Smoothness property for bifurcation diagrams

Publicacions Matem`atiques, Vol 41 (1997), 243–268. SMOOTHNESS PROPERTY FOR BIFURCATION DIAGRAMS R. Roussarie Abstract Strata of bifurcation sets related to the nature of the singular points or to connections between hyperbolic saddles in smooth families of planar vector fields, are smoothly equivalent to subanalytic sets. But it is no longer true when the bifurcation is related to transition near singular points, for instance for a line of double limit cycles in a generic 2-parameter family at its end point which is a codimension 2 saddle connection bifurcation point. This line has a flat contact with the line of saddle connections. It is possible to prove that the flatness is smooth and to compute its asymptotic properties. 1. Structures of bifurcation diagrams Let Xλbe a smooth (C∞) unfolding of planar vector fields, with parameter λ∈Rk. In the parameter space Rk, the bifurcation diagram Σ is the set on which one has a variation of the topological type of Xλ, and inversely Xλis structurally stable for λ∈Rk−Σ. In the simpler situations, Σ has a semi-algebraic structure (up to some differentiable diffeomorphism). For instance, for the codimension k saddle-node unfolding, up to a smooth equivalence (which can be chosen Cfor an arbitrarily ), the unfolding is equivalent near 0 ∈R2, to: ±y∂ ∂y +xk+1 + k−1  i=0 ai(λ)xi∂ ∂x where ai(λ) are Cfunctions [D]. The bifurcation diagram is induced by the map a(λ)=(a0(λ),... ,a k−1(λ)) from the semi-algebraic diagram for the roots of the polynomial xk+1 + k−1  i=0 aixi. 244 R. Roussarie As long as the bifurcations are related to analytic functions (up to smooth equivalences), the bifurcation diagram will have a sub-analytic structure (up to smooth diffeomorphisms). But in general, the bifurcation diagram will not be smoothly equivalent to a sub-analytic set. We want to discuss the more simpler situation where this phenomenon occurs, the generic saddle connection unfolding of codimension 2. We suppose that Xλ,λ∈R2is defined for λnear 0, and that X0has a hyperbolic saddle s0with a connection Γ of some stable with some unstable separatrix. Moreover, if −λ1(0), λ2(0) are the eigenvalues of s0(with λ1(0), λ2(0) >0), one supposes that r(0) = λ1(0) λ2(0) =1. Let σbe some segment transverse to Γ and σa half segment in σ, on the side of Γ where a return map P0can be defined for X0, from σto σ. Under the condition r(0) = 1, the map P0is C1. As a first generic condition, one supposes that P 0(0) = 1 (we parametrize σby x∈[0,X[, 0 corresponding to σ∩Γ). Up to the change of Xλby −Xλ, one can suppose that P 0(0) >1 (expanding saddle connection). For λnear 0, one has a saddle point sλof Xλnear s0with eigenvalues −λ1(λ), λ2(λ). Let be r(λ)=λ1(λ) λ2(λ)and α1(λ)=1−r(λ). Also, for λnear 0, the unstable separatrix near Γ cuts σat a first point b(λ) and the stable one at a first point a(λ) (in the sequel, one chooses a smooth parametrization of σsuch that a(λ)≡0). Let be α0(λ)=b(λ)−a(λ). As a second generic condition, one supposes that the smooth map λ→α(λ)=(α0(λ), α1(λ)) is of maximal rank at λ= 0 and so, one can suppose that λ≡α=(α0,α 1). The bifurcation diagram is well understood [DRS1]. Of course the axis {α0=0}is a line of saddle connection bifurcations. Next, one has an half line D={(α0(α1),α 1)| α1≤0}, graph of some function α0(α1) defined for α1≤0 and |α1| small enough. Along this line one has a (double) semi stable limit cycle near Γ. The number of limit cycles is as follow: for α0<0 one has one unstable limit cycle. For αin the tongue between {α0=0}and D, one has two limit cycles: a stable and an unstable one. These two limit