Loïc Teyssier

Coalescing complex planar stationary points

Keywords: Coalescing complex planar stationary points Saddle-node bifurcation, normal forms, holomorphic vector fields, unfolding of singularities, modulus space, inverse problem

tion). We particularly show that in general the invariant cannot depend holomorphically on the parameter over a full neighborhood of the bifurcation value. Among all bifurcation behaviors of parametric families of real planar vector fields Z • = (Z λ ) λ∈Λ , those that stand out most prominently are confluences of distinct stationary points. The qualitative change is so drastic that in some classes of families (e.g., foldlike bifurcations) the stationary points generically annihilate each other in the process (Sotomayor's theorem). The simplest example of such a behavior, an instance of saddle-node bifurcation, is the polynomial family (X ∞ λ ) λ∈R given in the canonical basis of R 2 by X ∞ λ (x, y) :=

x 2 + λ y .

(1.1)

The bifurcation value occurs at λ = 0: for negative λ the system has two stationary points located at ± √ -λ, 0 which collide as λ reaches 0, while none remain for λ > 0. The stationary points have left the real plane, true enough, but only to slip into the complex domain. Let us elaborate a bit on this observation in order to motivate the need for complexifying the whole setting, even in the context of real dynamics.

The trajectories t → (x (t) , y (t)) of X ∞ λ appear naturally as solutions of the autonomous flow-system of X ∞ λ :

ẋ (t) = x (t) 2 + λ ẏ (t) = y (t)
and can be implicitly expressed by solving the associated non-autonomous differential equation. This equation is obtained by eliminating the time in the flow-system using the rule ẏ ẋ = dy dx :

x 2 + λ y ′ (x) = y (x) .

Separation of variables yields multivalued complex solutions

y λ : z -→c z -i √ λ z + i √ λ 1 /2i √ λ , c ∈ C . (1.2)
On the one hand, if λ < 0 real solutions are given on appropriate intervals by

y λ : x -→c x - √ -λ x + √ -λ 1 /2 √ -λ , c ∈ R , (1.3)
from which we deduce that -√ -λ, 0 is a saddle-point and √ -λ, 0 a node-point. On the other hand, for λ > 0 we have

y λ : x -→c exp 1 √ λ arctan x √ λ , c ∈ R . (1.4)
The latter is a perfectly honest real-analytic function on R. One might wonder why, despite the fact of being so regular a function, its Taylor expansion at 0 does not have infinite convergence radius instead of √ λ. One can explain the discrepancy by, say, direct use of Cauchy-Hadamard formula, although one cannot understand its source without noticing the imaginary singularities ±i √ λ quietly sitting on the boundary of the disk of convergence. Also it is hard to understand why, when playing the movie backwards starting form positive values of λ and reaching negative ones, a stationary point somehow pops out of nowhere. One can see the singularity coming only when looking along the imaginary axis.

At a less commonplace level, when λ < 0 both stationary points organize the dynamics of X ∞ λ and there is no reason why they should stop when λ > 0, or even when λ is not real, and we will present how.

Class of parametric families

The present chapter deals with germs of a parametric family of vector fields in the complex plane, enjoying a saddle-node bifurcation of codimension 1 and corresponding to firstorder non-linear differential equations. The basic examples addressed by the text are affine families perturbing X ∞ • X • : X λ (x, y) :=

x 2 + λ y -x 2 + λ a λ (x) ,

where (λ, x) -→ a λ (x) is a given analytic function near the origin of C 2 . Although for such elementary families all computations can be performed explicitly (variation of constant), some natural questions and non-trivial answers arise already in this case-study.

Generalizing the constructions and objects introduced in that simple situation to arbitrary bifurcation-preserving analytic perturbations of the model family X ∞ • is the main concern of the rest of the text.

Reducing the setting to analytic parametric families may seem rather restrictive. Yet the geometric approach we present here could be inherited by less regular situations, or could give insights as to where sources of peculiar behavior may lie. On the other end of the argument, the obvious added benefit stemming from this restriction is the rigidity of holomorphic functions and diffeomorphisms of complex (compact) manifolds. Also the analytic class comprises polynomial vector fields, of special interest for planar vector fields, e.g., regarding Hilbert's 16 th problem on the number / position of limit cycles, or Poincaré's problem on the existence of rational first integrals.

Scope of the study

In the sequel we investigate the links between local dynamics on the one hand, local classification (i.e., up to local changes of analytic coordinates and parameters) on the other hand, while at the same time hinting at how they can help measuring divergence of some class of «summable» power series. We particularly explain the role of complex geometry and analysis in understanding saddle-node bifurcations. We wish to underline that the two objects Z 0 and (Z λ ) λ =0 are intertwined, as dynamical properties for one can be deduced from studying objects attached to the other and vice versa.

C. Rousseau has pioneered the classification of some non-linear (discrete or continuous) dynamical systems having a saddle-node bifurcation [START_REF] Mardešić | Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms[END_REF][START_REF] Rousseau | Modulus of orbital analytic classification for a family unfolding a saddle-node[END_REF][START_REF] Rousseau | Normal forms for germs of analytic families of planar vector fields unfolding a generic saddle-node or resonant saddle[END_REF][START_REF] Rousseau | Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle[END_REF][START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF][START_REF] Rousseau | The moduli space of germs of generic families of analytic diffeomorphisms unfolding of a codimension one resonant diffeomorphism or resonant saddle[END_REF]. She has also contributed to the study of families of vector fields corresponding to linear differential systems in finite-dimensional complex linear spaces, with Fuchsian singularities merging to an irregular singularity [START_REF] Lambert | The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation[END_REF][START_REF] Lambert | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1[END_REF][START_REF] Lambert | Moduli space of unfolded differential linear systems with an irregular singularity of Poincaré rank 1[END_REF][START_REF] Klimeš | Analytic classification of families of linear differential systems unfolding a resonant irregular singularity[END_REF][START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF]. The Stokes matrix of the irregular system is recovered as the limit of well-chosen monodromy matrices of the Fuchsian systems. The linear / non-linear and discrete / continuous settings all share the same basic idea, which can be summarized as the following recipe: ⊛ find a finite decomposition of both parameter space and dependent-variable space in «sectors» over which the family is conjugate to some known, simple normal form; ⊛ form the classification invariants as transition maps between overlapping normalization sectors.

We do not wish to emphasize too much the link between local orbital classification of Z • and local classification of its strong holonomy h • , the family of holomorphic firstreturn map of Z λ on a fixed horizontal disc which crosses x 2 + λ = 0 . The connection is very clearly explained by C. Rousseau, for instance, in [START_REF] Rousseau | Modulus of orbital analytic classification for a family unfolding a saddle-node[END_REF] for saddle-node bifurcations, or again in [START_REF] Rousseau | The moduli space of germs of generic families of analytic diffeomorphisms unfolding of a codimension one resonant diffeomorphism or resonant saddle[END_REF] for deformations of a resonant saddle stationary point. Although both objects encode somehow the same dynamics, and are classified by the same invariants under local analytic equivalence and change of parameters, we take advantage of the extra dimension the complex plane C 2 offers to deploy more geometrical constructions à la Martinet-Ramis [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF][START_REF] Teyssier | Analytical classification of singular saddle-node vector fields[END_REF]. Instead of simply deducing the classification of vector fields from that of holonomies, which would frankly spoil all the fun, the present text is focused on building objects specifically from the continuous nature of the dynamics of Z λ . Although both moduli spaces end up with the same presentation, some formulations for vector fields yield different characterizations of, e.g., the «compatibility condition» as compared to holonomies [START_REF] Rousseau | Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle[END_REF].

A by-product of that approach is an explicit family of normal forms for bifurcations Z • having persistent heteroclinic connections, generalizing [START_REF] Schäfke | Analytic normal forms for convergent saddle-node vector fields[END_REF] to the case λ = 0 as done in [START_REF] Rousseau | Analytic normal forms and inverse problems for unfoldings of 2dimensional saddle-nodes with analytic center manifold[END_REF]. There is as yet no such known explicit universal family for holonomies h • (not even for h 0 ).

Contents description

The text begins with two preliminary sections devoted to covering basic examples, objects and tools, as well as fixing notations.

⊛ The example of affine families (1.5) is presented in Section 1.1. The thread of the exposition is the link existing between the (lack of) analytic center manifold of X 0 and the (lack of) persistence of heteroclinic connections in the family X • . In the process of revealing this bond through the use of elementary complex analysis, we perform the analytical classification of all affine families and present a collection of normal forms. ⊛ The other sections are framed in a geometric setting, with its own standard terminology. We present in Section 1.2 basic objects attached to singular holomorphic vector fields: directional derivative, flow, change of coordinates, and most importantly singular foliations, first integrals, leaves space and normal forms. We recall related basic results of differential geometry. Readers familiar with these concepts should skim briefly through this section mainly to fix notations.

The next six sections form a survey on formal and analytical classification of saddlenode vector field bifurcations. The choice has been made to focus mainly on precise constructions, while providing sketches of proof whenever doing so helps the exposition.

The missing technical details are to be found mostly in [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF][START_REF] Rousseau | Analytic normal forms and inverse problems for unfoldings of 2dimensional saddle-nodes with analytic center manifold[END_REF].

⊛ Section 1.3 is devoted to an introductory text, giving a brief historical overview of the emergence of moduli à la Martinet-Ramis for saddle-node vector fields and their deformations. The dynamical nature of these invariants is explained from their very construction as transition maps of a rigid analytic atlas of the corresponding leaves space. ⊛ We present a more detailed account of the formal and local classifications in Section 1. [START_REF] Bruno | Groupes d'automorphismes de (C, 0) et équations différentielles ydy + • • • = 0[END_REF], where the main theorems are stated and the structure of the construction is presented. Each one of subsequent sections develops a particular aspect.

⊚ The formal normalization is performed in Section 1.5 by reducing the problem to solving a couple of well-chosen cohomological equations. ⊚ Sections 1.6 and 1.7 contain the precise construction of the normalization sectors in parameter-space and dependent-variable-space, respectively. Ascertaining the rigidity of the corresponding leaves space atlas is literally what shapes the normalization sectors. ⊚ Section 1.8 contains the material needed to perform the sectorial normalization, by solving on such sectors the cohomological equations. The classification theorem is finally proven: to each family Z • = (Z λ ) λ corresponds a functional invariant m (Z • ), and the fiber of m over m (Z • ) consists precisely of the conjugacy class of Z. The general inverse problem (to determine the range of m) is still open.

The rest of the chapter is mostly concerned with the inverse problem and its dynamical ramifications. The last two sections involve more recent (and perhaps more difficult) material, coming with full proofs.

⊛ Section 1.9 is concerned with the dynamical interpretation of the invariant of classification m (Z • ), from which is formulated the «orbital compatibility condition». This condition is expected to solve the orbital inverse problem. The fact that (the orbital part of) m is not onto the natural candidate is established. The proof is based on the characterization of those Z • for which m (Z • ) is analytic in the parameter, which turns out to seldom happen. ⊛ Section 1.10 is concerned by partial answers to the inverse problem. First we formulate the «temporal compatibility condition» and prove it completely characterizes the range of m for given orbital part, hence describing the moduli space of saddle-node bifurcations inducing the same given bifurcation of foliations. At last we provide analytical normal forms in the case of persistent heteroclinic connections. The combination of both temporal and orbital compatibility conditions is proved to solve the inverse problem completely in that (non-generic) case.

Affine saddle-node bifurcations

The study of affine saddle-node vector fields was initiated in the second half of the nineteenth century by C. Bouquet and C. Briot [START_REF] Briot | Recherches sur les propriétés des fonctions définies par des équations différentielles[END_REF], starting with a collection of examples of invariant manifolds existing at a formal level but not at an analytic one, generalizing the famous behavior displayed by Euler's differential equation x 2 y ′ = y -x. Their first significant result is the existence of a formal weak separatrix for X 0 , that is an invariant formal curve {y = ŝ (x)} with ŝ ∈ C [[x]]. They obtained an explicit criterion for convergence of s in terms of the Taylor coefficients of a 0 , which we recover in Proposition 3 after a study aimed at understanding how the trajectories of X λ , λ = 0, evolve into those of X 0 . We begin our study of the affine collection Affine [START_REF] Birkhoff | Déformations analytiques et fonctions auto-équivalentes[END_REF], whose members are given by (1.5), by presenting the easiest instance a • := 0. Standard results describing the regularity of parametric solutions state that solutions y λ in (1.3) and (1.4) converge to corresponding solutions for λ = 0

x 2 y ′ (x) = y (x)

(1.6)

y 0 : x -→ c exp - 1 x , c ∈ C
uniformly on compact subsets of R =0 as λ → 0 (it suffices to wait until the stationary points have left the compact set). Now, can we say something about the convergence near 0? Obviously the question only makes sense for families of solutions bounded near 0 as λ → 0. For λ = 0 the limiting objects are center manifolds of the saddle-node stationary point (0, 0) of X ∞ 0 . As a real vector field X ∞ 0 has infinitely many center manifolds passing through (0, 0), each one given by the graph {y = s (x)} of the smooth (meaning C ∞ ) function

s : R -→ R x ≤ 0 -→ 0 x > 0 -→ c exp - 1 x
for arbitrary c ∈ R. Those are the only bounded solutions of (1.6) at 0. Only one of them is analytic there, namely S 0 := {y = 0}, all others being non-zero flat functions. This property identifies uniquely a distinguished center manifold, called the weak separatrix of X ∞ 0 , with the most regular dynamics. The weak separatrix is the limiting curve of the family (S λ ) λ collecting the only smooth integral curve connecting both stationary points ± √ -λ, 0 for λ < 0. In this simple situation the only such heteroclinic integral curve is S λ = {y = 0}, since y λ in 1.3 is not of class C r+1 at the node, r :=

1 2 √ -λ , save for c = 0.
Consider next a quadratic perturbation of X ∞

• , the Euler family

E λ (x, y) := x 2 + λ y -x 2 + λ (1.7)
whose stationary points are again located at (±s, 0) where, for the sake of simplicity, we set:

s := √ -λ .
This is a special member of Affine (1) obtained by setting a • := 1, yet we are to prove that together with X ∞ • they somehow span all possible behaviors for members of the whole collection.

