I except (c. 3) ; i.e. the center-unstable manifold W*"(p(~)) is not transversal to W'(q(~)). To have a codimension-two bifur- cation, we assume that W~(p(~)) is transversal to W*'(q(~)). Since we also assume that Xg is C ~ linearizable near p(~), choosing a C ~ center-unstable manifold W*"(p(~)) which is linear in these coordinates, we suppose that along the orbit of tangency 7 the stable manifold W'(q(~)) is quasi-transversal to WC"(p(~)). Here we take Comment. -- Although center-unstable manifolds are not unique, this condition does not depend on the choice of a C"* center-unstable manifold, if m is sufficiently high. In fact, if N is a (u -t- 1)-dimensional invariant manifold of class C" for m as above such that T~o)N : E 0 @ T~0)W"(p(~)), E o being the eigenspace corresponding to the weakest contraction, then the contact between N and WCU(p(~)) along 7 is of order at least two. That is, for each point r ~ 7 there is a local diffeomorphism + in a neigh- bourhood U of r in M, j2 +(r) = 2-jet of the identity map, such that +(u n N) = U. Proof. -- Let (xl, xi,yl,yr,) be C ~ linearizing coordinates for Xg near p(0) such that W'(p(0)) = (xa, xx, 0, 0) and W"(p(0)) = (0, O,y, yL). We suppose that 7 is tangent BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 153 to the yl-axis (weakest expansion). Then N t~ {Yl = 1 } = { xl, Nz(xx,yL),yr.)} with N I of class C '~. Let P :{yl = 1 }-+{ xl = 1 } be the Poincar6 map P(xz, xI,yT.) = (x; ~'/~' x,, x~ g~', x~/",yL) ; then P(N c~ {yz = 1 }) is a C ~ manifold parametrized by (y~-~'/~' Nz~t,,~'/~'/a , y~.),ya,y~/~'y~.). Since each com- ponentyi -~g~' Nj(y~g~',yL) is of class C" with m/> max 3, [ ~1 -F 1, ~11 -+- 1 t' and there are no resonances between the eigenvalues, we must have dNj(0) = 0 and d * Nj(0) = 0. Thus, j2 Nz(0 ) = 0, proving our statement. 9 We also suppose the generic unfolding of the orbit y, so there is in the parameter space a curve I'QT containing ~ along which X~ exhibits a quasi-transversality between W"(p([L)) and W'(q([z)). We require the tangency between W~"(p(~)) and W'(q([x)) to unfold generically, so we also get a curve I" 0 containing ~, along which X~ presents a quasi-transversality between Wr and W"(q([x)). It is easy to see that I'Q~ and lr 0 are always tangent at the point ~. Therefore we require that PQT and I" 0 have a qua- dratic contact at -~. (5.A) Other generic assumptions. -- In addition, we assume that the family X~ satisfies the following generic conditions. First, let Wf"(p(~)) be a (u + 2)-dimensional center-unstable manifold, u = dim W~(p(~)), which we assume linear in the linearizing coordinates. Then, W~"(p(~)) is transversal to W'(q(~)). Now, let WS'(p(~)) and W~'(p(~)) be the invariant submanifolds of W'(p(~)) of codimension one and two, respectively, which corresponds to the eigenspaces of strongest contractions. For any singularity a(~) of Xg, we assume that W"(a(~)) is transversal to W"s(p(~)) and to W*'(p(~)). Moreover, let F' be the codimension-two foliation in W"(p(~)) having W*"(p(~)) as a distinguished leaf. If W"(a(~)) is not transversal to F *' and dim W*(a(~)) n W"(p(~)) /> 2, we require that the restriction of n*s (projection along F w) to W"(a(~))caW'(p(~)) has a fold singularity along one orbit. This last hypothesis is similar to the one used in w 4. If L is a leaf of the strong stable foliation and n~" is the projection along the leaves of P" contained in L, then we assume that ~* restricted to W'(e(~)) n L is a Morse function. It is easy to show the genericity of this hypothesis and that it does not depend on the leaf L. We also require that the points of tangency between W"(e(~)) n L and F *~ belong to distinct leaves. For each j < k, we denote by p~(~) the distinguished points of tangency between W~(a~(~)) n L and F *~. Since we are going to use a compatible center-unstable system, we assume that for each a(~) <~ p(~) there is the smallest contraction and that We"(a(~)) is transversal to F *' in WS(p(~)). We also require W"(a'(~)) to be transversal to W'"(a(~)) for all singularities a'(~)~< a(~) and W~"(q(~)) to be transversal to W'(~*@)) ifq(~) ~< a*(~). The genericity of these conditions follows exactly as in (4.A). 20 154 M. J. DIAS CARNEIRO AND J. PALIS (5.B) The bifurcation set. -- Let X. be a family satisfying the conditions described above at a bifurcation value ~. Let X~, CS = X~, [ WC~/a k k+l([~)) " We assume that ~ = 0 and that p(bt) = ak(~t) and q(~t) = a,+l([Z) in the usual ordering of the singularities of X.. Let us take ~z-dependent C m linearizing coordinates near %(~t), such that '~-~ 0 ~ 0 x~ ~ - Z ~,(~) x, + ~j(~)y~--, ,=I Ox~ ,=l Oy, with u = dim W~(a,(~t)), r = dim[W"(ak(~) ) c~ W""(%+1(~)]. Considering a cross-section S""(~z) C {Yl = 1 } with coordinates (va, v3, Vr, wr.), we get W"(%(~z)) c~ S'"(~z) = {(0, ..., 0, wz)}, W'"(%(~)) = {(vx, 0, ..., 0, wr.)} and Wf"(~,(~t)) = {(va, v3, 0, ..., 0, wL) }. The generic assumptions of (5.A) imply that W*(=k+l(~t)) n S""(~z) = {v3 = F(~t, vt, vi, wL)}, F being a C" function such that F(0) = 0 and F(0, v~, 0, wT,) is a Morse function with critical point at the origin. The condition of generic unfolding imply that From this we obtain the curve F,.,+ 1 of quasi-transversality between W'(a,(~)) and W"(% + ~(~)) by solving the system of equations F(~, 0, 0, wL) = 0, ~ (~, 0, 0 wL) = 0 . The curve of tangency F o between WC"(~k(~)) and W"(%+1(~.)) is given by F(~, v~, O, w~) = O, ~ (~, v~, O, w,.) = O, ~ (~, vx, O, w~.) = 0 ," We may write Fk, k+ ~ ={~z=0} and F o--= ~3=~-~ 9 (We are assuming that 8 3 F 8 3 F OF Ov---~ (0), Ov~ (0) and ~ (0) are all positive.) Although F 0 does not belong to the bifurcation set, it serves as a guide to obtain the other curves along which the family presents a quasi-transversality between W~(a~(~z)) and W'(%+x(~t)), j < k. Let us assume that W"(%(~)) has codimension one (if not, just restrict X~' to W~"(%(~t)) c~ W~*(%(~t))). As is w 4 it is easy to see that if W"(%(0)) is transversal to the foliation F "~ in W'(%(0)), then, for ~z near 0, W"(%(~t)) is transversal to W'(%+~(~z)). Hence, possible tangencies between these manifolds occur near the tangency points p~,(~z) between the unstable manifolds and the foliation F "*, see (5.A). We write the intersection of W"(aj(~)) with a cross-section Se'(~) C { xx = 1 } near p~(0) as a graph x 3 = G~(~, xx,y~,yL ) with x I ~ Gj~(0, x~, 0, 0) being a C "~ Morse function with critical point Xr(p~,(O)). Using this expression and the generic unfolding of the quasi-transversality, we obtain as in w 1 0~-------~ BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 155 a (31 curve F j, such that ~ ~ Fr if and only if X~ presents one orbit of quasi-transversal tangency between W"(a~(tz)) and WS(ak+l(~)). This curve l~i is tangent to F 0 at 0. It follows from the fact that the points of tangency Pr 4' and p~, belong to distinct leaves of P' if (j, i) # (j', i'), that Pr n Us, r ---- O. We stress the similarity between the present bifurcation and the one treated in w 1. In order to analyze all the secondary bifurcations simultaneously, we let F~'+l(tZ ) be a C '~ foliation in S~'(t~), compatible with W'(ak+l(~t)) and defined by ~*+a.~(Vl, v~, vi, wL) ----- (vx, v~ -- F(~, Vl, VI, W) -~- F(~, vl, 0, WL) , WL). Negative satu- ration of F~l(~) by the flow of X~' converges to the foliation F'. Since the tangencies between W~(as(~)) and W'(~k+l(~) ) in S~*(~) occur in the set of tangencies between F~8 t W"(a~(~)) and ~+~k~), we associate to each distinguished point pj~(0) a submani- fold Tji(~) in W~(aj([~))n S"*(~) such that U~=, ..... ~l~j~(~) contains all those tan- gencies. Using the above notation, the submanifold is obtained by solving the system OG~, OF e -'~t WL) e -c'~-'lt 0. Since the 0xI (tz, xx, e -~*t, e -~* w~) ~i (~z, e-'**, xx, . = points {p~(0)} belong to distinct leaves of F ~', the images Ta,(~z) = ~_~.~(T~,(tz)) are disjoint submanifolds ofcodimension one in L*(~z), the leaf space of F~+I(~) with coordinates (vl, vz, w~). All these manifolds are contained in a wedged shape region of the form Iv2 ] -< ~,. Iv1 ["/~' in L*(tz). The tangencies between W~ and W"(%(Vt)) in S*'(t~) correspond to tangencies between W'(,~+~(~t)) and Ta~(~z) in L~(~z) for some i e{ 1, ..., n(j)}. Proceeding as in w 1, we let ~ be a C l foliation of codimension two in L*(vt) which is compatible with W"(,,(~z)) and with all submanifolds Ts~(Vt), for jl>k-- 1 and i= 1,...,n(j). Since W*(~,(~))nL~(~) ={Vl=Vz=0} we may also choose ~ compatible with the " horizontal " foliation Vl = constant. As in w 1, we obtain a two-dimensional C 1 manifold S~(~) of class C 2 outside the origin, which is transversal to W~(,,(~))n L~(~) and to W"(~,+I(~) ) n L*([z). With this process we reduce the analysis of the bifurcation of type VI to the corresponding one for two- parameter families of gradients in a three-dimensional manifold. Thus, the bifurcation set is obtained by analyzing the following situations: a) the point p,(~) = W"(a,(~)) tn S~(~t) belongs to the curve T'(~) = W'(a, + I([Z)) (~ Sg(~t), i.e., W"(a,(~)) is tangent to W*(a,+l(~)), b) the curves T~,(~) ---- T~(~) tn S~(~) and Ts(~) are tangent, i.e. the manifold W"(%(~)) is tangent to W'(a,+x(~) ). The first situation yields the curve F,,,+ 1 obtained above. In the second one we have to consider two non-equivalent cases: a~(0) < 2~x(0) and a2(0) ~ 2a1(0) as in the three-dimensional case analyzed in [22]; if ~(0) = 2a1(0), the family is not stable in general. By parametrizing the curve T'(~) by (Vl, F(~, va, 0, ~,.([z, Vl))) and each curve T~j,(~) by (v~, M~,(~, Vl)) for Vl >/ 0 (or for Vl <~ 0), and letting S~ = U~ S~(~), OF the hypothesis 0~ (0) + 0 implies that the set M~ = {F(~, Vl, ~(~, v,)) M~(~, v~) } 156 M. J. DIAS CARNEIRO AND J. PALLS is a two-dimensional submanifold of S~. Hence, the bifurcation set F~ which corresponds to tangencies between W~(6~+a(~)) and W"(%(~)) is the image of the singular set of the map ~, restriction of the projection =(~, vt) = ~ to M~,. Each P~ is a branch of a C ~ curve which is tangent to the curve F 0 defined by F(~, v~, 0, ~(~, v~, 0)) = 0 = OF 0, 0)) and for (j', i') . (2, i) the branches are disjoint in a neighbourhood of 0. If~%(0) > 20r then all branches of F~ are on the same side of F~, ~+ 1; otherwise one may find branches in both sides. See Figure XI. l'k k-1 Fro. XI (5. C) Local stability. -- Let us construct an equivalence between X~ and a nearby 9 ., k-l~) and a family :~,. We take a compatible center-unstable system F~"(~), . F ~" ' ' stable system F~_I(~), F~+~($), ..., F~(~) for X~. In the discussion of the bifurcation set, we have already observed the similarity between this case and the one in w 1. As in that case the main point to prove stability is to obtain a homeomorphism h~ 8 on the cross-section SC*(~): S(~)nWCs(%+t(~) ) where S($) is a small neighbourhood of the tangency point p~(0) = g ta Bk(~) in a fence Bk(~). We now describe this homeo- morphism, beginning with a reparametrization q~ together with a homeomorphism h~ : S~(~) -+ S~(9(~)), S~($) as defined in (5.B) above. Each curve T~j~(~) = (vt, Mj,(~, Vl)) obtained at the end of (5. B) is a leafofa singular foliation defined by a one-form on S~(~), wj~(~) = -- ~1(~) Vl dv~ § [~(~) va § 0(v~+')] dv 1. Using a partition of unity, we may define a C t one-form w(~) such that restricted to a sector of the form l v~ -- M~(~, v~) I < ~tva {"/~' it coincides with wj,(~); outside the origin, w may be taken of class C ~. We can also take w(~)=- ~1(~)vldv~ + ~2(~)v~dvx for ~{ Vl{~'[~'~< { v, { and assume that the curve To(~) = W~"(~(~)) c~ S~(~) is a leaf of w(~) = 0. The set of tangency points between the curve T'(~) = W'(cr~+ 1(~)) ~ S~(~) and the leaves of w(~) = 0 is described by an equation OF -- ~1(~) 731 ~O 1 (~, Vl) ~- 0~1(~) F([~, ~71) -3u r([s ~)1, F(~, ~)1)), OF where F is as in (5.B), F is C 1 and T'(~) = graph F(~, .). Since ~ (0) . 0, this is the graph G of a C t function ~ = ~(~, vt) (which is even C ~ outside the origin), whose projection ~ : G -+ R a has a fold singularity along a C ~ curve ~. Thus, there exists a BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 157 homeomorphism a:G ~ G with fixed point set % such that r~(~l, v~) = n(~q, vi) if and only if b-~ = a(~t~, v~). The image of 9 by r~ is a C ~ curve tangent to I~,,+~ at the origin. Moreover, if M 0 = { F(~t, vi) = 0 ) represents the intersection of the curve T*(ti) with T0(~t), then -r is transversal to M 0 c~ T at the origin. Since all curves T~,(~) are tangent to T0(~t ) at the origin, the curves A~, =- M~, c~ G are also transversal to -r at the origin, where M~ = { F(t~ , vi) -- M~(~, v~) = 0 } = T'(~) n T~.~(~). Let A ~ be the foliation in G defined by the pull-back of o~ by the map ([11, ~)1) ~ ([11, [12([11, Vl), Vl, F([ll, [1~, ~)1), The leaves of A*, except two of them, are tangent to A~ ---- M 0 n G. We can take a homeomorphism from the space of leaves of A c to the space of leaves of ~c, sending A~ to As~ and A~ to .~. Let A C G be a closed conic region, with vertex at the origin, which contains all the distinguished leaves A~i and A~, intersects v at the origin and whose boundary is transversal to A c. This region A is taken so that v n A 4: O and also 7:-l(Fk, k_l) t% A = 0. We let q~(~x, K2) = (qh(~q), ~2(~i, ~2)) be a reparametrization that sends the n-image of the curves A c contained in A to the ~-image of ~e, ~(~) to ~(~) and F,,k +~ to Dk, k + 1" This induces a homeomorphism ~ : G --~ G by sending A to A, preserving the foliations A*, ~e and ~i = constant, and, by using the involutions a, ~, in such way that q~ o ~ ----- r~ o ~. By preserving the surfaces M e that represent the inter- section of Te(v~) with T*(~), we already have a continuous family of homeomorphislns v x ~ ~(vi) in the set ] F(~z, v~)[ ~< ~ [ v I 1. They are extended continuously outside this region by performing an extension on each fiber ~ = constant. This gives a homeo- morphism on the space of leaves of the foliation dvi = 0 in S~(~). The homeomor- phism h~ : S~(~z) -+ in the conic region [ v, [ ~< 8 a [ v x [ preserves the foliations dvl = 0 and T"(~). Also h~ automatically sends T*(~t) to ~'~(~?(~t)). We extend h~ arbi- trarily outside the conic region but preserving T*(~t). We now extend h~ to the tangency submanifold L~(~)C Se*(~t) (see (5.B)). This is analogous to the construction used in w 1; the difference, due to the tangency between W'(%+l(~t)) and WC"(a,(V)), is that we need a new process to define a two-dimensional foliation (SN)~ : like in (1 .C), this foliation has a saddle-node singularity along the curve T*([z). Since we preserve the foliation given by dvl = 0, we may define (SN)~ using once more a family of vector fields Y~,,,, now also parametrized by vi, which is compatible with W'(a,+l(~t)). For fixed (~t, vi) , Y~,,, presents a unique singularity of saddle-node type at { T'(~)c~ (vi = constant)}. Outside this point, the trajectories of Y~,,, are transversal to F~, (5.B). Hence, by applying a parametrized version of Lemma 1, we obtain a homeomorphism from Le(~t) to ~'e(~?(~)). We can now define a homeomorphism on a neighbourhood S([z) of the tangency point p,(0) = g ~ B,_~(~) in the fence B~_l(~). Since W~(~+i(~) ) is transversal to W~(~(~)), the cone- like method of Theorem A is applied to obtain a homeomorphism on = n 158 M. J. DIAS CARNEIRO AND J. PALIS which preserves the center-stable foliation F"'(ak+l(~z)). Hence, as in w 1, it is enough to construct a homeomorphism h~ * on S*'(~) -~ S(~) n ~ k+ik~)). We already have a homeomorphism on the space of leaves of Fff+~(~), a foliation of dimension (s -- 2), s = dim W'(cr,(~)), which is compatible with W~(ak+l(~)). It remains to construct a suitable center-unstable foliation F~(a,(~z)) and to adapt the cone-like construction to this case. (We recall that in the previous applications of this method (Theorem A) the foliation F~', dual to F~(ak(~) ) in S~*(~), had dimension equal to (s- 1).) We describe the (u + 1)-dimensional leaves of type F~(c~,(~z)) whose space of leaves cor- respond to closed discs D~(~z) in the fundamental domain A~0z ) = Ak(~z ) c~ W'(~,(~)), = ~ ,+1,,) (L*(~z)), where ~,+i.~ is the A~(~) being a fence in a level set. Let Y~(~z) f~'~ -~ "* projection along F~,~_l([Z), and consider P~':~(~) --~ A,([z) the restriction of the Poin- card map to ~([z). Using the linearizing coordinates and the fact that F ~* is of class C '~, it is easy to see that the image of this foliation is a codimension-one foliation in P~*(~(~)), which extends continuously to the strong stable foliation F~'(~) in the discs D~([z). We raise over each point of D~:(~) a one-dimensional foliation FI(~) in P~*(fl(~)), which is compatible with the center-unstable system and also has its inverse image by p~8 (~,+1,~) (T (~t)). We then raise over Fl(g. ) a u-dimensional continuous compatible with ~' - ~ r foliation F~(~) also compatible with the stable system and transversal to D~:([z). We define F~"(a,([~)) by taking the positive saturate of F~([z) by the flow of X~. The (u + 2)- dimensional leaves of type F~n(a,(Iz)) are obtained as in Theorem A, its leaf space is a sphere A~S(~t) W~'(~,(~t)) c~ A,(~). However, to avoid tangencies between ,+lk[~) and F~(a,(~)) in S~S([z) we go one step further and distinguish a new type of leaves, denoted by F~"(r which are (u + 3)-dimensional. Let Cff(~z) be a small tubular neighbourhood of the sphere A~*(~z) = W~'(a~([z)) c~ A~([z) in A~([z), which is bounded by two leaves of Fff(~)c~A~,*(V). Using the transversality between W~(%($)) and W~(~,(~z)), we can construct a one-dimensional foliation F~(~z) on (]~,'(~z) which is compatible with the center-unstable system. We let F~(a,(~z)) be the foliation whose leaves are of the form [-],~r F ~,,(,([z)), ~u where t~([z) is the leaf of F~([z) containing x ~ A~'(~). We construct homeomorphisms on the space of leaves of F~'(%([z)) and of F~(~,($)) and apply Lemma 2 to get a homeomorphism on the space of leaves of F~'(a,(~t)). In this case we need Lemma 2 in order to preserve F~,'([z). With these homeomorphisms together with the foliation F~,~_ ~([z) we obtain h~' as follows. We divide S~'($) into three conic regions: and C(~) = { 012 ~< 8[/)~ Mr_ [ VI [~]} ('~ { 8 [ V I ]2 ) [ V~. 19~ }, with ~> 0 small. On A0x) we preserve F~"(z~(~)) and Fff+~(tz); in each leaf of (r~+l,~) ( (tx)) these foliations are complementary. On BOx ) it is defined by pre- serving the complementary foliations F~"(z~(~t)) and Fff+~. On C(~) we preserve F~"(z~(~z)) and Fff+~(t~). Let F~,~(z~(~)) be a leaf of F~(z~(tx)). The intersection of F a,~(,(~z)) ~ ~ with OC(~) projects homeomorphically, via n*' ~+~(~), ~ " to L~(~). Hence, it BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 159 defines a homeomorphism on the leaf space of the leaves of type F~"(%(~z)) which are contained in F~a((rk([z)). So, h~* restricted to F~',d(%(~))\Int(C(~z)) preserves F~"(a,(~t)) and F~_~(~t). We then extend it to Int(C(~t))c~ F~.~a(~rk(~t)) arbitrarily but preserving F~.'+~(~). The definition of h~ ~ is now complete. As observed above, this is enough to obtain a homeomorphism on S(~x). We extend this homeomorphism to all of Bk(~z ) pre- serving the center-stable system and F~"(ak(~t)). By reasoning as in Theorem A we obtain an equivalence on a neighbourhood of the closure of the orbit of tangency -( preserving level sets off~. Proceeding by induction, we construct the equivalence on distinguished neighbourhoods of the singularities, i >/ 1, also preserving level sets off~, as it is done at the end of w 4. We conclude the result by extending these equivalences to all of M as in Theorem A. 9 w 6. Orbit of tangency of codimension two (6.A) Generic assumptions. -- We consider in this paragraph a family X~, such that for a value ~ there is a unique orbit -( contained in the intersection of an unstable manifold W"(p(~)) and a stable manifold W~(q(~)) of two hyperbolic singularities of Xg such that dim[T, WU(p@)) +T,W"(q(~))] -----n--2 for r~, (dimM =n). We assume X~, to be C"* linearizable, m t> 3, nearp(~) and q(~) and that the eigenvalues of the linear part of X~ have multiplicity one at these points. We also assume that for any hyper- bolic singularity a(~) 4: p(~) the unstable manifold W~((r(~)) is transversal to W'(p(~)) and to W'(p(~)), the strong-stable manifolds of p@) of codimension one and two. We suppose that W"(a(~)) has at most a quadratic contact with the very strong-stable foliation F~(p(~)) in a leaf L of the strong stable foliation F'(p(~)) (see w 5). Dually, we require transversality between W"(cr'(~t)) and W~"(q(~)) and between W"(~'(~)) and W*"(q(~)) and quadratic contact between W"(a'(~)) and F~,~(q(~)), the very strong- unstable foliation, in a leaf L' of the strong-unstable foliation F""(q(~)). Let W~'(p(~)) and W~"(p(~)) be (u + 1)- and (u + 2)-dimensional C "~ center-unstable manifolds of p(~) (u = dim W"(p(~))); then, W'(q(~)) is transversal to W["(p(~)) and dually W"(p(~)) is transversal to W~"(q(~)), a C ~ center-stable manifold of dimension (s q- 2). We also suppose that W"(p(~)) is transversal to W~'(q(~)) and the generic unfolding of the orbit of tangency y. This means that if ~", ~" :R ~ -+ M are immersions with ~(~) = a~(~) = r e y and ~'(~t) C W~(q(~t)), a~(~t) C W~(p(~t)), then the restriction of the projection T, M ~ T, M/T, W"(p(~)) + T,W'(q(~)) to Im[d~*(~) -- da~(~)] is an isomorphism. Actually, a generic family X~ presents an orbit of tangency of codi- mension two when there is lack of dimensions, that is u-t-s = n- 1. Proposition 5. -- There is an open and dense subset ~' C ;(g(M) such that if X~ e fr and for some value ~ the vector field presents an orbit of tangency y C W~(p(~)) n W"(q(~)) with dim[T, W"(p(~)) + T, W'(q(~))] = n -- 2 for r ~ V, then dim W"(p(~)) § dim W"(q(~)) = n -- 1. 160 M. J. DIAS CARNEIRO AND J. PALLS Proof. -- Let u = dim W"(p(~)) and s = dim W~(q(~)). We take ~x-dependent C m coordinates (Xl, ..., x,_,,yl, ...,y,_~, z) (m >>. 3) in a neighbourhood U of r in M such that X~ [ U = 0z and W"(p(~t)) ~ U = { x~ ..... x,_, =- 0 }, W~(q(~x)) t~ U = { x~ = F~(~x,y~, ...,yt), x, = F,(~x,y~, ...,Yt),Yt+I ..... y~_x = 0 }. We are assuming g/> 1, where / = dim[T, W"(p(~)) n T, WS(q(~))] -- 1. Hence, we can associate to X~ a two-parameter family of C" maps F:ll* � R t -+R ~, F([~,yx, ... ,Yt) ----= (FI(~,Y~, ... ,Y~), F~(~x,y~, ... ,Yt)). Ifj~ F(~,y) denotes the one-jet with respect to the variables (Yl, .-.,Ye) =.~, then dim[T, W"(p(~)) + T, W"(q(~))] = n -- 2 is equivalent to j~ F(~, 0) = (0, 0) e R ~ � L(R t, R ~) ~ J~(R t, R~)(g,0). But, since t/> 1, we have dim(R ~ � R t) = 2 -k- t < 2 -t- 2t = codima,(lr ~,)(0, 0) and the transversality theorem implies that (0, 0) is generically avoided. That is, with a small perturbation of F we get j~ F(~, 0) 4= (0, 0). This proves the proposition. 9 (6.B) The bifurcation set. -- Assume that p(~x) = ,~(Ex) and q(~) = ,k+l(~x) in the usual ordering of the singularities of X~. Let us describe the bifurcation set associated to tangencies between W"(,j(~)) and W"(%+1(~)) for j ~< k -- 1 and between W"(ej,(~)) and W~(*k(Ex)) for j' t> k + 2. It is easy to see, as in w 5, that these tangencies cor- respond to criticalities of W"(,i(~) ) (resp. WS(er(~))) with respect to F~S(%(~x)) (resp. F~(*k+l(~))). Let W~'(%+1(~)) be a (s + 2)-dimensional center-stable mani- fold of %+x(~) extended as in w 1 to a neighbourhood of %(~x), and consider the restric- tion X~' = X~I W~'(*k+l(~)). Assume that there are C" linearizing coordinates (Xl, x2, xi,Yl) for W~S(,k+l(Ex)) near ek(~) such that "-~ 0 0 X 2 = -- Z ~,(~) x, + ~I(~)Y~-- (u = dim W"(%(~)), 0 < ~,(~) < ~+~(Ex), [3~(~) > 0). In a cross-section Z~' contained in {yx = 1 } and with coordinates (v~, v2, vi) such that W"(%([x)) tn ]s ---- {(0, 0, 0)}, W~"(~;k(~)) t3 Z~ s ---- {(vl, v,, 0)}, we have a = { = = with FI(~, 0) ---= F2(~, 0)-=--0. The hypothesis of generic unfolding of the family X~ implies that the map ~z w, (FX(~, 0), F~(~z, 0)) is a local diffeomorphism near ~ and, hence, after a change of coordinates in the parameter space, we may assume ~ = 0, F~([x, 0) = ~ and F~(~z, 0) = ~tz. To "reduce dimensions ", we consider the C" folia- tion F~+a(~) in Z~' whose main leaf is WS(ffk+l([.~))("I Z$, and which is defined =~, t FZ(~t, v,) -k- V~(~, 0)]. Let by ,+a~, vx, vz, v~) = [v~ -- Fa(~, v,) -t- F~(~, 0), v z -- [J~<,~<,(~)?,~(~) be the image in W~"(%(~))t% Z~' of the set of points of tangency between F~'+x(~t) and W"(~(~)). Each ~(~) is a branch of a C ~ curve tangent to the v~-axis and corresponds to the distinguished point P~i(~) in the cross sections BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 161 S~(~) C {[ x x [ = I }. As in w 5, pa([,) is a point oftangency between W"(%(E*)) and F"(E*) in W*(%(k)) c~ W'(%(bt)). Since these critical points belong to distinct leaves of F'(~,), we obtain X~.(~) c~ X~.,,,(~) = 0 for (i,j) =t= (i',j') and for tz in a neighbourhood of (0, 0). These curves are contained in the region AI(~z ) = {] v 2 ] ~< ~ [ v x [~" =' } and are tangent to a vector field Z~ which has a hyperbolic singularity at the origin and is equal to O O --%(~t) vl Ova =2(~*) ve ~-v~v, outside A~(b~). It is clear from this construction that an orbit of quasi-transversality between W~(%(Ez)) and W'(%+1([,)) will occur in Z~' if and only if the point rk+l(~)=W'(%+x(~))c~W~(%([,))c~X~ ' belongs to the curve X~(V), for some 1 <~ i <<. n(j). We now apply the same reasoning to X~= X~ [ W~(%(~)), by taking a codi- mension-two foliation F~"(E, ) in a cross-section ]By, near the singularity %+~(t*), having W*(%(b0) n Z~" as a distinguished leaf. In this way we get a vector field Z*+l_~ on Z~c~W~'(%+~(~)) with distinguished trajectories X~.(~), j>~k+2, such that W"(%(g.)) is quasi-transversal to W'(%(~,)) in Zy if and only if r~(~t) = W~(%([z)) r~ lii~uc~ W~'(%+~(lz)) belongs to X~(b~ ) for some l<~i<<.n(j). W~ 1(~)), we are reduced to Therefore, by taking X~ = X~ I X~'(%(V)) r~ 1 ~ ,+ consider the three-dimensional case with the corresponding singular foliations in the cross sections Y.~(~) = ~*'(~) n W~(%(~)) and ~;~(~) ~ Y.~(~) t~W~'(%+~(~)). Let P~ : Z~(~) -> Z~(~) be the Poincard map, P~(v~, re) = (P~(~t, v~, re) , Pg(~, Vl, v2)), -~-- 7/r+ i and consider the induced field (P~)~-~ ~.~+~_~ _~Z t+~. Since the integral curves of _~ W ~ ~ 0~ (except for two of them) are tangent to (~+t(~)) ~ 2;~ , using the transversality between W~(%(~)) and W~'(%+~(~)) and restricting Z~ and ~.~+~_~, to the regions and A2([*) ---- {I P~(~*, v~, re) [ 4 a [ P~(~t, v~, re)l, I P~(b t, v,, re) [ ~ r } for 0 < 8 < 1 small, the trajectories of Z~ and of ~.*+*_~, are transversal to each other for [z close to 0. In the parameter space we obtain the corresponding regions B, = {[ Fe(j, , 0)[ ~< 8 [ F~(~,, 0)[, [ F,(~,, 0)[ ~< ~ } and B e ={] P~(~z, 0)l ~< a I P~(~,, 0)], } P~(~, 0)l .< ~}, which contain the bifurcation set. Note that B, n B e = {(0, 0)} and the bifurcations are characterized by the fact s n~e '~b + 1 that (0, 0) belongs to the integral curve X~(~,) ~ _~ , or r~+,(~,) belongs to the integral curve X~t(bt) of Z~. Since the map ~z ~-~ r~+~(~,) is a local diffcomorphism, we get a finite number of integral curves r~ of (r~-~ ~). Z~ in the region B~. Similarly, we obtain finitely many trajectories P~ of (P.~)-* ~+*_~ in the region B e. Thus, the bifurcation set is as in the picture. 21 162 M. J. DIAS CARNEIRO AND J. PALIS ~p.u. Fro. XII (6.C) Stability. -- The construction of an equivalence between X~ and a nearby family X~ is similar to the one in w 5 and we will describe only its main steps. The impor- tant point is to obtain a homeomorphism on a cross-section E contained in a distinguished neighbourhood of the orbits of tangency y and ~. For that, we first obtain a homeo- morphism h~:Z~(Ez ) -+,~(q~(~t)) between cross-sections in the center manifold W~"(cr~(~t)) nW~(~k+l(~t)) and W~(~(~(~t)) nW~(~k+l(q~(~t)). We start by taking a homeomorphism from 0Ax(0) to &~l(0) which sends X~(0) n 0Al(0) to ~,~(0) n &~l(0) ; this induces (via r k + 1 defined above) a homeomorphism from ~B 1 to 0B 1. We also consider a homeomorphism from ~A~(0) to &~(0) sending X~(0)n 0Az(0 ) to X~.~(0)n OA~(0), which induces a homeomorphism from 0B 2 to OB2 (via P~ defined above). Let q~ : (u, 0) ~ (q~(u), 0) be a homeomorphism sending B~ to B~, i = 1, 2, that extends the above homeomorphisms and preserves the trajectories of the fields (r[_~), Z~ and (p,e)-I ~+1 The homeomorphism hi: E~(~t) ~Z'(~(~t)) is defined in such way --I/, * ^le that it sends trajectories of Z~ to trajectories of Z,I~ inside Al(~) and .