UNFOLDING OF RESONANT SADDLES AND THE DULAC TIME

In this work we study unfoldings of planar vector fields in a neighbourhood of a resonant saddle. We give a C normal form for the unfolding with respect to the conjugacy relation. Using our normal form we determine an asymptotic development, uniform with respect to the parameters, of the Dulac time of a resonant saddle deformation. Conjugacy relation instead of weaker equivalence relation is necessary when studying the time function. The Dulac time of a resonant saddle can be seen as the basic building block of the total period function of an unfolding of a hyperbolic polycycle. Introduction. In this work we study unfoldings of planar vector fields in a neighbourhood of a resonant saddle. We give a C normal form for the unfolding with respect to the conjugacy relation. This generalizes the known orbital normal form with respect to the equivalence relation [4] and [13]. Using our normal form we determine an asymptotic development, uniform with respect to the parameters, for the Dulac time of a resonant saddle. Our asymptotic development of the Dulac time is of a similar nature as the asymptotic expansion of the Dulac map given in [13]. It generalizes our previous work [7] dealing with the Dulac time of orbitally linearizable families, but without being as explicit on the coefficients. Our initial motivation was the problem of finite “cyclicity” (i.e., existence of a local uniform bound) for the number of critical points of the period function of polynomial vector fields on hyperbolic or more general polycyles. The condition of non-criticality of the period appears for instance in the bifurcation theory of subharmonics. Under the non-criticality of the period, zeros of appropriate Melnikov functions guarantee the persistence of a subharmonic periodic orbit of a Hamiltonian under a periodic non-autonomous deformation (see Theorem 4.62 of [3]). 2000 Mathematics Subject Classification. Primary: 37G05; Secondary: 34C07, 37C10, 37G15.

Introduction. In this work we study unfoldings of planar vector fields in a neighbourhood of a resonant saddle. We give a C k normal form for the unfolding with respect to the conjugacy relation. This generalizes the known orbital normal form with respect to the equivalence relation [4] and [13].
Using our normal form we determine an asymptotic development, uniform with respect to the parameters, for the Dulac time of a resonant saddle. Our asymptotic development of the Dulac time is of a similar nature as the asymptotic expansion of the Dulac map given in [13]. It generalizes our previous work [7] dealing with the Dulac time of orbitally linearizable families, but without being as explicit on the coefficients.
Our initial motivation was the problem of finite "cyclicity" (i.e., existence of a local uniform bound) for the number of critical points of the period function of polynomial vector fields on hyperbolic or more general polycyles. The condition of non-criticality of the period appears for instance in the bifurcation theory of subharmonics. Under the non-criticality of the period, zeros of appropriate Melnikov functions guarantee the persistence of a subharmonic periodic orbit of a Hamiltonian under a periodic non-autonomous deformation (see Theorem 4.62 of [3]).
We see our asymptotic development of the Dulac time as the basic building block in establishing an asymptotic development of the total period function (Poincaré time), which we hope to study in a subsequent work. In its turn, such a uniform asymptotic development should be the main ingredient in the proof of finite "cyclicity" for critical points of the period function on hyperbolic polycycles.
For a fixed vector field several results are known: An asymptotic development of the Poincaré time was obtained in [16,17]. Non-accumulation of critical periods of a fixed polynomial vector field on hyperbolic polycycles has been recently proved in [9]. In [2] Chicone and Dumortier show that the Poincaré time of a fixed vector field on a polycycle is non-oscillating if the polycyle has at least one finite saddle point.
Hence, special attention must be payed to the study of polycycles whose all vertices are at infinity in the Poincaré disc. For that reason, in our study of unfoldings of saddle points (2) we permit polar factors. They can come from the line at infinity in a saddle at infinity or, more generally, appear in a divisor after desingularizing more general singular points at infinity in a polycycle. The case of lines of zeros in at least one of the separatrices is also allowed as it can appear after desingularizing a degenerate singular point at finite distance.
We think that our normal form is also of independent interest. Note that due to unfolding of resonances, one cannot hope for a C ∞ or analytic normal form in a neighbourhood of a resonant saddle. When studying unfoldings of polycycles of finite codimension a C k normal form should be sufficient. For studying unfoldings of infinite codimension, analytic normal forms in some domains unfolding sectors should be developed in the spirit of the unfoldings of saddle-node in [14].
This paper consists of two parts. The first part is dedicated to establishing the normal form Theorem A of an unfolding of a resonant saddle with possibly polar factors in the axes. In the second part we apply this normal form to obtain an asymptotic development Theorem B for the Dulac time.