cycles collide for α∈Dand one has no limit cycle above D(for α1≤0) and for α0≥0, α1≥0. The function α0(α1)isC∞for α1<0. It is easy to prove that there exist A, B > 0 such that α0(α1)∼−Bα1eA/α1(see [DRS1]). Here and in the sequel the symbol ∼means the equivalence of 1-variable functions at 0(f(u)∼g(u)⇐⇒ f(u)/g(u)→1 for u−→ 0±,±depending on the case). So the function α0(α1) has a topological flat contact with {α0=0}, Smoothness property for bifurcation diagrams 245 the line of saddle connections, and the diagram of bifurcation is not smoothly equivalent to an analytic one. Nevertheless we want to prove that α0(α1) is smooth at α0= 0. More precisely one has: Theorem 1.1. The equivalence α0(α1)∼−Bα1eA/α1can be differentiated indefinitely. Remark. As a consequence of the theorem dα0 dα 1 ∼(−1)+1BA α2−1 1 eA/α1 for any ≥0. This implies that each derivative of α0(α1) goes to 0 when α1→0−. So, the function α0(α1)isC∞at α1= 0 (it is “C∞-flat” at 0). The precise value of A,Bare given in the paragraph 3. Related problems concern the bifurcations of zeros of abelian integrals. For instance, let us suppose that a smooth function Hon R2, has one saddle point sand that the level of scontains a saddle connection Γ. Let be ω(m, λ) a smooth parameter family of 1-forms defined in some neighborhood of Γ. Let be, as above a half segment σ≃[0,X[ transverse to the closed cycles of the function Hnear Γ. We call γxfor x∈]0,X[ the cycle through x. Now, one considers the abelian integral: I(x, λ)=γx ω(m, λ). It is easy to prove that I(x, λ)=f(x, λ)−g(x, λ)Lnx for some smooth functions f,g. If we suppose that ω(s, λ)≡0, then g(0,λ)≡0 and the map x→I(x, λ) has a Dulac expansion at x= 0 in the monomials xi, xiLnx, with smooth coefficients in λ: (1.1) I(x, λ)=α0(λ)−α1(λ)xLnx +α2(λ)x+··· One looks at a singularity of codimension 2 of I, defined by α0(0) = α1(0) = 0. As a generic condition, one can suppose that the map λ→ α(λ)=(α0(λ), α1(λ)) is of maximal rank at λ= 0 and that α2(0) =0. So that, one can suppose that λ=α=(α0, α1) and also, by a smooth change of the parametrization, that α2(α)≡1. The bifurcation diagram for I(x, α) is very similar to the above one, with a line Dof double zeros for I, graph of a function α0(α1) defined for α1≤0, near 0. We have now the same result of smoothness as above: 246 R. Roussarie Theorem 1.2. There exist constants A,B>0such that α0(α1)∼ −Bα1eA/α1and this equivalence can be differentiated indefinitely. The Theorem 1.2 can be seen as a limit case of Theorem 1.2. In fact the function δ(x, λ)=P(x, λ)−x(where (P(x, λ) is the return map of Xλfrom σto σ) has a similar expansion as (1.1) where −Lnx is replaced by ω(x, α1)=x−α1−1 α1(see [R]), and one may notice that lim α1→0ω(x, α1)=−Lnx. The abelian integrals are directly related to bifurcations of limit cycles for perturbations of hamiltonian vector fields, or dually for perturbations of integrable 1-forms. Let be ωε,αa(ε, α)-parameter family (ε∈R), of smooth 1-forms, (1.2) ωε,α =dH +ενα+◦(ε) where ναis an α-parameter family of 1-forms. Then, the return map of ωε,α on a segment transverse to cycles of H, has the following expansion, in term of the hamiltonian value h: (1.3) Pε,α(h)=h+εγh να+◦(ε). If α∈R2, one can define generic 3-parameter unfoldings, prescribing that I(h, α)=γhναis a generic unfolding of abelian integrals. One finds back for δ(h, α)=(Pε,α(h)−h)/ε the situation of Theorem 1.1 when ε= 0 and of Theorem 1.2 when ε= 0. In fact, one has a smoothness which is uniform in ε: Theorem 1.3. Take δ(h, α, ε)=Pε,α(h)−h ε. The bifurcation diagram of {δ=0}has a surface of double zeros which is the graph of a smooth function α0(α1,ε)defined above for