~l(q~(V)), respec- q,+l inside Bx([z) and B~(q~([z)). We tively, and trajectories of Z *+~ to trajectories of ~,~ -.~ choose h~, to send X~,([z) to X~.i(~(~t)) and Z~.(~z) to X~i(q~([z)). Since Z'([z) is the space of leaves of F~_l(~z), in order to define a homeomorphism k~': Y."~(~) ~Z~'(q~(~t)) it is enough to define a center-unstable foliation F*"(cr,(~z)) as in w 5 and proceed exactly like in that case. It is important to observe that, by construction, the pull-back of the trajectories of Z~ via the projection 7r~(~t) gives a codimension-one foliation, singular along [rc~+~(~z)]-~[W"(~([z))n Z~(~t)], such that the foliations in the cross-sections r \ C {1 xt I -- 1 } induced by the Poincar6 map, extends continuously to the very strong-stable foliation F"'(%(~t)). We apply Lemma 2 again, to obtain a homeomorphism in the space of leaves of the center-unstable foliation. The same procedure also works to define a homeomorphism h~":Z~"(~) ~ Z'"(q~(~)) preserving the foliation F~*(~z) BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 163 and a center-stable foliation F"'(% + t([x)) as above. Finally, to get a homeomorphism h~ from Y.([x) to ~(?([x)), we take a u-dimensional continuous foliation G~" in ~(~) with C 1 leaves transversal to W~(e,(~)) n Z(~), which extends F~'+I(~ ) and is compatible with F**(a,+~(~))c~ Y~(~). Dually, we take an s-dimensional foliation G~ ~ in Y~(~x) which extends F;"(~t) and is compatible with F~(a,([x))c~ ~(~). Similarly, we define G~' and 0~, ~ in the section Y.(~x). Since u + s-----n- 1 and we already have defined the homeomorphisms h~' and h~" in the space of leaves of these foliations, the homeomor- phism h~ :~(~) ~ Z(?([x)) is defined by sending G~ to G,(~) and G~" to G,(~). The extension to the distinguished neighbourhood of the closure of the orbit of tangency y and to the whole manifold M is exactly like in w 5. w 7. Remaining cases: two saddle-nodes and codimension-one and two singularity In this paragraph we finish the proof of local stability for the codimension-two bifurcations by analyzing the two remaining cases. (7. A) Two saddle-nodes. -- We assume that the family X~ has a bifurcation value where the vector field Xg presents two saddle-nodes p(~) and q(~). We assume the existence of C", m/> 3, linearizations transversally to center-manifolds of p(~) and q(~) and transversality between all unstable manifolds and the strong-stable manifolds W88(p(~)) and W'S(q(~)) and between all stable manifolds and the strong-unstable manifolds W~(p(~)) and W"'(q(~)). The saddle-nodes unfold generically and do so independently. Hence, ~ belongs to the transversal intersection of two C 1 curves F 1 and F 2 with ~ E P 1 if and only if X~ presents one saddle-node near p(~) and Ix belongs to 1" 2 if and only if there is a saddle-node for X~ near q(~). In a neighbourhood U of the bifurcation set is the union of P 1 with P~. Let us prove the local stability of X~. Suppose, in the usual ordering of the singu- larities of X~, that p([x) = ~,i(~) = ~+1(~) for ix e I' 1 and q([x) ---- ~k(~t) = ~k+~(~) if ~x e P~. We have two possibilities: there is an intermediate singularity < < or not. We will construct an equivalence between X~ and a nearby family X~ for the first case; the second case is simpler and can be derived from the first. We begin by considering a reparametrization q~ : (U, ~) -+ (R ~, q~(~)) that sends P, to Px, F~ to P~ and it is defined so that there are conjugacies between X~ restricted to the center- manifolds Wc(p(~t)) and WC(q(~)) and X~(~I restricted to WC(p(q0(~))) and W~(q(q~([x))). We then consider a compatible unstable system 1~(~), ..., 1~_~(~) and construct a center-unstable foliation F~"(~) which is compatible with this system and has a center- unstable manifold W""(ak([x)) as its main leaf (see w 3). Since the singularities of F~(~) occur along C 1 manifolds which are transversal to alI intermediate manifolds W"(~j(~t)), 164 M. J. DIAS C, ARNEIRO AND J. PALLS we can proceed as in w 2 to get an unstable foliation 1~,(~) which is compatible with the system F~(~),...,I~,_I(v),F~"(~),...,F~_I(V) for k+Z<j<~i--1. Now we construct a compatible strong-unstable foliation 1~'~"(~) whose space of leaves is a center- stable manifold W*'(a~(~)). Dually, we let F~+~(~), ..., F~(~) be a compatible stable system, and construct a center-stable foliation F~*(~). Actually, the procedure here is not quite dual since we are going to use the strong-unstable foliation F~'"(V) as part of a system of coordinates near cr,(~). To do that, let K+(~) be a closed disc contained in a leaf of the strong-stable foliation F~'(~t) inside WC"(~([z)) and let V~(~) be a cross- section of the form V~(~) ---- [-J,e,r~r F~,~(~), where F~(~) is the leaf of F~" through x. Part of the leaves of F~'(~) is obtained by negative saturation by the flow X~, t of an W 8s o" si-dimensional continuous foliation F]*([~) in V~(~) (s~ = dim (i(~))), topologically transversal to We~(~5(~) ) c~V~*(&) and compatible with the stable system. The other leaves of F~'(~) are obtained exactly as in w 3. The process to construct an equivalence is now clear by previous arguments, but we briefly describe it as follows. We begin with a (compatible) family of homeomorphisms hg(~) : W"(a,(~)) -+ W*(a,($(~))), j = 1, ..., k -- 1. It induces a homeomorphism in part of the space of leaves of F~"(~), which can be extended to all of Wr by first extending it to a fundamental domain D~(~) t3 CZ(~) and then to all of W*"(~r,(~)) by preserving the strong stable foliation and the inter- sections of F~"(9) with Wr Next, we consider successively fundamental domains D](~) of W'(a~(~)) for k -t- 2 ~< j~ i -- 1 to get (compatible) homeomorphisms in the space of leaves of the unstable foliations F"(~r~(~)). We finally reach the domain D~(~) ~ C~(~), corresponding to the space of leaves of the strong-unstable foliation 1~"(~). Here, again, the equivalence restricted to the center-stable manifold W~"(~(~)) is a conjugacy preserving the strong-stable foliation F~*(~). Proceeding dually, a family of homeomorphisms h~'(~) : W"(a~(B)) ~ W"(8~(?(~))), for j = i + 2, ...,t, gives rise to a homeomorphism in the space of leaves of the center-stable foliation Ff"(~) and the equivalence near ai(~) is obtained as in w 3 by preserving the complementary folia- tions F~'"(~) and Ff*(B). We now extend this equivalence to a neighbourhood of each singularity ~(~), forj = k + 2, ..., i -- 1, by using the procedure say ofw 2 to construct compatible stable foliations F]~+~(~), ..., F~_l([Z ) and homeomorphisms in the space of leaves of these foliations. The equivalence in these neighbourhoods is a conjugacy preserving stable and unstable foliations. Proceeding by induction we reach the saddle- node ~,(~). We then construct a strong-stable foliation F~*(~) compatible with the stable system and extend the equivalence to a neighbourhood of ~,(B) by preserving F~(~) and F]~'(B). For the extension of the equivalence to all of M, we proceed as in previous paragraphs, concluding the proof of the local stability of this case. 9 (7. B) Codimension-one or two singularity. -- We consider here a family of gradients X~ such that the vector field Xg presents exactly one non-hyperboIic singularity r We BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 165 suppose that 0 is an eigenvalue of dXg(a(~)) of multiplicity one. Therefore, there exists a center-manifold WO(a(~)) containing e(~), which is of class C '~, m large (see [10]). It is well known that transversal to W~(~(~)) there are unique strong- stable and strong-unstable manifolds W~"(a(~)) and W"~(a(~)). We assume that Xg] W"(a(~)) ----- [(x * -k 0(] x I~+a)] 0 (i.e. the germ of X~ has finite codimension) and that the family X~ unfolds generically the singularity ~(~). This means that the potentialf~ is a versal unfolding offg. In addition, we require that all stable and unstable manifolds are transversal, and for each singularity a'@) its stable and unstable manifolds are transversal to W""(a(~)) and W~"(a(~)). These assumptions imply that there are no secondary bifurcations and, hence, the bifurcation set of X~ near ~ coincides with the catastrophe set off~. That is, it coincides with the set of values ~ such that f~ presents a degenerate critical point. In particular, let us consider ~ e llt~, ~ = 0 and k = 3. We then obtain the cusp-family which is equivalent to f(~, x,y) = x ~ -k ~-~ x ~ + ~xz x § Q(y) in a neighbourhood of the bithrcation of type IX described in Section I, and the bifurca- tion value ~ = 0 represents two collapsing saddle-nodes. Theorem. -- Let X~, be a family in X~(M) which unfolds generically a non-hyperbolic singularity of type IX as above. Then X, is stable at -~. Proof. -- We will actually proof that if X~ is a d-parameter family of gradients which unfolds generically a (k -- 1)-codimension singularity such that 0 is an eigenvalue of mul- tiplicity one ofdXg(a(~)) and d >/k -- 1, then X~ is stable at ~. For simplicity, we suppose ---- 0. From the theory of singularity of functions [7], if X~ is a nearby family with associated potential 9r there is a local diffeomorphism of the form [r ~(~, z)] defined in a neighbourhood of (0, a(0)) in R a > M such that at~ o ~(~t, z) =f~(z). Moreover, iff~(~, x) is the restriction off~ to the central manifold W~ then there exists a C "-2 diffeomorphism of the form [+(~t), ~([x, x)] such that 1r f:o W,z) + z Hence, since Wr is one-dimensional, in this new ~-dependent coordinate we can k--1 write X~ I W~(~(~)) ----- -~(~t, x) [(k + 1) x k + Z ~ x~-l], where -~(~, x) is a positive i~l C ~- ~ function defined in a neighbourhood of (0, ~(0)) in R ~ x M. Now, extending to all of R d X M so that 9 = 1 outside a neigbourhood of (0, ~(0)), we define a new family of vector fields y~ __ _1 X~ which is equivalent to X~. By performing the same construction for X~, we define a family ~r which is equivalent to X~ and is such that #Z~ [W~(a(~)) and Y~ ]W~'(a(Vt)) have exactly the same expressions in the respective coordinates s and x. Therefore, by taking h*(tx, x) = s we obtain a conjugacy between 166 M. J. DIAS CARNEIRO AND J. PALIS Y~ [ W~ and ~z [ W~(~(~)). We can now proceed exactly like in Theorem A of (]hapter III of [15] to extend h~, to a conjugacy h~ : M -+ M between Y~ and Y,~. In this way we obtain an equivalence between X~ and X,~, concluding the proof of the stability of X~. 9 Section IV. -- Globalization In Sections II and III we have obtained a finite number of open and dense subsets of ?(~(M), each one corresponding to the cases described in Section I, with the property that every family X~ contained in their intersection ~/1 is locally stable at every value of the parameter. Suppose now that ~x varies on a fixed closed disc D in R *. From our analysis of the bifurcation sets in previous sections, it follows that there exists a subset ~/C ~/1, also open and dense, such that there are no codimension-two bifurcations on 0 D and the curves that represent the codimension-one bifurcations are transversal to 0 D. Hence, for X~, ~ ~/1, the codimension-one bifurcations occur on isolated points in 0 D and there is a finite number, say r, of codimension-two bifurcation values in the interior of D. For each 1 <~ i ~< r, let D e be a small closed disc transversal to all branches of codi- mension-one bifurcations and containing a unique codimension-two bifurcation value in its interior. Let D* ---- D -- Ui Int D e. For X~ ~ ~tl, the intersection of the bifurcation set with D* consists of the union of a finite number of closed (]1 simple curves or intervals, I'~, ..., I',,, each one corresponding either to a saddle-node or to an orbit of quasi- transversality. We denote by ~1, -.-, ~, the codimension-two bifurcation values of X~ inside D. From the local stability of X~ at ~, there exist open neighbourhoods V of X~ in ~t 1 and U, of ~, in D such that any family X~ in V is equivalent to X~ for [x ~ U,. We are now going to piece together these equivalences. To do this, we take for each i= 1, ...,r a smooth function p~:R ~-+R such that 0~< p~<~ 1, supp(p~)CU~ and p~ = 1 in a closed disc D~ centered at ~, and define perturbations R~ = grad~ f~ where the metrics g"~ and the potentials f~ are defined inductively as follows: fr + Pl( ) = + - and f~ =f)-i + p~(bt) [f~,--a~ g~, = gr -t- p,([x) [ff~- gd-x]. Hence, )~[~ = ~2X -~ for ~x r Ui, X x = X~ for Ix r U~ =~ U, and R~ = ~2~ for ~ e U~=t D~. Using the remark concerning local stability made after the proof of Theorem A, and which applies to all bifurcation cases in Sections II and III, we obtain that X~ is equi- ^i--1 valent to X,i~ ~ with the reparametrization q~, satisfying q0~(~) = ~ for ~ r U s and the equivalence h~ = identity for ~x r U~. Therefore, by transitivity, X~ is equivalent to X' Now, let I'~ be the first curve ofcodimension-one bifurcation in D\U~.= 1 D~. We cover r~ by a finite number of domains of reparametrizations, UI, " " ", U}~, and starting with X" [L, define perturbations X~ , k = 1,...,t~, along I'~, as above, such that X~'*---= X~ BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 167 for tz r U~= 1 U 1 and X~'k = X~ for ~z e U~=~ D~, where D] is a closed disc inside U ] l x 1 and [-J~=l D~ D Fx. Performing again the modifications referred to above of the equivalences inside each domain U~, we show that the family 1'k-1 is equivalent to the family X~,k with an equivalence which is the identity outside U~. In this way, starting with the equivalence between )(~ and )~' 1 and proceeding by induction, we construct an equivalence between X~ and x~,tl in a neighbourkood of F 1. It is now clear that by covering each curve Fj, j = 1,..., m, with domains of reparametrizations U~,..., UtS~, we obtain inductively an equivalence between X~ and X~' tm (and therefore between X~ and X~) in a neigh- bourhood W of the entire bifurcation set in D. It is important to observe that all the reparametrizations that we perform preserve ~ D. Finally, we repeat the same procedure in each component of D\~u thus achieving a global equivalence between X~ and X~. The proof of the main theorem in the paper is complete. 9 Remark. -- In the arguments presented above, the closed disc D can be replaced by any compact surface as the parameter space. REFERENCES [1] V. I. ARNOLD, S. M. A. N. Singularities ofdifferentiable maps, vol. I, Birkh/iuser, 1985. [2] T. BR6I~SR, Differentiable germs and catastrophes, Cambridge University Press, 1975. [3] J. GUCKENHEIMER, Bifurcations and catastrophes, Dynamical Systems, Acad. Press, 1973, 95-109. [4] B. A. KHESIN, Local bifurcations of gradient vector fields, Functional Analysis and Appl., 20, 3 (1986), 250-252. [5] N. H. KUIPER, Cl-cquivalence of functions near isolated critical points, in Syrup. on Infinite Dim. Topology, Ann. of Math. Studies, 69, Princeton University Press, 1972, 199-218. [6] R. LABARCA, Stability of parametrized families of vector fields, in Dynamical Systems and Bifurcation Theory, Pitman Research Notes in Math. Series, 160 (1987), 121-214. [7] J. MARTINET, Singularities of smooth functions and maps, Lecture Notes Series, 58, London Math. Society, 1982. [8] J. MARTINET, Ddploiements versels des applications diff6rentiables et classification des applications stables, in Lecture Notes in Math., 535, Springer-Verlag, 1975, 1-44. [9] J. MATH~R, Finitely determined maps germs, Pubt. Math. I.H.E.S., 35 (1968), 127-156. [10] 0. E. LANFORD III, Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Wakens, in Nonlinear Problems in the Physical Sciences and Biology, Lecture Notes in Math., 322, Springer-Verlag, 1973, 159-192. [ 11] S. NEWHOUSE, J. PALIS and F. WAKENS, Stable families of diffeomorphisms, Publ. Math. L H.E.S., 57 (1983), 5-71. [12] R. PALAXS, Local triviality of the restriction map for embeddings, Comm. Math. Helvet., 84 (1962), 305-312. [13] R. PALAIS, Morse theory on Hilbert manifolds, Topology, 2 (1963), 299-340. [14] J. PALLS, On Morse-Smale dynamical systems, Topology, 8 (1969), 385-405. [I 51 J. PALIS and F. WAKENS, Stability of parametrized families of gradient vector fields, Annals of Math., 118 (1983), 383-421. [16] J. PALIS and S. SMAIm, Structural stability theorems, in Global Analysis Proceedings Syrup. Pure Math., 14, A.M.S., 1970, 223-231. [17] S. STERNBERG, On the structure of local homeomorphisms of euclidean space II, Amer. Journal of Math., 80 (1958), 623-631. [18] F. TAKENS, Moduli of stability for gradients, in Singularities and Dynamical Systems, Mathematics Studies, 103, North-Holland, 1985, 69-80. VARCHENKO, GUSEIN-ZADE, __~<A 168 M. J. DIAS GARNEIRO AND J. PALIS [19] F. TAr~NS, Singularities of gradient vector fields and moduli, in Singularities and Dynamical Systems, Mathematics Studies, 108, North-Holland, 1985, 81-88. [20] F. T~a~NS, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147. [21] R. TI-IOM, Stabilitd struaureUe et morphogen3se, Benjamin, 1972. [22] G. VZOTER, Bifurcations of gradient veaorfields, Ph.D. Thesis, Groningen, 1983. [23] G. VEOTER, Global stability of generic two-parameter families of gradients on three manifolds, in Dynamical Systems and Bifurcations, Lecture Notes in Math., 1125, Springer-Verlag, 1985, 107-129. [24] G. VEOTER, The C~-preparation theorem, C~-unfoldings and applications, Report ZW-8013, Groningen, 1981. Departamento de Matem~itica- I.C.E.X. Universidade Federal de Minas Gerais Belo Horizonte, M.G. Brdsil Instituto de Matem~itica Pura e Aplicada (I.M.P.A.) Estrada Dona Castorina, 110 Jardim Bot~inico Rio de Janeiro, R.J. Brfisil Manuscrit re~u le 30 janvier 1989. Rdvisd le 2 fivrier 1990. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals http://www.deepdyve.com/lp/springer-journals/bifurcations-and-global-stability-of-families-of-gradients-3XQEIa1kHA

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Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
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Abstract

BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS by M. J. DIAS CARNEIRO and J. PALIS Dedicated to Ren~ Thom on his sixty fifth anniversary. In 1967 it was proved that among gradient vector fields on compact boundaryless manifolds, the elements of an open and dense subset are structurally stable: under small perturbations they have their orbit structure unchanged up to orbit preserving homeo- morphisms [14], [16]. From this result it follows that the stability of a gradient flow is equivalent to the hyperbolicity of the singularities and transversality of their stable and unstable manifolds. At the end of that decade, Thorn was asking about the bifurcations and stability of families of gradients, specially about k-parameter families with k <~ 4. The question is very challenging and indeed it might ammount to a rather formidable program, since not even just locally near a singularity the question for k = 4 is solved (and this problem by itself is very interesting). A point to stress here is that the dynamic bifur- cations of a gradient family are in general considerably richer than those of the corres- ponding family of potentials; see [3], [4], [21], [22] for comments. Also from the global point of view this comparison is relevant to understand Thom's question: often near a bifurcating singularity there appear secondary bifurcations due to tangencies between invariant (stable and unstable) manifolds from far away singularities. Finer dynamic analysis is then needed to describe the bifurcation diagram and to prove stability of a generic family. In this line, in 1983, in a paper dedicated to Thom on the occasion of his sixtieth birthday, the question for k = 1 was settled [15]: among one-parameter families of gradients, the stable ones are dense. These stable families can be charac- terized up to high codimension degeneracies; see Section I. The purpose of the present paper is to provide a proof of a similar result for two- parameter families of gradients. New techniques, specially concerning singular invariant foliations, are introduced to study the bifurcation diagrams and to prove stability. Let us state our result in a precise way. Let M be a compact boundaryless C ~ 104 M. J. DIAS CARNEIRO AND J. PALIS manifold. Gradients of real functions can be considered either with respect to a fixed Riemannian metric or to all possible ones. Although our result is true in both cases, we will restrict ourselves to the last one. Let x~(M) denote the set of C ~~ two-parameter families of gradients endowed with the C ~ Whitney topology, the parameter being taken in the unit disc D in R ~, and denote by rc2:M � D-+D the natural projection. We say that X~, ~2~ e z~(M) are equivalent if there are homeomor- phisms H:M � D-+M X D and ~:D-->D such that ~2H=q0n, and, for each ~x e D, h~, is an equivalence between X~ and X~,~), where h~ is defined by H(x, ~) = (h~(x), ~([x)). That is, h~ sends orbits of X~, onto orbits of X,c~ for each ~ e D. The family X~ is called (structurally) stable if it is equivalent to all nearby elements in ;(g(M). Our main result can now be presented as follows. Theorem. -- There is an open and dense subset ~t in z~(M) whose elements are stable. Several comments are in order. First of all, as we observe in Section IV, the para- meter space in our theorem can be taken to be any compact surface. Second, while the result makes one hopeful of giving a similar positive answer about stability of k-parameter families for k = 3 and k = 4, it is known that this is not true for k/> 8 [18] ; actually, it is not true even locally near a singularity [19]. On the other hand, positive local results near a singularity were obtained for k = 3 and to some extent k = 4 in [22], [4]; however, the question for k = 4 is still essentially open and very interesting. We also point out that our result was obtained by Vegter [22], [23] for manifolds of dimension less than or equal to three. These papers and [15] were the starting point of our work. However, the analysis of codimension-two bifurcations in higher dimensions is considerably more elaborated and led us to introduce new kinds of singular invariant foliations (that might even be useful in other contexts) ; see, for instance, w 1 of Section III below. The paper is organized in the following way. The first two sections are prepara- tory ones, so the reader gets acquainted with some basic concepts and tools and the previous result for one-parameter families. To serve as references for other cases, already in Section II we use these tools to exhibit the bifurcation diagrams and to prove local (in the parameter space) stability of families with quadratic and higher order contact between invariant manifolds. In Section III we complete the definition of the subset of families ~ (up to a slight modification still to be performed in the last section) and prove the local stability of its elements, except for the ones already considered in the preceeding section. In Section IV we globalize the result to the whole parameter space. To be more specific, in Section I we recall the characterization of the stable one- parameter families and from it infer what shall be a corresponding characterization for two parameters. This leads to a list of cases that begins with codimension-one bifurcations, namely a saddle-node and a quasi-transversal tangency. We then have BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 105 combinations of these two cases, like the simultaneous occurence of two saddle-nodes. There are also the purely codimension-two cases: a codimension-two tangency (cubic contact or lack of dimensions) and a codimension-two singularity. Also, three cases arise from the degeneracy of one of the transversality conditions concerning center- stable and strong-stable, rep. unstable, manifolds that are required for one-parameter families. In Section II we present the basic concept of compatible systems of invariant foliations and provide a brief description of how they are constructed. This kind of foliations have been used in previous work like [6], [14], [15], [16] and [22]. Here, the concept has to be considerably extended to include several new types of singular foliations. Using these foliations, we treat in this section the initial cases of quadratic and cubic (or higher order) contact between invariant manifolds. Finally, in Section III we obtain ~ as the intersection of several open and dense subsets of x~(M), each one corresponding to families that present one of the bifurcations listed in Section I. We prove that every family in ~ is stable at every value of the parameter Ix 9 D; i.e., the family restricted to a small neighbourhood of ~ in D is globally stable on M. We then show in Section IV that we can piece together, in terms of the parameter space, our construction of the equivalence between two nearby families in ~/, thus proving the result. We are thankful to several colleagues, including Arnold, Takens and Thorn, for their interest and valuable comments. We are specially grateful to Khesin (see Theorem B, Section II) and noteworthy the referee of the present paper. Section I. -- Bifurcations of codimension two We first recall the bifurcations of codimension one and some generic conditions that are imposed in order to obtain stable one-parameter families as in [15]. Let X~, 9 R, be a family of gradients on M and dim M = n. a) Saddle-nodes. -- We say that X~ has a saddle-node singularity p if one of the eigenvalues of dX~(p) is zero and all the others nonzero. Moreover, restricted to a center manifold through p, X~ has the form Z~(x) = ax 20x -t- O(1 x [a) with a ~e 0 (about center manifolds see [10]). For each ~z near ~, there is a Ez-dependent center manifold restricted to which X~ has the form o ]8 X~(x) = (ax 2q-b([z-~))Ox +O([x + [x(~t--~)l + I(~ x-~)[~) with a + 0. The saddle-node unfolds generically if also b ~e 0. This condition is satisfied by the elements of an open and dense subset of families. b) Quasi-transversal orbits. -- Let p, q be hyperbolic singularities of Xg; that is, all eigenvalues of dX~ at p and q are nonzero. Let W"(p) and W'(q) be their unstable 14 106 M. J. DIAS CARNEIRO AND J. PALIS and stable manifolds. Suppose y is an orbit of tangency between them and assume that dim T r W"(p) + dim T r WS(q) = dim M -- 1 for r e y. In local coordinates near r, we have X~ -- Oxl" W"(p) -~ (xl, ..., x,, 0, ..., 0), u = dim W"(p), w'(q) = (xl, ..., 0, ..., 0, x,+l, ..