Part 1. Temporal normal form
This part is organized as follows. In Section 1 the theorem on normal form for conjugacy is formulated. In Section 2 tools necessary for its proof are collected. In Section 3 the normal form is proved modulo the tools. Finally Section 4 is devoted to prove these tools, the most important of them being the existence of solution of an adapted homological equation stated in Theorem 2.3.
1. Statement of Theorem A. Let us consider a C ∞ unfolding {X µ } µ∈U of a saddle point at the origin. More precisely X µ = a µ (x, y)x∂ x + b µ (x, y)y∂ y , with a µ (0, 0) = 1 and λ(µ) := −b µ (0, 0) > 0, (1) where a µ and b µ are C ∞ functions at the origin and U is an open subset of R m . We also consider the collinear family where v = x m y n and m, n ∈ Z.
In what follows we shall say that two vector fields (or germs of vector fields) Z and W are conjugated if there exists a change of coordinates Φ transforming Z to W , i.e., Φ ⋆ Z = W , where UNFOLDING OF RESONANT SADDLES AND THE DULAC TIME 1223 We shall say that two germs of vector fields Z and W are equivalent at a point p 0 , if they are conjugated up to a germ of a nonzero multiple: Φ ⋆ Z = f W with f (p 0 ) = 0. The two notions extend to germs of families of vector fields. Definition 1.1. Given µ 0 ∈ U, let us denote λ 0 := λ(µ 0 ). The orbital codimension κ ∈ N ∪ {∞} of the saddle of the vector field X µ0 is defined as follows. If λ 0 / ∈ Q, then we set κ := ∞. If λ 0 ∈ Q, then the infinite jet of X µ0 at the origin is C ∞ equivalent to In case that α i+1 = 0 for some i we set κ := min i ∈ N : α i+1 = 0 and, otherwise, κ := ∞. Remark 1. The orbital codimension does not depend on the particular equivalence used to bring X µ0 to a normal form (3) because the monomial (x p y q ) κ can not be annihilated by means of a smooth coordinate transformation preserving the normal form.
The main theorem proved in the first part is the following.

Remark 2.
In the definition of ℓ above, the symbol > stands for the partial order in Z 2 . Note that u ℓ /v is regular at (x, y) = (0, 0) and that if m 0 or n 0 then ℓ 1. The integer ℓ plays the role analogous to the orbital codimension in the bound of the degree of Q µ . However, a priori the order of Q µ0 (u) at u = 0 is a more natural notion of "temporal codimension", but it does not seem to have immediate applications.
2. Tools. In this section we collect some tools used in the proof of Theorem A. They will be proved in Section 4 except for Theorem 2.1, for which we give only a sketch of proof. This theorem is part of folklore. It appears, as we state it here, in [13] but referring to [1] for the proof. However, [1] deals only with a related problem of normal forms for diffeomorphisms. A proof of Theorem 2.1 appears in [4] but there is a delicate point concerning the elimination of the remainder term which is not dealt with in that paper. Later on we point it out in the sketch of the proof of Theorem 2.1. The mentioned delicate point can be overcome by applying the results of Samovol in a very technical paper [15]. We do not give a complete proof as it can be done along the lines of the proof of our Theorem 2.3.
Theorem 2.1. Let {X µ } µ∈U be a C ∞ unfolding of a saddle point as in (1) and consider some µ 0 ∈ U. Then for any s ∈ N the family {X µ } µ∈U is C s equivalent by a diffeomorphism of the form Φ(x, y, µ) = (Φ µ (x, y), µ) defined in a neighbourhood of (0, 0, µ 0 ) ∈ R 2 ×U to where (a) if λ 0 / ∈ Q, then P µ ≡ 0, (b) if λ 0 = p/q with (p, q) = 1, then P µ is a polynomial in the resonant monomial u = x p y q . Moreover, in case that X µ0 has orbital codimension κ < ∞ then deg P µ 2κ.