ε≥0,α1≤0. This function is C∞-flat along the line {α0=α1=0}. Remark. Perturbations of hamiltonian vector fields occur by blow-up of generic unfoldings. For instance the situation covered by Theorem 1.3 occurs in generic 3-parameter unfoldings to produce lines of saddle connections of codimension 2 (see [DRS1], [DRS2] for instance). The Theorem 1.3 shows that the diagram of bifurcation is smooth along these lines. So, for instance and as a consequence of Theorem 1.3, the bifurcation diagram of the cod. 3-Bogdanov-Takens bifurcation studied in [DRS1] is smooth. Smoothness property for bifurcation diagrams 247 In the case of analytic families it would be interesting to obtain more informations on the bifurcation lines. For instance it is reasonable to hope that the line Dwould be defined by a Pfaffian function (in the sense of [K]). Recent results about a preparation theorem for logarithmicexponential functions [LR], seem to be applicable to analytic unfoldings of abelian integrals to prove that the function α0(α1) is a convergent series in powers of α1and e1/α1.This would imply easily the Theorem 1.2 for analytic unfoldings. A similar result of convergence for Theorems 1.1 and 1.3 is more doubtful. We begin by the proof of the Theorem 1.2 in the next paragraph. It is the easiest case because the transcendent function Lnx does not depend on the parameter. The proofs given for the Theorems 1.1 and 1.3 in the paragraphs 3 and 4 will follow the same general lines. Anywhere in the text the symbol ∼is for equivalence of functions of xat x=0,orofα1at α1=0. 2. Smooth flatness in bifurcation diagrams of abelian integrals We will give a proof of Theorem 1.2 under some more general assumptions. Let be a function δ(x, α) (we no longer write the bar subscript and we will write δinstead of I), with parameter α=(α0,α 1) near 0 ∈R2 and variable x∈[0,X[. We will suppose that δis smooth for x= 0 and has Dulac expansions at any order in x, Lnx. This means that for any ∈N,one can write: (2.1) δ(x, α)=  0≤j≤i≤+1 αij(α)xiLnjx+Φ (x, α) where the αij are smooth in αand Φis a C-function in (x, α) which is -flat at x=0(Φ (0,α)=···=∂Φ ∂x(0,α) = 0 for any α). Moreover one supposes that α00(α)=α0,α11(α)=−α1and α10(0) = 0. Up to a change of variable depending on α, and if necessary the change of δin −δ, one can suppose that α10(α)≡1. This will be assumed, from now on. The set of parameters where δhas a double zero is given by: (2.2)    δ=α0−α1xLnx +x+···=0 ∂δ ∂x =−α1(Lnx +1)+1+···=0. 248 R. Roussarie The equations (2.2) can be solved to give a germ of curve D={(α1(x), α2(x)) |x≥0}with: (2.3) α1(x)∼(Lnx)−1 and: (2.4) α0(x)∼−x(Lnx)−1. It follows also from (2.2) that Dis the graph of a function α0(α1) for α1≤0 such that: (2.5) α0(α1)∼−e−1α1e 1 α1. We want to obtain similar equivalences for successive derivatives of α0(α1),which are defined for α1= 0. More precisely we want to prove that the equivalence (2.5) can be indefinitely differentiated (and so, the function α0(α1) will be smooth at α1= 0). The idea is to work with functions of x. So we postpone the study of α0(α1) until the end of this paragraph and begin to say more about equations (2.2). First, we can solve the first equation to obtain a function α0(x, α1): Proposition 2.1. The equation δ=0defines a function α0= α0(x, α1)where α0has Dulac expansions for any with smooth coefficients in α1. This means that there exists a sequence of smooth functions αij(α1),0≤j≤i, such that for any : α0(x, α1)=  0≤j≤i≤+1 αij(α1)xi(Lnx)j+ Φ(x, α1)(2.6)  Φ(x, α1),Cand -flat at x=0for any α1. Proof: For any K∈N, one can write δ(x, α0,α 1)=∆K(x, xLnx, α0,α 1) where ∆K(u, v, w, z)isaCK-function. Because ∂∆K ∂w (0,0,0,0) = 1, one can solve ∆K(u, v, w, z)=0asa CK-function w(u, v, z) near (0,0,0) with w(0,0,0) = 0. If one takes in account that ∆K(u, v, w, z) can be expanded in Taylor polynomials in (u, v) with smooth coefficients in (w, z), it turns out that whas Taylor polynomial of order Kin u, v, with smooth coefficients in z: (2.7) w= i,j≤Kwij(z)uivj+ψK(u, v, z) Smoothness property for bifurcation diagrams 249 where ψKis CKand K-flat at (u, v)=(0,0), for any z. To solve δ= 0, one has just to substitute z=α1,u=x,v=xLnx in (2.7). Remark that ψK(x, xLnx,α1)isCK−1and K−1 flat in x, α1. To obtain the expansion at order , it suffices to take K=+1. It is easy to obtain the first terms of α0. One has: (2.8) α0(x, α1)=α1xLnx −x+··· where + ··· means a finite sum of terms as in (2.7) of higher order, plus aC-flat function for an arbitrarily . Now we can substitute α0=α0(x, α1) in the second equation of (2.2) to obtain an implicit equation for α1(x). (2.9) δ1(x, α1)= ∂δ ∂x(x, α0(x, α1),α 1)=0. The function δ1admits expansions as (2.6) at any order, but now with j≤i+ 1. (This comes from the fact that δ1is obtained by one differentiation in x.) The first terms in δ1are: (2.10) δ1(x, α1)=−(Lnx +1)α1+1+β(α1)x(Lnx)2+··· for some smooth function β. The difficulty to solve the equation (2.9) is that ∂δ1 ∂α1=−(Lnx +1)+··· has a singularity at x= 0. So, we will not try to solve it. We will keep α1as an unknown function, and extend to negative powers of xor Lnx the polynomial expressions we will consider. Definition. Lis the ring of smooth functions on ]0,X[ with the following properties. For any f∈Lthere exists a 2-variable function F(x, α1) such that f(x)=F(x, α1(x)). For any k, this function Fcan be written F(x, α1)=Fk(x, α1)+ Φk(x, α1) with: 1) Fk(x, α1)∈C ∞(α1)[x, Lnx, x−1,(Lnx)−1], i.e. Fkis a polynomial in x±1,(Lnx)±1with smooth coefficients in α1. 2) Φk(x, α1)isCkand k-flat at x= 0 for any α1. 3) Fkis obtained by truncation of Fk+1. Let L0⊂Lbe the subset of functions in Lsuch that there exists a k for which Fkhas a leading term L(x)=xi(Lnx)j≡ 1 (i.e.: ior j=0) with coefficient a(α1) with a(0) =0. We will also consider the ring Fof fractions f/g with f∈Land g∈L 0. Any element in Fcan be written f gwith g=1+··· We define F0as the set of fractions f g∈F with f∈L 0and g=1+··· 250 R. Roussarie Remark. Any element h∈F 0can be written H(x, α1(x)) with H(x, α1)=a(α1)LF Gwith F=1+···,G=1+···∈L,a(0) =0 and L(x)=xi(Lnx)j≡ 1. It is clear that h(x)∼a(0)L(x)soa(0) doesn’t depend on the choice of k,ifkis large enough. On the contrary, the polynomial Fkin the above definition is not uniquely defined. This follows from the fact that α1(x) is implicit solution of (2.9). Nevertheless a formula as: f=aL +··· will mean that aand Lare the same for any k. Lemma 2.2. α1(x)∈F 0. More precisely: α1(x)=(Lnx)−1f 1+(Lnx)−1with f=1+···∈L. Proof: This is a direct consequence of the equation (2.9): δ1(x, α1)=−(Lnx +1)α1+1+···=0. Clearly, it follows from Proposition 2.1 that the sum f=1+··· belongs to L(in fact with monomials in x,Lnx) and that α1(x)= (Lnx)−1f 1+(Lnx)−1. Lemma 2.3. F0is closed by derivation: if h∈F 0then dh dx ∈F 0. More precisely, if: h(x)=a(α1(x))L(x)1+··· 1+··· then dh dx =a(α1(x))P(L)1+··· 1+··· where P(L)=ixi−1(Lnx)jif L=xi(Lnx)jwith i=0and P(L)= jx−1(Lnx)j−1if L=(Lnx)j. Remark. If L(x)=xi(Lnx)j,ior j=0: L(x)=dL dx (x)=ixi−1(Lnx)j+jxi−1(Lnx)j−1. So P(L) is the leading term of L(P(L)∼L). Proof: Let be f(x)=F(x, α1(x)) ∈F 0: (2.11) f(x)=∂F ∂x (x, α1)+ ∂F ∂α1 (x, α1)dα1 dx . We see that dα1 dx enters in the expression (2.11). The function α1itself belongs to F0by Lemma 2.2. So it is natural to begin the proof by