For λ = 0 infinitely many smooth center manifolds persist through (0, 0), given by the graphs of

s 0 : R -→ R (1.8) x < 0 -→ exp - 1 x ˆ0 x exp 1 u du 0 -→ 0 x > 0 -→ exp - 1 x c + ˆ1 x exp 1 u du .
A standard calculus exercise consists in checking for the smoothness of s 0 . Yet none of these functions can be analytic, as if one were it would possess a convergent Taylor series ŝ at 0 solving

x 2 ŝ′ (x) = ŝ (x) -x 2 .
(1.9)

A straightforward computation yields the unique formal power series

ŝ (x) = x n≥0 n!x n+1 , (1.10)
which has null radius of convergence. We say in that case that we encounter a divergent weak separatrix. It is worth mentioning that the Taylor expansion of each s 0 at 0 is ŝ.

Here we cannot distinguish a preferred center manifold in the class of analytic objects at (0, 0). Although the divergence of the weak separatrix can be explained computationally for the Euler family, the generic perturbation X • ∈ Affine (1) is impossible to deal with this way since no reasonable closed-form formulas for the coefficients of ŝ exist in general. Even so the basic formal approach, computing coefficients of ŝ one after the other, cannot prove nor disprove the power series convergence in finite time. We propose a dynamical approach instead to trace back the source of the divergence (Theorem 1), which leads to the semi-decidability of the convergence of s: there exists an algorithm taking a «computable» a • as input and stopping in finite time if and only if s diverges.

The key is to check whether the complex contour integral

ϕ n 0 := 1 2iπ ˛rS 1 a 0 (z) z + s z -s 1 /2s dz ∈ C
vanishes (meaning convergence), for r > 0 small enough. This viewpoint also allows us to find a complete collection of normal forms (Theorem 2).

When λ < 0 write S - λ the (analytic) stable manifold of the saddle-point located at (-s, 0) and, when it exists, S + λ the (analytic) unstable manifold of the node-point at (s, 0). What happens in the Euler family is that no heteroclinic connection between stationary points takes place: S - λ does not coincide with S + λ . We aim at establishing this property has a predominant bearing on the convergence of the weak separatrix.

Theorem 1. Consider a family X • ∈ Affine (1) as in (1.5). The implications (1)⇒(2)⇒(3) hold, and if moreover ∂a λ ∂λ = 0, then (3)⇒(1). 1. The vector field X λ has a heteroclinic connection for all λ < 0 sufficiently close to 0. 2. The vector field X λ has a heteroclinic connection for values of λ < 0 accumulating on 0. 3. The vector field X 0 admits a convergent weak separatrix (that is, an analytic center manifold).

Remark 1.

1.

In each item of the theorem the corresponding property is equivalent to the existence of an open interval I ∋ 0 such that the differential equation

x 2 + λ y ′ (x) = y (x) -x 2 + λ a λ (x) (1.11)
admits a solution analytic on I, for every corresponding values of λ. The solution is necessarily unique. 2. The practical usefulness of the theorem is by contraposition: if we happen to know that X 0 has a divergent weak separatrix, then any unfolding X • ∈ Affine (1) of X 0 eventually sheds all heteroclinic connections as λ goes to 0.

We prove this theorem for the Euler family E • in the next Section 1.1.1 for (2)⇒(3) and Section 1.1.2 for (3)⇒(1). After that step there are two ways to process the general case. On the one hand, the proof performed in Euler's case could be adapted straightforwardly to fit the more general setting. On the other hand, we can provide a collection of normal forms X κ • for Affine (1) on which the validity of the equivalences is easily read. This approach brings also the benefit of characterizing completely situations for which (3)⇒(1) holds.

Theorem 2. Consider a family X • ∈ Affine (1) as in (1.5).

1. There exists a unique

κ ∈ N := Z ≥0 ∪ {∞} such that X • is conjugate to one of the models X κ • X κ λ (x, y) := x 2 + λ y -λ κ x 2 + λ
where we conventionally identify λ ∞ with 0. This conjugacy can be chosen fibered in the variables x and λ. Moreover families X κ

• are mutually orbitally non-equivalent for differing values of κ. 2. The implication (3)⇒(1) in Theorem 1 holds if and only if κ ∈ {0, ∞}. Notice that the condition ∂a λ ∂λ = 0 implies κ ∈ {0, ∞}. This theorem, proved in Section 1.1.3 below, discriminates all three possible qualitative dynamical behaviors occurring in Affine (1).

κ = 0

Pure divergence. For every λ = 0 sufficiently close to 0 the vector field X λ has no heteroclinic connection while X 0 has a divergent weak separatrix. κ ∈ N >0 Sly convergence. For every λ = 0 sufficiently close to 0 the vector field X λ has no heteroclinic connection although X 0 has a convergent weak separatrix. κ = ∞ Pure convergence. For every λ = 0 sufficiently close to 0 the vector field X λ has a heteroclinic connection so that X 0 has a convergent weak separatrix.

Here the modulus space N for analytical orbital classification is discrete. The property no longer persists for families unfolding a more degenerate saddle-node, i.e. the coalescence of k + 1 stationary points with k > 1. We refer to [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF] for this more involved situation.

From heteroclinic connections to convergence

Forget for now that Euler's series (1.10) is divergent. We want to recover its divergence at λ = 0 by dynamical properties arising in the family when λ ∈ (C, 0) \ {0}. This story is told during the next two subsections 1.1.1 and 1.1.2. First, we must exclude values of the parameter λ for which there are no analytic unstable manifold through the node of the Euler vector field E λ . Although this phenomenon is not generic, it still turns up for an infinite discrete set of parameters accumulating on 0. The stable manifold is always unique, and the variation of parameters yields that it is given by the graph of

s - λ : ] -s, s[ -→ R x -→ s -x s + x 1 /2s ˆ-s x s + u s -u 1 /2s
du .

(1.12)

Proposition 1.

1. E λ admits a (unique) analytic unstable manifold S + λ if and only if

λ ∈ Λ := R <0 \ -1 4N 2 .
2. There exists a unique function c : -Λ → R such that for all λ ∈ Λ the manifold S + λ coincides with the graph of

s + λ : ] -s, s[ -→ R x -→ s -x s + x 1 /2s c (s) + ˆ0 x s + u s -u 1 /2s du .
3. c is analytic.

Proof. First notice that whatever the value of c (s) may be, the graph of s + λ is an integral curve of E λ even when λ / ∈ Λ. Swapping the order of summation and integration operations in the expansion

(u + s) 1 /2s =: ∞ n=0 α + n (s) (u -s) n ,
which converges uniformly on compact subsets of ] -s, 3s[, we isolate the candidate singular term of s + λ at s:

s + λ (x) = s -x s + x 1 /2s   c (s) - n+1 = 1 2s α + n (s) (-1) n s n+1-1 /2s n + 1 -1 2s -α * (s) ln s -x s   + (analytic at s) where α * (s) := 0 if 1 2s / ∈ N 1 otherwise .
Notice that (s -x) 1 /2s cancels out the non-integral exponent in the power series of the right-hand side. If 1 2s ∈ N no choice of c (s) ∈ C may yield an analytic s + λ . On the contrary for λ ∈ Λ we can only have

c (s) = ∞ n=0 α + n (s) (-1) n s n+1-1 /2s n + 1 -1 2s ,
which is an analytic function of s ∈ -Λ.

⊓ ⊔

A consequence of the proposition is the following: if 1 2s ∈ N, there is no heteroclinic connection, while a heteroclinic connection for λ ∈ Λ occurs exactly if

s - λ (0) = s + λ (0) , that is, if ϕ (s) := c (s) + ˆ0 -s s + u s -u 1 /2s du vanishes.
Corollary 1. If ϕ vanishes on a set accumulating on 0, then ŝ converges.

The proof requires our switching to complex analysis in order to use compactness of normal families of holomorphic functions. The main ingredient is therefore to show that s - λ -1<λ<0 extends to a uniformly bounded family of analytic functions on the slit unit disc

{z ∈ C\[s, ∞[ : |z| < 1} .
We need to slit the disc because the complex (multivalued) extension of s - λ is given by taking path integrals in the variation of constant method

s - λ (z) = s -z s + z 1 /2s ˆγ(z) s + u s -u 1 /2s du (1.13)
where γ is a piecewise smooth path linking z = s to -s. We choose the determination of the logarithm in such a way that s - λ (z) coincides with (1.12) on ] -s, s[. Remark 2. We will discuss the relevance of the multivaluedness of s - λ regarding the question of convergence of ŝ in the next section, when proving the converse of the corollary. Lemma 1. There exists C > 0 such that for every (λ, z

) ∈] -1, 0[× S 1 \ {1} we have s - λ (z) ≤ C .
Proof. Let us build an adequate integration path γ (z) for which bounds are easily obtained.

⊛ When ℑ (z) < 0 we first follow the shortest anticlockwise arc γ -(z) of S 1 joining z to -1, then the interval

I λ : = [-1, -s] .
⊛ Otherwise we follow the shortest clockwise arc γ + (z) of S 1 joining z to -1 before I λ .

For u ∈ I λ we have 0 < u+s u-s < 1 so that

ˆ-s -1 u + s u -s 1 /2s du ≤ (1 -s) .
Moreover there exists C 1 ≥ 0 for which

s - λ (z) ≤ s -z s + z 1 /2s (1 -s) + ˆγ± (z) s + u s -u 1 /2s du ≤ C 1 (1 -s) + π
because on the one hand, s+u s-u ≤ s+z s-z when z ∈ S 1 and u ∈ γ ± (z), while on the other hand, s+z s-z ≤ 1+s 1-s and lim s→0 1+s 1-s

1 /2s = e. ⊓ ⊔
We get on now with proving Corollary 1.

Proof. Each function s - λ , holomorphic on the slit unit disc, can be analytically extended to the whole D precisely when it is analytic near s. This happens precisely when the stable manifold of the node is analytic, in other words when ϕ (s) = 0. Let Ω ⊂ ϕ -1 (0) be a set accumulating on 0. Because of Lemma 1 and of the maximum principle we know that s - λ is bounded on D uniformly in -1 < λ < 0, i.e. the family s - λ λ∈Ω is normal. Thus by Montel's theorem we can consider an adherence value (for uniform convergence on compacts sets of D) which must be a solution of Euler's equation (1.9) with analytic Taylor expansion at 0. But there is only one such formal power series solving Euler's equation, namely ŝ.

⊓ ⊔ Remark 3. We can give a series representation for ϕ using the expansions

(s ± u) ± 1 /2s =: ∞ n=0 α ± n (s) (u ∓ s) n
where the determination of the logarithm on the left hand, side is chosen in such a way that the function is real on ] -s, s[. In particular

α + n (-s) = (-1) 1 /2s α - n (s) = α - n (s) exp iπ 2s .
In that setting

c (s) := ∞ n=0 α + n (s) n + 1 -1 /2s (-1) n s n+1-1 /2s
and it is easy to compute

ˆ0 -s s + u s -u 1 /2s du = ∞ n=0 α - n (s) n + 1 + 1 /2s (-1) n s n+1+ 1 /2s = -c (-s) , therefore ϕ (s) = c (s) -c (-s) . (1.14)

From convergence to heteroclinic connections

We just observed that if s - λ is uniform (that is, not multivalued) then ŝ converges. We want to establish the converse statement in the following way. When the formal solution of (1.9) converges it defines a real, entire holomorphic function, in particular for given x * > s

(x * , y * ) := (x * , ŝ (x * ))
is a well-defined point in R 2 . Consider the solution y λ of (1.11) with a = 1 and initial value (x * , y * ), and perform its local analytic (i.e. multivalued) continuation over

D λ :=D\ {±s} .
We are more particularly interested in the analytic continuation of y λ along the unit circle, which can be performed in the universal cover of D\ [-s, s] as we explain below. We identify the action of the deck transform of this covering with the symbolic multiplication of z by exp 2iπ, so that the analytic continuation of y λ along S 1 can be conveniently written y λ (x * exp 2iπ). Because (y λ ) λ converges uniformly on compact subsets of the preimage of S 1 in the universal cover as λ → < 0, if ŝ converges then its sum y 0 is uniform and we must have

lim λ→ < 0 y λ (x * exp 2iπ) = y * .
Let us see how this observation relates to the persistence of heteroclinic connections. Proposition 2. For every λ ∈ Λ and (x * , y) ∈ R >s × R write z → y λ (z, y) the solution of (1.11) with initial value (x * , y). The local analytic continuation of y λ (•, y) over S 1 follows the rule

y λ (x * exp 2iπ, y) = y -2i x * -s x * + s 1 /2s
s -s γ + 0 γ - Fig. 1.3: Generators of π 1 D λ , 0
In order to establish the proposition we need to understand the monodromy of

ĝs : z = ±s -→ s + z s -z 1 /2s
.

We fix a determination ĝ * s of ĝs on D\R ∪]s, s[ in such a way that ĝ *

s | ]-s,s[ coincides with the canonical real determination used previously. For any path γ, starting from 0 with image included in D λ , we define ĝs (γ) := ĝ * s (0) exp ˆγ du u 2 + λ .

Fix a system γ ± of generators of π 1 D λ , 0 whose index around ±s is 1 and 0 around the other point as in Figure 1.3. The monodromy of ĝs is multiplicative and given by ĝs γ ± γ = ĝs (γ) exp

˛γ± du u 2 + λ = ĝs (γ) exp ±iπ s ,
according to the residue formula and the identity

-1 z 2 + λ = 1 2s 1 s + z + 1 s -z .
In particular

ĝs γ -γ + γ = g s (γ)
so that ĝ * s is actually holomorphically extendable to D\ [-s, s], as claimed. We prove now the proposition.