-, g( 2, ..., where n = dim M and k = dim(T,W"(p) n TrWS(q)). We say that is quasi-trans- versal ifg is a Morse function. For each ~ near ~, we can write similar expressions for X~, W"(p~) and W"(q~), g being replaced by a ~-dependent function g~. We then say that y unfolds generically if ~- (r)[~=g ~e 0. c) Generic conditions. -- We now list a number of generic conditions concerning the stability of families of gradients. c. 1. Distinct eigenvalues. -- We assume that the eigenvalues of Xg at the singularities associated with an orbit of tangency have multiplicity one. Hence, there exists a smallest expansion (respectively contraction) and we can consider the strong unstable manifold W "" corresponding to all but the smallest positive eigenvalue (see [15]); similarly for the strong-stable manifold W". And, corresponding to the smallest positive eigenvalue and all negative ones, we have a C 1 center-stable manifold W c,, which is transversal to W "". Similarly for a center-unstable manifold W c". We observe that, in the presence of an orbit of tangency the assumption on the multiplicity one of the eigenvalues at singula- rities are generic (open and dense) for two-parameter families of gradients. In fact, a failure of these conditions gives rise to a subset of codirnension at least 3: an orbit of tangency corresponds to subsets of codimension at least one and a multiple eigenvalue of the linear part to subsets of codimension at least two (since it is a symmetric operator). c. 2. Noncriticality. -- We assume that the strong stable and strong unstable mani- folds of a saddle-node are transversal to the unstable, resp. stable, manifolds of all other singularities. Similarly for the singularities associated to a tangency. c. 3. Transversality of center-unstable and stable manifolds at a tangency. -- If ? is an orbit of tangency between W~(p) and W~(q), we require W~"(p) to be transversal to W~(q); similarly for W"(p) and W~"(q). c.4. Linearizability. -- For a family X~ with one of the bifurcations of type I through IX below, we assume that Xg is C" linearizable transversally to a center manifold of a saddle-node or near the singularities associated with an orbit of tangency. Actually, this linearization is also required for each ~ near the bifurcation value ~. The integer m is taken to be bigger than p + 2, where p is the maximum ratio of positive, resp. negative, eigenvalues of dX~ at the singularity (cf. c. 1). By [I7], [20], the linea- BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 107 rization may be taken to depend differentiably on the parameter and these conditions are generic for two-parameter families. We do not assume this hypothesis when we deal with strictly codimension-one cases: a quasi-transversal orbit of tangency, treated in Theorem A, Section II, or a saddle-node, treated in w 7.B, Section III. Given a family of gradients, a parameter value is called regular if it corresponds to a stable field, in this case a Morse-Smale gradient field (hyperbolic singularities and transversality between stable and unstable manifolds); otherwise, it is a bifurcation value. Now, for an open and dense subset of arcs each bifurcation value is isolated and it corresponds to a unique tangency or to a nonhyperbolic singularity, for which condi- tions a), b) and c) are satisfied. These arcs are stable [15]. What we prove in Sections II and III is the analogue of these results for two- parameter families of gradients. The subset of families we consider must now include codimension-two bifurcations; they are listed here and studied in detail in Section III except for case VIII (cubic contact) which is considered in Section II. We keep denoting a family of gradients by X~, but now ~ varies in the unit disc D in R ~. I. A quasi-transversal orbit of tangency with criticality. -- For some ~z = ~, there are singularities p, q such that W"(p) and WS(p) have a quasi-transversal tangency. However, unlike in c. 2 above, there is another singularity s such that W"(s) and WS'(p) are nontransverse along a unique orbit, which is quasi-transversal, or similarly there is such an orbit in W""(q). Except for that, all conditions in a), b) and c) above are satisfied. II. Two quasi-transversal orbits of tangency. ~ Two orbits of tangency may occur simultaneously, but they must satisfy the generic conditions b) and c) above. III. A saddle-node with criticality. -- The unstable manifold of some singularity is nontransverse to the strong stable manifold of a saddle-node along a unique orbit which is quasi-transversal, or similarly with respect to the strong unstable manifold of a saddle-node. All other generic conditions in a), b) and c) are satisfied. IV. Two saddle-nodes. -- Two saddle-nodes occur simultaneously and both satisfy the generic conditions a) and c) above. V. A saddle-node and a quasi-transversal tangency. ~ These two bifurcations may occur simultaneously; again we assume all generic conditions we have mentioned con- cerning hyperbolicity of the other singularities, linearizability and transversality in a), b) and c). VI. A quasi-transversal orbit of tangency along which the corresponding stable and a center-unstable manifolds are tangent. -- Since all center-unstable manifolds are tangent on each orbit of the unstable manifold, the condition does not depend on which center- unstable manifold we consider. We also require all singularities to be hyperbolic and the generic conditions in b), c. 1, c. 2 and c. 4 to be satisfied. We will show in w 5 of Section III that the orbit of tangency may be taken to have quadratic contact. 108 M. J. DIAS CARNEIRO AND J. PALIS VII. Codimension-two tangency originating from lack of dimensions. -- A tangency occurs between W"(p) and WS(q), for some singularities p and q, so that the sum of their dimensions is equal to (dim M)- 2. Several generic conditions are imposed including the ones already mentioned. VIII. Tangency corresponding to cubic contact. -- Similar to the previous case, but now W"(p) and W'(q) have cubic contact along a unique nontransversal orbit of intersection. IX. Codimension-two singularity. -- Xg has a unique nonhyperbolic singularity which has a single eigenvalue zero; restricted to the corresponding center manifold, X~ has the form Xg(x) = (x 8 + O(I x t 4) ~x" Actually, we will treat here the case of a codimension c singularity for all c >/ 1 under the hypothesis of a single eigenvalue zero. Section II. -- Invariant foliations and invariant manifolds A basic tool in the proof of stability of a family of gradients is to construct invariant foliations which should be globally compatible: they ought to be preserved so that we can fit together localized constructions of flow equivalences or conjugacies. It is also helpful to restrict the family to invariant submanifolds in order to " reduce dimensions ", for instance, to obtain the bifurcation diagram. The strategy has been successfully adopted in several previous papers [6], [11], [15] and we refer to them for more details. In this section we recall the notions of compatible system of (invariant) foliations and of center-unstable and center-stable foliations, applying them to prove local stability of families presenting either quadratic or cubic (or more generally simple) contact between stable and unstable manifolds of hyperbolic singularities as in b) and VIII of Section I. The first case has been proved in [15] since it corresponds to a typical codimension-one bifurcation. However, our treatment is different from the one in [15] and, in fact, it contains some of the main new and old arguments involved in the proof of several other bifurcations. For this reason it will be repeatedly quoted in subsequent cases. Definition H. 1. ~ Let ~ be a hyperbolic singularity for Xg, ~ e R 2, and U C R 2 a neighbourhood of ~ and V a neighbourhood of a in M such that for each ~ e U there is a unique singularity a(~t) of X~ in V, with ~(~) = ~. A (local) unstable foliation F~(a) for X~ is a continuous foliation such that a) The leaves are C" discs, m 1> 2, varying continuously in the C m topology, with distin- guished leaf F~(~(~x), ~x) = W"(~(~)) n V � { ~ }, b) Each leaf FU(x, ~) is contained in V � { ~x }, c) F~(a) is invariant: X~, t(F"(x, ~x)) 3 F~(X~. t(x), ~x), t/> 0, BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 109 d) For each ~t 9 U, the intersection of a leaf of F"(a(~t)) with WS(~r(~z)) is a point. A (global) unstable foliation F"(~) is just the positive saturation by the flow of X~, 9 U, of the local unstable foliation. Similarly we define a stable foliation Fs(cr). Let us suppose that the vector field X~ has only (finitely many) hyperbolic singu- larities with their unstable and stable manifolds intersecting transversally. We may order the singularities ~l(~z) <~ . .. ~< crt(~z ) for ~z 9 U, U a small neighbourhood of in 112, in such way that if WU(~(~)) c~ W~(%.(~z)) ~e ~, then ~([z) ~ cr~(~z), and if i 4: j, then i < j whenever ~,(tz) ~< crj(~t). For each singularity a~(~) which is not a sink, we consider an unstable foliation F"(ai(Ez)). Definition II.2. -- The foliations F"(al(~t)), ..., 1~(%_1([z)) form a compatible unstable system if whenever a leaf F of FU(a~(~z)) intersects a leaf F of FU(~(~z)), i<j, then F D F. A similar definition holds for a compatible stable system, F"(%+x(~z)), ..., F*(at(~)). The construction of such systems is detailed in [6], [14], [15]; in [6] a center-unstable foliation FCU(ak(~z)) is also obtained which is compatible with the system F"(%(~z)), ..., F"(ek_a(~)) in the above sense: a leaf of F"(a,(~z)) that intersects a leaf of F~"(%(~)) actually contains this leaf. Each leaf of the center-unstable foliation is a C 1 disc and is the union of leaves of an unstable foliation F~. For fixed ~ the foliation F~"(%(~x)) is tangent to the vector field X~. In order to construct Fr it is assumed that the linear part DX~(%(~x)) has a smallest contraction, that is a negative eigenvalue of smallest absolute value, and hence we may take a center-unstable manifold Wr as the distinguished leaf of Fr Actually, since in the bifurcations of type I to VII we assume the linearizability condi- tion c.4 for X~ near the singularity ~r,(~x), there is a natural choice for Wr in this special coordinates, namely, W~"(%(~t)) is linear. Another important tool that we have often been using in bifurcation theory, as in the present work, is the following parametrized version of the well-known Isotopy Extension Theorem (see [12]). Let N be a C' compact manifold, r >t 1, and A an open subset of IIL ". Let M be a C ~ manifold with dimM>dimN. We indicate by C~(N � A,M � A) the set of C * mappings f: N � A ---> M � A such that = = rdf, endowed with the C * topo- logy, 1 ~ k ~< r. Here, r~ and ~' denote the natural projections re:N � A-+ A, r~':M � A--->A. Let Diff[(M � A) be the set of C ~ diffeomorphisms q0 of M � A such that r~'= r~' % again with the C ~ topology. Isotopy Extension Theorem. -- Let i 9 C~(N x A, M � A) be an embedding and A' a compact subset of A. Given neighbourhoods U of i(N x A) in M x A and V of the identity in Diff~(M � A), there exists a neighbourhood W of i in C~(N x A, M � A) such that for each j 9 W there exists ~ 9 V satisfying ~i = j restricted to N � A' and ~(x) = x for all x r U. This theorem is used to extend homeomorphisms h which are defined on top dimension submanifolds with boundary N C M whose restrictions to the boundary 110 M. J. DIAS CARNEIRO AND J. PALIS are C 1 diffeomorphisms, C x close to the identity. Hence, by applying the above theorem to h [ ON we obtain an extension H to all of M and defining H : M ~ M such that H ] N = h and H ] M\N = H we get the desired extension. One needs this parametrized version in order to obtain such extensions on each leaf of an invariantfoliation. We refer to [15] and [11] for some applications of these ideas in very similar situations. The use of the above invariant foliations is illustrated in Theorem A below. Before that, we recall the definition of local stability. Definition H. 3 (Local Stability). -- A family X~ 9 )~g(M) is stable at ~ 9 R 2 if there is a neighbourhood U of ~ in R 3 and a neighbourhood ok' of X~ in x~(M) such that for each family X~ 9 q/there is a value ~ e U and a homeomorphism H:U x M-+R 3 x M of the form H(~x, x) ----= (q~(~), h(~, x)), with ? : (U, ~) -+ (112, ~) also a homeomorphism onto its image, such that h~ : M ~ M is a topological equivalence between X~ and X,I~I, where h~(x) = h(~, x) for x 9 M. Theorem A. -- Let X~, ~ 9 113, be a family of gradients and -~ a bifurcation value suck that Xg presents exactly one orbit of quasi-transversal intersection between stable and unstable manifolds, which unfolds generically as described in b) of Section I. Suppose that all singularities of Xg are hyperbolic and the conditions described in c. 1, c. 2 and c. 3 of Section I are satisfied. Then, X~ is stable at ~x. Proof. -- First we describe the bifurcation set for X~, ~z close to ~. For ~ in a neighbourhood U of ~ in 112, we order the singularities of X~, ~x(~) ~< --- ~< at(~t), as above, and assume that the orbit of tangency y belongs to the intersection of WU(~k(~)) and W~(~k+~(~)). Let us assume that dim W"(~k(~) ) § dimW~(%+~(~))>/ n § 1 for, otherwise, similar arguments apply. Let Z(~) be a small cross-section intersecting y and Z = U~ 9 1J Z(~z). From the assumptions ofquasi-transversality and generic unfolding, there are C oo coordinates (~z, xl, ..., x,,yl, ...,y,_,, zl, ..., z,_,, Wl) in N centered at p=ynZ(~) such that n = 0, 0), WS(ek+,(~.)) n Y, = (~, x, O, z, F(~., x)) with r = dim[T, W~(ak@)) n T r W'((~/c+l(~.)) ] -- 1 and x ~ F@, x) [ dV(r,,o) is a Coo M~ functi~ such that rank td [0F ~_(t -,0]l ~ ) = r + 1" Theref~ the tangency between W~(a,(~)) and W'(%+~(tx)), for [x near ~, is characterized by the equa- OF tions F(~t, x)= 0, Ox (~x, x)= 0. By the implicit function theorem, there is a Coo curve P in 113, containing ~, such that ~t 9 F if and only if W~(%(~)) is quasi-transversal BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 111 to W'(ak + 1 (V t))- Moreover, it is easy to see that the noncriticality and transversality condi- tions (c. 2, c. 3 of Section I) imply that these are the only bifurcations near ~. Let X~, be a nearby family so that all conditions described above are also satisfied. We get a curve P near F which represents the bifurcations of the family X~. Let Z~-- - (Vt, x, 0, 0, wl) be a normal section in Y.. From Morse's Lemma with para- meters (see [13] or [5]), there is a diffeomorphism M: Z~--+Z ~ of the form M(W, x, wx) = (~(~t), h~(~, x, wl) ) which sends WU(~k(~)) n ~c to Wu(~k(~?(~))) n Z ~, W*(c~+l(~)) n Z ~ to W'(~+~(~0(Vt)) ) n X ~. Here ~ is a diffeomorphism defined in a neighbourhood of ~ such that ~(F) ---- r and which is close to the identity. Let us assume thatf~, the potential function of X~, has distinct critical values for near ~. Iff~ is the potential function of )~, then, since f~ and J~ are C ~ close Morse functions, there are C ~~ families of diffeomorphisms H~ : M --+ M, ),~ : I --+ I such that f~oH;~= X~of~, and so grad~;,o~,f~ is equivalent to grada-f ~ ----X~. (Here, g~ and ~ are the respective metrics and I an interval.) Hence, there is no loss of generality if we assume that X~ and X,~) have the same potential. The equivalence between X. a and X,c~ wilt be a conjugacy outside a neighbourhood V of the closure of the orbit of tangency y in M x R 2. Inside V it will preserve the level sets off~. Let us now describe this distinguished neighbourhood V. Let ck(bt ) =f,(a,(b0 ) andf~(,k+l(bt)) = ck+l(b0. If, > 0 is small and V,(b~) is an open neighbourhood of ,,(b~) in M, we consider A,(tz ) =f~-l(c,(~) -- ,) n %(~) and B,(tz ) ----f~-x(c,(~) + ,) n %(~) fori-----k,k+ 1. Let V,(b0-----{x e'~,(bt); X~,,,(x) n B,(bt ) 4 = 13 for t> 0 or X~,(x) tnA,($) 4:0 for t< 0}u{a,(~)}, i:k,k + 1, be neighbourhoods of a,(~) and ~,+~(~), respectively. We connect V,($) to V,+~(~) along y in the following way. We consider D C B,(~), a small closed disc centered at y n B,(~) such that [.J,>~0X~.,(D) does not intersect the boundary of the closure of A,+~(~) in M, and define D(tz ) :{xe M;X~.,(x)eD for some t~< 0 and Xv, t(x ) e A~+l(bt ) for some t > 0 }. Let V(~) = V,(bt) u D(~,) u V~+I([,) and V -~ tJ~e~ V(bt). Observe that, in order to glue continuously a conjugacy in the com- plement of V(bt ) with a level preserving equivalence, we adjust the metric in a neigh- bourhood of the part of aV(bt ) which is a union of trajectories, in such way that 1[ X~, [[o~ = 1. Moreover, since the critical levels in V will be preserved, this ajustment is such that the time it takes to go fromf~-~(c,(~)) to A~(~,) and fromf~-~(c,(bt)) to Bi(b0 is constant. We now briefly describe the construction of a center-unstable foliation F~*(a,(W)) ; we refer to [15] for more details. Let A~(tz) denote the sphere A~(tz) n W'(a,([,)) which is transversal to Xu, and intersects every nonsingular orbit in W*(e,(~,)), i.e. it is a fundamental domain for W'(e~(t*)). It contains A~'(~) = A,(t, ) n W"(**(~)) as a codi- mension-one (equatorial) sphere. Recall that W"(a,(~)) is foliated by a unique codi- mension-one C ~ foliation F"(a,(t*)) -- the strong stable foliation. We can write 112 M. J. DIAS CARNEIRO AND J. PALIS A~(V) = D+(V)u C,(~)u D~-(~). Here Cs(~) is a small tubular neighbourhood of A~'(~) in A~(~) ; D+(~) and DT(~t ) are closed discs whose respective boundaries aD+(~) and a DT(~) are the intersection of leaves F + and F- of FS'(%(~)) with A~(~). The subset C~(Vt) is taken in such way that if W"(~+(~)) n C,(Vt ) 4: O, then W"(~+(~)) n A~(pt) 4= O and W"(a,(B)) n C,(B) is transversal to the induced foliation FSS(%(~)) n C,(B). This is possible because of the noncriticality assumption (c.2) of Section I. The condition also allows one to construct a one-dimensional C 1 foliation F~(%(~)) on G,(~) which is compatible with the system F"(al(~)), ..., F"(~k_l(~) ) and is transversal to F+"(%(~)) n C,(~) and to a D~-(~) u a D+(~). Let F~(~) be a u-dimensional continuous foliation with C 1 leaves on Ak(~) which is compatible with the system F~(~I(~)), ..., F"(%_1(~)) and is transversal to A~(~). We now point out the following key fact: if P~ : Ak(~)\A~(~) ---> Bk(~) is the Poincar6 map between the non-critical levels Ak(~) , Bk(~) and b~ : Bk(~) +-) is a homeomorphism preserving leaves of P~(F"(Gk(~))), then the induced map (P~)-I o b~ o P~ extends conti- nuously to a full homeomorphism of Ak(~). We observe that preserving leaves means that the map sends a leaf of the foliation into another one. This motivates the definition of a center-unstable foliation as F~"(%(~)) ----- U+/>0 X~.,(F~(~)), a distinguished leaf being a center-unstable manifold W~"(%(~)). We distinguish two parts in F ~" with different types of leaves. One, denoted by F~"(%(~)), has (u + 1)-dimensional leaves, with u = dim(W~(%(~))), each leaf corresponding to a point of D+(~) w D~-(~). The other part of the foliation, denoted by F~(%(~)), has a typical leaf of the form U~eF~ F~(%(~))~, where F~ is a leaf of F~(~(~)) and Fr is the leaf of F~"(%(~)) through the point x. Notice that the ]eaves in F~"(~(~)) have dimension u + 2. The fact pointed out above concerning extensions of homeomorphisms together with the existence of a weakest contraction (see condition (c. 1)) imply the following stronger property. A homeomorphism b~ : B~(~) +-~ that preserves F~"(%(~)) and the portion of F~"(%(~)) inside a conic region which contains the center-unstable manifold induces, as before, a homeomorphism on all of A~(~). Our conic region corresponds to a bundle of solid cones, with constant width over the sphere B~(~) n W~(%(~)). The construction of a center- stable foliation F~'(% + ~(~)) is dual to this one. We proceed in the same way to construct a center-unstable foliation F~"(~(q~(~))) and a center-stable foliation Fe'(~+,(q~(~))) for the vector field X~(~). Let us now construct an equivalence between X~ and :~(~1" We start by sending sources to sources and sinks to sinks. Then, proceeding by induction on j and using the Isotopy Extension Theorem as in [15], pages 413 and 414, we obtain a continuous family of homeomorphisms and k~,: U~>~+~ W"(a~(~)) + U~;~++~ W"(+~.(9(~))). On each step, say from i -- 1 to i, we use the unstable system in order to go from a fundamental domain of W*(a,_I(~)) to a fundamental domain of W*(a,(~)), and use BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 113 also the fact that Ws(,i(~)) and WS(~i(~(~))) are ~1 close on compact parts. By imposing that the equivalence preserves the unstable system F"(%(~)), ..., F"(%_1(~)), we see that h~ induces a homeomorphism in part of the sphere A~'(~). By using induction on the indices j, starting withj = k -- 1, and using again the Isotopy Extension Theorem, we extend this homeomorphism to the whole of A~*(~) (space of leaves of F~U(ak(~)). By preserving the central foliation FC(%(~)), we define a homeomorphism on 0 D+(~) ~J 0D~-(~). Using once more the Isotopy Extension Theorem and induction we extend this homeomorphism to D+(~) w D~-(~). In this way, we obtain a homeo- morphism on the space of leaves of the center-unstable foliation FCU(,k(~) ). We proceed dually to get a homeomorphism in the space of leaves of the center-stable foliation F~ ). Our next task is to construct a homeomorphism in the cross sections Z(~) (which we assume to be contained in a level set off~) preserving the center-unstable and the center-stable foliations. We first construct this homeomorphism on the section X~(~) = WC~(~+l(~))nZ(~). Let F~C Zcs(~) be a C 1 foliation compatible with W'(ak+i(~)) n Z(~) whose leaves have complementary dimension and are transversal to Z~(~), where as above Z~(~) is a smooth cross section which is tangent to WCS((Tk..bl(~s ('~ Weu((~k(~) ) I~ Z(~). Let (Vl, vi, w,,) be coordinates for Zr such that W"(Gk_I(~) ) n E"~(~) = {Vl = vi = 0} and W~"(%_i(~t)) nZ~'(~) ={v I=0} where v I= (v2, ...,V~_,);WT,= (Wl,..,W,) and s = dim W~(%+l(~)). FIG. I 15 114 M. J. DIAS CARNEIRO AND J. PALIS In Z"'(~), we consider two conic regions C(ai) = { v~ -- 8~(~) II ~ It ~ ~> 0 ) and C(a~) = { v~ -- ~(~)[[ v~ ]1~< 0 }, where 0 < aa(~)< a~(~) are chosen so that 00(aa) and 0C(a~) are transversal to F~ The intersection of F~"(a~(~)) with Z"'(~) gives rise to a foliation in C(a~) which is singular along W"(a~(~)) n y o,(~). This foliation is extended continuously on each leaf, say F~, of F~"(~(~)) n Z"'(~) in such way that OC(a~) n F~ is a leaf and it is non-singular in the interior of C(a~) n F~. We denote by F~"(a~(~)) this new foliation. By construction Fg"(.~(~)) is topologically transversal to F~2, so the projection Eo'(~) ~ E"(~) along the leaves of F~', restricted to each leaf of Fg~(a~(~)), is a homeomorphism. Hence, by performing the same cons- truction for X~,), since we already have a homeomorphism h~:Z~ -+ Z~ we can define h~' by sending F~2 to F~(~ and F0"(bt ) to ~'~"(~0(~)) preserving leaves of the center-unstable foliations. The main property of h~' is that it preserves the leaves of type F~"(~(~)) inside the conic region C(aa). Therefore, as we pointed out above, the induced homeomorphism ~(~t~' ~-~ o h~C~ o P~* automatically defines an extension to the fundamental domain A~(~). By proceeding analogously in the section = n we obtain a homeomorphism h~" which preserves a foliation F~ u compatible with WC~(,k(~) ) and the center-stable foliation F~ ) n Zc"(~). Finally, we match h~' and h~, ~ to obtain a homeomorphism on the whole section X(~). We do this by first considering a C a foliation FC"(~) transversal to Z*(~) and of complementary dimension, such that Fs"(~)n E~(~)= F~ ' and F'"(~)n Z~"(~)= F~ ~. We then require the homeomorphism to preserve this foliation, as well as the center-unstable and center- stable foliations. The homeomorphism extends to V(~)n (UtE, Xt,~(Z(~))) just by preserving the level sets off~ and by sending orbits of X~ to orbits of )(,(~. In particular, it defines a homeomorphism on a closed disc D,(~) contained in the level set Bk(~) and on a closed disc Dk+a(~) contained in the level set Ak+~(~). By construction, near the boundary of these discs the homeomorphisms are actually C 1 diffeomorphisms close to the identity. Hence, since all stable manifolds W'(~(~)), k + 1 ~< i ~< t are transversal to Wu(~k(~)) outside D,(~), we can proceed by induction on i and apply again the Isotopy Extension Theorem to get a homeomorphism on all of Bk(~) which preserves the intersections of the stable system F"'(ak+l(~)), F'(ak+2(~)), ..., F~(~t(~)) with B,(~). Similarly, we obtain a homeomorphism on the level Ak+~(~) which preserves the unstable system. We complete the definition of the equivalence inside V(~) by preserving the level sets off~ and of course sending orbits of X~ to orbits of X,c~. Thus, we have defined two families of homeomorphisms which depend conti- nuously on ~, and : BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 115 Let F~ be the level set off~ that contains the cross-section Z(~) and suppose that f~ is nondecreasing with respect to the ordering of the singularities. Observe that if xr [Uj~<k+iWS(a~([z))] u [U~>kW"(cr~(~z))], then the orbit of x intersects F,. Therefore, to extend the equivalence to all of M, we just construct a continuous family of homeomorphisms on F~ which are compatible with h~ and h~, and which preserves the stable and unstable systems. This is done by using once more the Isotopy Extension Theorem and induction on the dimension of the stable manifolds that intersect Fc, exactly as in [15]. 9 Remark. -- Let 9 : (U, -~) ---> (q~(U), ~(~)) be the reparametrization obtained in the proof of Theorem A. If D i C D2 C U are closed discs centered at ~ and p : R 2 -+ R is a C = function with supp p C U, O I D1, 0 ~< p ~< 1 and p ---- 0 outside U, then by defining F~ ---- (1 -- O(~))f~ + p(~)f~, g~ ---- [1 -- p(~)] g~ 4- p(~) ~ and X~ ---- gradgt, f~ , we obtain a two-parameter family such that X~,----X~ for ~ r U, :X~ = X~ for [~ E D1. Since 9 is C 1 close to the identity, there is another reparametrization + : (U, ~) --+ (+(U), +(~)) such that + restricted to Di is equal to 9 and is equal to the identity outside a neighbourhood U2 of the disc D2. Observe that the system of foliations constructed in the proof of Theorem A can be taken to be the same for X~ and X~ when r U 2. Hence, having the identity as the homeomorphism on the space of leaves of these foliations and repeating the proof of Theorem A, we obtain an equivalence h~ between X~ and X~(~ such that h~ ---- ida, for ~ r U2. This fact # very relevant in order to prove global stability, see Section IV, and it applies to all bifurcation cases treated in this and the next sections. We now prove local stability of bifurcations of type VIII, an orbit of tangency with cubic contact. Actually, as M. Khesin pointed out to us, using the theory of V- equivalence (or contact equivalence) as in [1], [8], [9] and the arguments in the proof of Theorem A, one can show the local stability of a much wider class of families of gradients. Recall that two germs fl and f~ : (R', 0) --+ (R, 0) are V-equivalent if there is a germ of diffeomorphism h : (R', 0) ---> (R', 0) and a smcoth germ M : (R', 0) --+ R such that fi(x) ---- M(x) .f,(h(x)) (so, h sends the "variety "fl-~(0) to f2-~(0)). Let ~ be a bifurcation value for a family X~ such that ~, C W"(a(~)) tn WS(a'(~)) is an orbit of tangency along which the manifolds have simple contact of type Ak, Dk, E6, E~, E 8 as in Arnold's list [9]. By that we mean that for p ~ ~, dim[T~ W~(a(~)) + T, W'(a'(~))] ----- dim M -- 1 and if 2] is a smooth cross section at p, there are ~-dependent coordinates (x,y, z, wl) centered at p such that W"(a(~)) nY.={z-----0, wi----0},W'(o'(~)) nE={y----0, Wl----F(~, x)} with f(x)= F(0, x) being equivalent to one of the following normal forms: A, ---- ~+~ + Q, k 1> 1, Q.(x,, ..., x,) a non-degenerate quadratic form; + + 0_, E~ + 0_; + + U_; + + O_, ~D(x~, ..., x,) a non-degenerate quadratic form. We require F(~, x) to be a V-versal 116 M. J. DIAS CARNEIRO AND J. PALLS unfolding off(x) (~ = 0) [9]. So, for a nearby family X~ with corresponding unfolding ~'(tx, x) we have F(F , x) = M(F , x) F(q~(F), h(F, x)) with M(0, 0) 4: 0, ~ and h~ being local diffeomorphisms. Therefore, there is a local diffeomorphism Z * -+ ~,~ of the form (q~(~), h(~z, x), M(F, x) -1 wl) which sends W~(~(tx)) n Z ~ to W"(~(q0(F))) n ~,~ and W~(&(F)) n Z ~ to W"(~'(q0(tx))) n ~c (as in the proof of Theorem A, X ~ denotes a smooth cross-section tangent to WC~(~(~)) n W"(~'(~)) n Z). Hence, from the non-criticality condition c.2 and the transversality between W""(~(~)) and W'(~'(~)) and between W""(&(~)) and W"(~(~)) (condition c.3), we construct compatible unstable and stable systems and proceed exactly as in Theorem A to get an equivalence between X~ and X~. 9 Thus, we have the following Theorem B. -- Let ~ be a bifurcation value for a family X~ such that X~ presents exactly one orbit of tangency with cubic contact, or more generally simple contact, which unfolds generically. Suppose that all singularities are hyperbolic and conditions c. 1, c. 2, c. 3 of Section I are satisfied. Then, X~ is stable at ~. We will see in w 6 of Section III that if dim[T~ W"(,) + T~ W"(,')] <~ dim M -- 2, the family may present other tangencies (secondary bifurcations) besides the tangencies between W"(,(tx)) and W"(*'(~)). This will impose several delicate adjustments in order to extend a local equivalence (in a neighbourhood of the orbit of tangency) to an equivalence on all of M. Section III. -- Local stability In this section we continue to prove local stability of the bifurcations mentioned in Section I; in the previous one we have already studied the families that exhibit one orbit of simple contact between stable and unstable manifolds. As mentioned before, by local we mean that we only consider the parameter varying in some small neigh- bourhood in R ~ of an initial bifurcation value ~. In each case, we start by requiring several additional generic assumptions for the family X~, then we obtain the bifurcation set near ~ and finally we prove stability. w 1. Bifurcations of type I: one orbit of tangency with criticality (1 .A) Generic conditions describing the bifurcation. -- A family X~ in z~(M) has a bifurcation value ~ of type I if the following holds: BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 117 (1.1) The vector field Xg presents a unique orbit y of quasi-transversal intersection between say, the unstable manifold WU(p(~)) and the unstable manifold W'(q(~)) of hyperbolic singularities p(~), q(~), (1.2) Linearizability of the family near p(~) and q(~) with the respective linear part with distinct eigenvalues (conditions c. 1 and c.4 of Section I), (1.3) There is a unique orbit y' of quasi-transversal intersection between the unstable manifold Wu(a@)) of a hyperbo]ic singularity ~(~) and the strong stable manifold ofp(~), W'8(p(~)), (1.4) Let W"8(p(~)) C W"'(p(~)) be the codimension two submanifold of W"(p(~)), invariant by X~, which corresponds (in linearized coordinates near p(~)) to the negative eigenvalues of dXg(p(~)) except for the two ones of smallest norm a!