this function. Smoothness property for bifurcation diagrams 251 (a) The case of α1: Let us write again: δ1=−(Lnx +1)α1+1+g(x, α1) with g(x, α1)=β(α1)x(Lnx)2+··· as in (2.10). If we differentiate the equation δ1= 0, we obtain: (2.12) −(Lnx +1)+ ∂g ∂α1dα1 dx =α1 x−∂g ∂x. Now, clearly enough, ∂g ∂α1and ∂g ∂x ∈Lwith: ∂g ∂α1 =dβ dα1 (α1)x(Lnx)2+··· and ∂g ∂x =β(α1)(Lnx)2+··· Multiplying (2.12) by (Lnx + 1) and taking in account that (Lnx + 1)α1=1+···∈L one obtains: (2.13) −(Lnx +1) 2+···dα1 dx =x−1+··· So, the factor of dα1 dx is in L0.If one put x−1(Lnx)−2in factor, one has finally: dα1 dx =−x−1(Lnx)−21+··· 1+··· which belongs to F0.Remark that −x−1(Lnx)−2is the derivative of (Lnx)−1, the leading term of α1∼(Lnx)−1. (b) The general case: To begin with, let us suppose that h=f∈L, with F(x, α1)= a(α1)L(x)+···,L(x)=xi(Lnx)jior j= 0. As above it is clear that ∂F ∂x (x, α1(x)) and ∂F ∂α1(x, α1(x)) belong to L. Because dα1 dx ∈F 0it follows that f∈F. It remains just to check the leading term in (2.11). We have: dα1 dx ∂F ∂α1 =−∂a ∂α1 (α1)xi−1(Lnx)j−2+··· ∂F ∂x (x, α1)=iaxi−1(Lnx)j+··· if i=0 ∂F ∂x (x, α1)=jaxi−1(Lnx)j−1+··· if i=0. 258 R. Roussarie with: c(α1,α 1Lnx)=−α2(α1,0)(1 −α2(α1,0)) (1 −α1)2Φ(α1Lnx) (so that α0(x)∼c(0,v 0)x(Lnx)−1with c(0,v 0)=−(1 −α2)v0). Proof: From the equation (3.8) for α0and the Lemma 3.3, it follows easily that α0∈F ω. To prove that α0∈F ω 0, we bring the expression: −α1ω=α2(α1,0) −α1 1−α1 +··· coming from (3.10) in the expression: α0=−α1xω −α2(α1,0)x+··· We obtain that: (3.13) α0=α2(α1,0) −α1 1−α1 x−α2(α1,0)x+··· α0=−α1(1 −α2(α1,0)) 1−α1 x+··· The expression (3.13) is not an expression in Fω 0of order (1,0) because the coefficient of the leading term is zero for α1= 0. To obtain the expression (3.12), it suffices to substitute the term α1in this coefficient by its expansion (3.11). (We do not modify the other occurrences of α1 in (3.13)). Lemma 3.5. dα1 dx ∈F ω 0and dα1 dx (x)∼−v0x−1(Lnx)−2. Remark. The equivalence for dα1 dx (x) is just given by the formal derivation of the equivalence α1(x)∼v0(Lnx)−1. Proof: The equation (3.10) for α1writes: (3.14) α1((1 −α1)ω−1) = −α2(α1,0) + ··· Let Q(x, α1) be the right hand member of (3.14). It is a function in Lω 0. By differentiation in x, we obtain: (3.15) (1 −α1)ω−1−α1ω+α1(1 −α1)∂ω ∂α1 −∂Q ∂α1dα1 dx =−α1(1 −α1)∂ω ∂x +∂Q ∂x . Smoothness property for bifurcation diagrams 259 We look first at the factor Gin front of dα1 dx . Because ∂ω ∂α1= −Φ(α1Lnx)(Lnx)2and ∂Q ∂α1are in Lω, the same holds for G. The leading term of Gis given by: (3.16) −(1 −2α1)Φ + α2(α1,0) ΦΦLnx. (Here Φ(v) and Φ(v)=dΦ dv (v) must be evaluated at v=α1Lnx). The coefficient of this leading term is equal to a(α1,α1Lnx) with: (3.17) a(u, v)=−(1 −2u)Φ(v)+α2(u, 0)Φ(v) Φ(v). We have: −a(0,v 0)=Φ(v0)+ α2 Φ(v0)Φ(v0) which gives: (3.18) a(0,v 0)=−(1 −α2)=0. So, we have that G∈L ω 0. Look now at the right hand term of (3.15): (3.19) F=−α1(1 −α1)∂ω ∂x +∂Q ∂x . One has: (3.20) ∂ω ∂x =−(α1ΦLnx +Φ)x−1. If in this expression, we replace α1, coefficient of ΦLnx by its expansion (3.11), we obtain, taking in account that ∂Q ∂x =O((Lnx)2): (3.21) F=α2(α1,0) Φα2(α1,0)(1 −α1)Φ Φ+ψ1 ψ2 Φx−1(Lnx)−1ϕ1 ϕ2 where ϕ1=1+···,ϕ2=1+··· ∈L ωas well as ψ1,ψ2. Finally, using (3.17) and (3.21), one obtain the expression for dα1 dx as a function of Fω 0: (3.22) dα1 dx =F G=t(α1,α 1Lnx)x−1(Lnx)−21+··· 1+··· 260 R. Roussarie with: (3.23) t(α1,α 1Lnx)=−α2(α1,0) Φ (1 −α1)α2 Φ Φ+Φ (1 −2α1)Φ + α2 ΦΦ. We find that t(0,v 0)=−v0, which gives: dα1 dx ∼−v0x−1(Lnx)−2. We can now consider a general function in Fω 0: Proposition 3.6. Let be f(x)=c(α1,α 1Lnx)xi(Lnx)j1+··· 1+··· a function in Fω 0(i.e.: c(0,v 0)=0)and such