Proof. The method of variation of parameters yields the following expression for the monodromy

y λ (x * exp 2iπ, y) -y = - 1 ĝ * s (x * ) ˛S1 ĝ * s (u) du .
It can be computed by deforming S 1 into the concatenation γ -γ + of generators of π 1 D λ , 0 given by

γ -: t ∈ [0, 1] -→ -s + s exp (2iπt) γ + : t ∈ [0, 1] -→ s -s exp (2iπ (t -1))
using the relation

˛S1 ĝ * s (u) du = ˆγ- ĝs (u) du + ˆγ+ ĝs (u) du
(notice that we do not use the symbol ¸for the paths γ ± because the integration is actually performed in the universal cover of D λ and the lift of γ ± is not a loop). Using the notations and formulas presented in Remark 3 we compute

ˆγ- ĝs (u) du = ∞ n=0 α - n (s) n + 1 + 1 /2s z n+1+ 1 /2s s exp 2iπ s = 1 -exp iπ s c (-s)
with z := u + s, then

ˆγ+ ĝs (u) du = ∞ n=0 α + n (s) (-1) n+1 n + 1 -1 /2s z n+1-1 /2s s s exp(-2iπ) = exp iπ s -1 c (s)
with z := s -u. The conclusion follows from

1 ĝ * s (x * ) = x * -s x * + s 1 /2s exp -iπ 2s
and from (1.14).

⊓ ⊔

We end the story by an explicit computation which settles the question of the divergence of ŝ. Proof. We just proved

ϕ (s) sin π 2s = π × exp -iπ 2s × 1 2iπ ˛S1 ĝ * s (u) du = π × 1 2iπ ˛S1 u + s u -s 1 /2s
du .

The latter integral can be evaluated using the residue formula at ∞ since z → z+s z-s

1 /2s
is holomorphic at this point. Setting w := 1 u we compute

z + s z -s 1 /2s = 1 + ws 1 -ws 1 /2s = 1 + w + o (w)
and

1 2iπ ˛S1 u + s u -s 1 /2s du = 1 2iπ ˛S1 exp (w + o (w)) dw w 2 = 1 .

Normal forms

We just established the equivalence in the Euler family between ⊛ divergence of ŝ, ⊛ absence of heteroclinic connections (non-vanishing of ϕ), ⊛ non-vanishing of the integral

ϕ n s := 1 2iπ ˛S1 u + s u -s 1 /2s du = 1 π ϕ (s) sin π 2s = 1 .
Remark 5. Be careful that the exponent «n» refers to «node», and is not meant to be thought of as a variable. The terminology choice will be explained in the next part of the chapter.

In order to establish the classification Theorem 2 we need to find an (almost) invariant quantity under changes of coordinates. This invariant turns out to be ϕ n • . One can argue that it suffices to consider ϕ instead, which is somehow nicer because of its dynamical flavor. Yet ϕ is afflicted of serious drawbacks: ⊛ ϕ presents an accumulation of poles as s -→ > 0, and there is no hope of extending it analytically at 0, ⊛ ϕ is not even: there is no hope of extending it holomorphically on any open annulus surrounding 0 as a function of λ.

None of these shortcomings hinder ϕ n • , even in the more general setting of affine unfoldings.

Proposition 3. For X • ∈ Affine (1) as in (1.5) we may find ρ > 0 such that (λ, x) → a λ (x) is holomorphic on (a neighborhood of ) ρ 2 D × ρD. For s ∈ ρD\ {0} define

ϕ n s := 1 2iπ ˛ρS 1 a -s 2 (u) u + s u -s 1 /2s
du .

(1.15)

1. The holomorphic mapping s → ϕ n s can be continued to an even germ of a holomorphic function at 0 satisfying

ϕ n 0 = 1 2iπ ˛S1 a 0 (u) exp 1 u du . 2. Write a λ (x) = ∞ n=0 φ n (λ) x n . Then for s ∈ ρD\ {0} ϕ n s = ∞ n=0 φ n -s 2 (n + 1)! × 1 2 n p+q=n n p p j=1 (1 + 2sj) q j=1
(1 -2sj) , with limit

ϕ n 0 = ∞ n=0 φ n (0) (n + 1)! .
3. The formal solution ŝ with ŝ (0) = 0 of

x 2 y ′ (x) = y (x) -x 2 a 0 (x)
converges if and only if ϕ n 0 = 0. The third statement of the proposition is actually Briot-Bouquet's result [START_REF] Briot | Recherches sur les propriétés des fonctions définies par des équations différentielles[END_REF]. Remark 6. We deduce the determination of g s := The multiplicative monodromy of g s is the same as that of ĝs .

Proof.

1. Although it is a consequence of 2. we can prove directly the property. First notice that ϕ n -s = ϕ n s . Also z → a -s 2 (z) g s (z) converges uniformly to z → a 0 (z) exp 1 z on ρS 1 as s → 0, so that s → ϕ n s is bounded on a pointed neighborhood of 0. Riemann's removable singularity theorem yields the conclusion. This is a trick used extensively in this text. 2. For n ∈ Z ≥0 let us evaluate

t s (n) := 1 2iπ ˛S1 u n g s (u) du .
The residue formula used in Lemma 2 to compute t s (0) sure works here, yet one would have to formally derive a closed-form for the Taylor coefficients of z → g s (z) at ∞, which is no trivial task. We relate instead the computation at hands to the Beta function, more precisely its integral representation along a Pochhammer contour around 0 and 1. Introduce first the contour around -s and s

P := γ + γ -γ + -1 γ --1 (1.16)
where γ ± are generators of π 1 D\ {±s} , 0 as described in Figure 1.3. The identity

˛P u n g s (u) du = exp -iπ s -1 ˛S1 u n g s (u) du
holds because the value of g s above γ + γ -is multiplied by exp -iπ s as compared to that above

(γ + ) -1 (γ -) -1 .
We invoke now the standard formula

(1 -exp 2iaπ) (1 -exp 2ibπ) B (a, b) = ˛P z a-1 (1 -z) b-1 dz , (1.17)
where P is a Pochhammer contour around 0 and 1. We can take for P the image of P under the change of variable

z := 1 2s (s -u)
which transforms u -s into -2sz (maps s on 0) and u + s into 2s (1 -z) (maps -s on 1). It is therefore relevant to work with the expansion

u n = 1 2 n p+q=n n p (u + s) p (u -s) q .
From (1.17) we compute, for p + q = n non-negative integers,

t p,q := ˛S1 (u + s) p+ 1 /2s (u -s) q-1 /2s du = 1 exp -iπ s -1 ˛P (u + s) p+ 1 /2s (u -s) q-1 /2s du = (2s) n+1 1 -exp -iπ s (-1) q exp -iπ 2s ˛P (1 -z) p+ 1 /2s z q-1 /2s dz = (-1) q (2s) n+1 1 -exp -iπ s exp -iπ 2s 1 -exp -iπ s 1 -exp iπ s B 1 + q - 1 2s , 1 + p + 1 2s = (-1) q 2i (2s) n+1 sin π 2s B 1 + q - 1 2s , 1 + p + 1 2s = (-1) q 2i (n + 1)! (2s) n+1 sin π 2s Γ 1 + q - 1 2s Γ 1 + p + 1 2s .
Since Γ (z + 1) = zΓ (z) and Γ (1 -z) Γ (z) = π sin πz we deduce finally

t p,q = 2iπ (n + 1)! p j=1 (1 + 2sj) q j=1 (1 -2sj) and t s (n) = 1 2 n (n + 1)! p+q=n n p p j=1 (1 + 2sj) q j=1 (1 -2sj) .

Because ϕ n

s is obtained by integrating a holomorphic 1-form on a compact loop we can swap the order of summation operators:

˛S1 a -s 2 (u) g s (u) du = ∞ n=0 φ n -s 2 t s (n) .
3. After applying a convenient linear scaling of the x-coordinate we can assume that a 0 is holomorphic on D. For z ∈ D\ [0, 1] consider a path γ (z) joining 0 directly to -1, then reaching z within the domain. The function Theorem 3. Take two families X • and X • of Affine [START_REF] Birkhoff | Déformations analytiques et fonctions auto-équivalentes[END_REF]. The following properties are equivalent.

s - 0 : z ∈ D\ [0, 1] -→ exp -1 z ˆγ(z) a 0 (u)
1. There exists a germ of a holomorphic function λ → c (λ) with c (0) = 0 such that for all s sufficiently close to 0

φn s = c -s 2 ϕ n s .
2. X • and X • are conjugate.

Any conjugacy between the two families must fix λ, and in that case a change of coordinates

Ψ • such that Ψ * • X • = X • exists in the form (λ, x, y) -→ (λ , x , yc (λ) + φ λ (x)) .
Proof.

1. ⇒ 2. We find a germ of a holomorphic function (s, x) → ψ s (x) such that

Ψ -s 2 (x, y) := x, yc -s 2 + ψ s (x) satisfies Ψ * -s 2 X -s 2 = X -s 2 .
We prove next that s → ψ s is even, so that there exists a holomorphic function (λ, x) → φ λ (x) with φ -s 2 = ψ s . By definition we need to solve the equation

DΨ λ (X λ ) = X λ • Ψ λ where λ := -s 2 , that is z 2 ψ ′ s (z) = ψ s (z) -δ λ (z) z 2 + λ where δ λ (z) := ãλ (z) -c (λ) a λ (z) .
Without loss of generality we can assume that c is holomorphic on D. Suppose first that 0 < s < 1. The method of variation of the constant yields

ψ s (z) = 1 g s (z) ˆ-s z δ λ (u) g s (u) du , which is holomorphic on D\ [s, 1]. Because φn s = c -s 2 ϕ n
s the function ψ s extends to a uniform (holomorphic) function on D\ {s}. As in Lemma 1 it is easy to prove that ψ s is bounded on S 1 (uniformly in s). Using the maximum modulus principle and Riemann's removable singularity theorem we deduce that ψ s extends holomorphically to D. Montel's theorem ensures that (ψ s ) s converges uniformly on D to some function ψ 0 for which Ψ * 0 X 0 = X λ . The above construction can be holomorphically continued for all s ∈ D\R , in that case the graph of ψ s coincides with the invariant manifold of the collection ∆ • ∈ Affine (1),

∆ λ (x, y) := x 2 + λ y -δ λ (x) x 2 + λ ,
passing through the hyperbolic point (-s, 0) and transverse to the line {z = -s}. This manifold is unique, as other non-vertical trajectories of ∆ λ are multivalued. This property guarantees that a heteroclinic connection occurs in ∆ λ , otherwise ψ s would not be uniform near (s, 0). Therefore the local graph of ψ s near (s, 0) = (-(-s) , 0) coincides with that of ψ -s . From the analytic continuation principle we derive ψ -s = ψ s for s ∈ D\R, which allows to extend holomorphically (s, x) → ψ s (x) to D × D to an even function of s, as expected. 2. ⇒ 1. Take an orbital equivalence

Ψ : (λ, x, y) -→ (φ (λ) , Ψ λ (x, y)) ∈ Diff C 3 , 0 between X • and X • .
Assuming that Ψ is holomorphic on D × D × D does not lessen the generality of our argument. We prove that φ = Id. The key ingredient is the following classical fact. Lemma 3. Take p ∈ C 2 a stationary point of a holomorphic vector field X, and consider the linear part of X at p, i.e. the linear mapping DX (p). Let L p (X) denote the equivalence class of its spectrum under the equivalence

{λ 1 , λ 2 } λ1 , λ2 ⇐⇒ (∃c ∈ C =0 ) : {λ 1 , λ 2 } = c λ1 , λ2 .
Then L p (X) is invariant under orbital equivalence.

In our situation for given λ the diffeomorphism Ψ λ maps p ± := ± √ -λ, 0 to p± = ± (-1) ℓ -φ (λ), 0 for some integer ℓ. Because the spectrum of the linearization of

X λ at p ± is ±2 √ -λ, 1 we must have      √ -λ = -φ (λ) (-1) ℓ or 1 = √ -λ -φ (λ) (-1) ℓ .
The former identity yields λ = φ (λ) while the latter λφ (λ) = 1 cannot hold on a neighborhood of 0. Also Ψ λ must fix each stationary point (±s, 0). We will not prove that Ψ λ can be taken to act identically on the x-variable, although it is the case (see, e.g., [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF][START_REF] Rousseau | Analytic normal forms and inverse problems for unfoldings of 2dimensional saddle-nodes with analytic center manifold[END_REF]). The other claims can be recovered by a formal computation.

⊓ ⊔

Corollary 2. For X • ∈ Affine (1) there exists a unique

κ ∈ N := Z ≥0 ∪ {∞} such that X • is conjugate to one of the models X κ • X κ λ (x, y) := x 2 + λ y -λ κ x 2 + λ
where we conventionally identify λ ∞ with 0. Moreover families X κ

• are mutually orbitally non-equivalent for differing values of κ.

Proof. There exists a unique κ such that

ϕ n s = s 2κ c (-s 2 )
for a germ c of a holomorphic function at 0 satisfying c (0) = 0. Observe that the invariant φn s associated to X κ λ equals s 2κ . Using Theorem 3 we obtain the first claim. The theorem also implies that if

X κ • is orbitally equivalent to X κ • , then λ κ = λ κ c (λ)
for some holomorphic function c with c (0) = 0 and every λ close enough to 0. Therefore κ = κ.

⊓ ⊔

Basic objects and notations

Standard notations

In this paragraph n is a positive integer. All rings are commutative and unital.

⊛ We conventionally use N := {1, 2, . . .}. By putting expressions as index of N, Z, Q, R or C we build subsets of the space satisfying said expressions, e.g.,

R <-1 =]∞, -1[ or Z ≥0 = {0} ∪ N. ⊛ The open unit disc of C is written D := {z ∈ C : |z| < 1}
and we denote by D := adh (D) the closed unit disc. Also

S 1 : = D\D = ∂D = {z ∈ C : |z| = 1}
stands for the unit circle of the complex line. ⊛ A complex number z ∈ C has real part ℜ (z) and imaginary part ℑ (z). ⊛ The standard Riemann sphere is written C := C ∪ {∞}. ⊛ The multiplicative group of invertible elements of a ring R is written R × . ⊛ The ring of polynomials in the multi-variable (z j ) 1≤j≤n over a ring R is written

R [z 1 , . . . , z n ] ,
while for ⋆ ∈ {<, ≤, =, ≥, >} and d ∈ N the notation R [z 1 , . . . , z n ] ⋆d stands for the set of such polynomials of homogeneous degree δ satisfying δ ⋆ d. ⊛ The ring of all formal power series in the variables (z j ) 1≤j≤n over R is written

R [[z 1 , . . . , z n ]] . ⊛ For p ∈ C n the notation (C n , p)
should stand for the set of domains of C n containing p, but by a standard and convenient abuse of notations we actually write (C n , p) to mean some small enough such domain, much like the usage for Landau's o (•) and O (•) notations. ⊛ The algebra of holomorphic functions on an open set U ⊂ C n is written Holo (U). We say that a function is holomorphic on A ⊂ C n if it belongs to some Holo (U) for A ⊂ U.