(~), a~(~). Then W"(a(~)) is transversal to WW(p(~)); in particular, the orbit of tangency y' does not belong to Wv*(p(~)), (1.5) The orbit of tangency 5" unfolds generically as in b) of Section I, so there is a C 1 curve F in the parameter space such that [X s F if and only if W~(p([X)) is not transversal to WS(q([X)), (1.6) The orbit of tangency 5"' also unfolds generically so that there is a C 1 curve F' in the parameter space such that [X e F' if and only if W"(a([X)) is tangent to W"'(p([X)). Moreover, the curves I' and F' intersect transversally at K, (1.7) The vector field Xg satisfies the linearizability conditions near ~(~) and the eigenvalues of dXg(a(~)) have multiplicity one so that if we take a C 1 center-unstable manifold WC"(a(~)) for a(~), then it is transversal to W"~(p(~)). We also assume the non-criticality conditions c. 2 of Section I, and that every unstable manifold is trans- versal to WS'(a(~)) and every stable manifold is transversal to W"U(q(~)) and the hypothesis c.3 which says that W~(p(~)) is transversal to W"(q@)) and WC"(q@)) is transversal to W~(p(~)), (1.8) Let n~_~ = dim[T, W"(a(~)) n T, W'(p(~))], for r e 5"', and consider the invariant manifold VsC W~(p(~)) of dimension nk_ 1 corresponding to the nk_ 1 nega- tive eigenvalues of dX~(p(~)) of biggest norm. There is a subspace E(r) C T, M such that lira dXg ,(r).E(r) = Toc~, W~"(a(~)) and lim dXg ,(r).E(r) = T~, V ~, t ---~-- oo ' ~ --~ ~- 0o , (1.9) According to (1.2) there are C m ix-dependent coordinates (xl,...,x~, Yl, -..,Y~) in a neighbourhood of p(~) in M such that X~ = - ~i([x) x~ + --, ,-, - oy, where 0 < ~([X) < ... < a,(~) and 0 < ~([X) < ... < [3,([X). 118 M. J. DIAS CARNEIRO AND J. PALIS The manifold W'(q(~)) is transversal to the plane (0, x,,0, ...,0,y~, ...,y~). This can be formulated intrinsically by saying that W'(q(~)) is transversal to any C" invariant manifold that contains the orbit of tangency y' and the unstable manifold W~(p(~)). Similarly, the unstable manifold W"(a(~)) is transversal to the plane (0, x2, ..., x~,yl, O, ..., 0). (1.B) The bifurcation diagram. -- Let zx(~t) ~< ... ~< %(~t) ~< %+1(~t) ~< ... ~< zt(~t) be the ordering of the singularities of the vector field X~ for ~t near ~ as in the previous section. We assume p(~) = %(~t), q(~t) = %+a(~t) and without loss of generality we may also assume that z(~t) = z~_l(~t). First observe that since W~"(%(~)) is transversal to Wg(gk+l(~) ) and W~*(%+1(~)) is transversal to W~(%(~)) the only possibility for non-stability of the vector field X~ comes from either the tangency between W*(a,(~)) and W'(a,+l(~)) or between W"(~,_l(~Z)) and W~(%+1(~)). This follows from the fact that if W"(%(~)) is transversal to W"(a,(~)), then W"(%([z)) is transversal to W~(%+~(~z)) for ~z near ~. Proposition. -- Let X~ be a family presenting a bifurcation value ~. of type I as described above. Then, there is a neighbourhood U of -~ in R ~" such that the bifurcation set for X~ in U is the union of two C ~ curves F u P0, such that ~t e P if and only if X~ presents a unique orbit of quasi-transversality y~C W~(~k(~t)) c3 W'(e~+l(~t)) and bt ~ F0 if and only if there is a unique orbit of quasi-transversality ~,~C W"(%_~(~t)) ~ W~(%+l(bt)). The relative position of F and F 0 is illustrated in Fig. III. Proof. -- Using a ~t-dependent C ~ (m/> 3) linearization for X~ near %(~t) and the transversality between WU(%(~)) and W~(%+1(~)), we may construct a C a sub- C$ C8 -- manifold W e' C M � R 2, Wk+ 1 = [J~wr � { ix} such that W (r k+l contains the closure of y. Moreover, W~+I and WC'(zk+l(tz)) admit smoothing C ~ struc- tures, r/> 3 (see Chapter II. 1 of [15]). In the sequel, we shall take r = m 1> 3. Analo- gously, from (1.7) the center-unstable manifold W~u(%_~(~)) can be extended to a C 1 manifold W~_I that contains the closure of the orbit y' which also admits a C r smoothing structure. Let W e-- W~k-1 n W~+I and consider the restriction of Xa to W~(~t)----W~"(zk_~(tx))nWr In this setting both W"(%_,(tz)) and W~(z~+I(Ez)) have codimension one and we may also assume that X~, ] Wc(Ez) is of class C m-1 having a C ~ tz-dependent linearization near %+1(~z) for ~ near ~: '*k-1 0 '~k 0 ,=1 Oy, with 0 < ~x(~t) < ... < ~,k_~(bt), 0 < ~l(0t) < ... < ~,k(bt) and n k_~ = dim[T,, W"(z) c~ T,, W*(p(~))] for z' ~ y', n k -= dim[T, W~(p(~)) c~ T, W'(q(~))] for z e y. BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 119 If Z * C { x~ ----- I } and S * C {Yl = 1 } are two cross-sections with coordinates (ix, xl, xi,yX,yL) and (a, vl, v,, vi, wry), respectively, such that y' n E*(~) = (~, 0, a~, 0, 0), 7nSC(~) = (~,0,0,0) where x I= (x3, ...,x,k_l ) and Yr.= (Y,, ...,ynk) then w'(~+,(~)) n s(~) = { vl = F(~, v~, ~i, w~)} and W~(ak_l(~)) ('~ E~(~t) --{ x I = C(~t, xi,yl,yr. ) } where F and G are C m functions, m >/ 3. The quasi-transversality assumptions mean that the functions wr. ~ F(~, 0, 0, wr.) and x I v-~ G(~, xi, 0, 0) have non-degenerate (Morse) critical points at WL = 0 and x I ---- ai, respectively. Hence, W"(%(~t)) is tangent 9 9 D to W*(%+a(~t)) if and only if F(~t, 0, 0, ws) = 0 and ~ (~, 0, 0, wr) = 0 forj = 2, n k 9 By the generic unfolding of the orbit of tangency Y and the implicit function theorem, we get a C m-1 curve P in the parameter space such that the corresponding vector field X~, ~ e F, presents an orbit of quasi-transversality between W"(,k(~x)) and W~(%+l(~t)). Therefore, F belongs to the bifurcation diagram. Analogously, solving 0G the equations G(~, xi, 0, 0) = 0, ~ (~z, xi, 0, 0) = 0, we obtain a curve F' containing such that, for ~t e F', the vector field X~ has an orbit of quasi-transversal intersection between W"(a,_~(~z)) and W"(%(~x)). The condition (I.6) says that F and P' are transversal at ~ (the tangencies have independent unfolding). We may suppose that ~=0, r={~=0}and r'={~x=0 }. To study the tangencies between W~(%_1(~)) and W"(%+~(tz)), we write in the above coordinates WU(~_~(~)) n S~ = {(e -~l'~,t G(~, XI, g--~l(g) t, e--~L(g) t /I)L), e--a2(~)t, e--at(g)t(Xi ~_ ai), WL)} with e- ~)t = diag(e- ~(~)t, .--, e- ~,~(~)~) and e- ~(~)~ = diag(e- %(~)~, . .., e- ~'~-,(~)~). Hence, the tangency between W~(%_I(~x)) and W'(g~+~(~z)) in S~(~) is expressed by the system of equations: e -=~tG(~) -- F(~) = 0 OG OF ~(~) -~(~).e",-~" =o ~= a, ...,n~_~ (El) e- =t t 0G OF Oy-~ (~)'e-~i* -- Ow~ (~) = 0 j = 2, ..., n~ -- al G(~) -- [3~(iz ) e-h'~" OG ~, e_~/~, , o.r--~ (~) - ~7~ 9 . (~). ~ 120 M. J. DIAS CARNEIRO AND J. PALLS -= (bt, xi, e -~ll~'~, e -~L~g~ wL), ~ -= (Vt, e -~c~, e-~t(xi -k ai), wz). Making t -+ q- oo OG OF we obtain F(~,O,O, wL) =- O, ~x~ (~'x~' 0,0) = O, Fw~ (~'O'O'wL) = 0 and G(~t, xi, O, O) = 0 which is non-singular due to the generic and independent unfolding of the orbits of tangency. Let ~ ---- min (~x(0), 0c1(0 ), 0~2(0) -- 0~1(0)} > 0 (SO ~j([s ---- %(Vt)/= /> 1, "~j(~t) = ~(tt)/0t >/ 1, for Vt near 0). By setting e-=~= z, we may extend the system to a neighbourhood of the origin in a C x fashion to apply the implicit function theorem and get a C ~ curve F o in the parameter space, tangent to the curve I' at 0, such that ~t ~ F 0 if and only if X~ presents a tangency between W"(~k_x(~t)) and W'(~k+l(Vt)). It also follows from the above equations that along P0 the manifolds W"(~_l(Vt)) and W'(~+x(Vt)) present an orbit of quasi-transversal intersection. For the singularities ~(~) with i < k -- 1, the transversality between W=(~,(0)) and W'(~(0)), and between WC~(~(0)) and W'(~+x(0)), guarantee the transver- sality between W"(~(~z)) and W'(%+1(~)) for tz near 0. 9 V 1 VI Lc(pL) FIG. II We now proceed to describe the above equations (El) in terms of tangencies between foliations: this geometric interpretation will be useful in the proof of the stability of the bifurcation of type L Let F~+l(t~) be the C" foliation in SO(or) which is compatible with W~(~k +l([z)) defined by ~l(tz, vl, v~, VI, WL) = (V 1 -- F([L, v2, /JI, WL) q- F(~L, V2, 0, WL) , 02, WL). It follows BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 121 from the quasi-transversality between W"(%_~(O)) and W'(ak(O)) that the restriction of n~_~ to W"(ak_x(~)) n S~(tz) is a submersion with fold along the set { e-"' G(~, xi( ~, e-", wL), e -~'', e -~'t, wr.), e-"*, e -~`t Xl([L , e-", w,), WL} where xi(~x , z, w~) is a C 1 function which is of class C m- ~ for z > 0. Hence, the subma- nifold L*(I~)C Sr defined by (v a, v2, v~X/~" xi/(~. , ~,,'/", w~), w~) v~ >>. 0 is transversal to F~x(tx ) and, to W"(*~-I(t~)) ~ S"(~) and it contains the locus of tangency between W~(%_x(~)) c~ S*(~) and F~_~(~); see Fig. II. Let F"(tx) be a codimension-two C x foliation in L*(vt) which is compatible with W"(%_~(~))n L~(t~) and with W"(.~(~)) ~ L~ Then, from the quasi-transversality between W"(%(0)) and W'(.~+x(0)), we get that W*(.~+I(~) ) c~L*(~) is tangent to F"(~) along a two- dimensional C ~ manifold Sg(~) which is the graph of a C 1 map w~, = f~L(~, V~), v~ >t 0. In short, the second set of equations in (El) above defines L*(~) and the second and third ones define S~(vt). In this way the bifurcation set is described as follows: a) The point W~(%(Vt)) n Sg(~x) belongs to the curve Wu(o'k+l(~L)) ('~ Sg(~L) if and only if there is a quasi-transversal orbit of tangency between W"(.~(tx)) and W*(.~+ a(~)), b) The curves T~(~) = W~(~+~(Vt)) c~ S~(~) and T"(~) = W"(~_I(~)) n S~(tx) are tangent in Sg(~) if and only if there is a quasi-transversal orbit of tangency between W~(%_l(~X)) and W*(%+x(~)). It corresponds to the curve I' 0 in the parameter space while condition a) corresponds to 17. I" o FIo. III We now further analyze the bifurcation diagram, specially the second condition above, in order to initiate the proof of the local stability in this case. First, notice that the curve T~(~) is a leaf of a foliation pg defined by a C 1 one-form + O(4+') dv2' > 0, in the region R~ = { v~ >/ ~ I Vl l ~'/'}" Notice that if ~1 q- el/> ~ then 0 is a hyper- bolic singularity of sink type with linear part equals to ~ v2 dvl -- ~1 vl dv2. If e2 > ~1 -/- ~1, then by setting w = v~ ~' + ,,>/-, we obtain, after dividing by ~2 w- 1 + ,,/c,~ + ~,~, the expres- sion of a C 1 one-form with a hyperbolic sink at the origin with linear part equals to ( 0o) (~x + ~1) w dw -- ~1 vl + ~1 W~y 1 (0) dv2. Hence, in both cases, the foliation p~ = 0 ~lv~l+~l'm'OG P~=~2v~dvl--~lVldV2--[ 122 M. J. DIAS CARNEIlZO AND J. PALIS has a singularity of sink type at 0 and is such that all leaves, except one, are tangent to the axis v2 = 0. Therefore, a tangency between p~ and T*(~z) occurs along a C I curve l~ aF v~, + ~,)/~, aG defined by ~2 V2~v 2 -- ~17)1 -- ~1 " Z 0Y + 0(V[ +') = 0, V 2 >/ 0. Using again the OF 0G hypothesis (1.9), which means that 0v-~ (0) 4:0 and Oy---- 1 (0) 4: 0, we obtain that/~ is transversal to T*(~) and to { 9~ ---- 0 } and the contact along t~ is quadratic. Thus, the bifurcation set is characterized by the position of the points Pk+I(V) ----- T~(~) n t~, pk_l(Ez)---= T~(~)r~t~ and pk(~z)= W~(a~(~))raS~(~z). By taking ~z in small neigh- bourhood V of R ~ and shrinking S(~z), we can modify p~ in a neighbourhood of the boundary of R~ in order to include the curve v~ = ~ I vl I (~'/~(~) as a leaf of p~ and to extend it linearly by setting p~ --= o~ 2 v 2 dv x -- ~1 vl dv2. The inverse image of p~ by the Poincar6 map between level sets P~ : Ak(~) -+ B~(tx ) gives a continuous one-dimensional foliation with C 1 leaves which are topologically transversal to A~(~z): this remark is easy to check by using the linearization of X~ and it will be important in the proof of local stability at the end of the paragraph. Suppose now that X~ is a nearby family. Let q0 : V -+ R ~ be a local homeomorphism which sends the bifurcation set of X~ to the bifurcation set of X~. Then, we define a homeomorphism h~: S~([z) -+ S~(q0(~z)) which sends p~ to ~(g) and the curve T*(tx) to T*(q~(~z)) as follows. We first define a homeomorphism between the lines of tangency tg, "[~,~) such that p~_l(tZ) is sent to/~_x(q~(~)) and P~+I(&) goes to/~+~(~?([x)). This is only necessary in the region tXl i> 0 or ~xz/> 0 since, otherwise, the curves T*(tz ) and T"(~z) do not intersect ~. This gives a homeo- OF morphism on part of the space of leaves of a foliation -~ defined by dv~ -- ~v z (~z, v~) dr. z = 0 and which has T*(~) as a distinguished leaf (T*(~z) is defined by v~ = F(~z, v~)). We extend such a homeomorphism to the axis v~ =: 0, v~ ~< 0, also preserving its intersection with T~(~z), to complete the definition of a homeomorphism on the space of leaves of ~. This yields a homeomorphism also on the space of leaves of ~. By the reparametrization above, when p~_l(~X) coincides with p~+x(~z), the same occurs with /~e_l(q~(~)) and ~+l(~?(t~)). Therefore, since ~ and ~ have quadratic contact along IF, we define h~ by sending leaves of ~ to leaves of 9"~(~) and leaves of -~ to leaves of $,(~). In the sequel we will fully develop the proof of local stability. (1.s The local stability of the bifurcation of type L -- Let )~, be a family near X~ so that all conditions described in (I. A) are satisfied with respect to a bifurcation value near ~ where )~ presents one orbit of quasi-transversality with a criticality. The equivalence between X~ and X,(~) (after choosing an appropriate repara- metrization q~) will be a conjugacy outside a neighbourhood of the closure of y ~3 y' in M. This neighbourhood U is the union of two distinguished neighbourdhoods U,_I and U, +a of the orbits of tangency y' and y, respectively, which were constructed in the previous section (Theorem A). Inside U the equivalence will preserve the level sets BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 123 of the potentials of X~ and X,(~, which can be assumed to be the same for both families. We suppose that a compatible unstable system F"(%([Z)), ..., F"(ak_2([z)) together with a center-unstable foliation F~U(ak_l([z)) have been already constructed and also a compatible stable system FS(~k+2(~)), ..., F'(~t([Z)) with a center-stable foliation F9 We consider similar compatible systems for the nearby family { X~} and assume that we already have homeomorphisms defined on the space of leaves of these foliations. We first construct a center-unstable foliation FCU(ak([z)) which is compatible with the unstable system and with FC"(ak_l([z)). Besides the presence of criticality, this last compatibility condition with a singular foliation is the novelty here. As in the proof of Theorem A, we begin by describing the central foliation Fc(%(Ez)) in a neighbourhood of the point of tangency rk(0 ) = y' c~ A~(0). Recall that A~([Z) is the sphere Ak([Z ) c~ W'(%([Z)), where Ak(~z ) is the part of the boundary of Uk_l([z) c~ Uk+a([Z) which is contained in a non-critical level of the potentialf~. The central foliation is constructed in C~([Z), a tubular neighbourhood of W"(%([Z))c~ A],(~) in A~([Z) which is bounded by two spheres 0D+([Z) and 0 D[([z). We take [z-dependent coordinates (x~, xx, xa) for the cylinder C],([Z) centered at rk(0 ) such that = { = 0 }, Wc"(%_~([Z)) c~ C~([Z) ----{(xl, xx, xz ---- G*"([Z, x~, xi) ) } and WU(6k_I([Z)) (~ C~([Z) = { X 1 = G([Z, xi) , x]. = G~(~, x,, xx) }. The leaves of F~(%(~t)) inside W'"(ak_I([Z)) c~ C~([z) are the integral curves of the vector field Z~ defined by A1 = [xl -- G(~, xi)]* + Z 0G ([Z, xi) , 0G = ([z, Hence, F~(ok(V)) has a "saddle-node " singularity at rk([z), the point of tangency between W"(ok_~(~z)) c~ C~(~z) and FS'(%([z)) n CZ([Z ). f~ Fro. IV 124 M. J. DIAS CARNEIRO AND J. PALIS Now, since F~"(%_I(~)) n C~(~t) is transversal to F8~(%(~)) c~ C~(~t) we are going to lift this foliation to each leaf of F""(%_1(~)). Recall that there are two types of leaves of F~(ak_l(~)) which are denoted by F~"(%_l(~X)) and F~"(%_I(EX)) with dimensions equal to (dim W"(%_x(~)) -k- 1) and (dimW"(%_a(~)) -~ 2), respectively. Let 0 < 81 < 8~ be small numbers and consider the following two conic regions in a neigh- bourhood of r~(~) in C~(~t): C(Sx) = (I xt -- G(~, xx)t2/> 8~ II x~ -- G~(pt, xl, xx)ll2 ) and C(~2) ={1 x~ -- G(V, x~)[2~< ~211 Xa -- G""(V, x~, x~)l]2 }. The numbers 8~, 8, are taken so that the boundaries of these regions are transversal to F~"(ek_l([x)). Inside C(~1) , we just lift the foliation F~(ek(V)) C W~ to each leaf of F~"(%_I(V) ) via the projection (Xl, xi, xj) ~-* (xl, xj). Hence, all leaves of type F~"(%_I(V)) are subfoliated by a one-dimensional foliation diffeomorphic to the one described above. In the region C(~2) we take the codimension-one foliation defined by the non-positive level sets of the map (x~, x2, xa) ~-* " ] xa -- G(~, x~)] 2 -- ~, [] x a -- G~"(~, x~, x~)]] ~ and intersect it with F~"(%_~(~x)) to obtain a codimension-one foliation transversal to F*~(%(~)). The central foliation, in the intersection of F~"(a,_~(~)) with C(82), is given by the one-parameter unfolding of Z~ lifted to intersections of levels of ~h with leaves of F~"(ak_a(V) ). That is, ifF denotes a leaf, then in F r~ n-l(0) we just lift the vector field Z~ and in F n r~i-a(-- a) we lift the perturbed field Zi. ~ = Z i + a Oxt The region between C(~1) and C(~2) is used to match continuously the above foliations. To do that, we need to modify the intersections of F~"(%_~(~)) with the complement of C(8~) in order to include the boundary of C(82) as a new leaf. In doing so we can glue a singular central foliation near the tangency point r,(0) with a non- singular foliation F~(a,(~)) in C~(~t) which is compatible with the unstable system F"(~(~)), ..., F"(%_~(~)), F~ see Theorem A. Before concluding the construction of Fo~(%(~)), we indicate how to extend a homeomorphism hl, defined on a neighbourhood of the tangency point r,(0) in Wo"(%_l(~))r~ C~(~t) and which preserves the central foliation F~(a,(~)), to a full neighbourhood of r~(0) in C~(~). Inside the conic region C(8~), we just use the homeomorphism in the space of leaves of (~_~(~)) and the projection (xa, x~, xa)w-~ (x~,x~) to lift h i to each leaf of F cu F cu (~ (~_~(~)) c~ C~(~). Then, since F~(a~(~)) is non-singular outside C(8~) we extend it to a neighbourhood of C(~2) t~ 0 D+(~) in the sphere O D+(~). This neighbourhood is taken to be bounded by non-singular levels of n~ and in this boundary the homeo- morphism is actually a diffeomorphism C 1 close to the identity. Therefore, using the Isotopy Extension Theorem we can proceed as in Theorem A to extend it to all of 0 D+([x) in a way that it is compatible with the homeomorphism defined on the space of leaves of the unstable system. In the following lemma, we show how to obtain this homeomorphism h~. Lemma 1. -- Let F~(~) be a central foliation for the family X~(~) defined by a vector field Zi which has a saddle-node at ~(~) as above. Then, there is a homeomorphism h i defined BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 125 on a neighbourhood of r,(O) and depending continuously on ~, which sends leaves of F~(B) to leaves of and to Proof.- Using the above coordinates to describe WC"(ek_a(F))nWCS(~,(~)), we observe that the projection (Xl, xi) ~ xl, along the leaves of FS"(ak(~)), gives a Liapunov function for Z~ and that Z~ restricted to W"(ak_l(F) ) n C~([x) has a hyper- bolic singularity at rk(~). Therefore, the proof consists in showing that there exists an equivalence between Z~ and Z,(~) near saddle-nodes which preserves the level sets of a Liapunov function. To simplify the notation, we drop the parameter ~ in the following arguments and denote by W~(Z~), j = u, s, cu, cs, the invariant submanifolds of Z c. For a > 0, we let D- = { X 1 = -- ~; } and D + = { x 1 = * } be non-critical levels of such that for small tz the singularities rk(F) and ;k(q0(tz)) are contained in Ix, 1< e. We first obtain an equivalence on W*"(Z~), starting by taking a C 1 diffeomorphism close to the identity from the closed disc D-(Z c) = W"(Z c) n D- to W"(Z ~) n D-. Then, we take a tubular neighbourhood of D-(Z ~) in W*8(Z ~) n D- with fibers forming a radial foliation ~8. Each fiber of ~" is a C 1 curve transversal and exterior to the boun- dary 0 D-(ZC). Positive saturation of ~ by the flow of Z r intersected with D +, gives rise to a one-dimensional foliation in W~"(Z *) n D + which is singular at the point Wc(Z c) n D +. Hence, performing the same construction for 7,~, we define a homeo- morphism from We'(Z ~) n D + to W""(Z e) n D + which preserves this foliation and it is a diffeomorphism outside the point We(7, r n D +. By preserving the level sets of and the trajectories of Z ~ inside WeS(Ze), we obtain an equivalence between Ze 1 Wc"(Z c) and ~c j W~,(~c). Proceeding dually, starting now at the level D +, we get an equiva- lence between Z~[ Wc~(Z ~) and 2~ w~-(2o). The corresponding radial foliation in Wc"(Z c) n D + is denoted by b". We are now going to match these two equivalences. In D-, we raise over each point of the disc D-(Z c) a continuous foliation g" with C 1 leaves, transversal to Wc~(Z~), which is compatible with W"(ok_I(F)) n D-. Each leaf of g" has dimension equals to d, = dim W""(Z"), and Wc"(Z c) n D- is taken as a distinguished leaf. In the complement of the component of D-\W*(ak_l(bt)) which contains Wc"(ZC), we take a (d, -t- 1)-dimensional continuous foliation g~, with C 1 leaves, which is trans- versal to W~"(Z ~ n D- and such that the boundary of each leaf of g~ is a leaf of g" in W~(ak_~(F)) n D- and g~ n Wc"(Z ~) is the radial foliation ~". Dually, we construct in D + the foliations g" and g[ with dimension d 8 = dim W"~(Z e) and (d, + 1), respec- tively. Still denoting by g~ the intersection of the positive saturation of g~ by the flow of Z e with D +, we observe that for each leafg~, ~ ofg~ the intersection g~ n g[, ~ is a one- dimensional foliation which is singular at the point b = g[,~ n 0D+(Z *) C W"(,,_I(B)) n D +. Hence, as in the proof of Theorem A, we take two families of closed conic regions E C g~, ~ with vertices at b, such that We*(Z c) n gl, ~ is contained in the interior of E~. We then modify g"n g~, ~ to get a new one-dimensional foliation oct ~ such 126 M. J. DIAS CARNEIRO AND J. PALIS that ~bn E~ = gun E~ and outside E b it is non-singular and transversal to g~ ----- g~, b n W~(%_l([z)). Clearly, this can be done depending continuously on b and Ix. Once the same construction for Z~ is performed, we are prepared to define a homeo- morphism h + on D + and conclude the proof of Lemma 1. The basic property of h + is that by preserving 9ff b it induces (via projection along the trajectories of the respective vector fields) a homeomorphism on D- which is a continuous extension of the homeo- morphisms already defined on D- n W~(Z ~) and D- n W~8(Z~). The definition of k + goes as follows. Let A+ be the closure of the component of D+\W"(%_I(~Z)) that contains the disc D+(Z ~) (i.e., the set x 1 = ~; G(~, xi) t> ~). In A +, we take a continuous foliation by C 1 closed discs compatible with g~ (the positive saturation ofg~), which is transversal to Wc*(Z ~) n D + with complementary dimension, such that D+(Z ") is a special leaf. The boundary of each of these discs is a sphere in W"(%_x(&)) n D + which is trans- versal to the foliation g~ n W*(ak_~([z)) , with complementary dimension. Thus, by preserving this family of discs and the foliation g8 and using the homeomorphisms already defined on W~'(Z ~) n D + and on W""(Z ~ n D + = D+(Z ") (space of leaves of these foliations) we obtain a homeomorphism on A+. This gives a homeomorphism on the space of leaves of 9ff b which are outside the conic region E b. Since in the space of the leaves of3r ~b which are inside E b is the disc D-(Z ~) (where we also have defined a homeomorphism), we have obtained a homeomorphism in the total space of leaves of 3r ~b. To complete the definition of h +, it is enough to preserve ~b and a codi- mension-one foliation whose space of leaves is W~'(Z ") n g~. b, which can be defined by G(~z, xi) = a, a ~< r This concludes the construction of h +. As observed above, the equivalence between Z ~ and Z~ is obtained by preserving levels of n, using h + and preserving trajectories of the fields. 9 A center-unstable foliation F""(ak(~)) can now be defined as in the proof of Theorem A except in a neighbourhood of rk(0 ). In this neighbourhood, we want to distinguish a leaf that contains the tangency point rk(~z ) and contains the possible tan- gencies between W'(%+1(~)) and W"(ak_a([Z)). So, we let Z~ be a neighbourhood ofrk(0 ) inAk([z ) n W"~(%+x(~)) n W""(%_~(~s and consider coordinates (xl, xi,yl,y~), as in (1 .B), to define a continuous family of vector fields Y~ in ~"(~) fory x >1 0 by Xl = [Xl -- G(Ez, XDyl,Y~/~'YT.)] 2 -? y" OG ,=8 ~ (~' xI'yl'Y~J~'YT') ~'' OG OF (y~,./~,,y~/~, x~,yT.) Y~-- ""/~' 4 = ~ ([z, xi,yl,y~J~'_yL) -- ~ j~=O, )t=O. Y~ is tangent to W"(%_I(~Z)) n Y~([z) and its restriction to A~(~z) is equal to Z~, the vector field which defines the central foliation F"(%(~t)). For fixed (~.,y~,y,), Y~ also has a singularity of saddle-node type and its singular set is a C ~ manifold contained in W"(%_a(~)) n Y~"(~). Moreover, the image of this singular set by the Poincar6 BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 127 map P~,: Z*([z)--~S*(~t) coincides with the set of points of tangency between W~(%_I([z)) n S*(~t) and the foliation F~+l(~t) defined in (1.B). In particular, this image contains the tangencies between W~(%_l(~t)) and WS(%+l(~t)) in the level set Bk(~x ). Therefore, over each leaf of F*(%(~z)) we can raise a (u + 1)-dimensional sin- gular foliation in Ak(~t ) ca W~ such that its intersection with Z*(~t) is tangent to Y~. The positive saturation of this foliation gives part of the leaves of F*~(%(~)) which are contained in WCU(ak_l(~x)). The process to define FC~((rk(~)) inside the other leaves of F*u(%_~(~t)) is analogous to the one described above to obtain the central-foliation F*(%(~t)), i.e. one uses conic regions and projections onto W*~(%(~t)) ca A~(~t). Since outside a neighbourhood of the tangency point rk(0), F*"(%(~t)) is exactly as in Theorem A, we have completed the definition of F*~(%(~t)). Now comes the main step in proving the stability of the bifurcation of type I: to define a homeomorphism in the level Bk(~t ) which preserves the intersections of leaves of the stable system and the center-unstable foliation FC~(ak(~)). Since the stable system is transversal to the singular set of F*~(%(~t)), it is transversal to all leaves of F*~(%(~t)) outside a neighbourhood of the tangency point r~+x(lz). Hence, it is enough to obtain a homeomorphism in a small neighbourhood S([x) of rk+l(~Z ) in Bk(~t ) and proceed with a cone-like construction as in Theorem A outside S(~t). The same is valid in the section S*~(~t) = S(~t) ca W~ since WC"(%(~z)) is transversal to W"(%+~(~t)). The novelty here is to obtain a homeomorphism on the dual section S*S(~t) = S(~) ca W~*(%+~(~)). Fol- lowing the methods of the non-critical case (Theorem A, Section II), we want to preserve the intersections of FCU(ak(~)) with S~"(~z) and a C '~ foliation F~*, which is compatible with W'(%+x(~)) and transversal to W*~(%(~)) ca S*"(~) with complementary dimen- sion, m >t 3. However, due to criticality, this process must be modified in a neighbourhood of L*(~t). Let us recall the notation used at the end of (1 .B). First, ~u 0" v8 S~(~) ---- W (k_~(~t)) ca S~s(~t) and F~+~(~t) is a C" foliation in S~(~) which is compa- tible with W"(~e+a(~)) and has codimension equals to dim [W~(%(~t)) ~ S"(~t)] q- 2. Further, L~(~t) is a submanifold of S~ which contains the set of tangencies between F~_l(a) and W"(v~_~(~z)). In L'(~) there is a C 1 codimension-two foliation F~, which is compatible with W"(%_x(a)) ca L~(~) and with W"(%(~t)) ca L~(a). Finally, S~(a) is a two-dimensional manifold of L~(a) which is transversal to W"(%(~t)) and contains the set of tangencies between W'(%+l(~z)) and F~,. The curve T"(~x) = W"(%_a(~Z)) ca Sg(~t) is a leaf of a singular foliation defined by the one-form ~ and the curve T'(~t) = W'(%+1(~)) ca Sg(~t) is a leaf of a foliation %. These foliations are defined in (1. B). We now start constructing a homeomorphism h~ s : S~"(~t) ~ S~"(q0(~)). Using the reparametrization q~(~x), also obtained in (1. B), we define a homeomorphism from Sg(~) to Sg(~(~t)) that preserves the singular foliation p~ and the curves T'(~t) and T~(~t). Next, we extend the homeomorphism to L'(~t). We take a two-dimensional foliation (SN)~ in L~(~x) which is compatible with W'(%+I([z)) and is singular along the curve T~(~t). 