that i=0. Then df dx ∈F ω 0 and: df dx =ic(α1,α 1Lnx)xi−1(Lnx)j1+··· 1+···. Remark. Again, the equivalence for df dx is just the formal differentiation of the equivalence f(x)∼c(0,v 0)xi(Lnx)j. Take any function g(x)∈L ω. At any order , there exists a function G(x, u, v) (we omit !) such that g(x)=G(x, α1(x),α 1Lnx) where the dependence on α1and α1Lnx is located in the smooth coefficients and the remaining term of the expansions introduced above in Definition 3.1. Now, we have: (3.24) dg dx =∂G ∂x +α1 ∂G ∂v x−1+∂G ∂u +∂G ∂v Lnxdα1 dx . Using this expression it is clear that dg dx ∈L ω. Now, we consider any f(x)∈F ω 0,f(x)=c(α1,α 1Lnx)xi(Lnx)jg h, with g=1+···,h=1+···∈L ω. Because g,h= 1 + 0((Lnx)−1) one has that: (3.25) dg dx,dh dx =0(x−1(Lnx)−2) and also that: (3.26) d dx g h=x−1(Lnx)−2g 1+··· Smoothness property for bifurcation diagrams 261 with g,1+···∈L ω. Let us consider now the derivation of the principal part of f: (3.27) d dx [cxi(Lnx)j]= d dx(xi(Lnx)j)c+xi(Lnx)jd dxc(α1,α 1Lnx) (3.28) d dx(xi(Lnx)j)=ixi−1(Lnx)j+···(because i=0) (3.29) d dxc(α1,α 1Lnx)=dα1 dx ∂c ∂u +∂c ∂vLnx+x−1α1 ∂c ∂v. Using Lemmas 3.4, 3.5, one can write this function in Lωwith leading term: x−1(Lnx)−1. Comparing with (3.28), we see that d dx [cxi(Lnx)j)∈ Lω 0with leading term xi−1(Lnx)jand coefficient ic(α1,α 1Lnx). (3.30) d dx [cxi(Lnx)j]=icxi−1(Lnx)j+··· If we return now to df dx , one has: (3.31) df dx =d dx(cxi(Lnx)j)g h+cxi(Lnx)jd dx g h. And, it follows from (3.26) and (3.30) that df dx can be written in Fω 0: df dx =ic(α1,α 1Lnx)xi−1(Lnx)j1+··· 1+···. Now, as in paragraph 2, one introduces for any k≥0, the function ϕk(x)=dkα0 dαk 1 (α1(x)). One has ϕ0(x)=α0(x)∈F ω 0, and from 3.12: (3.32) ϕ0(x)∼−(1 −α2)v0x(Lnx)−1. For k≥0,the ϕkare related by the recurrence relation: (3.33) ϕk+1(x)=dϕk dx (x)dα1 dx (x)−1 . From this, we deduce the: 262 R. Roussarie Proposition 3.7. For any k≥0,ϕk∈F ω 0and: (3.34) ϕk(x)∼(−1)k+1(1 −α2)v−k+1 0x(Lnx)2k−1. Proof: For k= 0 one has already noticed that ϕ0=α0and (3.34) reduces to (3.12). Now, by recurrence, suppose that ϕk∈F ω 0and that ϕk(x)∼(−1)k+1 (1 −α2)v−k+1 0x(Lnx)2k−1. From Proposition 3.6, one obtains that: dϕk dx ∈F ω 0and dϕk dx (x)∼(−1)k+1 (1 −α2)v−k+1 0(Lnx)2k−1. From (3.5), one has that dα1 dx ∈F ω 0and dα1 dx ∼−v0x−1(Lnx)−2and finally: ϕk+1 ∈F ω 0with: ϕk+1 ∼(−1)k+2(1 −α2)v−(k+1)+1 0x(Lnx)2(k+1)−1. We can now finish the proof of Theorem 1.1. From δ1(x, α1)=(1−α1)x−α1+(α2(α1,0)−1)+0(xω2) = 0 we deduce that: x−α1(1+0(x1+α1ω2)) = 1−α2 1−α1 =1−α2+o(1). This gives: (3.35) x(α1)∼e−1e v0 α1. Recall that, inversely α1(x)∼v0(Lnx)−1. From this, one can deduce that: (3.36) dkα0 dαk 1 (α1)=ϕk(x(α1)) ∼(−1)k+1(1 −α2)e−1vk 0 α2k−1 1 ev0/α1. As it was announced in Theorem 1.1, this means that one can differentiate indefinitely the equivalence: α0(α1)∼−(1 −α2)e−1α1ev0/α1. Smoothness property for bifurcation diagrams 263 4. Smooth flatness in bifurcation diagrams for perturbations of hamiltonian vector fields (Theorem 1.3) We consider now a perturbation of hamiltonian vector fields. In the proof of Theorem 1.3, one can replace the parametrization by the hamiltonian value h, by any smooth parametrization x∈[0,X[ like in paragraph 3. In the present case the 2 parameters α0,α1can be divided by the perturbation parameter ε:α0=εα0,α1=εα1. Next, the function δ and its (x, ω)-expansions at any order , can also be divided by ε: (4.1) δ(x, α0, α1,ε)=εδ(x, α0, α1,ε). The function δhas also (x, ω)-expansions at any order : (4.2) δ(x, α0, α1,ε)=α0+α1xω +α2(α0, α1,ε)x+··· One has α2(0,0,0) = 0. We can suppose and we will suppose that α2(0,0,0) >0. We consider ωas a function of x,α1,ε: (4.3) ω(x, α1,ε)=x−α1−1 