The algebra of all such germs of a function is denoted by Holo (A). ⊛ In the special case A = {p} ⊂ C n we more conventionally refer to Holo ({p}) as Holo (C n , p) , the algebra of germs at p of a holomorphic functions. The group Holo (C n , p)

× consists of all germs U ∈ Holo (C n , p) such that U (p) = 0. ⊛ If moreover p = 0 we identify Holo (C n , 0) with the sub-algebra

C {z 1 , . . . , z n } of C [[z 1 , . . . , z n ]
] consisting in formal power series which are absolutely convergent on a neighborhood of 0. ⊛ For a domain D ⊂ C n+1 of the form s∈Σ {s} × D s with Σ ⊂ C and D s ⊂ C n we define the functional space

Holo c (D) := f • ∈ C 0 (adh (D)) : f • ∈ Holo (D) , (∀s ∈ adh (Σ)) f s ∈ Holo (D s ) .
⊛ The Holo (U)-module of all holomorphic vector fields on U is written X (U). The Holo (C n , p)-module of all germs at p of a holomorphic vector field is written

X (C n , p) .
= (Ψ j ) 1≤j≤n ∈ C [[z 1 , . . . , z n ]]
n is a formal diffeomorphism when Ψ (0) = 0 and Ψ is invertible for the composition of formal power series (that is,

DΨ (0) ∈ GL n (C)).
The group of all such formal diffeomorphisms is written

Diff (C n , 0) .

Lie derivative

Till the end of Section 1.2 we are given a vector field Z = 0 holomorphic on a domain U ⊂ C 2 , which we understand as a section

U -→ TU = U × C 2 p -→ (p , Z (p))
of the tangent bundle of U. We write vector fields Z = A B as derivations expressed in the canonical basis ∂ ∂x , ∂ ∂y , say

Z = A ∂ ∂x + B ∂ ∂y
for two functions A, B ∈ Holo (U) not both identically zero.

Example 1. The simplest saddle-node encountered in (1.1) can be written

X ∞ 0 (x, y) = x 2 ∂ ∂x + y ∂ ∂y .
The associated Lie (directional) derivative on functions f (or formal power series) is defined by

Z • f := A ∂f ∂x + B ∂f ∂y .
Considering Z (p) as an element of the tangent space of U at p we have

(Z • f ) (p) = D p f (Z (p)) .
The Lie bracket of two vector fields X and Y is the vector field whose action by derivation is

[X, Y ] • f := X • Y • f -Y • X • f . We write for short [X, Y ] = X • Y -Y • X
, which makes sense component-wise and endows the space of vector fields with a Lie algebra structure. When [X, Y ] = 0 we say that X and Y commute.

We define inductively for m ∈ Z ≥0

Z • 0 f := f Z• m+1 := Z • (Z • m f ) .
The action is extended component-wise to vectors or matrices of functions. Any holomorphic function H ∈ Holo (U) such that

Z • H = 0 is called a first integral of Z. Example 2. The function H : (x, y) → y exp 1 x is a first integral of the saddle-node X ∞ 0 on C × × C.

Flow, integral curves and singularities

The local flow of Z at p ∈ U is the germ of a mapping

Φ • Z : C 2 , p × (C, 0) -→ C 2 (x, y, t) -→ Φ t Z (x, y)
defined as the unique local solution of the flow-system of Z

dΦ t Z (x, y) dt = Z • Φ t Z (x, y) Φ 0 Z (x, y) = (x, y) .
The Lie formula gives a series expansion, normally convergent near p × {0}, in the form

Φ t Z = ∞ m=0 t m m! Z • m Id , (1.18)
where Id : (x, y) → (x, y) is the identity of the complex plane. More generally for any G ∈ Holo (U) the Lie identity holds (locally for all t ∈ (C, 0))

G • Φ t Z = ∞ m=0 t m m! Z • m G . (1.19)
In particular G is a first integral of Z if and only if G is constant along every integral curves of Z. If 0 ∈ U the formula also holds for any formal power series

G ∈ C [[x, y]], the right-hand side belonging to C [[x, y, t]].
Example 3. We compute easily

Φ t X ∞ 0 (x, y) = ∞ n=0 t n n! n!x n+1 , y = x 1 -tx , y exp t
For fixed p we perform the maximal analytic continuation of t → Φ t Z (p) by patching in the appropriate fashion well-chosen solutions to nearby flow systems. The result is a curve parameterization Φ • Z (p) : S p → U from a connected Riemann surface S p onto the integral curve of Z passing through p. One encounters also the terminology «orbit of p under (the flow of) Z», which is not used as such here but helps explaining some terminology we employ below for changes of coordinates. The parameterization itself may be referred to as the trajectory of Z passing through p. It is the natural parameterization of the integral curve by the time of Z. Two vector fields Z and X on U have same integral curves if and only if

Z = U X for some U ∈ Holo (U)
× . Notice that according to (1.19) the following identity holds (locally for all t ∈ (C, 0))

Z • Φ t Z = Z • Φ t Z .
A singularity (or stationary point) of Z is a point p ∈ U such that Z (p) = 0. The set of singular points of Z is written Sing (Z). Outside Sing (Z) we say that Z is regular. Singularities of Z are the only constant trajectories.

Example 4. The only singularity of X ∞ 0 is located at 0. It is therefore isolated. Any other integral curve, distinct from {x = 0 , y = 0}, coincides with a level {H = cst} of the first integral H : (x, y) ∈ C × × C → y exp 1

x . For arbitrary p = (x, y) ∈ C 2 , the trajectory of the integral curve passing through p is defined for times belonging to S p = C\ 1

x . See also Example 3. We wish to describe the holomorphic singular foliation F = F Z associated to Z on U. Roughly speaking it is the partition of U into singular points and leaves, the latter corresponding to nonconstant integral curves (without referring to a particular parameterization). There is a small catch, though, when Z is singular at p but the singularity is not isolated. In that case we can factor out a greatest common divisor in the components of Z, yielding a (local) decomposition Z = U X, where U ∈ Holo C 2 , p vanishes at p and X ∈ X C 2 , p either is regular or has an isolated singularity at p. All such eventually isolated singularities p ∈ Sing (X) form the singular set Sing (F) of F. By each point p / ∈ Sing (F) passes a unique leaf L p of the foliation, which is the maximal connected smooth complex curve tangent to Z and containing p. It is obtained by gluing integral curves of corresponding local vector fields X.

Holomorphic foliations

Two foliations F Z and F X are identical if and only if there exist V, W ∈ Holo (U) \ {0} such that V Z = W X. If Z has only isolated singularities in U, then the conditions boils down to Z = U X for some U ∈ Holo (U)

× . The restriction of F to a subdomain V ⊂ U is the foliation F| V of V, with singularities located at points of V ∩Sing (F) and whose leaves are the connected components of V ∩ L p for each p ∈ V.

Example 5. Take U := C 2 and Z : (x, y) → yX ∞ 0 (x, y). the vector field Z has the line {y = 0} for singular set. Yet F Z = F X ∞ 0 has only one singularity at 0, all other leaf is either of the form y = h exp -1

x , x = 0 for some h ∈ C, or coincides with {x = 0 , y = 0}. See Figure 1.4. Some leaves play a special role for the foliation. A separatrix of F at the singularity p ∈ U is a leaf whose adherence in U is a perhaps singular analytic curve containing p. Example 6. The foliation induced by X ∞ 0 on C 2 , 0 has exactly two separatrices, which are the connected components of {xy = 0} \ {0}.

Changes of coordinates

We define the action of Diff Ũ → U by change of coordinates on vector fields. On the source space U of the vector field, Ψ ∈ Diff Ũ → U acts as a usual mapping by composition. The action on the range space T Ũ is induced by the direct product Ψ ⊕ DΨ , sending (p, v) ∈ Ũ × C 2 to (Ψ (p) , DΨ (p) (v)). We write Ψ * Z the element of X (U) defined in such a way that the following diagram commute

Ũ Ψ / / Ψ * Z U Z T Ũ Ψ ⊕DΨ / / TU that is Ψ * Z = (DΨ ) -1 (Z • Ψ ) . (1.20)
The vector field Ψ * Z is called the pullback of Z by Ψ . In that situation trajectories of Ψ * Z are mapped to trajectories of Z, leaving the natural time unchanged (locally for all t ∈ (C, 0)):

Ψ • Φ t Ψ * Z = Φ t Z • Ψ . (1.21)
We say that Z ∈ X (U) and Z ∈ X Ũ are analytically conjugate if there exists

Ψ ∈ Diff Ũ → U such that Z = Ψ * Z.
This is equivalent to the conjugacy equation

Z • Ψ = Z • Ψ (1.22)
being satisfied.

We say that Z and Z are analytically orbitally equivalent when there exists Ũ ∈

Holo Ũ × such that Z is conjugate to Ũ Z.
This means that Z is conjugate to a vector field with same integral curves as Z, in other words that integral curves of Z are mapped under Ψ onto integral curves of Z, yet the natural time changes in general. Naturally all these notions can be germified. We then speak of local conjugacy and local orbital equivalence. If (1.22) holds at a formal level for some

Ψ ∈ Diff C 2 , 0 , then Z is formally conjugate to Z. If Z is formally conjugate to some Ũ Z with Ũ ∈ C [[x, y]]
× , then Z is formally orbitally equivalent to Z.

In case Z is (analytically, locally) orbitally equivalent to Z, a bijection Ψ realizing the equivalence maps Sing (F Z ) onto Sing (F Z ) and sends each leaf of F Z onto a leaf of F Z . We say the foliations F Z and F Z are (analytically, locally) conjugate and define

Ψ * F Z := F Ψ * Z .
We extend the terminology in the obvious way for formal diffeomorphisms, speaking of formal conjugacy between foliations.

Example 7. If one lets Ψ be (x, y) → (x, ys (x)), where s is the formal solution of (1.9), then

E 0 = Ψ * X ∞ 0 since X ∞ 0 • Ψ (x, y) = x 2 ∂ ∂x + (y -s (x)) ∂ ∂y E 0 • Ψ (x, y) = x 2 ∂ ∂x + y -x 2 s ′ (x) -x 2 ∂ ∂y .

Leaves spaces

The leaves space of F on U is, as a set, the quotient

Ω F := (U\Sing (F)) / F
where two points of U\Sing (F) are equivalent when they belong to the same leaf of F . It is endowed with the quotient topology, which (generally) is non-Hausdorff.

: (x, y) → lx ∂ ∂x +y ∂ ∂y with l ∈ C × . Both components of {xy = 0} \ {0} is a separatrix of F Z ; set L x := {x = 0 , y = 0} and L y := {x = 0 , y = 0}.
The foliation F Z has a singularity at 0, whose dynamical type and leaves space structure depend on the rationality of l. l = p q ∈ Q >0 : resonant node. Each leaf corresponds to a level {H = cst} of the rational first integral H : (x, y) ∈ U\ {0} → x q y -p (which is a connected complex curve). The leaves space Ω is homeomorphic to the Riemann sphere H (U) = C, where L x corresponds to 0 and L y to ∞. Taking U = C 2 , 0 instead of the whole plane does not change Ω. l ∈ R >0 \Q: quasi-resonant node. The vector field Z has no meromorphic first integral, although the multivalued function H : (x, y) → xy -l satisfies Z • H = 0 algebraically. Apart from L x and L y , each leaf corresponds to a «level» {H (x, y) = h} of H, h ∈ C × , which cannot be closed in U\ {(0, 0)}. The leaves space Ω is homeomorphic to the quotient of C by the action of the irrational rotation h → h exp 2iπl, whose orbits are dense in circles {|h| = cst}. The quotient cannot be Hausdorff, and actually no two leaves in the same circle {|h| = cst} can have separating neighborhoods in Ω. Taking U = C 2 , 0 instead of the whole plane does not change Ω. l = -p q ∈ Q <0 : resonant saddle. Each leaf corresponds to a level of the polynomial first integral H : (x, y) ∈ U → x q y p (which is a connected complex curve) save for

H -1 (0) = {0} ∪ L x ∪ L y .
The leaves space Ω is isomorphic as a set to the punctured line C × joined to two abstract points {0 x , 0 y }. As a topological space it is homeomorphic to C × ∪ {0 x , 0 y } equipped with the following topology: a non-empty subset

U ⊂ C × ∪ {0 x , 0 y } is open if and only if U ⊂ C × or U ∩ C ×
is an open, punctured neighborhood of 0 ∈ C. This space is not Hausdorff as 0 x and 0 y has no separating neighborhoods. Taking U = C 2 , 0 results in a smaller leaves space Ω where the role played by C × is replaced by (C, 0) \ {0}. l ∈ R <0 \Q: quasi-resonant saddle. It is a composite situation, that can be obtained from resonant saddles by quotienting out the action of the irrational rotation, as for quasi-resonant nodes. Details are left to the reader.

l / ∈ R: hyperbolic singularity. Apart from L x and L y , each leaf corresponds to a «level» xy -l = h , h ∈ C × , whose adherence in U\ {0} contains L x ∪ L y . The punctured leaves space Ω\ {L x , L y } is homeomorphic to the quotient of C × by the action of the linear map h → h exp 2iπl, which is a torus. Yet Ω is not Hausdorff as L x and L y cannot be separated from any other leaf. Taking U = C 2 , 0 instead of the whole plane does not change Ω.