Remark 3. For fixed m, n ∈ Z, the diffeomorphism Φ µ in Lemma 2.2 depends only on the initial data {Y µ } and {f µ }. Since we shall apply it several times, changing both data (vector fields and functions), we introduce the notation Let V be an open subset of R n and consider a smooth function f : V −→ R. We define f V = sup{|f (x)| : x ∈ V }. If I = (i 1 , . . . , i n ) is a multi-index with i j ∈ N ∪ {0} then we use the notation i = |I| = i 1 + · · · + i n and Thus, dealing with partial derivatives, we shall use the convention that if J is a multi-index then the small letter j stands for |J|. Moreover, given p ∈ V, we denote by D i f (p) the total differential of order i of f at p, which is defined as the symmetric i-linear form where the sum is taken over all the multi-index I = (i 1 , . . . , i n ) with |I| = i and x (j) = (x We extend these definitions to vector functions in the usual way. More concretely, if f = (f 1 , . . . , f m ) is a vector function from V ⊂ R n to R m then From now on we must distinguish between parameters µ ∈ R m and phase variables (x, y) ∈ R 2 when considering a smooth function f : V ⊂ R 2+m −→ R. We say that such a function is N -flat with respect to (x, y) if it is C N +1 and verifies the estimates in some neighbourhood of (0, 0, µ 0 ) ∈ R 2+m and for some constant C > 0. The flatness with respect to x or y is defined analogously by replacing (x, y) by |x| or |y| respectively. Theorem 2.3. Let {X µ } µ∈U be a family of vector fields as in (1) and consider some µ 0 ∈ U. Then for any k ∈ N there exists a natural number N = N k, λ 0 , m, n such that if {h µ } is a C N family of N -flat functions, then the homological equation has a C k family of solutions {f µ } defined in a neighbourhood of (0, 0, µ 0 ) ∈ R 2 ×U. More precisely, we can take where ν 0 = max{1, λ 0 } and [ · ] denotes the integer part.

Remark 4.
Analogously to the definition of ℓ in Theorem A, the natural number N k, λ 0 , m, n can be written as Note that the above formula is not symmetric with respect to m and n. This is so because in the proof we first show that it suffices to consider a vector field in normal form and we choose one containing all the resonant monomials in the ∂ y direction. It is important to mention that N depends only on the linear part of X µ0 .
Before proving our main theorem in which we give a normal form for conjugacy, we sketch the proof of Theorem 2.1 that deals with orbital normal form in order to see which kind of ideas are involved in this type of results. Theorem 2.1 is part of folklore (see [4,13]) but we want to point out a delicate point, which we think did not receive the required attention in the literature.
One uses first the Takens normal form theorem [18] (see also [3]). Let H h be the space of polynomial vector field families in the (x, y) plane depending on the parameter µ and homogeneous of degree h in (x, y). Let L = L(µ) = x∂ x − λ(µ)y∂ y be the linear part of X µ and for each h, consider the action of the Lie bracket [ where g h is a homogeneous vector field family belonging to G h ⊂ H h , for h = 1, . . . , N and the remainder term R(x, y) is a vector field R(x, y) = o(|(x, y)| N ). Moreover, That is, if λ 0 is irrational, then for λ sufficiently close to λ 0 , the family is linearizable up to an N -flat term for any N . If λ 0 = p/q, with p, q positive, relatively prime integers, then for λ(µ) sufficiently close to λ 0 up to an N -flat term, all monomials can be eliminated except for the resonant monomials: u k x∂ x and u k y∂ y . When working with the equivalence and not conjugacy relation, it is legitimate to divide (7) by the component of x∂ x . Hence, for any N there exists a polynomial change of coordinates transforming orbitally the vector field family X µ to with X N F as in (3) and R(x, y), N -flat, with respect to |(x, y)|. One next applies the second step in the normalization process, eliminating the N -flat term R by means of a C k diffeomorphism. We use here the homotopic method (see for instance [4,12]). As the dependence with respect to the parameter µ is inessential, we omit mentioning it. In general, the homotopic method says that vector fields X and X + R are C k smoothly conjugate if the homological equation has a C k solution Z t . The time-one flow of the vector field Z t realizes the conjugation (if it exists). In [12] it is proved that for X hyperbolic and R infinitely flat, the homological equation (8) has a solution in the class C ∞ . The proof is done first in the semihyperbolic case. That is, one decomposes the remainder R = R 1 + R 2 where R 1 is flat with respect to the y variable and R 2 flat with respect to the x variable. One uses first the contractibility of the flow of X in the y direction for solving the equation [X + tR 1 , Z t ] = R 1 (9) and hence proving that X is conjugated to X + R 1 . Next, one proves that X + R 1 and X + R 1 + R 2 are conjugated by solving the equation using the contractibility of X + R 1 + tR 2 for negative time. The two equations being of the same type, we comment only on (9). In order to solve it one globalizes first the vector field. That is, one modifies the ∂ y component of X in a complement of a small neighbourhood of the origin in such a way that the flow of the modified vector