α1 =x−εα1−1 εα1 . We want to study the surface Dof double zeros for δ. It is defined by the equation: (4.4)    δ=0 ∂δ ∂x =α1((1 −εα1)ω−1) + α2(α)+···=0. The surface Dis the graph of a function α0(α1,ε) defined for ε≥0 and α1≤0, small enough. This surface may be looked at as union of lines Dε, graphs of functions: α1→α0(α1,ε). For each ε= 0 we find back the situation studied in the previous paragraph. So that α0(α1,ε) is smoothly flat at α1=0. Forε=0, we find back the situation studied in paragraph 2 and α0(α1,0) is also smoothly flat at α1= 0. Here, we want establish the flat smoothness along the line {α1=0}in the (ε, α1)−plane for the 2-variable function α0(α1,ε). Because α0(α1,ε) is smooth for α1= 0, it will suffice to prove that any partial derivative: ∂i,j α0=∂i+jα0 ∂αi 1∂εj 264 R. Roussarie goes to zero, uniformly in ε, for α1−→ 0. As in the previous paragraph, we can parametrize Dby x∈[0,X[, the position of the double zero: D={(α0(x, ε), α1(x, ε)) |x∈[0,X[, ε small enough}. The function α1(x, ε) has an inverse function x(α1,ε) and α0(α1,ε)is equal to α0(x(α1,ε), ε). Also, like in paragraph 3, one introduces the x-parametrization for partial derivatives of α0(α1,ε): (4.5) ϕij(x, ε)=∂i,j α0(α1(x, ε),ε). The smooth flatness of α0(α1,ε) will follow from: Proposition 4.1. For any i, j ≥0,ϕij(x, ε)=O(x|Lnx |2i+j−1) uniformly in εin the sense that there exist constants Mij >0such that: |ϕij(x, ε)|≤ Mij x|Lnx |2i+j−1 for all x, ε. Before proving this proposition, we restate without proof the Lemma 3.1: Lemma 4.2. Let be α2(ε)=α2(0,0,ε). The functions x−εα1, α1ω(x, α1,ε)and α1Lnx, with α1=α1(x, ε),are continuous along {x= 0}. More precisely: (1) x−εα1−→ 1−εα2(ε) (2) −α1ω−→ α2(ε) (3) α1Ln(x)→v0(ε)=−Ln(1−εα2(ε)) ε=α2(0) + 0(ε)for x−→ 0+, uniformly in ε. Now, we can introduce sets of functions, similar to those in the previous paragraph: Definition 4.1. Lω,ε is a ring of C∞functions, defined for (x, ε) small and positive (x>0, ε≥0). Apart from the dependence on ε, the definition is similar to the Definition 3.1: we just suppose that the coefficients in Fkare smooth functions of α1,α1Lnx and εand that the rest Φkis Ckin x,α1,α1Lnx,εand k-flat at x=0. Let be Lω,ε 0, the subset of functions in Lω,ε such that the leading term L(x)=xi(Lnx)jis different from 1 (ior j= 0), with a coefficient Smoothness property for bifurcation diagrams 265 a(u, v, s) such that a(0,v0(ε), ε)=0,εsmall enough (it suffices that a(0,v0(0),0) = 0). Fω,ε will be the ring of fractions f/gwith f∈L ω,ε and g∈L ω,ε 0, and Fω,ε 0the subset of Fω,ε of fractions f gwith f∈L ω,ε 0and g=1+··· ∈ Lω,ε. Remark. If h=c(α1,α 1Lnx, ε)xi(Lnx)j1+··· 1+··· ∈F ω,ε 0then h(x, ε)∼ c(0,v0(ε),ε)xi(Lnx)j. The function c(0,v0(ε),ε) is continuous in ε.To prove Proposition 4.1, it will be sufficient to prove that ϕi,j(x, ε) belongs to Fω,ε 0with x(Lnx)2i+j−1as leading monomial. Now the proofs of Lemmas 3.3, 3.4 and 3.5 can be easily generalized here. The only changes is that one has to replace the derivative ∂ω ∂α1by ∂ω ∂α1=∂ω ∂α1·ε. This implies that the variation of ωwith α1is continuous at ε= 0, where ωis replaced by −Lnx. So, one has that α1(x, ε), α0(x, ε) and dα1 dx (x, ε) belong to Fω,ε 0with: α1(x, ε)∼v0(ε)(Lnx)−1, α0(x, ε)∼−(1 −εα2(ε))v0(ε)x(Lnx)−1 and: (4.6) dα1 dx (x, ε)∼−v0(ε)x−1(Lnx)−2. Moreover, if f(x, ε)=c(α1,α1Lnx,ε)xi(Lnx)j1+··· 1+··· is a function in Fω,ε 0,(i.e.: c(0,v0,0) = 0) with i= 0, then ∂f ∂x(x, ε)∈F ω,ε 0and: (4.7) ∂f ∂x =icxi−1(Lnx)j1+··· 1+···. This implies that the equivalence f(x, ε)∼c(0,v(ε), ε)xi(Lnx)jcan be formally differentiated: ∂f ∂x(x, ε)∼ic(0,v(ε),ε)xi−1(Lnx)j. By an argument similar to the one in paragraph 3, these