It is not always possible to endow the leaves space with an analytic atlas, although Ω is locally homeomorphic to an open set of C. Indeed around a regular point p / ∈ Sing (F) we can apply the rectification theorem to some regular X ∈ X C 2 , p defining the foliation: there exists a local diffeomorphism

Ψ : C 2 , 0 → C 2 , p such that Ψ * X = ∂ ∂x . Hence the leaves of F| (C 2 ,p) are images of small «horizontal» discs included in {y = cst}. A pair (D, Ψ ) of a domain D = C 2 , 0 and a map Ψ ∈ Diff (D → (C n , p)) sending X to ∂ ∂x is called a rectifying chart (or flow-box) for F. Since level sets of H : Ψ (D) -→ C Ψ (x, y) -→ y
coincide with leaves of F| Ψ (D) , the local leaves space has a holomorphic parameterization

Ω F| Ψ (D) ≃ H (Ψ (D)) = (C, 0) .
Yet transition maps between two rectifying charts may fail to be either bijective or holomorphic.

Instead of trying to force a superfluous analytic structure on Ω F , it will be sufficient for our purposes to use the sheaf FirstIntegral (•) of first integrals of F. Any holomorphic function H ∈ Holo (V) on a subdomain V ⊂ U, which is a first integral of any vector field locally defining F, is called a first integral of F on V. Level sets of H are saturated by F| V . When a connected level set of a non-constant first integral H does not contain a singularity of F, then it coincides with a single leaf of F| V . We say that H has connected fibers when it is non-constant and every level set is connected. (Notice this property is fulfilled and used in the previous Example 8.) First integrals with connected fibers will play a central role in the sequel, as the algebra FirstIntegral (V) is functionally generated by some (and in fact any) first integral H with connected fibers. The mapping The equation

Holo (H (V)) -→ FirstIntegral (V) f -→ f • H
X • F = 0 has formal solutions F (x, y) = n,m≥0 f n,m x n y m satisfy- ing f n,m = 0 if 2n = m, while each f n,2n is free to chose in C. Therefore F (x, y) = n≥0 f n,2n xy 2 n = f xy 2 where f (t) := n≥0 f n,2n t n .

Moduli spaces, normal forms

The local rectification theorem asserts the existence of a single equivalence class for local conjugacy near a regular point. One important goal in the theory of vector fields is therefore to understand qualitative behaviors near singular points up to diverse conjugacy notions (and their orbital counterparts for foliations). This means to describe the quotients, called moduli spaces, of X C 2 , 0 under the action by conjugacy of Diff C 2 , 0 or Diff C 2 , 0 respectively, i.e. to perform the (local, formal) classification by identifying a complete set of objects invariant under conjugacy. We call such objects (local, formal) invariants.

An important invariant is the following. Take p ∈ Sing (Z) and consider the linear part of Z at p, i.e. the linear mapping DZ (p). Then its spectrum, written Spec (Z, p) for the sake of simplicity, is invariant under formal conjugacy:

∀Ψ ∈ Diff ((C n , p) → (C n , p)) Spec (Ψ * Z, p) = Spec (Z, p) .
Besides, let Λ (Z, p) be the equivalence class of Spec (Z, p) under

{l 1 , l 2 } l1 , l2 ⇐⇒ ∃c ∈ C × : {l 1 , l 2 } = c l1 , l2 .
Then Λ (Z, p) is invariant under formal orbital equivalences.

Remark 7. The quotient space is naturally isomorphic to the double cover of

P 1 (C) ram- ified over [1 : 1] and [-1 : 1], obtained under the obvious Z /2Z-action [l 1 : l 2 ] → [l 2 : l 1 ] in homogeneous coordinates.
The quotient is a smooth, compact Riemann surface of genus 0 parameterized by [l 1 : l 2 ] → l 2 1 + l 2 2 : l 1 l 2 , therefore itself a conformal projective line P 1 (C).

Generically this is the only invariant, as if l 1 l 2 = 0 and l1 /l2 / ∈ R then the vector field is hyperbolic and Poincaré's theorem guarantees that Z is locally linearizable: there exists

Ψ ∈ Diff C 2 , 0 such that Ψ * Z = l 1 x ∂ ∂x + l 2 y ∂ ∂y .
This particularly means that the local leaves space near an hyperbolic singularity is a conformal torus with two points in the adherence of every others, corresponding to the two local separatrices passing through p (see Example 8).

To be altogether correct, we need to mention that the group Diff C 2 , 0 does not really act on X C 2 , 0 . If Ψ ∈ Diff C 2 , 0 and Z ∈ X C 2 , 0 , there is no reason why Ψ * Z should be a holomorphic vector field, even though (1.20) defines a perfectly valid vector field with formal power series components. Yet being formally conjugate defines an equivalence relation, and we write resulting quotients as if they were quotients of a group action, for convenience sake.

The complete invariants we seek should differ in nature from simply stating «the equivalence class in the quotient». We particularly wish to build non-trivial bijective mappings between the various flavors of moduli spaces and some functional spaces. Classifying vector fields is out of reach in such a general form, although it can be carried out for smaller classes of vector fields. We take F ⊂ X C 2 , 0 and write respectively

Mod loc (F) := F / Diff C 2 , 0 Mod for (F) := F / Diff C 2 , 0
the corresponding moduli spaces. Since formal conjugacy is weaker than local conjugacy there is a canonical onto mapping Mod loc (F) ։ Mod for (F), and this is why in practice we fix a formal equivalence class within which the local classification is conducted. The notation

[Z] ⋆ ∈ Mod ⋆ (F)
, ⋆ ∈ {for, loc} stands for the equivalence class of Z ∈ F with respect to corresponding class of conjugacy.

Let Mod ⋆ (F) = F / G stand for one of the above quotients and let Ω be a set. We call

⊛ an injective mapping C : Mod ⋆ (F) → Ω a classification of Mod ⋆ (F) (it is complete when surjective), ⊛ a surjective mapping R : Ω → Mod ⋆ (F) a realization of Mod ⋆ (F) .
The best way to realize a moduli space Mod ⋆ (F) in a concrete form is to find a collection of (local, formal) normal forms NF ⋆ (F) ⊂ F satisfying the first two following properties:

Versality The natural map NF ⋆ (F) / G → F / G is bijective Uniqueness There exist ν ∈ N and a smooth C ν -action on NF ⋆ (F) such that for any Z ∈ NF ⋆ (F) the whole equivalence class [Z] ∈ Mod ⋆ (F)
is included in a single orbit. Simplicity Although the notion of «simple» expression is primarily opinion-based, it is generally expected that elements of NF ⋆ (F) have «simple» expressions in some «natural» basis of the tangent bundle.

Remark 8. The clause of uniqueness states that a normal form Z ∈ NF ⋆ (F) is unique «up to a finite-dimensional space». A notion of smoothness on spaces of germs (endowed with a convenient locally convex topology) adapted to this context can be found for instance in [START_REF] Teyssier | Analyticity in spaces of convergent power series and applications[END_REF]. Once these normal forms are given it is in general straightforward to refine the study and pinpoint unique representatives for a given equivalence class. This work can be messy, though in practice seldom reaching further than linear algebra. In this text we stick to finite-dimensional uniqueness.

Example 10. Let H := Z ∈ X C 2 , 0 : Z is hyperbolic at 0 . Then C loc : Mod loc (H) -→ (C × ) 2 /Z /2Z [Z] loc -→ Spec (Z, 0)
is a classification for local conjugacy, which is not complete. Injective realizations are given by

R loc : (C\R) × C × -→ Mod loc (H) (ρ, l 2 ) -→ l 2 ρx ∂ ∂x + l 2 y ∂ ∂y loc .
Normal forms are given by

NF loc (H) := l 1 x ∂ ∂x + l 2 y ∂ ∂y : (l 1 , l 2 ) ∈ C × 2 , l1 /l2 / ∈ R .

General saddle-node bifurcations

From now on we deal with the general case of a holomorphic germ of a planar saddlenode bifurcation. For the bifurcation value of the parameter λ ∈ Λ, which we conveniently locate at the origin of a complex affine space of which Λ is a (sufficiently small) domain, the vector field Z 0 is of saddle-node type near, say, the origin of C 2 , that is:

⊛ 0 is an isolated singularity of Z 0 , ⊛ the differential at 0 of the vector field has exactly one non-zero eigenvalue or, with notations introduced earlier, Spec (Z, 0) = {0, l 2 } for some l 2 ∈ C × (the singularity is elementary degenerate).

To stick to general terminology, a (holomorphic germ of a) parametric family of (germs at 0 ∈ C 2 of) vector fields Z • = (Z λ ) λ∈Λ is called a holomorphic germ of an unfolding of Z 0 .

We study in details only «generic» unfoldings, those which possess the «right number» of parameters to encode the bifurcation structure. Let us be more specific.

Definition 1. Let k ∈ N be given. A generic unfolding of codimension k is a germ of an unfolding Z • for which the following conditions hold.

⊛ There exist Λ = C k , 0 and U = C 2 , 0 such that (λ, x, y) → Z λ (x, y) is holomorphic on Λ × U, and the vector field Z 0 has only one singularity in U. ⊛ For a dense open set Λ ⊂ Λ and all λ ∈ Λ the vector field Z λ has exactly k +1 (distinct) singularities in U, which are all hyperbolic and merge to 0 as λ → 0.

⊛ λ ∈ Λ → Sing (Z λ ) is injective.
The limiting saddle-node Z 0 has codimension k.

Families with k > k parameters can be dealt with by changing the parameters (say, using the implicit function theorem, after desingularization if need be) in such a way that the first k components govern the location of the singularities, while the rest (seen as extra parameters) do not move them around. To be more specific, we mean that each fiber of λ ∈ Λ → Sing (Z λ ) is included in a single fiber of the natural projection on the first k components. All results presented here hold also for these extra-parametric generic unfoldings, as it will appear clearly that the constructions depend holomorphically on extraneous parameters. We postpone a formal definition of conjugacy / orbital equivalence between unfoldings till the end of this section. Just keep in mind that we do not allow parameter changes involving the spatial coordinates (x, y) .

Affine unfoldings, as detailed in Section 1.1, suggest that dynamical questions regarding saddle-node bifurcations can be understood from the local classification of generic unfoldings. The classification of unfoldings contains the classification of Z 0 by specialization, which is why we deliberately elude presenting this degenerate situation in details. Yet as the strategies adopted to address saddle-node singularities will serve us well, we present them briefly as a steppingstone to the unfolded case. We refer the reader to the works cited below for a comprehensive study of the subject.

The analytic unstable manifold of Z 0 , tangent at 0 to the eigenspace associated to l 2 , is called the strong separatrix. The other eigenspace corresponds to a «formal separatrix» called the weak separatrix (generically divergent [START_REF] Remoundos | Contribution à la théorie des singularités des équations différentielles du premier ordre[END_REF], always summable in the sense of Borel [START_REF] Hukuhara | Equations différentielles ordinaires du premier ordre dans le champ complexe[END_REF]). We say that a saddle-node is convergent or divergent according to the nature of its weak separatrix.

The formal orbital classification was performed by H. Poincaré & H. Dulac [START_REF] Dulac | Recherches sur les points singuliers des équations différentielles[END_REF][START_REF] Dulac | Sur les points singuliers d'une équation différentielle[END_REF], yielding polynomial normal forms. It was known from the very beginning that the formal conjugacy cannot always converge, and as a matter of fact divergence is the rule. After some inspiring works by D. Birkhoff [START_REF] Birkhoff | Déformations analytiques et fonctions auto-équivalentes[END_REF] on local classification of resonant diffeomorphisms, a complete local orbital classification was achieved in the early 1980's by J. Martinet & J.-P. Ramis [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF]. At about the same time A. Bruno [START_REF] Bruno | Groupes d'automorphismes de (C, 0) et équations différentielles ydy + • • • = 0[END_REF] presented formal normal forms for saddle-node vector fields. These works were complemented with a complete local classification in the early 2000's simultaneously by Y. Meshcheryakova & S. Voronin [START_REF] Sergeï | Analytic classification of germs of holomorphic vector fields with a degenerate elementary singular point[END_REF] (k = 1) and by L. Teyssier [START_REF] Teyssier | Analytical classification of singular saddle-node vector fields[END_REF] for the general case. These studies reveal that classifying vector fields can be dissociated into two independent process: 1. classify the orbital part (the foliation), 2. classify the «time» (the vector field for fixed foliation). Item 2. is a linear problem, simpler to deal with. Hence in the current introduction we only present how orbital classification is achieved. Also, for the sake of keeping notations light, only the case k = 1 is presented below.

The foundational viewpoint introduced in [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF] bridges the gaps between classification on the one hand, dynamical and analytical properties on the other hand, . It consists in interpreting the orbital invariants as transition maps defining a holomorphic atlas of some analytic space Ω 0 closely related to the space of leaves of F Z0 . The conformal class of Ω 0 is reasonably a complete orbital invariant of Z 0 . As illustrated in Figure 1.6, this space is obtained by a conformal gluing between two Riemann spheres Ω sn 0 and Ω ns 0 . The spheres correspond to a decomposition of C 2 , 0 in two overlapping fibered sectors. Each Ω ♯ 0 is the union of C, the range of a holomorphic sectorial first integral with connected fibers, and ∞, standing for the strong separatrix. It can be arranged that the first integral maps the sectorial weak separatrix of Z 0 to 0. The transition map near 0 (resp. ∞) is a germ of diffeomorphism ψ s 0 (resp. a translation ψ n ) induced by the inclusions of the sectors in

V ns V sn V s V n 0 0 ψ s ψ n Ω ns Ω sn node-part saddle-part first integral first integral

In geometrical space

In leaves space C 2 , 0 . The resulting complex 1-manifold is Martinet-Ramis's chapelet de sphères, which we rather refer to as the orbital necklace of Z 0 .