field is well defined for positive time and all solutions tend to the x axes as t → +∞. By abuse, we keep the same notation for the modified vector field X. A solution of (9) is given by where D(X + tR 1 ) is the solution of the first variational equation of the modified vector field X + tR 1 and φ is its flow (see [4]). Using the flow-box theorem for the vector field X +tR 1 , it is easy to see that if the integral (10) is uniformly convergent, then it verifies (9). Further dominated convergence estimates are needed to assure the differentiability of Z t . In [12], these estimations are given in the C ∞ smoothness case. It is easy, following this proof, to see that a solution Z t of class C k of (9) exists provided that R 1 is sufficiently flat with respect to the y variable. The difficulty is that the required flatness as it appears in the proof of Proposition 2.2.11 in [12] depends on the norm of X. In our application, the vector field X appears as a result of Takens normal form procedure. It could happen that when obtaining higher flatness of R 1 as a result of applying Takens normal form procedure, the norm of X grows and an even higher flatness of R 1 would be required. It is sufficient to show that the required flatness N (k) of R 1 assuring the existence of a C k solution of (9) depends only on the linear part of the vector field X (which in not modified by the Takens normal form procedure). This is proved by Samovol in [15] where he proves that the required flatness N (k) in the homological equation (9) depends only on the linear part of the vector field (but only the case of vector field without poles is considered). An explicit estimates of N (k) appears also in [10] in the case of linear X, but the proof is very sketchy. In general, the independence of N (k) on higher order terms of X can be proved along the lines of our proof of Theorem 2.3. Yakovenko informed us that equation (9) can be reduced to (6) with v = 1. A detailed proof of an analogous problem for diffeomorphisms appears in [1]. Finally, in the finite orbital codimension case, the polynomial normal form can be improved using the Weierstrass preparation theorem (see [4]). We perform a similar construction concerning the temporal part in the next section.

Proof of Theorem A.
Proof of Theorem A. Fix k ∈ N and µ 0 ∈ U, and let N = N (k, λ 0 , m, n) be the integer given by Theorem 2.3. Take any s > N. By Theorem 2.1, there exists a C s change of coordinates Φ 0 µ such that , where X s µ = x∂ x + −λ(µ) + P s µ (u) y∂ y and R s µ is a C s function vanishing at the origin. (Here Φ 0 µ is the equivalence between X µ and X s µ that provides Theorem 2.1 and we took into account that it is tangent to the identity and preserving the axes.) Next we shall "simplify" the function R s µ by means of a conjugation and to this end we apply Lemma 2.2. Thus, recall Remark 3, the idea is to take the diffeomorphism Φ 1 .
The vector field X s µ acts linearly on the vector space (v)R[x, y] and note that its image contains all the monomials of (v)R[x, y] which are not inside R(u) because In other words, Hence we can choose f µ (x, y) as a polynomial so that .
We point out that the vector field Z s µ can be written as in (2), i.e., it is of the form 1/v times a smooth vector field at the origin because u ℓ /v has the same property. Therefore we can apply Lemma 2.2 and consider the coordinate transformation .
Our goal is to annihilate vh s µ by choosing an appropriate g µ . The problem reduces to solving the homological equation X s µ (vg µ ) = −vh s µ . Since h s µ is a s-flat function with s > N, by applying Theorem 2.3 we can assert that there exists a C k function g µ verifying the aforementioned homological equation. In short we have that It is important to mention that N in Theorem 2.3 depends only on the linear part of the vector field X s µ , which is independent of s. This enables us to fix in advance the required flatness of h s µ (x, y) in order to get the C k conjugacy Φ 2 µ that annihilates it. This constitutes the key point in all the process because X s µ does depends on s. Assume finally that the original vector field X µ0 has orbital codimension κ < ∞. In this case, by applying Theorem 2.1 we have that X s µ = x∂ x + −λ(µ)+P s µ (u) y∂ y , where P s µ is a polynomial in u with deg(P s µ ) 2κ for µ ≈ µ 0 and such that P s µ0 has order κ at u = 0. Again, on account of the definition of ℓ, W s µ can be written as in (2) because As before we consider Φ 3 µ := Φ W s µ , τ µ where τ µ is a smooth function to be determined. However now we want it of the form τ µ (x, y) = τ µ (u)/v. The reason for this will be clear in a moment but note that if u ℓ |τ µ (u) then, by the definition of ℓ, τ µ will be regular at (x, y) = (0, 0). By Lemma 2.2 we can assert that .
Then, since v τ µ = τ µ depends only on u, the above denominator becomes v + u ℓ Q s µ (u) + τ ′ µ (u)X s µ (u) and an easy computation shows that X s µ (u) = u p − λ(µ)q + P s µ (u) . Thus, since λ 0 = p/q and P s µ0 (u) has order κ at u = 0, by applying the Weierstrass Preparation Theorem, we have that X s and so we seek for a function τ µ such that u ℓ Q s µ (u) + uτ ′ µ (u)A s µ (u)B s µ (u) has few monomials. This "simplification" depends on weather ℓ is positive or negative.