results imply that: (4.8) ϕi,0(x, ε)∈F ω,ε 0with an order x(Lnx)2i−1. Next, we have to consider derivatives in ε. We will say that h∈F ω,ε has an order xi(Lnx)jif it can be written h=f gwith g=1+··· ∈ Lω,ε and f=a(α1,α1Lnx,ε)xi(Lnx)j+··· with a smooth coefficient a(u, v, ε) which may be equal to zero at (0,v0, 0). Of course this order is not an intrinsic property of hbut depends on the choice of an expansion for f. 266 R. Roussarie Lemma 4.3. If h∈F ω,ε has an order xi(Lnx)jthen ∂h ∂ε ∈F ω,ε and has a same order. Proof: We begin to prove this result for the function α1(x, ε). Like in paragraph 3, we can solve the first line of (4.4) to obtain α0=α0(x, α1,ε) and next eliminate α0in the second line of (4.4) to obtain an implicit equation for α1(x, ε): δ1(x, α1,ε)= ∂δ ∂x(x, α0(x, α1,ε),ε) = 0 which writes: α1((1 −εα1)ω−1) = c(α1,ε)+···(4.9) where c(α1,ε)=−α2(α0(0, α1,ε), α1,ε) is smooth with c(0,0) =0. Taking the derivative in εof (4.9) one obtains: (4.10) (1 −2εα1)ω−1+α1(1 −εα1)∂ω ∂α1 −∂c ∂α1 +···∂α1 ∂ε =O(1) where the right hand term of (4.10) is a function of Lω,ε of order 1. Now, taking in account that α1(x, ε)∈F ω,ε 0of order (Lnx)−1, that ω(x, α1,ε)=−LnxΦ(εα1Lnx) (where Φ(u)=1−e−u u, for u= 0, like in paragraph 3), it is easy to verify that the bracket, coefficient of ∂α1 ∂ε in (4.10), is in Lω,ε 0of order Lnx. It follows that: ∂α1 ∂ε ∈F ω,ε with the same order (Lnx)−1as α1. Consider now some f(x, ε)=c(α1, α1Lnx, ε)xi(Lnx)j+··· in Lω,ε with order xi(Lnx)j, with c(u, v, z) smooth. One has that: (4.11) ∂f ∂ε =∂c ∂u +Lnx ∂c ∂vdα1 dε +∂c ∂zxi(Lnx)j+··· Using the above result for ∂α1 ∂ε , one obtains that ∂f ∂ε ∈L ω,ε of order xi(Lnx)j. Finally, let us consider h=f gwith f=cxi(Lnx)j∈L ω,ε of order xi(Lnx)jand g=1+···∈L ω,ε of order 1: (4.12) ∂h ∂ε = ∂f ∂ε g−∂g ∂εf g2. Because the order of ∂f ∂ε and fis xi(Lnx)jand the order of gand ∂g ∂ε is 1, one has that ∂h ∂ε ∈F ω,ε of order xi(Lnx)j. Smoothness property for bifurcation diagrams 267 We can now finish the proof of Proposition 4.1. More precisely, we will prove inductively on j, that: (4.13) ϕi,j(x, ε)∈F ω,ε of order x(Lnx)2i+j−1. The equation (4.8) gives the result for j=0. Suppose now that (4.13) was true for some j≥0. If one differentiates in εthe equation (4.5), one obtains: (4.14) ϕi,j+1(x, ε)=∂ϕij ∂ε (x, ε)−ϕi+1,j ∂α1 ∂ε . Now, it follows from Lemma 4.3 and the induction hypothesis that each function in the right hand term of (4.14) is in Fω,ε and that ϕi+1,j has order x(Lnx)2i+j+1,∂α1 ∂ε has order (Lnx)−1and ∂ϕij ϕε has order x(Lnx)2i+j−1. So finally, ϕi,j+1 (x, ε)isinFω,ε with order x(Lnx)2i+(j+1)−1. This concludes the proof of Proposition 4.1. References [D] F. Dumortier, Local study of planar vector fields: singularities and their unfoldings, in “Structure in Dynamics-Finite dimensional deterministic Studies,” (H. W. Broer, F. Dumortier, S. J. Van Strien and F. Takens, eds.), Studies in Mathematical Physics 2, North Holland, 1991, pp. 161–241. [DRS1] F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of planar vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case, Ergordic Theory Dynam. Systems 7(1987), 375–413. [DRS2] F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear part, in “Bifurcation of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals,” (F. Dumortier and als., eds.), Lect. Notes in Math. 1480, Springer-Verlag, Berlin-Heidelberg-New-York, 1991, pp. 1–141. [K] A. Khovanskii,“Fewnomials,” Amer. Math. Soc., Providence, RI, 1991. [LR] J.-M. Lion and J.-P. Rolin,Th´eor`eme de pr´eparation pour les fonctions logarithmico-exponentielles, Pr´epublication du Laboratoire de Topologie, Universit´e de Bourgogne 98 (1996).