Remark 9. The necklace Ω 0 is not exactly the space of leaves Ω 0 of F Z0 . For one thing, most leaves accumulate on the strong separatrix so that Ω 0 is far from being Hausdorff: ∞ is not topologically separable from any other point of Ω 0 . Secondly Ω 0 is obtained from Ω 0 by modding out the action of the global (analytic) monodromy of the necklace ψ n 0 • ψ s 0 . This fact is explained in details later on for unfoldings (Section 1.9). That the pair of transition maps (ψ s 0 , ψ n 0 ) classifies, up to a finite dimensional space, the foliation F Z0 is a consequence of the conformal rigidity of orbital necklaces: their group of diffeomorphisms is small (the automorphism group of C is PGL 2 (C) acting by homography). Since any conjugacy between foliations induces a diffeomorphism between the respective necklaces, corresponding transition maps must be C × -conjugate. Indeed, with the choice of leaves for 0 and ∞, and yet another canonical normalization, the only remaining degree of freedom for the choice of (ψ s 0 , ψ n 0 ) is the linear action of C × , simultaneously on all spheres.

The first technique based on a deformation of a saddle-node vector field in order to recover Martinet-Ramis invariants was presented in the early 2000's, after an analogous work by J. Martinet [START_REF] Martinet | Remarques sur la bifurcation noeud-col dans le domaine complexe[END_REF][START_REF] Fauvet | Résurgence et bifurcations dans des familles à un paramètre[END_REF] for unfoldings of parabolic diffeomorphisms. A. Glutsyuk [START_REF] Glutsyuk | Confluence of singular points and the nonlinear Stokes phenomenon[END_REF] embedded Z 0 in a generic unfolding of codimension 1. Restricting λ to Λ, so that both singularities are hyperbolic, he let the singularities merge as λ → 0. He proved that the domains D ± λ of linearization of Z λ (near the respective singular point ± √ -λ) overlap and their union D λ contains a domain C 2 , 0 independent on λ. The «orbital link» of F Z λ | D λ is built by gluing the two spaces of leaves

Ω ± λ of F Z λ | D ± λ , which are (rigid) conformal tori ψ G Ω - Ω + first integral first integral D - D +
In leaves space

In geometrical space iR When λ → 0 each torus gets pinched more and more sharply along a meridian, converging towards a sphere C with the points 0 and ∞ identified. This continuous process lifts to the family of connections ψ G λ λ∈ Λ : Martinet-Ramis transition maps (ψ s 0 , ψ n 0 ) can be recovered from the limiting Glutsyuk connection ψ G 0 . We refer to Figure 1.8. The natural continuation of A. Glutsyuk's use of the saddle-node bifurcation would be to classify all generic unfoldings. Yet this approach is doomed to fail, because Λ is not connected. Glutsyuk's construction critically depends on the local behavior of hyperbolic singularities, and as such cannot be extended to Λ\ Λ. Indeed most (resonant) saddle points are not linearizable and, even if they were, their local leaves space would be a conformal disc, which is not rigid: the automorphism group of germs at 0 of conformal discs coincides with the infinite dimensional group Diff (C, 0). Roughly speaking, one cannot turn a saddle or node singularity into a hyperbolic singularity. But one can the converse, as done by C. Rousseau and L. Teyssier in [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF]. By cutting «sectors» attached to hyperbolic points with a special, asymptotically spiraling shape, the restricted foliation can be forced to behave locally very much like a node or a saddle. The process yields an orbital necklace Ω λ obtained exactly in the same way as for λ = 0, with an additional linear identification. This approach allows to cover the whole parameter space. The conformal structure of the necklace Ω λ depends locally analytically on λ and is continuous as λ → 0 (in particular sectors for Z λ converge toward standard sectors for Z 0 in the Hausdorff distance). Therefore unfoldings of foliations are locally classified by families of gluing mappings unfolding the local orbital invariants of Z 0 . 0, ∞ Fig. 1.8: Transition from Glutsyuk's torus to Martinet-Ramis's sphere at the limit λ → 0

The problem of giving a complete local classification of generic unfoldings (identifying the total image of the classification) is still open, except for the case k = 1 if one compiles the results of [START_REF] Rousseau | Modulus of orbital analytic classification for a family unfolding a saddle-node[END_REF] and [START_REF] Rousseau | Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle[END_REF]. The realization problem is double:

1. realize, for fixed λ, a given necklace as the leaves space of some Z λ , 2. glue all Z λ for λ ∈ Λ to form an unfolding of Z 0 . Item 1. poses no specific problem and can be dealt with in the usual manner using tools borrowed from complex geometry, by building an abstract almost-complex realization, then invoking Newlander-Nirenberg theorem to incarnate it as a germ of a foliated analytic manifold. On the contrary Item 2. is linked with the combinatorial structure of the covering of the parameter space by (contractible) open cells on which the invariants λ → m λ of the unfoldings are holomorphic. This decomposition is not trivial: it is indeed impossible to perform the previous sectorial decomposition uniformly for all values of λ. The reason is the following: the transfiguration of a saddle point into a sectorial saddle-like hyperbolic point cannot be pursued after a certain point, corresponding to parameter values for which a saddle-like singularity tends to a genuine node. A node cannot be tricked into believing it behaves like a saddle. One must therefore deal with finite families (m λ ) λ∈Λ ℓ ℓ of unfoldings of invariants of Z 0 , where Λ = ℓ adh Λ ℓ . On neighboring intersections Λ ℓ ∩ Λ ℓ all singularities are hyperbolic and the configuration is that of a Glutsyuk deformation. The invariants m ℓ

• and m ℓ • must relate to Glutsyuk tori decomposition, since all three objects encode the same leaves space and the same underlying dynamics. Expressing this identity yields necessary compatibility conditions (Section 1.9.3) that m ℓ

• and m ℓ • must obey, as explained clearly in [START_REF] Rousseau | The moduli space of germs of generic families of analytic diffeomorphisms unfolding of a codimension one resonant diffeomorphism or resonant saddle[END_REF]. In the case k = 1 the compatibility conditions guarantee that Newlander-Nirenberg theorem applies in parameter space. Corresponding conditions for k > 1 have not been written down and proved sufficient, although there is little doubt that they are. We do not present further details of this approach.

Another way of achieving a complete classification would be to describe a collection of normal forms for generic unfoldings. Normal forms have been devised recently by R. Schäfke & L. Teyssier [START_REF] Schäfke | Analytic normal forms for convergent saddle-node vector fields[END_REF] for convergent saddle-nodes. These families can be unfolded to families of normal forms [START_REF] Rousseau | Analytic normal forms and inverse problems for unfoldings of 2dimensional saddle-nodes with analytic center manifold[END_REF], in the case of pure convergence: every member Z λ of the unfolding has as a heteroclinic connection or, equivalently, sectorial weak separatrices patch continuously for all λ ∈ (C, 0). This method leaves open the generic case where there are no homoclinic connection in the unfolding, although general normal forms are expected to be worked out soon.

To conclude this introduction we give a precise definition of what changes of variables and parameters we allow between unfoldings. Section 1.2 recalled the diverse notions of (formal, local) conjugacy and orbital equivalence between vector fields in X C 2 , 0 . We need to precise the corresponding notions for unfoldings in order to perform their classification.

Definition 2. We say that two unfoldings (Z λ ) λ∈Λ and Z λ λ∈ Λ of codimension k are (formally, locally) conjugate (resp. orbitally equivalent) if there exists an association

Ψ : (λ, x, y) -→ (φ (λ) , Ψ λ (x, y))
in the corresponding regularity class, such that:

1. λ ∈ C k , 0 → λ = φ (λ) has invertible derivative at 0, 2. the identity Z φ(λ) • Ψ λ = Z λ • Ψ λ is satisfied at a formal level.
When Ψ is analytic this is equivalent to the property that, for each λ ∈ C k , 0 , the component Ψ λ is a conjugacy (resp. orbital equivalence) between Z λ and Z φ(λ) .

If the above conditions are fulfilled we write

Ψ * (Z λ ) λ = Z λ λ .
We wish to describe the (formal, local) classification in the set SNU (1) := {(Z λ ) λ : (Z λ ) λ generic unfolding of multiplicity 1} .

Definition 3. We use the notations Mod for (1) and Mod loc (1) to stand for the moduli spaces of SNU (1) under corresponding conjugacy. The orbital moduli spaces under orbital equivalence is written Mod orb for (1) and Mod orb loc (1) respectively. The same notational convention is used for the class of an unfolding:

[(Z λ ) λ ] ♯ ⋆ ∈ Mod ♯ ⋆ (1)
.

We need to slacken a little the clause of uniqueness for normal forms: we require that there exists ν ∈ N such that the equivalence class of [Z λ ] ♯ ⋆ is contained in the orbit of a C {λ} ν -action.

Every step of the way

For the sake of clarity we present only the case of codimension k = 1. Unlike saddlenode singularities, where a general complete classification is not harder to obtain than for k = 1, unfoldings for k > 1 are more difficult to deal with this way, due to the need of splitting the parameter space into many cells. The specific problems and corresponding results are detailed in [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF].

Let us summarize the different steps leading to the classification of generic saddlenode unfoldings Z • of codimension 1. The section ends by the statement of the main theorems. Most important items in the list below are developed later on in the course of the chapter. There we will outline the precise setting of and problems occurring in the upcoming constructions, while referring to [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF] for all details concerning actual proofs.

Preparation of the family (Section 1.5.1)

We can choose local conformal coordinates (λ, x, y) in which Z • can be put under prepared form A formal conjugacy between prepared unfoldings must fix the parameter λ: it is a formal invariant. In the sequel we only consider prepared unfoldings, which allows us to work for fixed λ. To lighten notations we omit to mention the subscript «λ» in all the following items.

Z λ = U λ X ∞ λ + (• • • ) ∂ ∂y where X ∞ • , called
Sectorial decomposition (Section 1.7) The canonical sectors are fibered over squid sectors V ♯ in the x-variable, displayed in Figure 1.9. The intersection V ns ∩ V sn has three components:

s n s s
⊛ a saddle-part V s having only s s in its adherence, ⊛ a node-part V n having only s n in its adherence,

V ns V sn V s V n s n s s V g (a) λ = -1 V ns V sn V s V n s n s s V g (b) λ = exp -iπ 8 V ns V sn V s V n s n s s V g (c) λ = exp iπ 8
Fig. 1.9: A sectorial decomposition of the x-variable when k = 1 for three values of the parameter ⊛ a gate-part V g having both points in its adherence.

We write

V ♯ := V ♯ × (C, 0) , ♯ ∈ {n , ns , s , sn , g}
the corresponding fibered sectors. There are two transitions to consider for orbital equivalence (happening in the saddle-and node-part), and one more to account for conjugacy (over the saddle-part). We denote by F the (germ of a) singular foliation induced by Z. Each canonical sector is attached to both singular points s n and s s . The boundary of a sector is carved in such a way that the leaves of F| V ♯ near s n behave like those of a node: every leaf accumulates on the singular point. Near s s the foliation is similar to a saddle: every leaf but one (the local invariant manifold) stays far away from the singular point (Figure 1.10). This topological configuration allows us to mimic constructions performed in Section 1.1 for affine unfoldings. In particular the sectorial space of leaves is a conformal line

Ω ♯ = H ♯ V ♯ = C , where H ♯ ∈ Holo V ♯ is the canonical first integral of F| V ♯ , having connected fibers.

Straightening of the weak separatrices

If a singularity p of X is not a node, there exists only one integral curve with smooth analytic closure passing through p (a separatrix, or invariant manifold) and transverse to the vertical lines {x = cst}. It is given by the graph of a holomorphic function x → s (x), with holomorphic continuation over every canonical sector. We call such an analytic continuation a sectorial weak separatrix. It corresponds to the level 0 of the canonical first integral H ♯ . In particular both sectorial weak separatrices coincides in saddle-and gate-parts.

Applying the change of coordinates ψ : (x, y) → (x, y + s (x)) to Z straightens the sectorial weak separatrix into {y = 0}. Notice that s cannot (in general) be analytically continued on a whole neighborhood of the other singularity, as a given sectorial weak separatrix may not coincide with the other one when continued (the typical s is multivalued). When s does extend holomorphically on a neighborhood of all singular points we say that a heteroclinic connection occurs between s s and s n .

Normalization strategy (Section 1.5.2)

After straightening, we can write:

Z = U X X = X ∞ + yR ∂ ∂y
for some (sectorial) holomorphic functions U and R. It turns out the conjugacy equation O * X ∞ = X is equivalent to the following orbital cohomological equation

X • O = -R (1.24)
if one seeks a conjugacy in the form O (x, y) = (x, y exp O (x, y)) .

Also the conjugacy equation T * X = U X takes the form of a temporal cohomological equation

X • T = 1 U -1 (1.25)
for changes of coordinates

T (x, y) = Φ T (x,y) X (x, y)
obtained by taking a dependent time in the flow Φ t X (x, y) of X. Therefore the normalization process has been reduced to solving two cohomological equations.

Formal classification (Section 1.5.3)

The straightening step can be realized at a formal level. The orbital cohomological equation (1.24) can be solved formally, more or less by construction of the model X ∞ and of the prepared form. The temporal cohomological equation (1.25) needs some adjustment since 1 U -1 may not belong to the image of the Lie derivative associated to X. We can find an affine function u ∈ C [x] ≤1 , relatively prime with P , such that 1 U -1 u belongs to that image. Hence uX ∞ is formally conjugate to Z, finally yielding a polynomial formal model

Z u := uX ∞ .
This complete formal classification of generic unfoldings ranges in the space of germs C {λ} 3 .

Local classification (Section 1.8)

The normalizing cohomological equations (1.24) and (1.25) admit a bounded solution on each canonical sector. They are obtained by integrating the right-hand side against dx P along asymptotic paths tangent to F, ending at the point (x, y) and accumulating on s n in backward time. The node-like nature of s n guarantees that every point in the sector can be reached by asymptotic paths, and that both solutions coincide in the node-and gate-part. Therefore Z is conjugate to the formal model Z u on each canonical sectors. The local class of Z • is thus completely determined by the following data.

⊛ Its orbital class ϕ := (ϕ n , ϕ s ), obtained by comparing:

⊚ the sectorial weak separatrices in the node-part, measuring how far the vector field is from having a heteroclinic connection and encoded in a translation h → h + ϕ n , ⊚ the sectorial solutions to (1.24) in the saddle-part, measuring how far the continued sectorial solution of the orbital cohomological equation is from uniformity (that is, continuity).