128 M. J. DIAS CARNEIRO AND J. PALLS The foliation (SN)~ is tangent to a family of C a vector fields Y~ with singularity of saddle-node type defined by OF ~ = [v~ - F(~, v~, O, w~,)] ~ + ~ ~w~ (~' v~, O, wd w~, OF OG ,,1/~ .... ~ "~"~" @'~'wL) v~'+~Y% ~v~ Ow~ (~' v~, o, w~) ~ (~, x~(~, P2 ~ 0. S~(tt) is a distinguished leaf of (SN)~, and F~ is transversal to (SN)~ except along the curve T~(~). Hence, the intersection of (SN)~ with (~)-1 (v~) is a continuous one- dimensional foliation transversal to WC"(a,(~)) ca SC'(~), where ~ is the projection into S~(~) along the leaves of F~. We can apply Lemma 1 to obtain a homeomorphism WC~(ak(~)) n SC"(~t) to WC~(~k(~(~t))n S~'(9(~t)) which sends trajectories of Y~ to trajectories of ~',(~) and also preserves W~(ak(~)) n S~S(~). This gives a homeomorphism in the space of leaves of (SN)~. We now have homeomorphisms defined on the space of leaves of two complementary foliations: S~,(~) (whose leaf space is S~(~z)) and (SN)~. Thus, we have a homeomorphism from L*([z) to [,*(~?(~z)) which preserves W"(%([z)), W~(%_x([z)) and W'(,k+a(~z)). We extend this homeomorphism to all of S*'(~z). We recall that there are two types of leaves of F*"(,,([z)) which are denoted by F~"(,k(~) ) and F~"(a,(~)) such that dim F~"(%(~z)) = dim W"(%(Ez)) + 1 and dim F~'(%(~z)) = dim W"(%(~)) + 2. The foliation F~"(,k(~) ) has a saddle-node type singularity along a u-dimensional sub- manifold, which contains the tangencies between W"(,k_x(~)) and W'(%+l(~z)). Outside this submanifold, F~"(ak(~z)) is transversal to WS(,k+l(~Z)). Let us construct a foliation F~ * in SC*([z) which is compatible with W~(%+x([z)) and transversal to W*"(%(~z)) ca S**([z) with complementary dimension. We first take a C m foliation F~, ~, m/> 3, which is compatible with W~(,,(~z)) and transversal to S*(~z) such that F~ *' ca S*(~) = F,+i([z )*~ ; the foliation F~,*s is defined by a submersion r:~ *~. We then define F~, ~ by taking the pull back via ~" of the one-dimensional foliation in L*(~z) defined by (~,)-~ ('~) ~ (SN)~. By construction, L~(bt) is a typical leaf of F~"(%(~t)) ca S~(~), i.e. the restriction of r~' to each leaf of type F~"(a,(~)) ca SC"(~z) near L~(~) is a homeo- morphism onto L~(~t). In particular, this is valid for all singular leaves of F~"(~,(~t)) . Therefore, we can define a homeomorphism on a neighbourhood of L~(~t) using the two complementary foliations: F~"(a,(~t)) ca SC"(~t) and F~ s. Such a homeomorphism has one important property: it induces a homeomorphism in the level set A,(~t)\A]~(~t) which extends continuously to the sphere A~.(~t) = A,(~t) ca W"(a,(bt)). Let us explain this point. Denote by P~" : S~"(~t) ~ A,(~z)\A~(tx) the restriction of the Poincar6-rnap. Since F~ ~ is of class C", m/> 3, and the singular foliation p~ is preserved on its quotient space, the image by P~" of the singular foliation induced on each leaf of F~"(a,(~)) by z:~" is a continuous foliation with C ~ leaves which are topologically transversal to A~(~t). Therefore, a homeomorphism which preserves F~"(a,(~t)) and F~," extends automatically to the sphere A~(~t). As we pointed out above, the idea is to construct h~,' by preserving F~ ~ BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 129 and FC"(,k(V)) outside a neighbourhood A,(V) of Lc(V). Inside A2(~) it preserves F~ 8 and F*~(ek(~) ). We detail this construction. Let AI(V)C A~(~) be two wedge-shaped regions in SC*(~) that contain LC(~) in their interior and which are bounded by non- singular leaves of type F~(ak(~)). We also require that all singular leaves of FC~(%(~)) are contained in the interior of AI(~). Each Ai(~) , i = l, 2, is the image by the Poin- card-map Ak(~)-+S(~) of a solid cylinder transversal to A~(~). Inside the subset VS\-- I ~ ) (R~) n AI(~) , R~ as in (1. B), we preserve the two complementary foliation F~ ~ and F~(ak([z)) n S~8([z). In the complement of this set in AI([~ ) we preserve the comple- mentary foliations F~(~k(~)) c~ SC*(~) and F'*([~). Since the intersection of the boundary of / w\--I (rcv) (R~) with each leaf of F~"(%([z)) n S~"(~x) is a leaf of type F~"(%([x)) and F~ ~ is a codimension-one foliation in F~", we obtain a homeomorphism on Al([x). In the complement of A2([x ) in S"'(~t) we proceed with the cone-like construction of Theorem A to define a homeomorphism preserving F~ ~ and F~"(ak(~)). The region A2([x)\AI(~t ) is now used to match these homeomorphisms. Notice that each non-singular leaf of type F~"(ak([z)) is parametrized by a point in the sphere A~*(a) = Ak(~x ) n W"8(%(a)). We assume that the boundaries of AI([x ) and A,(~t) correspond to codimension-one spheres S~ and S~ in A~"(~x) centered at the point of tangency r,_~(0). Hence, the matching is done as we move radially from S~ to S~ preserving F~* in a subset R, of R,. This subset is bounded by the pre-image of two leaves of p~ whose distance gets smaller as we approximate the outer sphere S~. Finally, when we reach a point in S~, this region collapses into the unique leaf of a~ which is transversal to the axis v2 = 0. The picture illustrates this process in a section complementary to W"(%([x)) in S~*(~x). It shows how the region R~, foliated by leaves of p,, shrinks to a curve. leof of P,u. - R x x inA1(~) y inA2(zU)\ A|(/U,) neor ~2(~) ze a~(y) Fro. V 17 130 M. J. DIAS CARNEIRO AND J. PALIS In this way we have obtained a continuous family of homeomorphisms h2 : -+ As pointed out above, this is enough to get a homeomorphism on the neighbourhood S(~) of the tangent point rk+l(~X ) in the fence Bk(~). We can now, using the methods in Theorem A, obtain an equivalence between X~ and IK,~) on the neighbourhood U(~) = Uk_l([x) w U~+I(Ex ) of the orbits of tangency 7' and 7 preserving level sets of the potential f~: this is possible because we have preserved the center-unstable foliation F~"(%_t([z)) throughout the process. The extension of this equivalence to all of M is done as in Theorem A: outside the distinguished neighbourhood of the orbits of tangency we obtain a conjugacy between X~ and X,~. w 2. Bifurcations of type II: two orbits of quasi-transversality (2. A) Description of the bifurcation. -- This is a codimension-two bifurcation presented by families { X~ } in ?(~(M) such that, for ~ ~ 112, the vector field Xg presents exactly two orbits Yx and Yz of quasi-transversal intersection between stable and unstable mani- folds of hyperbolic singularities: Y1C W"(pl(~) ) n W'(ql(~)) and y, C W~(p~(~)) n W'.(q~(~)). In addition, we assume the following conditions: (2.1) C" linearizability of X~ near each of these singularities with the eigenvalues of dXg at these points having multiplicity one, m being sufficiently large as specified in Section I, (2.2) Non-criticality of any other singularity with respect to the strong-stable or the strong-unstable manifolds: if p e M is a singularity of Xg different from Pl(~), P2(~), ql(~) and q~(~), then W"(p) is transversal to WS"(pl(~)) and to WS"(p,(~)) and W"(p) is transversal to W""(ql(~) ) and to W~"(q2(~)), (2.3) W~"(pi(~)) is transversal to WS(q~(~)) and W~'(q~(~)) is transversal to W"(p,(~)) for i= 1, 2, (2.4) Generic and independent unfolding of the orbits of tangency of the family X~, so that there exist two C 1 curves F 1 and P 2 in the parameter space crossing each other transversally at the point ~ such that ~ e F ~ if and only if W"(p~(v)) is not transversal to W"(q~(~)), for i --= 1, 2. We distinguish two possibilities, (II. a) and (II. b), that will be treated separately: a) two of the above singularities coincide, namely ql(~) = Pz(~) or qz(~) = Pl(~) (which are dual) or the easier case Pl(~) = P~-(~), b) all singularities above are distinct. BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 131 (2.B) The bifurcation diagram of type (II.a). -- Let us first assume that Pl(~) =~k--l(~), ql(~) =P,(~)=%(~) and q,(~) =~k+t(~) in the ordering al(~) ~< ... ~< %(~) of the singularities of X~ described in section I. We begin by ana- lyzing the restriction X~ of X~ to the center manifold W cuff w~ = (k-l(~)) n w0,(~k+i(~)) near %(t~). Let (xl, xi,yl,yT.) be C" linearizing coordinates for X~ near ~k(Vt): 0 0 0 0 ~ = Z ~,(~) x, + ~l(~)yl + y- ~(~)yj Oy~ with 0 < ~l(t~) < ... < ~,(~), 0 < ~1(~) < ... < ~,(~), x1 = (Xl,.-., xs), Y*. = (Yx,.--,Y~), u = dim[W"(ek(~)) n W*'(.~+I(~))] and s = dim[W'(ek(~) ) n WC"(%_l(Vt))]. By the quasi-transversality assumption in a cross section l~c(Ex)C{x 1 ----1 }, we have W~(%_l(~))r3Xc(W)={(xI, G(~,xI,yL),yL)} , with 13 being a C '~ function such that x I --~ O(~, xi, 0) has a non-degenerate critical point at 0. Hence, we get from the generic unfolding of the orbit Y1 that the map Vt ~ G(~, xt(~) , 0) is a submersion at ~, where xi(vt ) is the solution of 0G Ox~ (~' xD O) = O. Also, by taking coordinates (vx, vi, wL) in a cross-section S*(t~)C {Yl = 1 } such that W"(~k(~) ) n S~(tz) = {(0, O, wr.)}, we have ws(~+~(~)) n s0(~) = {(F(~, v~, w~), ~i, w~)}, where F is a C '~ function such that w L ~* F(~, 0, WL) has a non-degenerate critical point at 0. Hence, the conditions of generic and independent unfolding imply that the map (~, xi, w,~ / ~ G(~, ,~, 0), ~ (~, xi, 0), F(~, x. 0/, ~ (~, O, ~,./ is a local diffeomorphism at the point (~, O, 0). Therefore, if ~x ~ WL(~X ) is the solution OF of (~x, 0, w~) = O, then the curves Ow L r~ = { G(tz, xi(~) , 0) ---- 0 } and Fz ---- { F(~, 0, WT.(~)) -- 0 } belong to the bifurcation diagram near ~. Also, Vt e Fx if and only if W~(%_~(~)) is not transversal to W'(%(~)) and ~ e Fz if and only if W~(%(tz)) is not transversal to W*(%+~(iz)). Furthermore, the intersection of W"(%_l(iZ)) with S*(~) is dcscribed by the equations v~ ---- e -~''l')t, vx = e -~'(g)t xi, e -~"~)~ -- G([s XI, e -~(~)~ Wr,) ---- O. Using that the bifurcation unfolds generically and the implicit function theorem, we obtain a third C 1 curve I'3 in the parameter space tending to ~ (but disjoint from F 1 M. J. DIAS CARNEIRO AND J. PALIS and F~ outside this point), such that ~ ~ Pn--{~ } if and only if W"(%_~([~)) is not transversal to W"(a,+a(~)). Along the curve F a the family X~ presents one orbit of quasi-transversality between W~(%_x({x)) and WS(a,+x(~)). 1"1 1"~ 1-, 2 FIG. V[ (2.C) The stability of the bifurcation of type (II.a). -- As in Theorem A, we focus our attention to a neighbourhood U(~) of the closure of the orbits of tangency Ya and 7~ in M which is constructed by glueing together distinguished neighbourhoods of these orbits. We construct in U(~) flow equivalences that preserve compatible systems of foliations, so that they can be extended to flow equivalences on all of M. Suppose we have already constructed a compatible unstable system F"(el(~), ..., F"(~k_2(~)), F*"(ak_l(~)) and a compatible stable system Fr , ..., FS(%+2(~x)), ..., FS(%(~x)), together with a homeomorphism in the space of leaves of these foliations. We start by constructing a center-unstable foliations F*"(%(~)) com- patible with the unstable system whose main leaf, W*"(%(~)), is a C 1 invariant manifold contained in W*"(ak_l(~)) and transversal to W"(%_a(~X)) and which contains all possible tangencies between W"(~,_ 1 (~t)) and Ws(a, + 1 (~x)). This construction resembles very much the one done in w 1 for the orbit of tangency with criticality. In the cross section Z~(~) C { x I = 1 } consider coordinates (xi,yl,yT.) centered at rk(0 ) ----- 71 n Z~(0), as in (2.B) above. Let the vector field Z~, tangent to W"(ak_l(~t)) , be defined by 0G Jl = (Y~ -- G(~t, x~, y~IL/~yL)) 2 + Z ~, (~x, xI,y~L/~'y~) ~, 9, = ~ (~, x I, y~l~/'yL) -- OF yl,,_,,,/0, "~v, ' j~ = 0, 1 -- ~ (Y~'/~' "DYT.) eI X /~=0, i---- 1,...,s, fory~>0. Since [~j(~)>~(~) and ~,(~)>e~(~) for i/> 2 and j1> 2, this extends to a (I ~ vector field in N~(~) which has for each (~,y,.) a singularity of BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 133 saddle-node type. The singular set of Z~, Sing(Z~), is a submanifold of dimension u of W"(ak_x(F) ) c~ E~(~) which is topologically transversal to Ws(a,(~))nZ~([x). Its image by the Poincarf-map P~:Y,'(~)~S~(~) contains the tangencies between W~(ak_~(F)) and W'(a,+~([x)) in S'(~), a cross-section in {Yx = 1}. We consider a foliation F~([x) in Z~(~t) which is tangent to the vector field Z~ and singular along Sing(Z~) having C 1 leaves of dimension (u + 1). We distinguish a leaf M~([x) which is transversal to W"(ak_~(~) ) and such that M~(F)c~W"(ak_l([X))= Sing(Z~). Let F~(~t) be a uk-dimensional (u k --dimW"(ak(~) ) foliation in W'"(a~_~(~))nZ(~) which is compatible with W"(ak_a(~) ) n Z(~) and such that F~([x) n Z'(~x) = F~(~). Positive saturation of F~(~x) gives part of the center-unstable foliation F~"(~(~)) inside W'"(~,_I(F)), which has a distinguished leaf denoted by W~"(~). In the next figure we see these leaves in a slice complementary to W"(~,(~)). The construction of the other leaves of F'"(~(F) ) corresponding to points near the singular set Sing(Z,) follows as in w 1 of the present section. Dually, we obtain an s,-dimensional singular foliation F~(~), (s~ = dim W'(~(~t))) in the level set B~(~) = [f~l[f~(ak([X)) -F ~]] which is compatible with the stable-system F~'(~+~([x)), F"(~+~(~)), ..., F"(~t(~x)). We denote by M~(~) the distinguished leaf of F~(F) that contains the point p~+l(~X) = yz(~x) n B~(~). ()a~ n s c" ()LI) " (oK_l(F) )n s Cs (p) w" (ok ()a))ns 9 I)~) FIG. VII Let Xc(~t) be the C 1 curve defined by X'(~) := W~([z) n M~(F) n S~(~) and consider the points pe_l([Z) = W"(~e_l(~z)) n V(~t), W~(ak(F)) n V(~) =Pk(F) (= ( 0 }) and Pk+ I(B) = W"(ak + l(~t)) c~ V(B). Notice that p~_ 1([z) is only defined for B on a connected 134 M. J. DIAS CARNEIRO AND J. PALIS component of V -- Pl, where V is a neighbourhood of ~ in R ~. The curves P~ and F 3 of the bifurcation set correspond to {p~+l(t~)=pk(tz)} and {pk+l(tz)=p~_l(~z)} respectively. We start the construction of an equivalence between X~ and a nearby family X~. Consider a reparametrization ~0: (V,~)-+ (R 2, ~), such that q~(Fi)= Fi for i = 1, 2, 3 and which sends the regions A,~ between the curves onto corresponding regions 2~ as in the picture (Fig. VIII). To obtain an equivalence between X~ and X~(~, we first want to define a homeo- morphism on the level set B~(W) which preserves the foliations F~(W) and F~(~) ~ B~(tz). The main step is the construction of a homeomorphism on S(tz), a neighbourhood of p~+~([~) in B~(lz). Let M~"(~t) = W~"(tz) r~ W~(%+~(~z)) c~ S([z) and let (v~, w~) be a system of coordinates for M*"(~t) such that ( vl - ~(~, w~) = o } = w~(~+~(~)) r~ M0~(~), { v~ = 0 ) = W"(%(tz)) c~ M~"(tz) and WU(6k_l([s n M""(t~) = { v~ ,/~', -- G(t~, v~"/~".w,,) = O, v~ > 0 }, where F, G are of class C 2. Hence, by construction the foliation F~(B) r Mr is tangent to the vector field Y~ defined by eg ~ = (vl - Y(~, w~))~ + Ow~. (v., wd .~,~, - ~14"" + Oy~ (~' v~'~'" w~). ~,~, w for v 1 > 0. We extend it to v I x< 0 by setting ~ 0F /?1 = (Vl -- W(~, wL)) 2 -Jr- ~w (~' wE) /i/L' w~ = ~w~ (~' w~). Let L"(~, vl, w~) = [v~ ''~''(~' -- G(~, v~ L/~''~' WL) + G(~, 0)] ~'("'/~'~' be a C 1 sub- mersion defined for vl~> 0; observe that (L~)-I(G([x, 0) ~'/~'(~l) = W~(%_l(tz)) and (L~)-x(0) = W~(%(~t)). For vl < 0, we extend it as L*(~t, Vl, //JL) = Vl" It is easy to check that -- L"(~t, vl, wr) is a Liapunov function for the vector field Y~. We apply Lemma 1 of w 1 to get a homeomorphism M~"(~x) -+ l~IC"(q~(~x)) which is a topological equivalence between Y~ and ~r(~) preserving the level sets of the respective functions L~ and I,~. The same procedure is used in order to get a homeomorphism on the cross section M"*(~)= W~*(~)n W""(%_1(~))r~ Y~(~), where E(~)is a neighbourhood of %(~) in Ak(~). Now, to complete the definition of the homeomorphism on the cross-section BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 135 r ~ r 3 A12 / J Az3 r 2 A21 A12 FIG. VIII S(~) n W"(ak(~t)), which corresponds to part of the space of leaves of the foliation F~([z), we take a C 1 foliation on W*(%([z)) n S(~t) transversal to Wes(O'k+l(~)) ('~ WU(qk(~)) ('~ S([~) and of complementary dimension. We make the same construction for W"(~(~(~))) n S(~(~)) and obtain the extension of the homeomorphism by requiring that this foliation be preserved and also the intersections of the center-stable foliation F"(~k+l(~) ) n W"(%(~)) n S(~). This homeomorphism is further extended to the whole sphere (fundamental domain) W"(~(~t)) n Bk(~t ) by preserving the stable system FS(%+2(Ez)), ..., FS(~rt(~t)) in a compatible way with the homeomorphisms on the space of leaves of these foliations. Similarly we obtain a homeomorphism on the space of leaves of the center-unstable foliation F'"(~k([z)), which is compatible with the unstable system F"(~x(~t)), ..., The homeomorphism on the level set B~(~t) is then well-defined since we want it to preserve the complementary foliations F~(~t) and F~"(~t) n Bk([z ). As we saw in Theorem A, Section II, this is enough to obtain the equivalence in the neighbourhood of the singularity %(~z). The arguments to define the equivalence in B~_a(~) and, afterwards, its extension to all of M are now very similar to those in Theorem A. (The corresponding facts in case (II. b) are somewhat more delicate and will be treated in more details in the sequel.) 9 (9.. D) The case of two orbits of quasi-transversality corresponding to disjoint pairs of singu- larities (type II. b). -- This case goes much in parallel with the previous one: the main difference consists in a more careful construction of an equivalence for the two nearby 136 M. J. DIAS CARNEIRO AND J. PALIS families. This is due to the existence of intermediate singularities between the ones corresponding to the orbits of tangency. Let X~ be one of these families and let us order the singularities at the bifurcation point ~, so that Y1 is an orbit of quasi-transversality between W~(a,(~)) and W"(a, + 1(~)) and Y2 is a similar orbit between W~(a,+,(~)) and W"(a,+,+~(~)) for some k/> 2. We first observe that the construction for ~x near ~ of compatible unstable and stable systems, which are now denoted by F~(~z), ..., F~"(~), Fg~(~z), F~'+~(~z),..., Fg~_,(~z) and F~ ,+~k~z), , , F~+~(~), ~ .. ., F ~+,~), ~s , , F~+~+~(~z), ~ . .., F}(~z), respectively, is very similar to the previous case. The difference is that now we have to construct F~+~(~z) for 2 ~<j~< k -- 1. To do this, we just note that although like before F~_l(~Z ) is a singular foliation, its singular set, Sing(F~_ t(~x)), is the union of two manifolds which are trans- versal to W~(a~+~(~)) for j~> 2. Moreover, since each leaf of F~_~(~) accumulates in a C a fashion on Smg(F,+l(~x)), any foliation on this singular set can be extend to the leaves of F~_~(~t) in a continuous way. Thus, when constructing F~+~(~), in a compa- tible way with Fr it is enough to do so in Sing(F~_~(~z)), and then extend it to each leaf of Fg~_~(~). The same reasoning applies to F~'+~(~), 2 ~<j~< i -5 k -- 1 and to V;~_~(~). In the construction of the above foliations, we can also require W~(~(~x)) to be foliated by leaves of F~(~). In particular, since W'(e~+~+,(~)) is transversal to We"(~i+~(~)), we conclude that W'(~h+~+l(~)) is transversal to W~(~(~)), ~x near ~. Thus, the bifurcation set of X~ near ~ consists exactly of two C 1 curves r 1 and Pz that intersect transversally at ~: ~ e F~ if and only if W~(e~(~x)) is quasi-transversal to W"(%+~(~x)) and ~ e Fg if and only if W"(~;~+~(~)) is quasi-transversal to W'(%+~+~(~)). (2.E) Local stability. -- Let X~ be a family of type (II.b) and let X~ be a nearby family with main bifurcation value ~ near ~. Let (F, G) : (V, ~) ~ (R 2, 0) be a C 1 map defined in a neighbourhood U of ~ in R * such that F-l(0) = { ~z ~ V; W"(a~+k(~z)) is quasi-transversal to W"(a~+k+l(~)} and G -1 ={ ~x E V; WU(a~(~)) is quasi-transversal to WS(a~ + l(~x)}. By the hypothesis of independent unfolding, (F, G) is a local diffeomor- phism. Therefore, if (~', G) is the corresponding map associated with the family { ~2~ }, we can define the reparametrization q~ = (~', G)-lo (F, G). To prove that X~ is equivalent to X,(~) we take two distinguished neighbourhoods U~(?) and U~ +k(~t) of the closure of the two orbits of tangency, ~'1 and 72, as constructed in Theorem A. Inside these neighbourhoods the equivalence h~ will preserve the level sets of the potential function f~ and outside them it will be a conjugacy. The idea of the proof is to first define a continuous family of homeomorphisms on the space of leaves of the unstable F ~" ' ' The important point here is to preserve the leaves system from F~(~z) up to ,+kk~z). of the stable system which are contained in the stable manifolds. Dually we define a family of homeomorphisms on the space of leaves of the stable system from F}(Ez) to F~_ ~ (Ez). We then obtain a homeomorphism on the fence Bi + k(~) C f~- ~ (f~(~ + k(~) ) -+- s), preserving F~k(~z ) n B~+,(Ex) and the stable system. At this point we obtain an equi- BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 137 valence on a full neighbourhood of the singularity a~ + k: we use the cone-like construction in Theorem A, Section II, and preserve the level sets of the potential f~. The equi- valence is extended to the distinguished neighbourhood U,+ k by preserving level sets of the potential and repeating the cone-like construction near a~+k+ 1. With this we define homeomorphisms on the space of leaves of F ~+lk~z) c8 " ' and complete the definition on the fence B~+I(~)Cf~-l(f~(a~+l(~)) § ~) also preserving Fg~_l(t~ ). Since the folia- tion Fg"(~) is preserved in this process, the equivalence can be extended to the second distinguished neighbourhood U,(~) again by the methods explained in Theorem A. Let us give more detail on this construction. We assume that we already have homeomorphisms on the space of leaves of F[(Vt), ..., F~'_I(~) and F~"(~) as well as on the space of leaves of F~k+~(~), ..., F~(~). The homeomorphism on the space of leaves of the foliation F~_~(t~) is obtained as in (II.a) using Lemma 1. Next, we obtain homeomorphisms h~+ ~, ~ on the space of leaves of the foliation F~'+ j(t~) forj = 2, ..., h -- 1, and, also, of F~_k(~). We will perform the construction for j = 2, since the general case can be done by induction in a similar way. Construction of h~+2,~. -- Let us suppose that W"(~i+l(~) ) nWS(~+2(~))4: O. We denote by F~_2(a~+~(~) ) the set of leaves of F"'(a~+I(Vt)) which are contained in W*(a,+2(~)). We recall that A,+I(~) and B,+I(~) are two small fences contained in the non-critical levels f~-~(f~(a~+a(~))+ ~) for 9 > 0 small, respectively. We are going F ~s (~ to define a homeomorphism on B~+I(t~ ) n W"(~,+2(~) ) which preserves ,+~(~+1(~)). So, we first construct a homeomorphism on the space of leaves of this foliation: this is done leaf by leaf using the Isotopy Extension Theorem, as in the previous cases. Since we already have a homeomorphism on the space of leaves of the foliation F~_I(t~), we obtain a homeomorphism on A~+~(~)t~W"(a~+~(~)) which preserves Fr ,+2ka,+~(~)), / / Fg~ ~(t~) and a complementary foliation F~: this is exactly like in the proof of Theorem A when we restrict ourselves to W*(a,+z(~)). Therefore, through the Poincar6 map P,+a,~ : A,+~(~)\W"(a~+~(~)) ~ B~+a(~)\W"(a~+~(t~)) , we get the required homeo- morphism on W~(a~+~(B)) c~ B~+~(~). Let D'(i + 2, t~) be a fundamental domain for WS(a,+2(~)) which is contained in the non-critical level set ffa(f~(a~+l(Vt)) + ~). Using the Isotopy Extension Theorem and the compatibility of the homeomorphisms on the space of leaves of the foliations F~(~), ..., F~"(B), we obtain the extension of the homeomorphism W*(~+2(~) ) ~ B,+I(~) -+ W~(~+~(q~(~))) ~ B,+~(q~(~)) to D'(i + 2, ~), finishing the construction of h~ + 2, ~. 9 As mentioned before the construction of the other homeomorphisms hf+~,~, for 3 ~< j ~< k, is analogous to the one described above: we proceed by induction, using the $ $ leaves of the stable system Ff~_~(~), F~+~(t~), ..., F~+~_a(~) which are contained in We are now prepared to define an equivalence on the distinguished neighbourhood U~+,(~) of the orbit of tangency u To do that we again apply Lemma 1 to obtain a homeomorphism on the space of leaves of F~_,(t~ ). The construction is dual to the one 18 138 M. J. DIAS CARNEIRO AND J. PALIS used to obtain a homeomorphism on the space of leaves of F~_l([Z ). This homeomor- phism, together with the homeomorphism h~+k, ~ constructed above on the space of leaves of F~_k(~t), yields the definition of an equivalence on the neighbourdhood U~ + k(~t) according to the methods in Theorem A. We conclude our arguments with the cons- truction of an equivalence in the distinguished neighbourhood Ui(~z ). We have already defined a homeomorphism on the set [J2~<j~<k+xW*(~r~+i(~t))('~]~i-1-1(~) which pre- serves the foliation F~_t(~z). We can then extend this homeomorphism to the remaining part of the space of leaves of F~_a(~z) corresponding to the leaves contained in W~+~+2(~t), ..., WI(~z). This extension, which is by now standard, is compatible with the homeomorphisms already defined on the space of leaves of the corresponding stable foliation. With this, since we also have preserved the foliation F~"(Ez) throughout the process, we can define the equivalence on the neighbourhood U~(~t) again by the methods in Theorem A. To obtain the globalization of the equivalence to all of M, we just choose the non-critical level F, =ffX(c) where fg(a,+,(~)) < c (A(6i+/r and proceed as it was done at the end of Theorem A. 9 w 3. Bifurcations of type 111: saddle-node with criticality In this paragraph, which is similar to w l, we treat the case of a saddle-node with criticality. Let X~ be a family in z~(M) such that for a value ~ ~ R *, the vector field Xg presents a unique nonhyperbolic singularity p(~) which is a saddle-node unfolding generically, as defined in Section I. Suppose that there is one hyperbolic singularity q(~) such that the unstable manifold of q(~) is transversal to the stable manifold of p(~), but there is one orbit y of quasi-transversal intersection between W~(q(~)) and W"(p(~)), the strong stable manifold of p(~). In addition we assume the following conditions to hold for the family X~. (3. A) Other generic conditions. (3.1) The pair (p(~), -((~t)) unfolds generically at ~t =~. This means that, provided that the saddle-node unfolds generically, there is a C I curve Fs~ in the para- meter space such that ~ e Ps~ r if and only if the vector field X~ exhibits a saddle-node singularity p(~z), and an orbit of tangency between W"(q(~z)) and W~*(p(~)) occurs only for the isolated value ~ in rs~. This is equivalent to say that, if ~t e I~sN ~ a"(~) and [z ~Pst~-*a~([z) are two C 1 curves in M such that a"([z)~W"(q(~z)), a~([x) ~ W~(p(~t)) and a~(~) = &*(~.) = r e y, then the projection of ~r~(~) -- ~8~(~) onto T, M/T, W"(q(~)) § T, W""(p(~)) is not zero, (3.2) X~ is C" linearizable near q(~) and partially linearizable near the saddle- node p(~) as described in Section I (c.4), its linear part having distinct eigenvalues at these points and m 1> 3, (3.3) W~ is transversal to W"~(p(~)), BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 139 (3.4) Let W'*(p(~)) be the invariant manifold of codimension one in W*'(p(~)) whose tangent space at p(~) is complementary to the eigenspace corresponding to the weakest contraction for XgIW"*(p(~)). Then, W"(q(~)) is transversal to W"*(p(~)), (3.5) There are no other criticalities: for any singularity a different from p(~) and q(~), W"(a) is transversal to W"(q(~)) and to W*"(p(~)) and W'(z) is trans- versal to W"(q(~)) and to W""(p(~)). All other invariant manifolds intersect trans- versally. (3.B) Tke bifurcation set. -- The hypothesis of generic unfolding of the saddle- node implies that there exists a C 1 curve I's~ r near ~ in the parameter space, such that along rs~ the family X~ presents a saddle-node bifurcation. Fs~ is the image of the singular set of the restriction of the projection (x, V-) ~ ~ to the manifold ~ (x) = 0 , where (x, V-) are C ~ coordinates in a neighbourhood of (p(~.), ~.) in M � I! ~" and f~ is the potential function associated to the family X~. Let W~*(p(~t)) be a C 8 center-stable manifold. From the linearizing assumptions, we can write 0 0 X 2 = X~ ] W~*(p(~t)) = B(~t, x) Ox + A,~(~t, x)y,- j= a 021 in a neighbourhood of p(~t) in M, where s = dim W"(p(0)) and (x, yl,...,y,) are ~t-dependent C" coordinates, m >/ 2, such that the eigenvalues of the matrix A(~t, x) ~- (A~j(~t, x)), � are distinct and negative. The ordering (Yl, ...,Y,) corres- ponds to the ordering al(~t) < ... < %(~t) of the absolute values of the eigenvalues of 0B A(~t, x). The generic unfolding of the saddle-node implies that B(0, 0) = ~x (0, 0) = 0, 0 3 B 0B Ox ~ (0, 0) ~e 0 (say positive) and ~ (0, 0) 4 0. Therefore, there is a diffeomorphism r x) = (r ~2(~t, x)) such that B o ~(~t, x) -~ x ~ + ~t x. Using the change of coor- dinates x ~- ~2(~t, ~), y~ =.