(The last equality follows from taking (11) into account.) That is, .
Since the proof of Theorem 2.3 is very technical, we begin by giving first its idea omitting the dependence on µ to simplify the exposition. Let ϕ t : (x, y) −→ ϕ(t; x, y) be flow at time t of a given vector field X and consider also a given function H. In this case, if OF RESONANT SADDLES AND THE DULAC TIME   1231 is a well-defined smooth function then it is a solution of the homological equation X(F ) = H. Indeed, by making the change of variables τ = t + s we obtain Our goal is to solve the homological equation (6), where recall that v = x m y n with m, n ∈ Z. Note that it coincides with the above one taking H = vh and F = vf . The strategy consists in modifying conveniently X and h in order to make F welldefined and f = F v to be of class C k . Taking this into account, let us introduce the functions that will appear in the proof of Theorem 2.3.
So let us consider the homological equation X(vf ) = vh. Since h is N -flat, denoting by M the integer part of N/2, we can decompose it as a sum h = h 1 + h 2 , with h 1 and h 2 being M -flat with respect to x and y respectively (see [15]). The first step in the proof will be to show that there is no loss of generality in assuming that the homological equation is X N F (vf ) = vh, where X N F is the vector field in normal form provided by Theorem 2.1. Accordingly we consider where ϕ t is the flow of X N F µ . In order to study F we must control the function v • ϕ t , which satisfies the differential equation Consequently and therefore f = F v is given by In order to prove that f is a well-defined C k function we must bound the derivatives of I i and, in particular, the derivatives of the flow ϕ t with respect to (x, y, µ) ∈ R 2+m . To this end some technical lemmas are needed. From now on, if g is a symmetric l-linear form on R n and v 1 , . . . , v l ∈ R n , then we shall write g(v 1 , . . . , v l ) ∈ R as gv 1 · · · v l . The following result provides an expression for the chain rule of higher order. (Its proof, being straightforward, is omitted for the sake of shortness.) Here J = (J 1 , . . . , J l ) is any l-tuple of vectors in (N∪{0}) m verifying J 1 +· · ·+J l = I and {C I J } is a collection of constants with C I I = 1. We shall also use the well-known Gronwall's Lemma (see for instance [19]).  ν. Then, for each i 1, there exists a constant K i > 0 such that the total i-differential of the flow ϕ t of X verifies D i ϕ t V K i e iν|t| for all t ∈ R.
Proof. We proceed by induction on i. Due to D i ϕ t V = max ∂ i I ϕ t V : |I| = i , it is clear that it suffices to prove the inequality for any partial derivative of order i.
Let us prove the result for i = 1. To this end let I 1 , I 2 , . . . , I n be the canonical basis of R n and consider some ∂ Ij ϕ t . This partial derivative verifies the first variational equation, namely d dt ∂ Ij ϕ t = DX •ϕ t ∂ Ij ϕ t with initial condition ∂ Ij ϕ t t=0 = I j . Accordingly Consequently, since DX V ν by assumption, the function u 1 (t) = ∂ Ij ϕ t V satisfies Applying Gronwall's Lemma to each inequality we obtain respectively u 1 (t) 1 + Hence u 1 (t) e ν|t| and so this proves the result for i = 1.
Assume now that the result is true for j < i and fix some multi-index I with |I| = i. Since ϕ t is the flow of the vector field X, we have that d dt ∂ i I ϕ t = ∂ i I (X • ϕ t ) with ∂ i I ϕ t t=0 = 0. We expand the right hand side of the above equality by applying (a) in Lemma 4.1 to the each component and, after integration, we obtain Note that the second summation above is taken over all the l-tuples J = (J 1 , . . . , J l ) with J 1 + · · · + J l = I. Therefore we can split it up as Then, denoting u i (t) = ∂ i I ϕ t V and taking C I I = 1 into account, by using the inductive hypothesis we obtain Here the positive constant depends continuously on D j X V , j = 2, . . . , i. Finally, by applying Gronwall's Lemma, it follows that where we take K i = max KI i(i−1)ν : |I| = i . This completes the proof of the result.
The following result will be used to bound the derivatives of I k (x, y, µ, t) with respect to x, y and µ. Note that it refers to the functions J k (x, y, µ, t) such that I k = e (m−λn)t J k .
Let h 1 and h 2 be M -flat functions on R 2 × U with respect to x and y respectively. In addition, for k = 1, 2, define where ϕ t is the flow of X. Then, for each i = 0, . . . , M , we have , for all (x, y, µ) ∈ V δ and some positive constant K (independent of x, y, µ and t).