⊛ Its temporal class f , encoded by comparing the sectorial solutions to (1.25) in the saddle-part.

In both cases where cohomological equations are involved, comparing sectorial solutions means to consider the period

g (p) := 1 2iπ ˆγ(p) G dx P
of the right-hand side G along an asymptotic cycle γ (p) circling around s s and tending asymptotically to s n both in forward and backward time, while passing through p ∈ V s , as displayed in Figure 1.11. This gives an integral representation of the invariants over the saddle-part (Section 1.8.5.2). The value of the integral does not depend on the choice of p in a fixed leaf of the restriction of F to the sector

V nsn := C 2 , 0 \ (V n ∪ V g )
because the leaf is simply connected. Hence g is a first integral of X and for that matter factors as a (germ of a) holomorphic function of the canonical first integral H ns , holomorphic on V ns : Remark 10. For the period T X • (G) to be well-defined, we need to mod it out by the degree of freedom in the choice of the sectorial first integrals H ns . This freedom results from a faithful action of C × by linear changes of variables h → ch. We leave such essentially irrelevant technicalities out till subsequent sections.

g = T X • (G) • H ns , T X • (G) ∈ hC {h} .
For a fixed formal model we finally obtain a local classification ranging in a functional space

[(Z λ ) λ ] loc -→ (λ → m λ ) , m λ = (ϕ n λ , ϕ s λ , f λ ) ∈ C × hC {h} × hC {h} .
Notice how cowardly we shied away from discussing the dependence in the parameter λ until now. The result to come, proved in Sections 1.8 and 1.10.1, gives a precise description of that dependence.

Theorem 4. There exists a cover of (C, 0) by the adherence of two germs of sectorsΛ + , Λ - attached to 0 (called cells), for which we can find a complete local classification of generic unfoldings of codimension 1 with fixed formal class, ranging in the space of those collections

(m - • , m + • ) satisfying: ⊛ (λ, h) ∈ Λ ± × (C, 0) → m ± λ (h)
is holomorphic with continuous extension to adh (Λ ± ) × (C, 0), and m ± λ is holomorphic for every λ ∈ ∂Λ ± , ⊛ m ± λ ∈ C × hC {h} × hC {h}, ⊛ the compatibility condition, given explicitly in Definitions 16 and 17.

By construction the model u

• X ∞ • has local class m ± • = 0.
Normal forms for pure convergence (Section 1.10.2)

When ϕ n • = 0 (we say the unfolding is purely convergent) we provide normal forms for Z • . Remark both sectorial separatrices glue to form a holomorphic weak separatrix (heteroclinic connection) and therefore purely convergent unfoldings are locally conjugate to prepared unfoldings for which {y = 0} is a leaf, and vice versa.

Theorem 5. Let Convergent (1) be the space of all purely convergent, generic unfoldings of codimension 1. Define

τ := 0 if µ 0 / ∈ R ≤0 1 + ⌊-µ 0 ⌋ otherwise Section (1, τ ) := x τ +1 yC {λ, x τ y} .
Then the collection

u • 1 + u • Q • X ∞ • + yR • ∂ ∂y : Q • , R • ∈ Section (1, τ )
is a family of normal forms for Convergent (1). More precisely, two unfoldings in that form for

(Q • , R • ) and Q • , R • are locally analytically conjugate if there exists c • ∈ C {λ} × such that R λ (x, y) = R λ (x, c λ y) Q λ (x, y) = Q λ (x, c λ y) ,
amounting to a linear C {λ} × -action.

Preparation and formal classification

Preparation

Theorem 6. There exists local conformal coordinates (λ, x, y) in which Z • has the following prepared form

Z λ = U λ X λ X λ (x, y) = P λ (x) ∂ ∂x + (y (1 + µ λ x) + P λ (x) R λ (x, y)) ∂ ∂y (1.26)
where

         λ ∈ (C, 0) µ • ∈ C {λ} R • ∈ C {λ, x, y} U • ∈ C {λ, x, y} × are arbitrary.
Notice that every prepared form is a generic unfolding of codimension 1, for Sing (X λ ) = {y = 0} ∩ P -1 λ (0). It turns out that a formal changes of coordinates fixing the general form of the family (1.26) must leave the parameter λ invariant. This parameter therefore plays a special role for the unfolding, and we call it the canonical parameter. It thus suffices to work with fixed λ in order to perform the local classification.

From normalization to cohomological equations

The whole procedure relies on writing the conjugacy equation

Ψ * Z = Z as cohomolog- ical equations Z • F = G (1.27)
for well-chosen right-hand sides. The key computation is the following proposition. has same regularity as f , and satisfies

Ψ * X = X - X • f 1 + Y • f Y .
In particular, the following properties hold.

1. (Temporal conjugacy) U Z is conjugate to V Z by Φ T UZ if and only if

Z • T = 1 U - 1 V . 2. (Orbital conjugacy) Assume X ⋔ Y . Then X is conjugate to X + RY by Φ O Y if and only if X • O = R .
Proof. It is sufficient to perform computations at a formal level. We use Lie formula (1.19) so that, because

X • Y = Y • X, X • Ψ = n≥0 f n n! Y • n (X • Id) = n≥0 f n n! X • (Y • n Id) . Besides DΨ (X -RY ) = (X -RY ) • Ψ = n≥0 (X -RY ) • f n n! Y • n Id = (X -RY ) • f × n≥0 f n n! Y • n+1 Id + n≥0 f n n! (X -RY ) • Y • n Id = X • Ψ + (X • f -R (1 + Y • f )) × Y • Ψ . Therefore R = X • f 1 + Y • f .
To prove 1. it suffices to take X := U Z and Y := U Z.

⊓ ⊔

By taking (Z s ) s in prepared form (1.26) we can write

Z λ = U λ X λ X λ = X ∞ λ + P λ R λ ∂ ∂y X ∞ λ := P λ ∂ ∂x + y (1 + µ λ x) ∂ ∂y .
Notice that

X ∞ λ , y ∂ ∂y = 0
so we can apply Proposition 4 Item 2. as soon as R s can be factored by y, that is once the weak separatrix is straightened into {y = 0}.

Formal classification

The strategy is the following.

⊛ There exists a unique weak separatrix family, that is a formal family of curves {ys λ (x) = 0} such that

P λ (x) s ′ λ (x) = s λ (x) (1 + µ λ x) + P λ (x) R λ (x, s λ (x)) , s • ∈ C [[λ, x, y]] .
⊛ After applying the change of coordinates (x, y) → (x, ys λ (x)) to X λ we obtain the formal vector field

X λ = X ∞ λ + P λ R λ y ∂ ∂y where R λ (x, y) := R λ (x, y + s λ (x)) -R λ (x, s λ (x)) y .
(1.28)

⊛ The cohomological equation

X λ • O λ = -P λ R λ admits a unique formal family of solutions O • ∈ C [[λ, x, y]] such that O λ (0) = 0. Therefore X ∞ • is orbitally formally conjugate to X • by O • := Φ O• y ∂ ∂y ,
thus formally conjugate to X • by a (λ, x)-fibred formal conjugacy. Remark that the previous cohomological equation is equivalent to one involving X λ in the original coordinates:

X λ • O λ = -P λ R λ (x, y) -R λ (x, s λ (x)) y -s λ (x) . (1.29)
⊛ The cohomological equation with u

• ∈ C [[λ, x]] × X λ • T λ = 1 U λ - 1 u λ (1.30)
admits a formal solution if and only if

U λ (x, y) = u λ (x) + O x 2 + O (y)
where u λ is relatively prime with P λ in the factorial ring C [[λ, x, y]]. The holomorphic germ u • can therefore be taken as the remainder of the Euclidean division of U λ (x, 0) by P λ (x). In particular for each value of the parameter u λ is affine. Because O λ is x-fibred, the vector field u λ X ∞ λ is formally conjugate to u λ X λ , thus to Z λ . The last two claims derive from the following easy computational lemma.

Lemma 4. [24]Let G • ∈ C [[λ, x, y]] be given. The cohomological equation X λ • F λ = G λ admits a formal solution F • ∈ C [[λ, x, y]] if and only if G λ = O (P λ (x)) + O (y).
∂ ∂x + y (1 + µ • x) ∂ ∂y : µ • ∈ C {λ} .
Definition 4. An unfolding Z • in NF orb for (1) is now fixed, corresponding to the choice of the holomorphic germs µ • : λ → µ λ and u • : (λ, x) → u λ (x). It is referred to as the model.

The local classification is obtained by repeating the construction on sectors over which O λ and T λ have holomorphic «sums». The local invariant measures how far from converge these power series are.

Parameter space

To carry out the construction of the sectorial decomposition we need to follow singularities as λ varies. Although the set Sing (Z λ ) depends continuously on λ, it is not possible to mark and follow continuously singularities, as these get exchanged by turning around the bifurcation value of λ (corresponding to P 0 (x) = x 2 ). This phenomenon has a prominent bearing on the construction of sectors, since for the same value of λ one obtains dynamically non-equivalent coverings, as illustrated in Figure 1.12. We resolve the ambiguity in the labeling of the singularities by using the two-fold branched covering

P • : s ∈ C -→ (x -s) (x + s)
and take s, which determines completely the position of the roots ±s, as new parameter for the constructions. Remark 11. Notice that λ (Σ ∪ {0}) = (C, 0) so every original parameter is covered this way. The explanation as to why we cannot take Σ ∪ {0} = (C, 0) will be given in the course of the upcoming sections (especially Section 1.7.3).

s n s s V ns V sn V n V s V g s n s s V ns V sn V n V s V g Σ -s s
The next properties will be used without explicitly referencing the trivial lemma beneath.

Lemma 5.

1. The automorphism group of the covering λ is isomorphic to Z /2Z. Definition 6. In all the remaining text, we make the following notational conventions:

⊛ when an object Ω is subscripted with «s» we imply s → Ω s depends on s in a (holomorphic, continuous) way on Σ, ⊛ when an object Ω is subscripted with «λ» we imply λ → Ω λ depends (formally, holomorphically, continuously) on λ ∈ (C, 0).

Canonical sectors

Splitting vector fields

The boundary of a squid sector will be defined by real trajectories of vector fields

Ξ s (x) := ϑP λ (x) ∂ ∂x
for some suitable choice of a direction ϑ = ϑ (s) ∈ S 1 . In order to ensure the upcoming construction matches our needs, we must ensure that the vector field is sufficiently generic.

Let us explain what we mean by describing some dynamical data attached to the planar real-analytic foliation F induced by Ξ s on C. The most complete reference on the subject is [START_REF] Branner | Classification of complex polynomial vector fields in one complex variable[END_REF].

⊛ Sing (F) = P -1 λ (0), in particular F is regular near ∞. We call local separatrix of Ξ s any one of the two leaves of F| (C,∞)\{∞} accumulating on ∞. We call separatrix the corresponding leaf in F| C . ⊛ Because Ξ s is holomorphic, F is free from limit (poly)cycles. Therefore the fate of a non-singular trajectory Γ of Ξ s can only be one of the following (Bendixon-Poincaré theorem).

⊚ Γ is a separatrix and its adherence connects ∞ to either a singular point ±s of Ξ s , or ∞. In the former case we say that Γ lands at ±s. In the latter case we say Γ is a homoclinic connection (happening exactly when the continuations of both local separatrices meet en route). ⊚ Γ connects (asymptotically) both singular points s and -s. ⊚ Γ is a non-isolated simple loop. In that case both ±s are center points. Definition 7. Take s ∈ Param (1). We say that Ξ s is splitting if it admits no homoclinic connection. Lemma 6. The following conditions are equivalent. 1. Ξ s is splitting. 2. There exists a leaf connecting s and -s.

3. ϑP ′ λ (±s) / ∈ iR.
Let us briefly explain why this lemma holds. When s = 0 the vector field Ξ s is locally linearizable around each root ±s of P λ , and its linear part is given by ϑP ′ λ (±s) (x ∓ s) ∂ ∂x . Therefore ϑP ′ λ (±s) is purely imaginary (non-zero) if and only if ±s is a center point of Ξ s .

⊛ When ±s is a center, it lies within an open basin B ± of periodic trajectories. In particular no leaf can connect s and -s. Moreover ∂B ± must be a homoclinic connection, and Ξ s is not splitting. ⊛ When ±s is not a center, it is either a focus or a sink. In any case it lies within an open basin of attraction B ± (respectively in forward or backward time) and any integral curve of Ξ s crossing the basin must accumulate on ±s one way or the other. There are no trajectory accumulating on s (or on -s) in both directions.

Transvestite hyperbolic points

t ∈ R -→ x (t) := x * exp (ϑlt) , x * ∈ C × .
We can lift this path into F W through the projection Definition 8. For given l ∈ C × we say that ϑ ∈ S 1 is a saddle-direction (resp. nodedirection) for l if:

Π : (x, y) -→ x ,
⊛ ℜ (ϑ) > 0, ⊛ ℜ (ϑl) < 0 (resp. ℜ (ϑl) > 0).
Remark 12. Just pointing out the obvious.

1. l admits a saddle-direction if and only if l / ∈ R >0 (i.e. W is not a node). 2. l admits a node-direction if and only if l / ∈ R <0 (i.e. W is not a saddle).

For such a choice of ϑ, the curve t → x (t) is a spiral (specializing to a straight line when ϑl ∈ R). Consider the domain V := C\ exp ϑlR obtained by slitting the complex line along the adherence Γ of a real integral curve of ϑlx ∂ ∂x , and build the fibered domain of the complex plane

V := V × C .
Then F W | V is saddle-like (resp. node-like) in the sense that only one (resp. every) leaf accumulates on 0. Notice that a saddle-like (resp. node-like) singularity is reached in positive (resp. negative) time. Also, (any determination of) the first integral H on V is holomorphic. The following properties are immediate to establish.

Lemma 7.

1. F W | V is saddle-like if and only if for every U = C 2 , 0 we have H (U ∩ V) = (C, 0) and its diameter goes to 0 as that of U does. 2. F W | V is node-like if and only if for every U = C 2 , 0 we have H (U ∩ V) = C.