~j, ~ = ~l(~t), we have [ . = [ + Multiplying by the nonvanishing function 0~2 0~ (~' ~)' we obtain a family X~ equivalent 0 0 to X 2 near p(~.) such that X~ = (~ -t- ~-1) ~-~ + I~N~(~, ~).~ ~--~. From now on we drop the bars to simplify the notation. Let :~2 C {y~ = 1 } be a cross section such that 140 M. J. DIAS CARNEIRO AND J. PALIS W*'(p(0, [x~)) n Z~' -= {(0, 1,y~, ...,y,)}. Since W*"(q(0)) is transversal to W'?(p(0)), we may write W*"(q(v)) c~ Z~' = { (x, yL,y,= =Yx(~, x,y,.))} and W"(q(~)) n Z~ ~ = { (F(~,y~.),y~.,yK(~, F(Ex,y~.),y~)) }, with YL = (Y2, ...,Y,q),Yx = (Y,q+a, ...,Y,), 1 + s, + dim W""(p(0)) = dim W"(q(0)), F(0, 0) = 0, ~ (0, 0) = 0 and \Oy~ Oy~ (0, O)/2<<.j.~,q nondegenerate (we assume dim W~(q(0)) + dim W"~(p(0))/> n § 1). For ix x < 0, we have two distinguished hyperplanes in E~{, namely x ----- ~ ~, which correspond to WS"(pl([/,)) ~ Z~' and to W"(p2(~x)) c~ Z~', where pl([X) and p2(~x) are the two hyperbolic singularities that collapse to form the saddle-node. There- OF fore, W"(q([x)) is nontransversal to W'(p2(~) ) if and only if Oy---~ (~'Y~): 0 and ~v/~ ~x~ = F([x,y~.). From the hypothesis of quasi-transversality and the implicit function theorem, we obtain a C a curve P in the parameter space defined by ~ ~1 = F(~, D~(~x)), OF OF where Yr. = flL(~) is a C 1 solution of ~ (~, y~) = 0. Since --8~t2 (0, 0) 4:0 (by the inde- pendent unfolding hypothesis), we obtain that F is a C x curve tangent to Fs~ at 0. There are no other criticalities and W"(q(~)) is transversal to W~(p(~)), and, thus, the bifurcation diagram for the family X~ for ~x near 0 is exactly I' ~ Fs~. Remark. -- Along I' the field X~ presents one orbit of quasi-transversality between W"(q(~x)) and W'(p~(~)). If dim W"(q(0)) -t- dim W"(p(0)) -= n, then the above equa- tions simplify to x = F(~) and P is given by ~ ~x~ = F(~). (3. C) Stability. -- Let X~ be in z~(M) such that X~ presents a saddle-node with criticality and the family satisfies all the conditions described in (3. A). If X~ is close to X~ so that it also has a bifurcation of type III for ~ near ~, then we will show that { X~ }~ e v is equivalent to { ~2~ }~ e v', where U and U' are open neighbourhoods of and ~ in IR ~. We may assume in the usual ordering of the singularities of X~, al(0t) ~< a~(~x)~< ... ~< ~t(~), that %(~) : fl'k_bl([s :p([-~)for ~ ~ Fs~ and q([z) : ~k_l([.L). We will see at the end of this paragraph that there is no loss of generality in doing so. We consider a distinguished neighbourhood Uk_l(~) of %_1([x) as constructed in Theorem A and connect it along the orbit of tangency y to a neighbourhood V(~) of the saddle-node. As in previous cases, we construct an equivalence k~ that preserves the level sets off~ inside Uk_ ~ (~x). In V(~) it preserves two continuous invariant foliations with C 1 leaves and depending continuously on ~; these foliations, denoted by F~" and F~', have complementary dimensions. The leaves of F~" have dimension equal to dim W""(%(~)) and its space of leaves is the center-stable manifold W~'(%([~)). BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 141 We start constructing the equivalence between X~ and X~ on the neighbour- hood V(iz) by obtaining an equivalence between X~S=X~[W~ ) and )~8 ___ )~ ]W0,(~(~)). To do that, let us consider a compatible unstable system FI(&), ..., F~_~(~z), F~U_l(~) as before. We take a continuous family of C 2 cylinders C(~) in WCS(ak(t~)) and a continuous family of C 1 closed discs D([z) contained in some leaf F of the strong-stable foliation F~ s so that, for [z e Fs~, C(~) is transversal to WS~(%([z)) and C(~z) ~3 D(~z) contains a fundamental domain for W~(ak(~)). On C(Ez), we construct a C 1 foliation F~(V) of dimension one, which is compatible with the induced system F[(~z) n C(~z), ..., F~_2([z ) c~ C(~z), F~"_l([Z ) c~ C(~z). Also, F~(~z) is compatible with W"(%_t(~)) c~ C(~z) and has a unique singularity of saddle-node type which is the point of tangency between W"(a,_x(~z)) r~ C(~z) and F~ ~ c~ C([z); outside this point, F~(~z) is transversal to F~" c~ C(~z). The construction of F~ is exactly like in the pre- vious paragraph. Let M~ be a distinguished leaf of F~ namely the curve in W~ n C(~z) defined by yr. = ~r.([z), where ~z ~ U ~ ~r.(~z) is the C m-~ solu- OF tion of ~-~v ~ (~,y~) = 0 and, as above, (x,y~,y,r) are C ~ coordinates for C([z) near the point of tangency y(~t) c~ C([z), m t> 3. For ~h ~< 0 in this curve, there are three distin- guished points P~-I(~-) = Wu(~k-l([z)) ~ M~(~z), p,(~z) = W~(a~(~z)) n M~ and p,+~([z) = W~(%+~(~z)) n M~(~t), so that the curve p,_~(~z) =p~+~(~z) represents the values of the parameter such that W"(a,_~(~z)) is quasi-transversal to W~(%+~(~z)). Therefore, in the three-dimensional manifold M~ Uv~vM"([z), we have two C m-~ surfaces intersecting transversally at 0 defined by M~ = { x = F(~z, ~([z))} and M~ = {B([z, x)= 0}. So, let ~:M~ M ~ be a diffeomorphism of the form ~(~L, X) = (q?l(~s ho(~L, X)) such that ~(M~) = { x -- tx~ = 0 }, B o ~(~, x) = x ~ + ~. Then, it is clear that X~ * is topologically equivalent to ~ 0 ~ O X~' = (x ~ + ~1) Ox + ZA,~(~, x)y, Oy~ and the manifold i~I~(~x) = 2~~ ~ W"(,r is represented by { x -- tz2 = 0 }. If we repeat the construction for the nearby family X~, we obtain X~* equivalent to a family with the same normal form along the central manifold (still denoted W~ and with the same expression for the manifold ~I~(~t). Hence, X~" [ W~ is conjugate to X~ ] W~ and the conjugacy preserves the distinguished point x~_x(~), which is the projection via the strong-stable foliation F~ * of the point p~_ x(~x) = W~(o~_ ~(tx)) ~ M~ Thus, X~,s] WO(tx) is equivalent to X,,~, l W~ with ~: (U, 0) -+ (It ~, 0) being a homeomorphism that sends the region A i onto fi~, as in the picture (Figure IX). This gives a homeomorphism in the space of leaves of the strong-stable foliation F~ ~. We now define a homeomorphism h~":WO*(,~(tx))--~WeS(~(q)(tz))). Let us consider, as in previous paragraphs, a continuous family of compatible homeomorphisms h~, for i = 1, ..., k -- 1, defined on the space of leaves of the foliations F~(tz), . .., F~_~(tz), F~a([~). We define h~* in the same way as in Theorem A, Chapter III of [15], the only M. J. DIAS CARNEIRO AND J. PALIS difference arising from the singularity of the central foliation F~,. Hence, we begin by applying Lemma 1 to get a homeomorphism between W cu (k_X(~t))C~ (y C(~t) and W""(~_~(?(~t))) r~ (~(?(~t)) preserving the central foliation. We then proceed as in w 1 to extend this to a homeorr.orphism on C(~t) which is compatible with h~, i = 1, ..., k -- 1 and sends F"(~) to F"(~0(bt)). This induces a homeomorphism on the boundary of the disc D(~t) which is extended to its interior, the extension being compatible with the homeo~orphisms h~. Finally, we define h~ 8 by sending F~," to F~(~). A z / A! ~"SN Fxa. IX Now, over each point of C (~t) u D (~t) we raise a u-dimensional (u = dim W "" (o k (~t))) continuous foliation F~, ", with C 1 leaves compatible with the unstable system F~(~t), ..., F~_~(~t), F~_x(~t) and with W~((~_l(bt)). Positively saturating it by the flow Xr, ~ and adding the strong-unstable foliation restricted to W~(ak(~t)) for ~1~< 0, we obtain a strong-unstable foliation F~" whose space of leaves is W~8(ok(~) ). We then construct a complementary foliation denoted by F~,' compatible with a stable system Fr for i = k q-2, ...,L We start by constructing a compatible stable system F~+2(~t), ..., Fl(~t), together with a homeomorphism in the space of leaves of each of these foliations. Let L+(~) be a leaf of F~, 8 in W"'(~t) such that F ~8 n WC(~t) consists [J., of a point x, with coordinate r > 0 small. Over each point x of L+(~t) we take F.~." the part of the leaf of the strong-unstable foliation that contains x and is contained m the neighbourhood V(~t). If we let Dr, ~ = [-J~eT,§ then Dr, , is a C ~ disc of codi- mension one which is C" outside L+(tz). We also take a continuous family of C ~ cylin- ders CX(tz) in We~(~t) transversal to W~"(bt), so that CX(bt) t3 D c" contains a fundamental It, tg domain for W"(bt), where D*~r,, = Dr,, c3 WC"(ak(bt)), and the vector field X r is tangent to Cl(bt) n W c" In C~(bt), we let Fd(bt) be a one-dimensional central foliation compa- r,~ ~ tible with the stable system F~+~(tz), ..., F~(tz). Over each leaf of Fa(bt) we raise an (s + 1)-dimensional foliation compatible with the stable system. Over each point of Dr,. we raise an s-dimensional continuous foliation compatible with the induced system F'(a,(bt)) t3 Dr,., k + 2 ~< i~< t. The center-stable foliation F~ 8 is the (s + 1)-foliation obtained by saturating negatively the foliation and adding to it the center-stable mani- fold W*8(iz) for [z x < 0. We repeat the same constructions for }~,r~" BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 143 We can now get a homeomorphism h+ . +,+-+ D+<~+,+ compatible with the homeomorphism on the space of leaves of the stable system, by first constructing it on 0 D e" and then extending it to the interior of the disc. The equivalence between X~ and X+I+) in the neighbourhood V(~t) is finally obtained by preserving the complemen- tary foliations F~ ~ and F~,". Since we are preserving the center-unstable foliation F~"_t(~t), we may extend it to a neighbourhood Uk_t(bt ) of ++_a(~t) by preserving the level sets of the function f+. The globalization of the equivalence to all M follows exactly like in w 1 of this section or in Theorem A, Section II. Finally, if in the ordering of the singularities of X+, ~1(~) ~ 9 9 9 ~ (~k--l(~) ~ r ~ (Yk+l(~) ~ " " " ~ (Yl(~L), with cr,(~z) : ak+a(~t ) :p(~t) along the curve of saddle-nodes I~sz~, the orbit of quasi- transversality occurs in the unstable manifold of a singularity ~r~(~t), with j<~ k- 2, we proceed as in w 2, case II. b. That is, we construct a compatible system of unstable foliations F~(ax(~)), ..., F~"(~(~t)), F"(a~+~(~t)), ..., F~(%_~(~t)) and follow the same steps as above. Again, we connect the distinguished neighbourhood U~(~t) of aj(~t) to the neighbourhood V(~z) along the orbit of tangency. w 4. Bifurcations of type V: saddle-node with an orbit of tangency So far we have treated the cases which present at most one secondary bifurcation: in a neighbourhood of the bifurcation value ~, the family X~ presents for ~t 4: ~ at most one new bifurcation. Contrary to this, the bifurcations corresponding to types V, VI and VII of the list in Section I may present several secondary bifurcations. This lead us to analyze orbits of tangency between several invariant manifolds and a certain invariant foliation. For this reason, to prove stability, a globalization of Lemma 1 in w 1 (Lemma 2 below) will be necessary. In this paragraph we study the case where Xg presents a saddle-node p(~) and an orbit y of quasi-transversality. We assume that T belongs to the unstable manifold W"(q(~)) of a hyperbolic singularity and the stable manifold W+(p(~)) of the saddle-node. The case where the quasi-transversal orbit occurs between invariant manifolds of hyper- bolic singularities, will be discussed at the end of this paragraph. Besides the assumptions that we have already used in previous cases, like linearizability and partial linearizability for X~ near q(~) and p(~), generic and independent unfolding of the saddle-node and the orbit of quasi-transversality, and transversality between W+"(q(~)) and W+(p(~)), several others are required here. They are satisfied by generic families X~ + z~(M) which present a bifurcation of type V. (4. A) Other generic assumptions. (4.1) Let W+"(p(~)) be the codimension-one invariant submanifold of W""(p(~)) such that T+<g> W+"(p(~)) is complementary to the eigenspace corresponding to the 144 M. J. DIAS CARNEIRO AND J. PALIS smallest nonzero eigenvalue of dXg(p(~)) (weakest expansion). Then, for r e y, there exists a linear subspace E, C T, W"(q(~)) with dim E,=dimW'"(p(~)) such that lim~_~oo dXg.~(r).E, = T~(g)W""(p(~)). Moreover, if ~ is a singularity of Xg different from p(~) and q(~), then W'(~) is transversal to W"(p(~)), W""(p(~)) and W""(p(g)), and W"(~) is transversal to W""(q(~)), W~(p(~)) and W"(p(~)). (4.2) Let F~ " be the unique codimension-two invariant foliation in W"(p(~)) which has W*"(p(~)) as a distinguished leaf. F~- " is compatible with F u" each leaf L iz ~. ' of F"-- " is subfoliated by leaves of F~ ". Suppose L + W""(p(~)) and that n*" : L -+ R Ix is a submersion that defines F~ " in L. Then, the restriction of n~ to each stable manifold W'(a(~)) n L is a Morse function with distinct critical values. For any stable manifold such that W~(~(~)) c~ L is tangent to F~ ~, the eigenvalues of dXix(a(~)) are distinct. In this case the center-stable manifold W~*(a(~)) n L is transversal to F ~" g. Comments. -- Clearly, these conditions do not depend on the leaf L. Also, if W'(a(~)) c~ L is compact, it is easy to perturb Xg so that n "~ [ W~(a(~)) n L is a Morse function with distinct critical values and W~"(~(~)) c~ L is transversal to F~- " To get Ix " the genericity of these hypotheses, we use the ordering ,~(~) .< ....< ,,(~) .< ,,+o(~) .< ....< ,~(~) of the singularities of Xg such that p(~) ---= %(~), assuming that W""(a,(~)) is transversal to W'(%(~)) for k + 2 ~< i ~< l -- 1 ; i 4- 1 ~< j ~< t and proceed by induction using trans- versality arguments, in particular, transversality between W~"(%(~)) and F~'. (4.B) The bifurcation set. -- Assume that in the ordering of the singularities ~1(~) -< ..- -< ~k-l(~) -< ~k(~) -< ~+1(~) -< ... -< ~t(~) of x~, we have %(fx)----~k+l(tx)=POx) for ~ e I'sy , the curve of saddle-nodes, and that %_a(~) = q(~) ; also assume ~ = 0. Using the transversality between W~(%_a(~)) and W~(ak(~)) and the partial linearizability of X~ near ak(lx), we extend W~"(%_l(~)) to a neighbourhood of the closure of the orbit of tangency y so that it contains the saddle- node. We may suppose that we have a normal form for Xix I W~"(%_1(~)) near ~k(~) and, as in w 3, we can write 2 0 O X~ ~ = X~ I W~(%-1(~)) = (4- x 1 § ~a) ~ § Z~,~(xa, ~)Y' 0-~j + Z~(x~, ~) z~ Oz~ in a neighbourhood of p(0), with all eigenvalues of Ax(Xl, ~) = (ei~(xa, ~)) being nega- tive and of B~.(xl, ~) = (~(xl, ~.)) being positive. In these coordinates we assume that the zl-axis corresponds to the direction of the weakest expansion, u = dim W""(p(~)); we choose the positive sign in the above expression. Let Z~([x) be a cross-section inter- secting the orbit of tangency T. Then, the intersection of W*(ek_i([x)) with Z~_"(tx) is BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 145 {(YI, Zl, ZL) ] Zl = F([z,YI, Zr,)} with YI ~ F(pt, YI, 0) being a deformation of a Morse function. Moreover, the generic and independent unfolding of the orbit of quasi- transversality implies that the map (~h, ~t2)~-* (~Zl, F(~t, Yr(~t), 0)) is a local diffeo- OF morphism, where ~ ~ Yi(~t) is the solution of Oy--- I (~,yr, 0) = 0. Therefore, the curve Fk_x, k of quasi-transversality between W~(%_l(~)) and W"(ak([z)) is locally defined by { ~t 1 < 0 } n { F(~, YI(~), 0) = 0 }. By changing coordinates, we get Fk- 1, k = { ~2 = 0, ~t 1 < 0 }. Since there are no criticalities, other bifurcations may occur only in the region ~t 1 > 0, where the corresponding vector field does not present singularities near the point p(0). To analyze these possibilities, we let Y~.u(~) be a small closed disc contained in the section { xl = ~1 } such that Z~"(~) c~ W"(p(~)) is contained in a leaf of the strong unstable foliation. The positive number ~ is taken so that if W"(%(~)) n OZ~_"([z) + O, then W"(%(0))c~ W""(p(0))~e O. Hence, if there is an orbit of tangency between W"(ak_x([x)) and WS(%.(~)), then it necessarily intersects the interior of Z~u(~x). Moreover, from (4.1) and (4.2), these tangencies may occur only near the points of tangency between W8(%(0))c~ W~(p(0))n Z~u(0) and the foliation Fg ". For each j t> k -t- 2 we denote by Pj,1, ...,Pj, ,(j) these points. Let (vI, Wl, wL) be C m coordinates in Y~?(~t) such that W"(ak([x))n Y~_"(~)= (0, wl, wL), m>~ 3. We may assume that WS(%(0)) has codimension one in Z~?(0); if not, we just restrict ourselves to W~(%(0)). Then, from (4.2), near each point p~ we may write WS(6j(~)) ~ ~_u(~) = { Wl = Gji(~, VI, WL) } with Gj~(0, vi, 0) having a nondegenerate critical point at vt(pj,). Let us extend Fg", previously only defined on W*(p(0)), see (4.2). Let Zr t) = {(YI, Zl, zL) e ZL~(~ t) [zL = 0} and r~": Zc_u(~t) -+ ZL(~t) n~,"(Yx, z~, z~,) = (yx, Zl -- F(~,yI, zr,) q- V(~,yx, 0), 0) be a submersion that defines a C" foliation F~" compatible with W"(%_1(~)) n Z'_"(~t). For latter purpose, the flow saturation of this foliation will still be denoted by F~". Using the normal form for X~," near p(0) to get a linear expression for the Poincar~ map P~ . Y.'__~(~) -+ Z~?(~t) for ~ > 0, we obtain that the restriction of ~ to is singular along disjoint C "-1 manifolds M~i(~z) for i = 1,..., n(j), with dimension equal to I I l= dimW'(p(0))c~ W~"(q(0)) and which depend differentiably on a. As [z -+ 0, all these manifolds become C 1 close to they,-plane and for a~ = 0 they collapse into this set. Since the points {p~ } belong to distinct leaves of the foliation Fg ", the images Mj,(~t) = ~"(~I~,(~)) are disjoint submanifolds of codimension one in E~_(~). If W"(%(~t)) has minimal dimension (equal to dim W'(p(0))), then 19 146 M. J. DIAS CARNEIRO AND J. PALIS From this construction we conclude that W"(ak_l(~t)) is tangent to W'(%(~t)) in ZL"(~t) if and only if M~(~t) is tangent to W~(%_I(~)) n ZL(~) for some i = 1, ..., n(j). Hence, for each (j, i), we consider possible tangencies between the manifold Mt~(~) and the foliation defined by (YI, Zl) ~ Zx -- F(~t,yx, 0) + F(tz, 0, 0). Using now the hypothesis of quasi-transversality between W'(p(0)) and W*(q(0)), we obtain for each ~t in a neighbourhood of 0 in { ~tx ~ 0 } a unique point of tangency q~,(~) ~ M~(~). The map ~t ~ q~(~) is of class C 1 in a neighbourhood of 0 in { [z x t> 0 } and qi~(0, ~t,) = 0. Therefore, X~ presents a quasi-transversal orbit of tangency between W"(%_l(~)) and W'(%(fz)) if and only if q~(~t) belongs to W=(%_~(~t)) n ZL(~t). These values of ~t correspond to a finite number of disjoint C a curves I'~._ a, ~. tangent to the ~q-axis at 0. The bifurcation diagram is as in the figure. ~SN k-l,j T~k- ! ~r Fro. X (4. C) Stability. -- Let X~ be a family in ~(~(M) which presents a bifurcation of type V at ~ and satisfies all the assumptions described in (4. A). If X~ is a nearby family, with ~ as the corresponding bifurcation value, then we show that there are neighbour- hoods U and lJ of ~ and ~ in R ~ such that { X~ }~ E ~ is equivalent to { )(~ }~, ~ O. We assume that ~ = ~ = O. We start by taking a compatible unstable system F~(~), ...,Fk_~(~),Fk_l(~) and neighbourhoods Uk_I(V) of ak-x(~) and V(~) ofp(~) in M which are connected along the orbit of tangency y. From the description of the bifurcation set, each point of tangency between F~" and W*(%(0)) yields a quasi-transversal orbit between W~(%_~(~)) and W*(%(~)). So, we consider distinguished neighbourhoods U~(V) of each such singularity and connect them to V(~) with tubes along each orbit of tangency Ts~" The equivalence will preserve the level sets of f~ inside the neighbourhood Ui(~). Using the transversality between W~(~h(0)) and W~(%(0)) for i>j >/k + 2, and pro- ceeding as in w 2 of the present section, we construct a compatible center-stable system ~+~(~), F,+3(~) , .., F~'(~). It may happen that for some i i> k + 2, the stable manifold W'(a~(0)) is transversal to F~" (for instance, when a((0) is a sink). In this case we take, as in w 2, the stable foliation F~(~). BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 147 To obtain the equivalence between X~ and X~, on a neighbourhood V of the saddle-node p(~) in M � R *, we construct a center-unstable foliation FCU(~) compatible with the unstable system and the unstable manifold W"(~k_l([Z)). The method to cons- truct F~"(~) is similar to the one already used in w 1 and w 2. The main difference here is that we want the singular set of FZ"([z) to contain the points of tangency between W~(~_I(~)) and the manifolds WS(~(~)) for j = k + 2, ...,L Let E~_(Ez) be the leaf space of the foliation F~" constructed in (4.B). We recall that to describe the bifur- cation set we have obtained codimension-one submanifolds M~,(~z)C Y.~_(~z) such that W~(a,_l(~Z)) is tangent to WS(aj(~z)) if and only if W"(ak_l(~) ) n Z~_(~) is tangent to Mj~(~) for some i = 1, ..., n(j). For each pair (j, i) and ~ > 0, we let OF 0A~ Yj,(~,YI) = ~ ([z,YI, 0) -- 0YI (~,YI), where graph (A~)= M~(~z). Since lim Yj~(V, YI) OF ~,~,0 ---- ~ (~,YI, 0), we may extend OF this family to Yj~(~z,y~)=~v (~,y~, 0) for [Zl~< 0. For 0 < r < ~2 small, we let A~(~I)CAj~(r be open neighbourhoods of M~(~z) nW~(%_l(~)) such that Ai~(r ~ At,~,(~) = O for (j', i') # (j, i). We define a family of vector fields Y(~,y~) OF such that Y I Ai~(r ---- Yi~ and Y~ in the complement of U A~(~) is equal to Oy-- I (~,y~). As in w 1 and w 2, the central foliation which gives rise to the leaves of F~"(~z) inside W~(%_l(~)) is tangent to a vector field Z~ with a saddle-node singularity such that Z~ restricted to each Ai~ is equal to Y~. Associated to a central n=anifold of Z~ we have a special leaf denoted by W~"(~). This invariant manifold is completed for ~z~ ~< 0 by adding part of a center-unstable manifold which is linear in the above normal form coordinates. By construction, W~"(~) contains all tangencies between W"(~_a(~)) and W~(g~(~)), j >1 k + 2. The other leaves of F~"(V) are obtained exactly as in w 1. Complementary to F~"(~), we define a strong-stable foliation F~'. Since all stable manifolds Ws(~(0)) are transversal to W"(g~(0)), the method described in w 3 can also be applied here. However, since F~"(~) is a singular foliation, in order to have transversality between F~' and F~(~) outside W~"(~z) we modify F~' for ~q 7> 0 near the points of tan- gency psi(0). Let Z be two cross-sections such that Y.+(~z) c~ W~"(a,_~([z)) ---- ~([z) and suppose that F ~* is a strong-stable foliation in W*(a~(~z)) n Z+(~), as constructed in w 3, which is transversal to W~"(V). We can also assume that F". is transversal to F~"(~z) outside a neighbourhood of each point pi~(0). Let P~ : Z (V) -+ Z+(~z) be the Poincard map for ~ > 0. We modify F**. in a neighbourhood of pi~(0) in such way that each leaf of the induced foliation P~ -1 (F~,~) ~ ~ Y~e"([z) projects by n~, V~ onto a level set of a Liapounov function of the vector field Z~ in Z~_([z). Proceeding in this way for all stable manifolds W*(a~(~z)) and extending this modifyied foliation to each leaf of F~'(~) as in w 1, we get the required strong stable foliation F~*. By preserving F~/ and F~ ", we M. J. DIAS CARNEIRO AND J. PALIS can obtain an equivalence between the two families X~ and X~ on a neighbourhood of the saddle-node singularity similarly to w 3. Hence, to prove local stability of X~ we have now to obtain homeomorphisms on the space of leaves of these foliations. Let us frst construct a suitable reparametrization q~. Consider X~ restricted to W~(~) (the space of leaves of F~,8). Since W~(~) depends differentiably on ~, it is transversal to W8"(%(~)) for ~t e Fsz ~ and admits a C" smoothing structure, r>~ 3 (see [15]), we conclude that X~[W~(~) has a ~-dependent normal form near the saddle-node as in (4.B). In W~,"(~) we consider a codimension-two invariant foliation compatible with F~ ~ such that for ~ E FsN it has as special leaf W*"(%(~)), the codi- mension-two strong unstable manifold (see (4.1)). For ~1 > 0, F~, s is obtained by satu- rating the foliation used at the end of (4. B) and intersecting with the leaves of F~". This foliation is extended to a neighbourhood of p(0) for ~x 1 ~< 0 by adding to it a codi- mension-two linear foliation. In particular for ~ = 0 this gives the foliation defined in (4.A). For each ~ the leaf space of F~" is an invariant surface W(~) that contains a center manifold, and it is defined in the above coordinates by z L = 0. In W(~t) we take a fundamental domain C(~)w E ~, where E ~ ={Xl = ~, ]zx] ~< ~} and = c+ u c = {I I = [ I -< Let C ---- [J~ev C(~) and definc I x : U\Fk_~, k --> C t3 E c, the map that associates to each ~ thc point of intcrsection of W"(%_x(~) ) with C([~) u E~. If c#,(~) e E" reprcsents the leaf of F~," which contains the tangency point p#,(~), then the curve F~_~,j obtained at the end of (4.B) is defined by Ixl(Cj,(~)). Moreover, from the hypothesis of generic unfolding of the orbit of tan- gency y (0) ~e 0 we obtain that Iff~(E ") is a wedged shape region A C { [z 1 >/ 0 } with vertice at 0, which is bounded by two curves Ix1(4 - 8). We also have in A a singular foliation r defined by Ixl(x) for x ~ E ~ with special leaves Pik_l. ~. We define a repara- metrization q~ : (A, 0) -+ (A, 0) of the form (q~1(~1), q~,(~zl, ~z,)) which sends F to F. Since a conjugacy on a center manifold induces via the strong unstable foliation a homeo- morphism h ~ :C ~ C, we choose the reparametrization on U\A in such way that I~ o ~? = h ~ o I x. This gives a reparametrization on a full neighbourhood of 0. We now prove that X~ and 1~,(~) are equivalent. We begin by taking a continuous family of diffeomorphisms +~ :E~(~) ~ F2(q~(~z)) sending c~(V) to ?~i(q0(~)). Using a conjugacy we define a homeomorphism on the space of leaves of F~,". To define an equivalence between X~ ] W~"(~z) and X,(~)[W~"(~(~x)) we use a conjugacy which preserves F~" inside each leaf of F~,". Therefore, for [z~ > 0, it is enough to obtain a continuous family of homeomorphisms on a leaf Z~?([x), preserving F~" and the center- stable system, in order to get an equivalence on a neighbourhood of the saddle-node p(0) in W~,"(~). Contrary to this, for [xx <~ 0, the negative flow saturation of Y.