Proof. The flatness assumption on h 1 and h 2 means that, for 0 r M, for all (x, y, µ) ∈ R 2 × U. It is easy to show that the first two components of the flow ϕ t are given by

The combination of this with (14) yields
for each r = 0, . . . , M . The case i = 0 follows easily from the above inequalities with r = 0 and the bound for χ. On the other hand, from (a) in Lemma 4.1, if j 1 then It is important to note that the second summation above is taken over all the multi-indices L 1 , . . . , L ℓ such that L 1 + · · · + L ℓ = J. In order to avoid cumbersome notations, when there is no risk of confusion we use a "universal" positive constant K (meaning that it is something independent from x, y, µ and t). Taking this into account, by using Lemma 4.3 we get |∂ j J χ(x, y, µ, t)| K |t| 0 e jνs ds K e jν|t| for all t ∈ R.
Hence, from (b) in Lemma 4.1, it follows that Exactly in the same way as we bound |∂ j J χ|, the combination of Lemmas 4.1 and 4.3 shows that Now, the two last inequalities and the well-known formula imply that if i ≥ 1 then Finally, thanks to (15), we obtain the desired inequalities.
Proof of Theorem 2.3. Given {X µ } as in (1), we must prove that if {h µ } is a family of N -flat functions with N N (k, λ 0 , m, n), then the homological equation X µ (vf µ ) = vh µ has a solution {f µ } of class C k . We claim that it suffices to prove this taking the normal form family X N F µ that appears in (5) instead of the original {X µ }. Indeed, thanks to Theorem 2.1, there exists a family of diffeomorphisms which clearly is also a family of N -flat functions. Now, if the corresponding homological equation From now on and, as we have just shown, without loss of generality, we study the equation In order to construct a solution of the homological equation it is convenient that the flow of X µ is defined for all t ∈ R. This can be achieved by a "globalization process" using a suitable family of bump functions. More precisely, we consider a family of C ∞ bump functions {ψ ε } such that and verifying moreover Dψ ε < c ε for a fixed c > 2. Then, setting h ε µ = h µ ψ ε and which coincides with (17) on the ε/2-disk centered at the origin. Now, as we explained before, taking M = N 2 , we write h ε µ = h ε µ,1 + h ε µ,2 with h ε µ,1 and h ε µ,2 being M -flat with respect to x and y respectively. Then we define where I ε l (x, y, µ, t) = e (m−λ(µ)n)t h ε µ,l • ϕ t (x, y, µ) exp n t 0 P ε µ • ϕ s (x, y, µ) ds and ϕ t (x, y, µ) is the flow of X ε µ . It is important to mention that this flow is defined for all t ∈ R because X ε µ is linear outside a compact set. Now the key point is to prove that (19) is a well defined C k function because then, as we showed before, it is straightforward to verify that it is a solution of (18). The rest of the proof is dedicated to showing that this is the case provided that ε and µ − µ 0 are small enough.
By continuity of η(ε, δ) and ν(ε, δ), there existε,δ > 0 such that η = η(ε,δ) and ν = ν(ε,δ) are close enough to 0 and ν 0 respectively, in order that the inequalities hold for µ − µ 0 <δ. Thus, by applying Lemma 4.4, we can assert that the inequalities are verified for 0 i k. (To see this we also used the formula in (16) for the derivation of a product.) Therefore, since α 1 > 0 and α 2 < 0 by construction, the functions ∂ i I Iε 1 and ∂ i I Iε 1 are integrable with respect to t on (−∞, 0) and (0, ∞) respectively. Thus, fε µ is a well-defined C k function and, accordingly, it is a solution of the homological equation (17) for (x, y) <ε/2 and µ − µ 0 <δ.  [3]). Moreover, the problem of existence of a uniform bound for the number of critical points of the period function on a family of polynomial (or analytic) vector fields can be seen as a problem analogous to the second part of 16th Hilbert problem on limit cycles. We see our work as a contribution to establishing a finite "cyclicity" result in finite codimension (i.e., existence of a local uniform bound) for the number of critical points of the period function of polynomial vector fields on hyperbolic or more general polycycles.
Let U be an open set of R m and let {X µ , µ ∈ U} be a C ∞ family of vector fields defined in some open set U of R 2 . Assume that the vector field X µ has a hyperbolic saddle p µ as unique critical point inside U . In this situation it is well know that there exists exactly two smooth transverse invariant curves S µ and T µ through p µ (depending also smoothly on µ). We also consider a family Y µ proportional to X µ but having poles along S µ and T µ of order m and n respectively. We make the convention that if m (respectively, n) is a negative integer then Y µ vanishes along the invariant curve S µ (respectively, T µ ) with multiplicity −m (respectively, −n). We can take a coordinate system (x, y, µ) on U× U ⊂ R 2+m such that p µ = (0, 0, µ), S µ = (x, y, µ) : x = 0 and T µ = (x, y, µ) : y = 0 .
In the coordinates mentioned above X µ and Y µ can be written as in (1) and (2) respectively. Our goal is to study these two families in a neighbourhood of a parameter µ 0 ∈ U such that λ(µ 0 ) = p q with with (p, q) = 1.
By applying Theorem A, in a neighbourhood of (0, 0, µ 0 ) ∈ R 2+m there exists a C k diffeomorphism Φ such that Here recall that v = x m y n , P µ and Q µ are polynomials in the resonant monomial u = x p y q and ℓ = min β ∈ Z : β(p, q) > (m, n) .
Theorem B is the main result of this part of the paper. It gives an asymptotic development of T (s; µ) near s = 0, uniform with respect to µ, assuming that m and n are not both negative. (This assumption implies that ℓ 1.) After exchanging coordinates if necessary, we assume that n 0. In order to state the result we must introduce the so called Roussarie-Écalle compensator, namely, We also define α(µ) := p − λ(µ)q.
Theorem B. With the above notation and assumptions, for each K ∈ N, we have that where, for µ ≈ µ 0 , Ψ K (s; µ) is a K-flat function at s = 0 uniformly on µ. Moreover (a) a 0 = −q in case that (m, n) = (0, 0) and zero otherwise. (b) A µ (z) and B µ (z, w) are polynomials in z and w and their coefficients are rational functions in the coefficients of P µ and Q µ in (4) without poles at µ = µ 0 . (c) The order of B µ (z, w) at (0, 0) is min(n, qℓ) and, if mq − pn = 0, then A µ (0) = 1 λ(µ)n−m . Let us clarify that the above expression of T (s q ; µ) provides, after the substitution of s by s 1/q , the asymptotic development of T (s; µ) at s = 0 for µ ≈ µ 0 . We prefer to state it in this way for the sake of shortness and simplicity in the proof. In order to prove Theorem B let us first note that, by construction, if x t (s), y t (s) is the solution of X N F µ = x∂ x + −λ(µ)+ P µ (u) y∂ y with initial condition (x 0 , y 0 ) = (s, 1), then where recall that v = x m y n and u = x p y q . Thus T (s; µ) is a finite linear combination of terms Here and in what follows, in order to avoid long formulae we omit the dependence on µ when there is no risk of ambiguity. Clearly T 00 (s) = − log s and, in case that i = 0, T i0 (s) = s i −1 i . So it suffices to study T ij (s) for j = 0 and to this end we take advantage of some results of Roussarie in [13,Chapter 5]. For the sake of clarity we collect them in the following lemma: Lemma 5.1. For each t 0, u t (s) = x t (s) p y t (s) q can be expanded as a series in s as where g 1 (t) = e αt and g k (0) = 0 for k 2. In addition, for each r 0, we have that |∂ r g k (t)| C r C k e tk/3 for all t 0 and µ ≈ µ 0 for some constants C and C r (independent of t, µ and k). Finally, g k (t) = e αtḡ k−1 (t) withḡ k−1 (t) being a polynomial of degree k − 1 in Ω(t, α) := e αt −1 α . It is to be noted that the upper bound of ∂ r g k in Lemma 5.1 is slightly different from the one in [13] because there the exponential factor is e tk/2 instead of e tk/3 . This is only a technicality. Indeed, one can easily verify that if µ is close enough to µ 0 so that |α(µ)| < 1/3 then we can replace 1/2 by 1/3 in the exponent. Now, with the notation introduced in Lemma 5.1, it follows that Note that there are as many summands above as the number p(k) of partitions of k and it is easy to see that p(k) is the general term of a convergent series in k provided that s and α are small enough. In short, we have shown that Our next goal is to bound the derivatives of the k-th term in the above series. More concretely, we prove the following result: Lemma 5.2. For each r 0 there exits a positive constant C r such that |∂ r s pk+i T ijk (s) | k r 4|j|C r k s (p−2/3−|α|)k−r+λj .