~_"([z) just fills a conic region A(~) with vertex at the singularity a,+~(~t). Therefore, to get an equi- valence on a full neighbourhood of p(0) in W~"(~), we construct a two-dimensional foliation F~(~) in the complement of A(~z) which is compatible with the center-stable system and transversal to F~,". Thus, the equivalence is defined by preserving F~," and BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 149 F~(V). Let us construct a foliation F~(~z): this construction resembles very much the one of a central foliation in [15]; the main difference here is that we want it to be transversal to the codimension-two foliation F~ ". Let K(~t) = U,~e(~D~"(~t), where D~"(~t) is a closed disc centered at x and contained in the leaf of F~" over x. As above, C(~t) is the intersection of the fundamental domain C with the plane ~t = constant. Let E""(~z) be a closed solid cylinder in the leaf Z~"(~t) which is bounded by two closed discs K~'"(~t) ---- K(~t) c~ Z~"(~t), i = 1, 2, and by a cylinder S(~t). Over each disc K~'"(~t) we raise a one-dimensional continuous foliation Zc(~t) in K(~t) which is compatible with the center-stable system. We can assume that C(~t) is a leaf of Zc(~t). We construct Z'(~t) in such way that the union of the leaves of Z'(~t) which are over the spheres 0Kx(~) and 0K~"(~t) is the closed cylinder U,~c~0D~"(~t). Since the tangencies between W'(~rj(~t)) and F~" occur in the interior of E""(~t), the cylinder S([z) C 0E""(~t) can be foliated by a one-dimensional foliation 8~(~), which is compatible with the center-stable system, and whose leaves are C ~ and transversal to F~ ~ n S(Ez). Over each leaf of 8~(~t) we raise a two-dimensional foliation Z~(~z) also compatible with the center-stable system, with each leaf of ~(~t) being bounded by two leaves of X~(~). Thus, F~(~) is obtained by taking the negative saturate of Xe(~t) and of k~(~t) by the flow of X~. This finishes the construction of Y~(Vt ) which has as space of leaves the boundary of E""(bt ). Since we already have defined a homeomorphism on the space of leaves ofF~ ", in order to conclude the construction ~ of the equivalence between X~ ] W~"(Vt) and W,(,)I W~"(q~(~t)) it is enough to obtain a continuous family of homeomorphisms h~": E~([z)-+E~(~(~z)) which preserves F~ ~ and the center-stable system. The idea to obtain h~," is to " project " E""(~z) onto E""(0) along the leaves of F~," and to construct a homeomorphism from E""(0) to E""(0) which satisfies the above requirements. We then pull back this homeomorphism to E""(~) to get h~ ". This process is achieved by constructing a continuous foliation 3~ ~ on E"~= O~vE""([z), with C ~ leaves of dimension two, which is transversal to E""(0) and compatible with both the center-stable system and with the foliation F~ ". The construction of ~ is easy except at neighbourhoods of the tangency points p~,(0). Near each point p~(0), ~ restricted to WC"(a~([z)) is defined by intersecting F~ " with a three- dimensional foliation given by a continuous family of vector fields, parametrized by ~, which has a saddle-node type singularity at p~,(tz). Therefore the surfaces of tangency (~, p~,(~)) are special leaves of ~. The extension of 3f ~ to the leaves of the system F~ ~ near p~,(0) is done as in w 1. The foliation j~o was conceived so that it may be used to tri- vialize the foliation F~" along the center-stable system. Suppose that h~" : E~"(0) -+ E""(0) is a homeomorphism preserving Fg" and the center-stable system. Then, we define h~," : E""(~z) ---> E""(q~(tz)) by sending ~ n E""([z) to ~,~ ~ E""(~(~)). The reparame- trization ~ obtained above guarantees that the point p~(~) is sent to the corresponding one ~.~(q~(tt)). Thus, to finish the construction of an equivalence between X~ [ W,""(~z) and X,(~) [ W,~"(~(~z)) it remains to prove the existence of h~". This is the content of the following key lemma. 150 M. J. DIAS CARNEIRO AND J. PALIS Lemma 2. -- There is a homeomorphism hg": E""(0) -+ F,""(0) that preserves the folia- tion Fo ~, the center-stable system F~'(%(0)) ~ E""(0) and the stable manifolds W'(%(0)) c~ E""(0) for j = k + 2, . ..,t. Proof. -- By using a diffeomorphism which preserves Fo" we may assume that E~"(0) = ~,""(0) and F;"= ~'g". We may also assume that ~*~l W'(%(0)) and ~"]W*(~j(0)) have the same critical values for j=-k-t-2, .,t. Let us assume that W~+2 = W'(ak+2(0)) c~ E""(0) is compact and disjoint from the boundary of E""(0). If ~;"(wl, w~.) = wl is the projection along the leaves of Fg" then n~-2 = ng"[ W~+2 is a Morse function with distinct critical values. Analogously, for rck+ ~^~" = %^*" [ Wk+ ^' ~. Let ~k + 2 : WZ + 2 -+ "vV~ + 2 be a diffeomorphism C ~ close to the inclusion map. Then, ~k ^~" + 2 o % + 2 and r~k+ ~" 2 are C ~ close Morse functions with the same critical values. Therefore, there exists a C 2 diffeomorphism h2: W i,+2-+'v~r~+2, close to the identity, t~Z~ At~t~ t*U $ such that ~k+2~176 We define the restriction of ho to W k+2 by ho"" = ~Pk+2 o h-Xk+2. The same is done, in a continuous way, for all leaves of FZ+ ~ =F"(a~+~(0))c~ E""(0) which are contained in the center-stable manifold W~'+2 = W"'(%+~(0))n E""(0). Since W~' is transversal to F~?(0), we let F~_ 2 be a C a foliation in a neighbourhood of W~+ 2 which is transversal to W"'k+2 and compatible c$ with Fg" such that dim F~,~_~ = codim~o ~ W~+ 2. The foliation F~_~ is defined on a tube % + 2 along the stable manifold W~ + 3. The intersection of % + 2 with each leaf F of F~" is a closed box B~'k+2 bounded by a cylinder transversal to F~_~ together with two closed discs ~[ u ~ contained in leaves of F~_ 2, such that any leaf of the center- stable foliation F~*+~, whose dimension is equal to the dimension of W ~'~+~, intersects transversally ~ ~ ~. To obtain v~ + 2, we first define local tubes % + ~,~ in a neigh- bourhood of each critical point p~+ 2,~ between two non-critical levels 0~-+ ~,~ and 0++ 2,~. Let Z~+2, ~ be a C 2 vector field (as constructed in w 1) tangent to each leaf of F~"+ 2 = F"'(a~+ 2(0)) n E""(0), whose restriction to W~"+ ~ has a saddle-node singularity atpe+~,~ and whose restriction to W'~+2 is the gradient of r~;~_.. In the leaf 0;-+2,~ we take a closed box B~-+ 2,~ as above and positive saturate it by the flow of Z~ + 2,~ in the strip between ~-+2,~ and 0~+2,~. We add to this set the stable manifold W"(Z~+2,i) in order to obtain the local tube %+2,i- The local tubes %+2,~ for i = 1, ..., n(j) are then connected along W]~+ ~ by using the integral curves of Z~+ 2, a C ~ extension of Z~+2,i along the leaves of F~+~ in a neighbourhood of W~- such that Zi restricted to W~ is a Morse-Smale vector field. Therefore, by preserving the two complementary foliations F ~'~+~ and F~_~, we obtain a homeomorphism h~ ~ on the tube %+~. Observe also that E""(0) is bounded by a cylinder S(0) transversal to Fg" and two closed discs, each one contained in a leaf of Fg ". Since W"(%) is transversal to Fg ~ at the boundary of E""(0), the cylinder S O can be foliated by one-dimensional leaves 8~(0) compatible with the stable system and transversal to Fg ~. Hence, if W],+2 c~ 0E""(0) 4 = O, we take the diffeomorphism described above also preserving the foliation 8~ + ~ (leaves of 8~ which are contained in W~+ 2 c~ 0E"~(0)). Next, suppose that W~+~ = W'(%+~(0))c~ E~"(0) BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 151 is nonempty and does not intersect the boundary of E""(0). If W~+ 3 is compact, we repeat the above argument. If not, then WU(%+~(0))n W'(a~+8(0))4:0 and we consider the foliation F~+ 2, k + 8, consisting of the leaves of the center-stable foliation F~+ 2 which are contained in W'(,k+3(0)). We also take a diffeomorphism on the space of leaves of this foliation. This is possible because the intersection of WS(,k+3(0)) with a fundamental domain of W"(ak+2(0)) is compact. By preserving F~+o-,k+3 and F~_o-, we get a homeomorphism from WT,+ 8 n %+.2 to X3(~ + 8 n % + 2. Using the Isotopy $ A $ Extension Theorem, we get a homeomorphism %+s: Wk+8-+Wk+8, which is a C ~ diffeomorphism on Wk+s\~e+ 2.' The functions nk+ ~*u and ne+8^*~ o ~k+8 have the same critical values and coincide in W'e+8 c~ %+2- Let n~+8 be the homotopy (1 -- t) zck~"+3+ tnk^~+8o ~k+8 for te [0, 1]. Then ~+8 [ %+~ n W ~e+8= z~k+8- ~ ~ By defining a family of vector fields ~k -t + 8 on Wk+ ~ with supp ~+8C W],+8\%+~ , such that ~t ~'~+8.',t , we obtain that ~**+3 is topologically trivial. That is, there exists a continuous family of homeomorphisms h~ + 3 : W~ + 8 ~ W'k+8 such that n~+3 oh~+~ = n0+~ = r~_~. Hence we define h~" restricted to W~+~ by h~" = q%+~oh~+~. We do the same on each leaf of F~ + ~ contained in W e + ~, to extend h 0 to a neighbour- hood of W~+n in W ~s~+~. Again, since W~+~ is transversal to F o , we take a C ~ folia- tion F~_~ in a neighbourhood of W~+o- which is transversal to W and compatible with F 0 and with F~+~ such that dimF~+ a = codzm~.~0)We+ a. This foliation is constructed in a tube ~ + a along W~ + a exactly as in the previous step of this induction. If W~+an0E"~(0)4: O, we take the homeomorphism %+a also preserving ~r in W~ + a ~ 0E~"(0) 9 Proceeding by induction on the ordering of the singularities, we obtain the homeomorphism h~ ~ as wished. 9 Thus, we have obtained a homeomorphism on W~(~), the space of leaves of F~'. By applying Lemma 1 and the methods described in w l, wc obtain a homeomorphism on the space of leaves of F~(~). These homeomorphisms define an equivalence on a neighbourhood V(~) of the saddle-node in M as in w 3, by imposing that the two comple- mentary foliations F~' and F~u(~) must be preserved. To extend this equivalence to a distinguished neighbourhood U~_~(~) of ~e-z(~), we connect it to V(~z) with an invariant tube Wk_l(~) along the orbit of tangency. In the fence B~_~(~)C OUk_l([.s we let D~_x(ix) be the intersection of [.J,<0X~.~(V~) with B~_ a ([z). We can assume, after a reparametrization of time, that D~_ ~ (~) is contained in X~,_v(Z (~t)) for some T. Hence, we have defined a homeomorphism on D~_a(~) which preserves the center-unstable foliation F~"(%_a(~)). The same arguments as for Theorem A are now applied to extend this homeomorphism to the fence Be_l(~) pre- serving F~"(%_a(~)) and the center-stable system. We define a homeomorphism on U~_z(~) by preserving level sets and trajectories. Inside the tube T~_~(~) the equi- valence is a conjugacy. Analogously, we get an equivalence between X~ and ~2~,(~) on a distinguished neighbourhood Ue+ ~([z) of %+ ~(~). Proceeding inductively and using the compatibility of the center-stable system, we construct equivalences on distinguished ~"~+O- 152 M. J. DIAS CARNEIRO AND J. PALIS neighbourhood Ui(~) of ~([x), j/> k -t- 3. Finally, as in w 2, we extend the equivalence to all of M as a conjugacy outside these neighbourhoods. 9 It remains to deal with the case where the vector field Xg presents a saddle-nodep(~) and one orbit 7 of quasi-transversality between W"(q(~)) and W'(q'(~)), q(~) and q'(~) being hyperbolic singularities. We assume the linearizability conditions and the non- criticality condition with respect to the strong-stable and strong-unstable manifolds ofp(~), q(~) and q'(~) and also the generic and independent unfoldings of the saddle- node and the orbit of quasi-transversality. Similarly to the case (II. b) of w 2, since there are no criticalities, we conclude that the bifurcation set near ~ is the union of two C ~ curves I~T t3 I'sN intersecting transversally at ~, such that for & ~ I'Q~ the field X~ presents one orbit of quasi-transversality between W~(q([x)) and W'(q'(~)) and for ~ I'Q~ a saddle-node p([x). The equivalence between X~ and a nearby family >2~ is obtained without much difficulty using a combination of the methods developed in (II. b) of w 2 and w 3. 9 w 5. Quasi-transversal orbit with tengency between center-unstable and stable msnlfolds In this paragraph we consider a family X~ ~ )(~(M) such that for a value ~ ~ R ~ the vector field Xg presents a bifurcation of type VI: there is an orbit of quasi-trans- versality between W"(p(~)) and W'(q(~)), p(~) and q(~) hyperbolic singularities, satis- fying all the generic conditions described in Section I except (c. 3) ; i.e. the center-unstable manifold W*"(p(~)) is not transversal to W'(q(~)). To have a codimension-two bifur- cation, we assume that W~(p(~)) is transversal to W*'(q(~)). Since we also assume that Xg is C ~ linearizable near p(~), choosing a C ~ center-unstable manifold W*"(p(~)) which is linear in these coordinates, we suppose that along the orbit of tangency 7 the stable manifold W'(q(~)) is quasi-transversal to WC"(p(~)). Here we take Comment. -- Although center-unstable manifolds are not unique, this condition does not depend on the choice of a C"* center-unstable manifold, if m is sufficiently high. In fact, if N is a (u -t- 1)-dimensional invariant manifold of class C" for m as above such that T~o)N : E 0 @ T~0)W"(p(~)), E o being the eigenspace corresponding to the weakest contraction, then the contact between N and WCU(p(~)) along 7 is of order at least two. That is, for each point r ~ 7 there is a local diffeomorphism + in a neigh- bourhood U of r in M, j2 +(r) = 2-jet of the identity map, such that +(u n N) = U. Proof. -- Let (xl, xi,yl,yr,) be C ~ linearizing coordinates for Xg near p(0) such that W'(p(0)) = (xa, xx, 0, 0) and W"(p(0)) = (0, O,y, yL). We suppose that 7 is tangent BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 153 to the yl-axis (weakest expansion). Then N t~ {Yl = 1 } = { xl, Nz(xx,yL),yr.)} with N I of class C '~. Let P :{yl = 1 }-+{ xl = 1 } be the Poincar6 map P(xz, xI,yT.) = (x; ~'/~' x,, x~ g~', x~/",yL) ; then P(N c~ {yz = 1 }) is a C ~ manifold parametrized by (y~-~'/~' Nz~t,,~'/~'/a , y~.),ya,y~/~'y~.). Since each com- ponentyi -~g~' Nj(y~g~',yL) is of class C" with m/> max 3, [ ~1 -F 1, ~11 -+- 1 t' and there are no resonances between the eigenvalues, we must have dNj(0) = 0 and d * Nj(0) = 0. Thus, j2 Nz(0 ) = 0, proving our statement. 9 We also suppose the generic unfolding of the orbit y, so there is in the parameter space a curve I'QT containing ~ along which X~ exhibits a quasi-transversality between W"(p([L)) and W'(q([z)). We require the tangency between W~"(p(~)) and W'(q([x)) to unfold generically, so we also get a curve I" 0 containing ~, along which X~ presents a quasi-transversality between Wr and W"(q([x)). It is easy to see that I'Q~ and lr 0 are always tangent at the point ~. Therefore we require that PQT and I" 0 have a qua- dratic contact at -~. (5.A) Other generic assumptions. -- In addition, we assume that the family X~ satisfies the following generic conditions. First, let Wf"(p(~)) be a (u + 2)-dimensional center-unstable manifold, u = dim W~(p(~)), which we assume linear in the linearizing coordinates. Then, W~"(p(~)) is transversal to W'(q(~)). Now, let WS'(p(~)) and W~'(p(~)) be the invariant submanifolds of W'(p(~)) of codimension one and two, respectively, which corresponds to the eigenspaces of strongest contractions. For any singularity a(~) of Xg, we assume that W"(a(~)) is transversal to W"s(p(~)) and to W*'(p(~)). Moreover, let F' be the codimension-two foliation in W"(p(~)) having W*"(p(~)) as a distinguished leaf. If W"(a(~)) is not transversal to F *' and dim W*(a(~)) n W"(p(~)) /> 2, we require that the restriction of n*s (projection along F w) to W"(a(~))caW'(p(~)) has a fold singularity along one orbit. This last hypothesis is similar to the one used in w 4. If L is a leaf of the strong stable foliation and n~" is the projection along the leaves of P" contained in L, then we assume that ~* restricted to W'(e(~)) n L is a Morse function. It is easy to show the genericity of this hypothesis and that it does not depend on the leaf L. We also require that the points of tangency between W"(e(~)) n L and F *~ belong to distinct leaves. For each j < k, we denote by p~(~) the distinguished points of tangency between W~(a~(~)) n L and F *~. Since we are going to use a compatible center-unstable system, we assume that for each a(~) <~ p(~) there is the smallest contraction and that We"(a(~)) is transversal to F *' in WS(p(~)). We also require W"(a'(~)) to be transversal to W'"(a(~)) for all singularities a'(~)~< a(~) and W~"(q(~)) to be transversal to W'(~*@)) ifq(~) ~< a*(~). The genericity of these conditions follows exactly as in (4.A). 20 154 M. J. DIAS CARNEIRO AND J. PALIS (5.B) The bifurcation set. -- Let X. be a family satisfying the conditions described above at a bifurcation value ~. Let X~, CS = X~, [ WC~/a k k+l([~)) " We assume that ~ = 0 and that p(bt) = ak(~t) and q(~t) = a,+l([Z) in the usual ordering of the singularities of X.. Let us take ~z-dependent C m linearizing coordinates near %(~t), such that '~-~ 0 ~ 0 x~ ~ - Z ~,(~) x, + ~j(~)y~--, ,=I Ox~ ,=l Oy, with u = dim W~(a,(~t)), r = dim[W"(ak(~) ) c~ W""(%+1(~)]. Considering a cross-section S""(~z) C {Yl = 1 } with coordinates (va, v3, Vr, wr.), we get W"(%(~z)) c~ S'"(~z) = {(0, ..., 0, wz)}, W'"(%(~)) = {(vx, 0, ..., 0, wr.)} and Wf"(~,(~t)) = {(va, v3, 0, ..., 0, wL) }. The generic assumptions of (5.A) imply that W*(=k+l(~t)) n S""(~z) = {v3 = F(~t, vt, vi, wL)}, F being a C" function such that F(0) = 0 and F(0, v~, 0, wT,) is a Morse function with critical point at the origin. The condition of generic unfolding imply that From this we obtain the curve F,.,+ 1 of quasi-transversality between W'(a,(~)) and W"(% + ~(~)) by solving the system of equations F(~, 0, 0, wL) = 0, ~ (~, 0, 0 wL) = 0 . The curve of tangency F o between WC"(~k(~)) and W"(%+1(~.)) is given by F(~, v~, O, w~) = O, ~ (~, v~, O, w,.) = O, ~ (~, vx, O, w~.) = 0 ," We may write Fk, k+ ~ ={~z=0} and F o--= ~3=~-~ 9 (We are assuming that 8 3 F 8 3 F OF Ov---~ (0), Ov~ (0) and ~ (0) are all positive.) Although F 0 does not belong to the bifurcation set, it serves as a guide to obtain the other curves along which the family presents a quasi-transversality between W~(a~(~z)) and W'(%+x(~t)), j < k. Let us assume that W"(%(~)) has codimension one (if not, just restrict X~' to W~"(%(~t)) c~ W~*(%(~t))). As is w 4 it is easy to see that if W"(%(0)) is transversal to the foliation F "~ in W'(%(0)), then, for ~z near 0, W"(%(~t)) is transversal to W'(%+~(~z)). Hence, possible tangencies between these manifolds occur near the tangency points p~,(~z) between the unstable manifolds and the foliation F "*, see (5.A). We write the intersection of W"(aj(~)) with a cross-section Se'(~) C { xx = 1 } near p~(0) as a graph x 3 = G~(~, xx,y~,yL ) with x I ~ Gj~(0, x~, 0, 0) being a C "~ Morse function with critical point Xr(p~,(O)). Using this expression and the generic unfolding of the quasi-transversality, we obtain as in w 1 0~-------~ BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 155 a (31 curve F j, such that ~ ~ Fr if and only if X~ presents one orbit of quasi-transversal tangency between W"(a~(tz)) and WS(ak+l(~)). This curve l~i is tangent to F 0 at 0. It follows from the fact that the points of tangency Pr 4' and p~, belong to distinct leaves of P' if (j, i) # (j', i'), that Pr n Us, r ---- O. We stress the similarity between the present bifurcation and the one treated in w 1. In order to analyze all the secondary bifurcations simultaneously, we let F~'+l(tZ ) be a C '~ foliation in S~'(t~), compatible with W'(ak+l(~t)) and defined by ~*+a.~(Vl, v~, vi, wL) ----- (vx, v~ -- F(~, Vl, VI, W) -~- F(~, vl, 0, WL) , WL). Negative satu- ration of F~l(~) by the flow of X~' converges to the foliation F'. Since the tangencies between W~(as(~)) and W'(~k+l(~) ) in S~*(~) occur in the set of tangencies between F~8 t W"(a~(~)) and ~+~k~), we associate to each distinguished point pj~(0) a submani- fold Tji(~) in W~(aj([~))n S"*(~) such that U~=, ..... ~l~j~(~) contains all those tan- gencies. Using the above notation, the submanifold is obtained by solving the system OG~, OF e -'~t WL) e -c'~-'lt 0. Since the 0xI (tz, xx, e -~*t, e -~* w~) ~i (~z, e-'**, xx, . = points {p~(0)} belong to distinct leaves of F ~', the images Ta,(~z) = ~_~.~(T~,(tz)) are disjoint submanifolds ofcodimension one in L*(~z), the leaf space of F~+I(~) with coordinates (vl, vz, w~). All these manifolds are contained in a wedged shape region of the form Iv2 ] -< ~,. Iv1 ["/~' in L*(tz). The tangencies between W~ and W"(%(Vt)) in S*'(t~) correspond to tangencies between W'(,~+~(~t)) and Ta~(~z) in L~(~z) for some i e{ 1, ..., n(j)}. Proceeding as in w 1, we let ~ be a C l foliation of codimension two in L*(vt) which is compatible with W"(,,(~z)) and with all submanifolds Ts~(Vt), for jl>k-- 1 and i= 1,...,n(j). Since W*(~,(~))nL~(~) ={Vl=Vz=0} we may also choose ~ compatible with the " horizontal " foliation Vl = constant. As in w 1, we obtain a two-dimensional C 1 manifold S~(~) of class C 2 outside the origin, which is transversal to W~(,,(~))n L~(~) and to W"(~,+I(~) ) n L*([z). With this process we reduce the analysis of the bifurcation of type VI to the corresponding one for two- parameter families of gradients in a three-dimensional manifold. Thus, the bifurcation set is obtained by analyzing the following situations: a) the point p,(~) = W"(a,(~)) tn S~(~t) belongs to the curve T'(~) = W'(a, + I([Z)) (~ Sg(~t), i.e., W"(a,(~)) is tangent to W*(a,+l(~)), b) the curves T~,(~) ---- T~(~) tn S~(~) and Ts(~) are tangent, i.e. the manifold W"(%(~)) is tangent to W'(a,+x(~) ). The first situation yields the curve F,,,+ 1 obtained above. In the second one we have to consider two non-equivalent cases: a~(0) < 2~x(0) and a2(0) ~ 2a1(0) as in the three-dimensional case analyzed in [22]; if ~(0) = 2a1(0), the family is not stable in general. By parametrizing the curve T'(~) by (Vl, F(~, va, 0, ~,.([z, Vl))) and each curve T~j,(~) by (v~, M~,(~, Vl)) for Vl >/ 0 (or for Vl <~ 0), and letting S~ = U~ S~(~), OF the hypothesis 0~ (0) + 0 implies that the set M~ = {F(~, Vl, ~(~, v,)) M~(~, v~) } 156 M. J. DIAS CARNEIRO AND J. PALLS is a two-dimensional submanifold of S~. Hence, the bifurcation set F~ which corresponds to tangencies between W~(6~+a(~)) and W"(%(~)) is the image of the singular set of the map ~, restriction of the projection =(~, vt) = ~ to M~,. Each P~ is a branch of a C ~ curve which is tangent to the curve F 0 defined by F(~, v~, 0, ~(~, v~, 0)) = 0 = OF 0, 0)) and for (j', i') . (2, i) the branches are disjoint in a neighbourhood of 0. If~%(0) > 20r then all branches of F~ are on the same side of F~, ~+ 1; otherwise one may find branches in both sides. See Figure XI. l'k k-1 Fro. XI (5. C) Local stability. -- Let us construct an equivalence between X~ and a nearby 9 ., k-l~) and a family :~,. We take a compatible center-unstable system F~"(~), . F ~" ' ' stable system F~_I(~), F~+~($), ..., F~(~) for X~. In the discussion of the bifurcation set, we have already observed the similarity between this case and the one in w 1. As in that case the main point to prove stability is to obtain a homeomorphism h~ 8 on the cross-section SC*(~): S(~)nWCs(%+t(~) ) where S($) is a small neighbourhood of the tangency point p~(0) = g ta Bk(~) in a fence Bk(~). We now describe this homeo- morphism, beginning with a reparametrization q~ together with a homeomorphism h~ : S~(~) -+ S~(9(~)), S~($) as defined in (5.B) above. Each curve T~j~(~) = (vt, Mj,(~, Vl)) obtained at the end of (5. B) is a leafofa singular foliation defined by a one-form on S~(~), wj~(~) = -- ~1(~) Vl dv~ § [~(~) va § 0(v~+')] dv 1. Using a partition of unity, we may define a C t one-form w(~) such that restricted to a sector of the form l v~ -- M~(~, v~) I < ~tva {"/~' it coincides with wj,(~); outside the origin, w may be taken of class C ~. We can also take w(~)=- ~1(~)vldv~ + ~2(~)v~dvx for ~{ Vl{~'[~'~< { v, { and assume that the curve To(~) = W~"(~(~)) c~ S~(~) is a leaf of w(~) = 0. The set of tangency points between the curve T'(~) = W'(cr~+ 1(~)) ~ S~(~) and the leaves of w(~) = 0 is described by an equation OF -- ~1(~) 731 ~O 1 (~, Vl) ~- 0~1(~) F([~, ~71) -3u r([s ~)1, F(~, ~)1)), OF where F is as in (5.B), F is C 1 and T'(~) = graph F(~, .). Since ~ (0) . 0, this is the graph G of a C t function ~ = ~(~, vt) (which is even C ~ outside the origin), whose projection ~ : G -+ R a has a fold singularity along a C ~ curve ~. Thus, there exists a BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 157 homeomorphism a:G ~ G with fixed point set % such that r~(~l, v~) = n(~q, vi) if and only if b-~ = a(~t~, v~). The image of 9 by r~ is a C ~ curve tangent to I~,,+~ at the origin. Moreover, if M 0 = { F(~t, vi) = 0 ) represents the intersection of the curve T*(ti) with T0(~t), then -r is transversal to M 0 c~ T at the origin. Since all curves T~,(~) are tangent to T0(~t ) at the origin, the curves A~, =- M~, c~ G are also transversal to -r at the origin, where M~ = { F(t~ , vi) -- M~(~, v~) = 0 } = T'(~) n T~.~(~). Let A ~ be the foliation in G defined by the pull-back of o~ by the map ([11, ~)1) ~ ([11, [12([11, Vl), Vl, F([ll, [1~, ~)1), The leaves of A*, except two of them, are tangent to A~ ---- M 0 n G. We can take a homeomorphism from the space of leaves of A c to the space of leaves of ~c, sending A~ to As~ and A~ to .~. Let A C G be a closed conic region, with vertex at the origin, which contains all the distinguished leaves A~i and A~, intersects v at the origin and whose boundary is transversal to A c. This region A is taken so that v n A 4: O and also 7:-l(Fk, k_l) t% A = 0. We let q~(~x, K2) = (qh(~q), ~2(~i, ~2)) be a reparametrization that sends the n-image of the curves A c contained in A to the ~-image of ~e, ~(~) to ~(~) and F,,k +~ to Dk, k + 1" This induces a homeomorphism ~ : G --~ G by sending A to A, preserving the foliations A*, ~e and ~i = constant, and, by using the involutions a, ~, in such way that q~ o ~ ----- r~ o ~. By preserving the surfaces M e that represent the inter- section of Te(v~) with T*(~), we already have a continuous family of homeomorphislns v x ~ ~(vi) in the set ] F(~z, v~)[ ~< ~ [ v I 1. They are extended continuously outside this region by performing an extension on each fiber ~ = constant. This gives a homeo- morphism on the space of leaves of the foliation dvi = 0 in S~(~). The homeomor- phism h~ : S~(~z) -+ in the conic region [ v, [ ~< 8 a [ v x [ preserves the foliations dvl = 0 and T"(~). Also h~ automatically sends T*(~t) to ~'~(~?(~t)). We extend h~ arbi- trarily outside the conic region but preserving T*(~t). We now extend h~ to the tangency submanifold L~(~)C Se*(~t) (see (5.B)). This is analogous to the construction used in w 1; the difference, due to the tangency between W'(%+l(~t)) and WC"(a,(V)), is that we need a new process to define a two-dimensional foliation (SN)~ : like in (1 .C), this foliation has a saddle-node singularity along the curve T*([z). Since we preserve the foliation given by dvl = 0, we may define (SN)~ using once more a family of vector fields Y~,,,, now also parametrized by vi, which is compatible with W'(a,+l(~t)). For fixed (~t, vi) , Y~,,, presents a unique singularity of saddle-node type at { T'(~)c~ (vi = constant)}. Outside this point, the trajectories of Y~,,, are transversal to F~, (5.B). Hence, by applying a parametrized version of Lemma 1, we obtain a homeomorphism from Le(~t) to ~'e(~?(~)). We can now define a homeomorphism on a neighbourhood S([z) of the tangency point p,(0) = g ~ B,_~(~) in the fence B~_l(~). Since W~(~+i(~) ) is transversal to W~(~(~)), the cone- like method of Theorem A is applied to obtain a homeomorphism on = n 158 M. J. DIAS CARNEIRO AND J. PALIS which preserves the center-stable foliation F"'(ak+l(~z)). Hence, as in w 1, it is enough to construct a homeomorphism h~ * on S*'(~) -~ S(~) n ~ k+ik~)). We already have a homeomorphism on the space of leaves of Fff+~(~), a foliation of dimension (s -- 2), s = dim W'(cr,(~)), which is compatible with W~(ak+l(~)). It remains to construct a suitable center-unstable foliation F~(a,(~z)) and to adapt the cone-like construction to this case. (We recall that in the previous applications of this method (Theorem A) the foliation F~', dual to F~(ak(~) ) in S~*(~), had dimension equal to (s- 1).) We describe the (u + 1)-dimensional leaves of type F~(c~,(~z)) whose space of leaves cor- respond to closed discs D~(~z) in the fundamental domain A~0z ) = Ak(~z ) c~ W'(~,(~)), = ~ ,+1,,) (L*(~z)), where ~,+i.~ is the A~(~) being a fence in a level set. Let Y~(~z) f~'~ -~ "* projection along F~,~_l([Z), and consider P~':~(~) --~ A,([z) the restriction of the Poin- card map to ~([z). Using the linearizing coordinates and the fact that F ~* is of class C '~, it is easy to see that the image of this foliation is a codimension-one foliation in P~*(~(~)), which extends continuously to the strong stable foliation F~'(~) in the discs D~([z). We raise over each point of D~:(~) a one-dimensional foliation FI(~) in P~*(fl(~)), which is compatible with the center-unstable system and also has its inverse image by p~8 (~,+1,~) (T (~t)). We then raise over Fl(g. ) a u-dimensional continuous compatible with ~' - ~ r foliation F~(~) also compatible with the stable system and transversal to D~:([z). We define F~"(a,([~)) by taking the positive saturate of F~([z) by the flow of X~. The (u + 2)- dimensional leaves of type F~n(a,(Iz)) are obtained as in Theorem A, its leaf space is a sphere A~S(~t) W~'(~,(~t)) c~ A,(~). However, to avoid tangencies between ,+lk[~) and F~(a,(~)) in S~S([z) we go one step further and distinguish a new type of leaves, denoted by F~"(r which are (u + 3)-dimensional. Let Cff(~z) be a small tubular neighbourhood of the sphere A~*(~z) = W~'(a~([z)) c~ A~([z) in A~([z), which is bounded by two leaves of Fff(~)c~A~,*(V). Using the transversality between W~(%($)) and W~(~,(~z)), we can construct a one-dimensional foliation F~(~z) on (]~,'(~z) which is compatible with the center-unstable system. We let F~(a,(~z)) be the foliation whose leaves are of the form [-],~r F ~,,(,([z)), ~u where t~([z) is the leaf of F~([z) containing x ~ A~'(~). We construct homeomorphisms on the space of leaves of F~'(%([z)) and of F~(~,($)) and apply Lemma 2 to get a homeomorphism on the space of leaves of F~'(a,(~t)). In this case we need Lemma 2 in order to preserve F~,'([z). With these homeomorphisms together with the foliation F~,~_ ~([z) we obtain h~' as follows. We divide S~'($) into three conic regions: and C(~) = { 012 ~< 8[/)~ Mr_ [ VI [~]} ('~ { 8 [ V I ]2 ) [ V~. 19~ }, with ~> 0 small. On A0x) we preserve F~"(z~(~)) and Fff+~(tz); in each leaf of (r~+l,~) ( (tx)) these foliations are complementary. On BOx ) it is defined by pre- serving the complementary foliations F~"(z~(~t)) and Fff+~. On C(~) we preserve F~"(z~(~z)) and Fff+~(t~). Let F~,~(z~(~)) be a leaf of F~(z~(tx)). The intersection of F a,~(,(~z)) ~ ~ with OC(~) projects homeomorphically, via n*' ~+~(~), ~ " to L~(~). Hence, it BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 159 defines a homeomorphism on the leaf space of the leaves of type F~"(%(~z)) which are contained in F~a((rk([z)). So, h~* restricted to F~',d(%(~))\Int(C(~z)) preserves F~"(a,(~t)) and F~_~(~t). We then extend it to Int(C(~t))c~ F~.~a(~rk(~t)) arbitrarily but preserving F~.'+~(~). The definition of h~ ~ is now complete. As observed above, this is enough to obtain a homeomorphism on S(~x). We extend this homeomorphism to all of Bk(~z ) pre- serving the center-stable system and F~"(ak(~t)). By reasoning as in Theorem A we obtain an equivalence on a neighbourhood of the closure of the orbit of tangency -( preserving level sets off~. Proceeding by induction, we construct the equivalence on distinguished neighbourhoods of the singularities, i >/ 1, also preserving level sets off~, as it is done at the end of w 4. We conclude the result by extending these equivalences to all of M as in Theorem A. 9 w 6. Orbit of tangency of codimension two (6.A) Generic assumptions. -- We consider in this paragraph a family X~, such that for a value ~ there is a unique orbit -( contained in the intersection of an unstable manifold W"(p(~)) and a stable manifold W~(q(~)) of two hyperbolic singularities of Xg such that dim[T, WU(p@)) +T,W"(q(~))] -----n--2 for r~, (dimM =n). We assume X~, to be C"* linearizable, m t> 3, nearp(~) and q(~) and that the eigenvalues of the linear part of X~ have multiplicity one at these points. We also assume that for any hyper- bolic singularity a(~) 4: p(~) the unstable manifold W~((r(~)) is transversal to W'(p(~)) and to W'(p(~)), the strong-stable manifolds of p@) of codimension one and two. We suppose that W"(a(~)) has at most a quadratic contact with the very strong-stable foliation F~(p(~)) in a leaf L of the strong stable foliation F'(p(~)) (see w 5). Dually, we require transversality between W"(cr'(~t)) and W~"(q(~)) and between W"(~'(~)) and W*"(q(~)) and quadratic contact between W"(a'(~)) and F~,~(q(~)), the very strong- unstable foliation, in a leaf L' of the strong-unstable foliation F""(q(~)). Let W~'(p(~)) and W~"(p(~)) be (u + 1)- and (u + 2)-dimensional C "~ center-unstable manifolds of p(~) (u = dim W"(p(~))); then, W'(q(~)) is transversal to W["(p(~)) and dually W"(p(~)) is transversal to W~"(q(~)), a C ~ center-stable manifold of dimension (s q- 2). We also suppose that W"(p(~)) is transversal to W~'(q(~)) and the generic unfolding of the orbit of tangency y. This means that if ~", ~" :R ~ -+ M are immersions with ~(~) = a~(~) = r e y and ~'(~t) C W~(q(~t)), a~(~t) C W~(p(~t)), then the restriction of the projection T, M ~ T, M/T, W"(p(~)) + T,W'(q(~)) to Im[d~*(~) -- da~(~)] is an isomorphism. Actually, a generic family X~ presents an orbit of tangency of codi- mension two when there is lack of dimensions, that is u-t-s = n- 1. Proposition 5. -- There is an open and dense subset ~' C ;(g(M) such that if X~ e fr and for some value ~ the vector field presents an orbit of tangency y C W~(p(~)) n W"(q(~)) with dim[T, W"(p(~)) + T, W'(q(~))] = n -- 2 for r ~ V, then dim W"(p(~)) § dim W"(q(~)) = n -- 1. 160 M. J. DIAS CARNEIRO AND J. PALLS Proof. -- Let u = dim W"(p(~)) and s = dim W~(q(~)). We take ~x-dependent C m coordinates (Xl, ..., x,_,,yl, ...,y,_~, z) (m >>. 3) in a neighbourhood U of r in M such that X~ [ U = 0z and W"(p(~t)) ~ U = { x~ ..... x,_, =- 0 }, W~(q(~x)) t~ U = { x~ = F~(~x,y~, ...,yt), x, = F,(~x,y~, ...,Yt),Yt+I ..... y~_x = 0 }. We are assuming g/> 1, where / = dim[T, W"(p(~)) n T, WS(q(~))] -- 1. Hence, we can associate to X~ a two-parameter family of C" maps F:ll* � R t -+R ~, F([~,yx, ... ,Yt) ----= (FI(~,Y~, ... ,Y~), F~(~x,y~, ... ,Yt)). Ifj~ F(~,y) denotes the one-jet with respect to the variables (Yl, .-.,Ye) =.~, then dim[T, W"(p(~)) + T, W"(q(~))] = n -- 2 is equivalent to j~ F(~, 0) = (0, 0) e R ~ � L(R t, R ~) ~ J~(R t, R~)(g,0). But, since t/> 1, we have dim(R ~ � R t) = 2 -k- t < 2 -t- 2t = codima,(lr ~,)(0, 0) and the transversality theorem implies that (0, 0) is generically avoided. That is, with a small perturbation of F we get j~ F(~, 0) 4= (0, 0). This proves the proposition. 9 (6.B) The bifurcation set. -- Assume that p(~x) = ,~(Ex) and q(~) = ,k+l(~x) in the usual ordering of the singularities of X~. Let us describe the bifurcation set associated to tangencies between W"(,j(~)) and W"(%+1(~)) for j ~< k -- 1 and between W"(ej,(~)) and W~(*k(Ex)) for j' t> k + 2. It is easy to see, as in w 5, that these tangencies cor- respond to criticalities of W"(,i(~) ) (resp. WS(er(~))) with respect to F~S(%(~x)) (resp. F~(*k+l(~))). Let W~'(%+1(~)) be a (s + 2)-dimensional center-stable mani- fold of %+x(~) extended as in w 1 to a neighbourhood of %(~x), and consider the restric- tion X~' = X~I W~'(*k+l(~)). Assume that there are C" linearizing coordinates (Xl, x2, xi,Yl) for W~S(,k+l(Ex)) near ek(~) such that "-~ 0 0 X 2 = -- Z ~,(~) x, + ~I(~)Y~-- (u = dim W"(%(~)), 0 < ~,(~) < ~+~(Ex), [3~(~) > 0). In a cross-section Z~' contained in {yx = 1 } and with coordinates (v~, v2, vi) such that W"(%([x)) tn ]s ---- {(0, 0, 0)}, W~"(~;k(~)) t3 Z~ s ---- {(vl, v,, 0)}, we have a = { = = with FI(~, 0) ---= F2(~, 0)-=--0. The hypothesis of generic unfolding of the family X~ implies that the map ~z w, (FX(~, 0), F~(~z, 0)) is a local diffeomorphism near ~ and, hence, after a change of coordinates in the parameter space, we may assume ~ = 0, F~([x, 0) = ~ and F~(~z, 0) = ~tz. To "reduce dimensions ", we consider the C" folia- tion F~+a(~) in Z~' whose main leaf is WS(ffk+l([.~))("I Z$, and which is defined =~, t FZ(~t, v,) -k- V~(~, 0)]. Let by ,+a~, vx, vz, v~) = [v~ -- Fa(~, v,) -t- F~(~, 0), v z -- [J~<,~<,(~)?,~(~) be the image in W~"(%(~))t% Z~' of the set of points of tangency between F~'+x(~t) and W"(~(~)). Each ~(~) is a branch of a C ~ curve tangent to the v~-axis and corresponds to the distinguished point P~i(~) in the cross sections BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 161 S~(~) C {[ x x [ = I }. As in w 5, pa([,) is a point oftangency between W"(%(E*)) and F"(E*) in W*(%(k)) c~ W'(%(bt)). Since these critical points belong to distinct leaves of F'(~,), we obtain X~.(~) c~ X~.,,,(~) = 0 for (i,j) =t= (i',j') and for tz in a neighbourhood of (0, 0). These curves are contained in the region AI(~z ) = {] v 2 ] ~< ~ [ v x [~" =' } and are tangent to a vector field Z~ which has a hyperbolic singularity at the origin and is equal to O O --%(~t) vl Ova =2(~*) ve ~-v~v, outside A~(b~). It is clear from this construction that an orbit of quasi-transversality between W~(%(Ez)) and W'(%+1([,)) will occur in Z~' if and only if the point rk+l(~)=W'(%+x(~))c~W~(%([,))c~X~ ' belongs to the curve X~(V), for some 1 <~ i <<. n(j). We now apply the same reasoning to X~= X~ [ W~(%(~)), by taking a codi- mension-two foliation F~"(E, ) in a cross-section ]By, near the singularity %+~(t*), having W*(%(b0) n Z~" as a distinguished leaf. In this way we get a vector field Z*+l_~ on Z~c~W~'(%+~(~)) with distinguished trajectories X~.(~), j>~k+2, such that W"(%(g.)) is quasi-transversal to W'(%(~,)) in Zy if and only if r~(~t) = W~(%([z)) r~ lii~uc~ W~'(%+~(lz)) belongs to X~(b~ ) for some l<~i<<.n(j). W~ 1(~)), we are reduced to Therefore, by taking X~ = X~ I X~'(%(V)) r~ 1 ~ ,+ consider the three-dimensional case with the corresponding singular foliations in the cross sections Y.~(~) = ~*'(~) n W~(%(~)) and ~;~(~) ~ Y.~(~) t~W~'(%+~(~)). Let P~ : Z~(~) -> Z~(~) be the Poincard map, P~(v~, re) = (P~(~t, v~, re) , Pg(~, Vl, v2)), -~-- 7/r+ i and consider the induced field (P~)~-~ ~.~+~_~ _~Z t+~. Since the integral curves of _~ W ~ ~ 0~ (except for two of them) are tangent to (~+t(~)) ~ 2;~ , using the transversality between W~(%(~)) and W~'(%+~(~)) and restricting Z~ and ~.~+~_~, to the regions and A2([*) ---- {I P~(~*, v~, re) [ 4 a [ P~(~t, v~, re)l, I P~(b t, v,, re) [ ~ r } for 0 < 8 < 1 small, the trajectories of Z~ and of ~.*+*_~, are transversal to each other for [z close to 0. In the parameter space we obtain the corresponding regions B, = {[ Fe(j, , 0)[ ~< 8 [ F~(~,, 0)[, [ F,(~,, 0)[ ~< ~ } and B e ={] P~(~z, 0)l ~< a I P~(~,, 0)], } P~(~, 0)l .< ~}, which contain the bifurcation set. Note that B, n B e = {(0, 0)} and the bifurcations are characterized by the fact s n~e '~b + 1 that (0, 0) belongs to the integral curve X~(~,) ~ _~ , or r~+,(~,) belongs to the integral curve X~t(bt) of Z~. Since the map ~z ~-~ r~+~(~,) is a local diffcomorphism, we get a finite number of integral curves r~ of (r~-~ ~). Z~ in the region B~. Similarly, we obtain finitely many trajectories P~ of (P.~)-* ~+*_~ in the region B e. Thus, the bifurcation set is as in the picture. 21 162 M. J. DIAS CARNEIRO AND J. PALIS ~p.u. Fro. XII (6.C) Stability. -- The construction of an equivalence between X~ and a nearby family X~ is similar to the one in w 5 and we will describe only its main steps. The impor- tant point is to obtain a homeomorphism on a cross-section E contained in a distinguished neighbourhood of the orbits of tangency y and ~. For that, we first obtain a homeo- morphism h~:Z~(Ez ) -+,~(q~(~t)) between cross-sections in the center manifold W~"(cr~(~t)) nW~(~k+l(~t)) and W~(~(~(~t)) nW~(~k+l(q~(~t)). We start by taking a homeomorphism from 0Ax(0) to &~l(0) which sends X~(0) n 0Al(0) to ~,~(0) n &~l(0) ; this induces (via r k + 1 defined above) a homeomorphism from ~B 1 to 0B 1. We also consider a homeomorphism from ~A~(0) to &~(0) sending X~(0)n 0Az(0 ) to X~.~(0)n OA~(0), which induces a homeomorphism from 0B 2 to OB2 (via P~ defined above). Let q~ : (u, 0) ~ (q~(u), 0) be a homeomorphism sending B~ to B~, i = 1, 2, that extends the above homeomorphisms and preserves the trajectories of the fields (r[_~), Z~ and (p,e)-I ~+1 The homeomorphism hi: E~(~t) ~Z'(~(~t)) is defined in such way --I/, * ^le that it sends trajectories of Z~ to trajectories of Z,I~ inside Al(~) and .~l(q~(V)), respec- q,+l inside Bx([z) and B~(q~([z)). We tively, and trajectories of Z *+~ to trajectories of ~,~ -.~ choose h~, to send X~,([z) to X~.i(~(~t)) and Z~.(~z) to X~i(q~([z)). Since Z'([z) is the space of leaves of F~_l(~z), in order to define a homeomorphism k~': Y."~(~) ~Z~'(q~(~t)) it is enough to define a center-unstable foliation F*"(cr,(~z)) as in w 5 and proceed exactly like in that case. It is important to observe that, by construction, the pull-back of the trajectories of Z~ via the projection 7r~(~t) gives a codimension-one foliation, singular along [rc~+~(~z)]-~[W"(~([z))n Z~(~t)], such that the foliations in the cross-sections r \ C {1 xt I -- 1 } induced by the Poincar6 map, extends continuously to the very strong-stable foliation F"'(%(~t)). We apply Lemma 2 again, to obtain a homeomorphism in the space of leaves of the center-unstable foliation. The same procedure also works to define a homeomorphism h~":Z~"(~) ~ Z'"(q~(~)) preserving the foliation F~*(~z) BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 163 and a center-stable foliation F"'(% + t([x)) as above. Finally, to get a homeomorphism h~ from Y.([x) to ~(?([x)), we take a u-dimensional continuous foliation G~" in ~(~) with C 1 leaves transversal to W~(e,(~)) n Z(~), which extends F~'+I(~ ) and is compatible with F**(a,+~(~))c~ Y~(~). Dually, we take an s-dimensional foliation G~ ~ in Y~(~x) which extends F;"(~t) and is compatible with F~(a,([x))c~ ~(~). Similarly, we define G~' and 0~, ~ in the section Y.(~x). Since u + s-----n- 1 and we already have defined the homeomorphisms h~' and h~" in the space of leaves of these foliations, the homeomor- phism h~ :~(~) ~ Z(?([x)) is defined by sending G~ to G,(~) and G~" to G,(~). The extension to the distinguished neighbourhood of the closure of the orbit of tangency y and to the whole manifold M is exactly like in w 5. w 7. Remaining cases: two saddle-nodes and codimension-one and two singularity In this paragraph we finish the proof of local stability for the codimension-two bifurcations by analyzing the two remaining cases. (7. A) Two saddle-nodes. -- We assume that the family X~ has a bifurcation value where the vector field Xg presents two saddle-nodes p(~) and q(~). We assume the existence of C", m/> 3, linearizations transversally to center-manifolds of p(~) and q(~) and transversality between all unstable manifolds and the strong-stable manifolds W88(p(~)) and W'S(q(~)) and between all stable manifolds and the strong-unstable manifolds W~(p(~)) and W"'(q(~)). The saddle-nodes unfold generically and do so independently. Hence, ~ belongs to the transversal intersection of two C 1 curves F 1 and F 2 with ~ E P 1 if and only if X~ presents one saddle-node near p(~) and Ix belongs to 1" 2 if and only if there is a saddle-node for X~ near q(~). In a neighbourhood U of the bifurcation set is the union of P 1 with P~. Let us prove the local stability of X~. Suppose, in the usual ordering of the singu- larities of X~, that p([x) = ~,i(~) = ~+1(~) for ix e I' 1 and q([x) ---- ~k(~t) = ~k+~(~) if ~x e P~. We have two possibilities: there is an intermediate singularity < < or not. We will construct an equivalence between X~ and a nearby family X~ for the first case; the second case is simpler and can be derived from the first. We begin by considering a reparametrization q~ : (U, ~) -+ (R ~, q~(~)) that sends P, to Px, F~ to P~ and it is defined so that there are conjugacies between X~ restricted to the center- manifolds Wc(p(~t)) and WC(q(~)) and X~(~I restricted to WC(p(q0(~))) and W~(q(q~([x))). We then consider a compatible unstable system 1~(~), ..., 1~_~(~) and construct a center-unstable foliation F~"(~) which is compatible with this system and has a center- unstable manifold W""(ak([x)) as its main leaf (see w 3). Since the singularities of F~(~) occur along C 1 manifolds which are transversal to alI intermediate manifolds W"(~j(~t)), 164 M. J. DIAS C, ARNEIRO AND J. PALLS we can proceed as in w 2 to get an unstable foliation 1~,(~) which is compatible with the system F~(~),...,I~,_I(v),F~"(~),...,F~_I(V) for k+Z<j<~i--1. Now we construct a compatible strong-unstable foliation 1~'~"(~) whose space of leaves is a center- stable manifold W*'(a~(~)). Dually, we let F~+~(~), ..., F~(~) be a compatible stable system, and construct a center-stable foliation F~*(~). Actually, the procedure here is not quite dual since we are going to use the strong-unstable foliation F~'"(V) as part of a system of coordinates near cr,(~). To do that, let K+(~) be a closed disc contained in a leaf of the strong-stable foliation F~'(~t) inside WC"(~([z)) and let V~(~) be a cross- section of the form V~(~) ---- [-J,e,r~r F~,~(~), where F~(~) is the leaf of F~" through x. Part of the leaves of F~'(~) is obtained by negative saturation by the flow X~, t of an W 8s o" si-dimensional continuous foliation F]*([~) in V~(~) (s~ = dim (i(~))), topologically transversal to We~(~5(~) ) c~V~*(&) and compatible with the stable system. The other leaves of F~'(~) are obtained exactly as in w 3. The process to construct an equivalence is now clear by previous arguments, but we briefly describe it as follows. We begin with a (compatible) family of homeomorphisms hg(~) : W"(a,(~)) -+ W*(a,($(~))), j = 1, ..., k -- 1. It induces a homeomorphism in part of the space of leaves of F~"(~), which can be extended to all of Wr by first extending it to a fundamental domain D~(~) t3 CZ(~) and then to all of W*"(~r,(~)) by preserving the strong stable foliation and the inter- sections of F~"(9) with Wr Next, we consider successively fundamental domains D](~) of W'(a~(~)) for k -t- 2 ~< j~ i -- 1 to get (compatible) homeomorphisms in the space of leaves of the unstable foliations F"(~r~(~)). We finally reach the domain D~(~) ~ C~(~), corresponding to the space of leaves of the strong-unstable foliation 1~"(~). Here, again, the equivalence restricted to the center-stable manifold W~"(~(~)) is a conjugacy preserving the strong-stable foliation F~*(~). Proceeding dually, a family of homeomorphisms h~'(~) : W"(a~(B)) ~ W"(8~(?(~))), for j = i + 2, ...,t, gives rise to a homeomorphism in the space of leaves of the center-stable foliation Ff"(~) and the equivalence near ai(~) is obtained as in w 3 by preserving the complementary folia- tions F~'"(~) and Ff*(B). We now extend this equivalence to a neighbourhood of each singularity ~(~), forj = k + 2, ..., i -- 1, by using the procedure say ofw 2 to construct compatible stable foliations F]~+~(~), ..., F~_l([Z ) and homeomorphisms in the space of leaves of these foliations. The equivalence in these neighbourhoods is a conjugacy preserving stable and unstable foliations. Proceeding by induction we reach the saddle- node ~,(~). We then construct a strong-stable foliation F~*(~) compatible with the stable system and extend the equivalence to a neighbourhood of ~,(B) by preserving F~(~) and F]~'(B). For the extension of the equivalence to all of M, we proceed as in previous paragraphs, concluding the proof of the local stability of this case. 9 (7. B) Codimension-one or two singularity. -- We consider here a family of gradients X~ such that the vector field Xg presents exactly one non-hyperboIic singularity r We BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 165 suppose that 0 is an eigenvalue of dXg(a(~)) of multiplicity one. Therefore, there exists a center-manifold WO(a(~)) containing e(~), which is of class C '~, m large (see [10]). It is well known that transversal to W~(~(~)) there are unique strong- stable and strong-unstable manifolds W~"(a(~)) and W"~(a(~)). We assume that Xg] W"(a(~)) ----- [(x * -k 0(] x I~+a)] 0 (i.e. the germ of X~ has finite codimension) and that the family X~ unfolds generically the singularity ~(~). This means that the potentialf~ is a versal unfolding offg. In addition, we require that all stable and unstable manifolds are transversal, and for each singularity a'@) its stable and unstable manifolds are transversal to W""(a(~)) and W~"(a(~)). These assumptions imply that there are no secondary bifurcations and, hence, the bifurcation set of X~ near ~ coincides with the catastrophe set off~. That is, it coincides with the set of values ~ such that f~ presents a degenerate critical point. In particular, let us consider ~ e llt~, ~ = 0 and k = 3. We then obtain the cusp-family which is equivalent to f(~, x,y) = x ~ -k ~-~ x ~ + ~xz x § Q(y) in a neighbourhood of the bithrcation of type IX described in Section I, and the bifurca- tion value ~ = 0 represents two collapsing saddle-nodes. Theorem. -- Let X~, be a family in X~(M) which unfolds generically a non-hyperbolic singularity of type IX as above. Then X, is stable at -~. Proof. -- We will actually proof that if X~ is a d-parameter family of gradients which unfolds generically a (k -- 1)-codimension singularity such that 0 is an eigenvalue of mul- tiplicity one ofdXg(a(~)) and d >/k -- 1, then X~ is stable at ~. For simplicity, we suppose ---- 0. From the theory of singularity of functions [7], if X~ is a nearby family with associated potential 9r there is a local diffeomorphism of the form [r ~(~, z)] defined in a neighbourhood of (0, a(0)) in R a > M such that at~ o ~(~t, z) =f~(z). Moreover, iff~(~, x) is the restriction off~ to the central manifold W~ then there exists a C "-2 diffeomorphism of the form [+(~t), ~([x, x)] such that 1r f:o W,z) + z Hence, since Wr is one-dimensional, in this new ~-dependent coordinate we can k--1 write X~ I W~(~(~)) ----- -~(~t, x) [(k + 1) x k + Z ~ x~-l], where -~(~, x) is a positive i~l C ~- ~ function defined in a neighbourhood of (0, ~(0)) in R ~ x M. Now, extending to all of R d X M so that 9 = 1 outside a neigbourhood of (0, ~(0)), we define a new family of vector fields y~ __ _1 X~ which is equivalent to X~. By performing the same construction for X~, we define a family ~r which is equivalent to X~ and is such that #Z~ [W~(a(~)) and Y~ ]W~'(a(Vt)) have exactly the same expressions in the respective coordinates s and x. Therefore, by taking h*(tx, x) = s we obtain a conjugacy between 166 M. J. DIAS CARNEIRO AND J. PALIS Y~ [ W~ and ~z [ W~(~(~)). We can now proceed exactly like in Theorem A of (]hapter III of [15] to extend h~, to a conjugacy h~ : M -+ M between Y~ and Y,~. In this way we obtain an equivalence between X~ and X,~, concluding the proof of the stability of X~. 9 Section IV. -- Globalization In Sections II and III we have obtained a finite number of open and dense subsets of ?(~(M), each one corresponding to the cases described in Section I, with the property that every family X~ contained in their intersection ~/1 is locally stable at every value of the parameter. Suppose now that ~x varies on a fixed closed disc D in R *. From our analysis of the bifurcation sets in previous sections, it follows that there exists a subset ~/C ~/1, also open and dense, such that there are no codimension-two bifurcations on 0 D and the curves that represent the codimension-one bifurcations are transversal to 0 D. Hence, for X~, ~ ~/1, the codimension-one bifurcations occur on isolated points in 0 D and there is a finite number, say r, of codimension-two bifurcation values in the interior of D. For each 1 <~ i ~< r, let D e be a small closed disc transversal to all branches of codi- mension-one bifurcations and containing a unique codimension-two bifurcation value in its interior. Let D* ---- D -- Ui Int D e. For X~ ~ ~tl, the intersection of the bifurcation set with D* consists of the union of a finite number of closed (]1 simple curves or intervals, I'~, ..., I',,, each one corresponding either to a saddle-node or to an orbit of quasi- transversality. We denote by ~1, -.-, ~, the codimension-two bifurcation values of X~ inside D. From the local stability of X~ at ~, there exist open neighbourhoods V of X~ in ~t 1 and U, of ~, in D such that any family X~ in V is equivalent to X~ for [x ~ U,. We are now going to piece together these equivalences. To do this, we take for each i= 1, ...,r a smooth function p~:R ~-+R such that 0~< p~<~ 1, supp(p~)CU~ and p~ = 1 in a closed disc D~ centered at ~, and define perturbations R~ = grad~ f~ where the metrics g"~ and the potentials f~ are defined inductively as follows: fr + Pl( ) = + - and f~ =f)-i + p~(bt) [f~,--a~ g~, = gr -t- p,([x) [ff~- gd-x]. Hence, )~[~ = ~2X -~ for ~x r Ui, X x = X~ for Ix r U~ =~ U, and R~ = ~2~ for ~ e U~=t D~. Using the remark concerning local stability made after the proof of Theorem A, and which applies to all bifurcation cases in Sections II and III, we obtain that X~ is equi- ^i--1 valent to X,i~ ~ with the reparametrization q~, satisfying q0~(~) = ~ for ~ r U s and the equivalence h~ = identity for ~x r U~. Therefore, by transitivity, X~ is equivalent to X' Now, let I'~ be the first curve ofcodimension-one bifurcation in D\U~.= 1 D~. We cover r~ by a finite number of domains of reparametrizations, UI, " " ", U}~, and starting with X" [L, define perturbations X~ , k = 1,...,t~, along I'~, as above, such that X~'*---= X~ BIFURCATIONS AND GLOBAL STABILITY OF FAMILIES OF GRADIENTS 167 for tz r U~= 1 U 1 and X~'k = X~ for ~z e U~=~ D~, where D] is a closed disc inside U ] l x 1 and [-J~=l D~ D Fx. Performing again the modifications referred to above of the equivalences inside each domain U~, we show that the family 1'k-1 is equivalent to the family X~,k with an equivalence which is the identity outside U~. In this way, starting with the equivalence between )(~ and )~' 1 and proceeding by induction, we construct an equivalence between X~ and x~,tl in a neighbourkood of F 1. It is now clear that by covering each curve Fj, j = 1,..., m, with domains of reparametrizations U~,..., UtS~, we obtain inductively an equivalence between X~ and X~' tm (and therefore between X~ and X~) in a neigh- bourhood W of the entire bifurcation set in D. It is important to observe that all the reparametrizations that we perform preserve ~ D. Finally, we repeat the same procedure in each component of D\~u thus achieving a global equivalence between X~ and X~. The proof of the main theorem in the paper is complete. 9 Remark. -- In the arguments presented above, the closed disc D can be replaced by any compact surface as the parameter space. REFERENCES [1] V. I. ARNOLD, S. M. A. N. Singularities ofdifferentiable maps, vol. I, Birkh/iuser, 1985. [2] T. BR6I~SR, Differentiable germs and catastrophes, Cambridge University Press, 1975. [3] J. GUCKENHEIMER, Bifurcations and catastrophes, Dynamical Systems, Acad. Press, 1973, 95-109. [4] B. A. KHESIN, Local bifurcations of gradient vector fields, Functional Analysis and Appl., 20, 3 (1986), 250-252. [5] N. H. KUIPER, Cl-cquivalence of functions near isolated critical points, in Syrup. on Infinite Dim. Topology, Ann. of Math. Studies, 69, Princeton University Press, 1972, 199-218. [6] R. LABARCA, Stability of parametrized families of vector fields, in Dynamical Systems and Bifurcation Theory, Pitman Research Notes in Math. Series, 160 (1987), 121-214. [7] J. MARTINET, Singularities of smooth functions and maps, Lecture Notes Series, 58, London Math. Society, 1982. [8] J. MARTINET, Ddploiements versels des applications diff6rentiables et classification des applications stables, in Lecture Notes in Math., 535, Springer-Verlag, 1975, 1-44. [9] J. MATH~R, Finitely determined maps germs, Pubt. Math. I.H.E.S., 35 (1968), 127-156. [10] 0. E. LANFORD III, Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Wakens, in Nonlinear Problems in the Physical Sciences and Biology, Lecture Notes in Math., 322, Springer-Verlag, 1973, 159-192. [ 11] S. NEWHOUSE, J. PALIS and F. WAKENS, Stable families of diffeomorphisms, Publ. Math. L H.E.S., 57 (1983), 5-71. [12] R. PALAXS, Local triviality of the restriction map for embeddings, Comm. Math. Helvet., 84 (1962), 305-312. [13] R. PALAIS, Morse theory on Hilbert manifolds, Topology, 2 (1963), 299-340. [14] J. PALLS, On Morse-Smale dynamical systems, Topology, 8 (1969), 385-405. [I 51 J. PALIS and F. WAKENS, Stability of parametrized families of gradient vector fields, Annals of Math., 118 (1983), 383-421. [16] J. PALIS and S. SMAIm, Structural stability theorems, in Global Analysis Proceedings Syrup. Pure Math., 14, A.M.S., 1970, 223-231. [17] S. STERNBERG, On the structure of local homeomorphisms of euclidean space II, Amer. Journal of Math., 80 (1958), 623-631. [18] F. TAKENS, Moduli of stability for gradients, in Singularities and Dynamical Systems, Mathematics Studies, 103, North-Holland, 1985, 69-80. VARCHENKO, GUSEIN-ZADE, __~<A 168 M. J. DIAS GARNEIRO AND J. PALIS [19] F. TAr~NS, Singularities of gradient vector fields and moduli, in Singularities and Dynamical Systems, Mathematics Studies, 108, North-Holland, 1985, 81-88. [20] F. T~a~NS, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147. [21] R. TI-IOM, Stabilitd struaureUe et morphogen3se, Benjamin, 1972. [22] G. VZOTER, Bifurcations of gradient veaorfields, Ph.D. Thesis, Groningen, 1983. [23] G. VEOTER, Global stability of generic two-parameter families of gradients on three manifolds, in Dynamical Systems and Bifurcations, Lecture Notes in Math., 1125, Springer-Verlag, 1985, 107-129. [24] G. VEOTER, The C~-preparation theorem, C~-unfoldings and applications, Report ZW-8013, Groningen, 1981. Departamento de Matem~itica- I.C.E.X. Universidade Federal de Minas Gerais Belo Horizonte, M.G. Brdsil Instituto de Matem~itica Pura e Aplicada (I.M.P.A.) Estrada Dona Castorina, 110 Jardim Bot~inico Rio de Janeiro, R.J. Brfisil Manuscrit re~u le 30 janvier 1989. Rdvisd le 2 fivrier 1990.

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Bifurcations and global stability of families of gradients

Bifurcations and global stability of families of gradients

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Bifurcations and global stability of families of gradients

Bifurcations and global stability of families of gradients

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