Proof. The case r = 0 follows directly from (21). To study the case r 1 let us introduce the functionh jk (s) =ḡ jk (− log s), so that ∂T ijk (s) = −h jk (s)s λj−i−1 . By (a) in Lemma 4.1, we have that for some collection of constants {C r i } i,r . Accordingly, by applying Lemma 5.1, there exist positive constants C and C r such that Here C is the same as in Lemma 5.1 whereas C r is not. Sinceḡ k (− log s) = s α g k+1 (− log s), on account of (16) we get and consequently Now, by using the above estimates in the r-th derivative of (20), ∂ rh jk (s) = = k l=1 i1+···+i l =k j/q l j1+···+j l =r C j1,...,j l ∂ j1 ḡ i1 (− log s) · · · ∂ j l ḡ i l (− log s) , we obtain that (In the two inequalities above, and in what follows, for the sake of simplicity C r stands for a "universal" constant not depending on k.) Hence, due to ∂T ijk (s) = −h jk (s)s λj−i−1 , from (16) we conclude that Finally, the inequality in the statement follows by applying the derivation formula in (16) once again. Proposition 1. Fix (i, j) ∈ I with j = 0. Then, for each K > 0, there exists M (K) > 0 such that is a K-flat function at s = 0 for µ ≈ µ 0 .
Proof. Since α(µ 0 ) = 0, there exists a constant β such that If s is small enough then ∞ k=M(K) k r 4|j|C r s β k is a convergent series for each r = 0, . . . , K. Denote by C the maximum of their values for r = 0, . . . , K. Then, by Lemma 5.2 and taking M (K) > K−λj This proves the result.
Proof of Theorem B. Set Ψ K i0 ≡ 0. For j = 0, consider the functions Ψ K ij given by Proposition 1 and define T K ij = T ij − Ψ K ij . Then it follows that where T K := (i,j)∈I T K ij and Ψ K is K-flat at s = 0. Here recall that I = (m, n) ∪ ν(p, q) : ν = ℓ, . . . , ℓ + deg Q µ .
Note moreover that T 00 (s) = − log s and, in case that i = 0, T i0 (s) = s i −1 i . So it suffices to study T K ij with (i, j) ∈ I and j > 0. To this end notice that where T ijk are the functions introduced in (22). We claim that the following is verified: (a) If (i, j) = (m, n) with mq − np = 0 and n > 0, then where B 0 (z, w) is a polynomial of order n at (0, 0).
We will show in addition that, for each ν 0, the coefficients of B ν are rational functions in the coefficients of P µ in (5) without poles at µ = µ 0 .
In view of (a) it is clear that to prove the result it suffices to study those terms arising from (i, j) = ν(p, q) with ν > 0. However this is easy because, once again from (24), we can write s pq(k+ν) T νp,νq,k (s q ) as a polynomial in s p and s p ω s, α(µ) , which contributes to the terms of B µ s p , s p ω s, α(µ) in T (s; µ). This concludes the proof of the result. 6. Perspectives. In this section we give some perspectives for future work. The principal motivation for this work was the study of asymptotic properties of the period function near a hyperbolic polycycle. We give an asymptotic development of the Dulac time near a hyperbolic singular point. It is important to note that our Dulac time T in Theorem B is measured between normalized transverse sections which are constructed using the diffeomorphism that brings to the normal form (4). In order to have a result on the Dulac time between arbitrary transverse sections one must add to the local Dulac time T in Theorem B the two times necessary to go from given transverse sections to the normalized ones. The times must be calculated in the coordinate on the source transversal. This leads to a composition problem. We postpone the solution of this problem to the general paper dealing with hyperbolic polycycles to which we hope to come in a near future. In any case we must study then the composition problem in detail.
Note that the monomials log s, s mq+kpq , s jp ω j appearing in the asymptotic development permit a process of derivation division generating a Chebyshev system (see [6,11]). One can hope that this can be generalized to the total period of a hyperbolic polycycle and that hence in finite codimension one can prove non-accumulation of critical periods on hyperbolic polycycles.
It would be useful to know the structure of the coefficients in the Dulac time and divide the Dulac time similarly as for the Dulac map in [11]. It seems out of reach for the moment.
Note that our study covers all the cases of the polar factors of the vector field (2), except for the case m, n < 0. This last case occurs when both separatrices are lines of zeros of the vector field Y . For studying the accumulation of critical periods we don't have to study this case, since in this case the derivative of the Dulac time tends to infinity uniformly on the parameters and hence no critical periods can appear. Nevertheless, in this case very interesting resonances between the order of poles (m, n) and the eigen-values (p, q) seem to appear, leading to higher order compensators. We hope to return to this problem later.
Some parts of the present study apply also to the saddle node case. However, in the treatment of the remainder term via Theorem 2.3 only the part corresponding to the strong variable can be eliminated.