Sectorial decomposition

We work here within a fixed formal class µ • . We find a covering of (C, 0) \ {±s} by two squid sectors V ns s and V sn s attached to ±s, such that near s n = (s, 0) (resp. s s = (-s, 0)) the linear part of X λ defines a node-like (resp. saddle-like) foliation over both sectors, except for forbidden values of s we shall describe afterward.

The linear part of X λ at the singularity (±s, 0) is

±2s (x ∓ s) ∂ ∂x + y (1 ± µ λ s) ∂ ∂y ,
while its local analytic invariant is given by

l ± s := ±2s 1 ± µ λ s . (1.31)
Both functions s → l ± s are holomorphic on (C, 0). Notice that

l + s = l - -s
and, when s = 0,

1 l + s + 1 l - s = µ λ .
Definition 9. The curves s = 0 : ±l ± s < 0 are called the forbidden curves associated to µ • . Values of the parameter for «+» (resp. «-») correspond to configurations where X λ is a saddle at s n (resp. a node at s s ).

Till the end of the section we consider the principal determination of the argument on C\R ≤0 . For each s ∈ Σ set

ϑ (s) := exp -i arg s 2 . (1.32) Since l ± s ∼ 0 ±2s
the following lemma holds.

Lemma 8. There exists ρ > 0 so that for all s ∈ Σ

arg ϑ (s) l + s < 3π 8 (1.33) arg ϑ (s) l - s > 5π 8 .
In particular ϑ (s) is a node-direction ( resp. a saddle-direction) for l + s ( resp. l - s ), and Σ meets no forbidden curve. Definition 10. We refer to Figure 1.15 (b). Let r > 2ρ > 0 and s ∈ Σ ∪ {0}. We recall the vector field

Ξ s = ϑ (s) P λ ∂ ∂x
where ϑ is given by (1.32). The squid sector V ns s of radius r is the domain bounded by

V ns V sn V s V n s n s s V g (a)
A sectorial covering by squid sectors 8 till it reaches 2 |s| at a point s 1 , then the complete backward trajectory of Ξ s , ⊛ the integral curve of Ξ s passing through the «outward» point of intersection g between the perpendicular bisector of [-s, s] and a circle of radius small enough not to meet the already built paths (of the order of |s| (1 + cos arg s)), ⊛ the circular arc C := r exp i -iπ 8 , 9iπ 8 . The squid sector V sn s of radius r is build in much the same way, replacing the circular arc C by r exp i -9iπ 8 , iπ 8 . We mention without proof the next descriptive lemma. Lemma 9. (See [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF])

s s s n V ns V s V n V g s 0 s 1 n 0 n 1 g C (b)
1. The intersection V ns s ∩ V sn s has three components if s ∈ Σ:
⊛ a saddle-part V s s having only s s in its adherence, ⊛ a node-part V n s having only s n in its adherence, ⊛ and if s = 0, a gate-part V g s having both points in its adherence. When s = 0 we define the saddle-or node-part as the components crossing R <0 or R >0 respectively. 2. As s → 0 in Σ, the squid sectors V ♯ s tends (for the Hausdorff distance) to the sector V ♯ 0 associated to the saddle-node Z 0 . 3. The length of ∂V ♯ s is uniformly bounded for s ∈ Σ. Remark 13. Item 3. above is really important in order to get uniform bounds in s ∈ Σ for functions obtained by integrating over spirals included in V ♯ s , which includes almost all the upcoming material.

Local classification

We work here within a fixed formal class µ • . We take r, r ′ > 0 and ρ > 0 sufficiently small so that

Σ × U := s : 0 < |s| < ρ , |args| < 2π 3 × (rD × r ′ D)
is a domain on which:

⊛ every data appearing in the preparation Theorem 6 is holomorphic and bounded, ⊛ (s, x) → P λ (x) 1+µ λ x is holomorphic and bounded, ⊛ Lemma 8 holds.

The actual values of ρ, r, r ′ may be implicitly decreased finitely many times in the course of the construction. For any s ∈ Σ we denote V ♯ s the squid sector of radius r associated to s built in Definition 10. We follow now the strategy introduced for the formal classification, only on canonical sectors

V ♯ s ⊂ V ♯ s × r ′ D , ♯ ∈ {n , ns , s , sn , g} .
Before building the sectors, let us first introduce the space

Holo c (D) := f • ∈ C 0 (adh (D)) : f • ∈ Holo (D) , (∀s ∈ adh (Σ)) f s ∈ Holo (V s ) (1.34)
where D is a subdomain Σ × C n of the form

D = s∈Σ {s} × V s .
The key point is to provide weak sectorial separatrices s ♯ • and solutions F ♯ • to cohomological equations (1.27) which belong to Holo c D ♯ and Holo c D ♯ respectively, with

D ♯ := s∈Σ {s} × V ♯ s , (1.35)
D ♯ := s∈Σ {s} × V ♯ s .

Sectorial weak separatrices

Theorem 8. (See [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF]) Up to decrease ρ, r, r ′ there exist two unique families of functions s ♯ • ∈ Holo c D ♯ , called in the following sectorial (weak) separatrices, such that for any s ∈ Σ ∪ {0}:

1. y = s ♯ s (x) ⊂ V ♯ s × r ′ D is an integral curve of X λ , 2. lim x→±s s ♯ s (x) = 0, 3. s ns s (x) = s sn s (x) for all x ∈ V g s ∪ V s s .
The work of M. Klimeš [START_REF] Klimeš | Confluence of Singularities of Nonlinear Differential Equations via Borel-Laplace Transformations[END_REF] offers an other, less geometric (but worth mentioning) approach to the question, based on Borel-Laplace transform of the formal normalizing series.

Asymptotic paths and canonical sectors

Definition 11. Let s ∈ Σ∪{0}. An asymptotic path over V ♯ s ending at p ∈ U ∩ V ♯ s × C is a regular, smooth curve γ : R ≤0 → V ♯ s × C meeting the next requirements. ⊛ γ (0) = p. ⊛ γ = cX λ • γ for some smooth function c. In other words, γ is tangent to X λ or, equivalently, its image is contained in a single leaf of F λ . ⊛ lim t→-∞ γ (t) = s n .
We abusively write γ : (s n → p) or (s n → p) to stand for such an asymptotic path ending at p. Theorem 9. (See [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF]) Up to decrease ρ, r, r ′ the following properties hold for s ∈ Σ ∪ {0}.

1. The sets (called canonical sectors)

V ♯ s := p ∈ U ∩ V ♯ s × C : ∃ (s n → p) , ♯ ∈ {ns , sn}
are domains containing smaller sectors V ♯ s × rD with r > 0 independent on s. The union

V s := V ns s ∪ V sn s is a pointed neighborhood of {x = ±s} ∩ U. 2. Each leaf of F λ | V ♯
s is simply connected, and if γ and γ are two asymptotic paths ending at p ∈ V ♯ s , then the asymptotic cycle

-γ γ := t ≤ 0 -→ γ (t) t ≥ 0 -→ γ (-t)
is trivial, in the sense that there exists an asymptotic tangential homotopy h : [-∞, 0] × R → V ♯ s between γ -1 γ and s n , a mapping such that:

⊛ h is uniformly continuous, ⊛ h (0, •) = γ -1 γ, ⊛ h (-∞, •) = s n , ⊛ for every t real h (•, t) : s n → h (0, t),
⊛ for every τ negative real h (τ, •) : s n → p.

3. Take p ∈ V ns s ∩ V sn s and consider an asymptotic cycle γ (p) obtained by concatenating two asymptotic paths ending at p, one of which lies in V ns s and the other one in

V sn s . a. If p ∈ V g s ∪ V n s , then γ (p) is trivial. b. If p ∈ V s
s , then γ (p) is trivial if and only if p belongs to the sectorial weak separatrix. Two such asymptotic cycles γ (p) and γ ( p) are tangentially homotopic if (and only if ) p and p are in the same (sectorial) leaf.

Proof.

Straighten the sectorial weak separatrix to {y

= 0} by applying Ψ s : (x, y) → x, y + s ♯ s (x) beforehand. Along the real trajectories t ≤ 0 -→ (x (t) , y (t)) of ϑ (s) Ψ * s X λ , a direct variational computation shows that φ (t) := |y (t)| 2 = y (t) y (t) satisfies the differential relation φ = 2φℜ (1 + • • • ) > 0 , (1.36)
where (• • • ) stand for a term smaller than 1 2 in modulus. Actually φ is exponentially increasing therefore t → y (t) tends to 0 as t → -∞. By construction of the squid sectors t → x (t) converges towards s n . The argument works well if x remains in a bounded region, which may not be always the case. This is countered by altering slightly ϑ (s).

Because of the previous variational estimate, we can prove that homotopies in the x-

space along the flow of -ϑ (s) P λ ∂ ∂x lift in the leaf through the projection Π : (x, y) → x. Indeed the only obstructions preventing the lift is the tangency between X λ and Π -1 (cst), which does not happen close to 0. Since φ is decreasing the trajectory remains close to 0. As in [START_REF] Teyssier | Germes de feuilletages présentables du plan complexe[END_REF], it is then possible to prove that any sectorial asymptotic cycle is trivial, by deforming it using such contracting homotopies. 3. The argument is similar to (2) save for the fact that the flow of -ϑ (s) P λ ∂ ∂x always drive the x-coordinate away while crossing saddle parts. Therefore candidate trivializing homotopies of γ (p) in the x-variable cannot be lifted all the way if p ∈ V s s .

Sectorial solutions to cohomological equations

Theorem 10. (See [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF]) For s ∈ Σ define

V nsn s := V s \ (V g s ∪ V n s ) D nsn := s∈Σ {s} × V nsn s and take G • ∈ Holo c (D nsn ) such that (s, x) → Gs(x,0) P λ (x) is bounded. For s ∈ Σ and p ∈ V ♯ s define F ♯ s (p) := ˆ(s n →p) G s dx P λ
where (s n → p) is an asymptotic path of X • ending at p. The following properties hold.

1. F ♯ s (p) is an absolutely convergent integral.

F ♯

• is the unique family of solutions of the cohomological equation X • • F • = G • which belongs to Holo c D ♯ and vanishes at each s n . Another such solution differs from F • by the addition of a function f • ∈ Holo c (Σ). 3.

F sn s (p) -F ns s (p) = 0 if p ∈ V n s ∪ V g s ´γ(p) G s dx P λ if p ∈ V s s
where γ (p) is an asymptotic cycle passing through p.

The following properties are equivalent.

a. There exists

F • ∈ Holo c (D s ) such that X • • F • = G • . b. For all s ∈ Σ and p ∈ V s s ˆγ(p) G s dx P λ = 0 .
Proof.

1. This follows from the estimate (1.36) since for t ≤ 0 close to ∞ dx (t) P λ (x (t)) = ϑ (s) dt

x (t) = s n + O 1 t y (t) = O (exp t) .
2. The fact that F ♯ s is a solution of the cohomological equation can be understood by covering (s n → p) with flow-boxes. In such a local rectifying chart Ψ : D → C 2 , γ (t) one has to solve

∂F ♯ s • Ψ ∂x = G λ • Ψ so that for p * , q * ∈ D in the same leaf of F ∂ ∂x | D we have F ♯ s • Ψ (q * ) -F ♯ s • Ψ (p * ) = ˆp * →q * G λ • Ψ dx and vice versa. 3. By construction F sn s (p) F ns s (p) = ´γ(p) G s dx P λ .
Since all asymptotic cycles are trivial when p ∈ V n s ∪V g s (Theorem 9), then the integral ´γ(p) G s dx P λ vanishes because of Cauchy formula. 4. As a consequence of the previous items, ´γ(p) G s dx P λ encodes the additive monodromy of the analytic continuation of F ♯ s so that a.⇒b. Riemann's removable singularity theorem yields the converse implication.

Sectorial normalization and space of leaves

From Theorem 8 and Proposition 4 we obtain a vector field X s and a right-hand side

G s := -P λ R ♯ s with R ♯ s : (x, y) ∈ V ♯ s -→ R λ (x, y) -R λ x, s ♯ s (x) y -s ♯ s (x) (1.37)
satisfying the hypothesis of Theorem 10. The orbital normalization equation (1.29) therefore admits solutions O ♯ • ∈ Holo c D ♯ . The situation is the same for the temporal normalization with X s := X λ and G λ := 1 U λ -1 u λ , and (1.30) admits solutions

T ♯ • ∈ Holo c D ♯ . Corollary 3. Z • is conjugate to its formal model u • X ∞ • on D ♯ . It is orbitally conjugate to X ∞ • by (λ, x)-fibred transformations. The model X ∞ λ has a first integral H ∞,♯ λ ∈ Holo V ♯ with connected fibers H ∞,♯ λ (x, y) := yP -µ λ/2 λ (x) x + s x -s 1 /2s , s = 0 H ∞,♯ 0 (x, y) := yx -µ exp 1 x , with lim λ→0 H ∞,♯ λ = H ∞,♯
0 uniformly on compact subsets; for the remaining of the section we only deal with s = 0 to lighten notations, although everything is valid for s = 0 by continuity. Recalling notations and conventions of Section 1.1.3, we choose determinations of the multivalued functions involved above:

⊛ •+s •-s 1 /2s
= g λ coincides with the principal determination on V n s (the one given in Remark 6 for real s); note that g λ ∈ Holo (C\V g s ), ⊛ P

-µ λ/2 λ
agrees with the principal determination on V n s . Invoking Corollary 3 above and formulas given in Section 1.5.3, we obtain canonical sectorial first integrals with connected fibers

H ♯ s (x, y) := H ∞,♯ λ x, y -s ♯ s (x) exp O ♯ s (x, y) (1.38) = y -s ♯ s P -µ λ/2 λ (x) g λ (x) exp O ♯ s (x, y) .
The next result can be proved by studying the linearized system on squid sectors as in Lemma 7.

Theorem 11. (See [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF]) For s ∈ Σ define the sectorial spaces of leaves as

Ω ♯ s := H ♯ s V ♯ s , ♯ ∈ {ns , sn} Ω ♭ s := H ns s V ♭ s , ♭ ∈ {n , g , s} .
1. Ω n s = Ω ns s = Ω sn s = Ω g s = C, the sectorial weak separatrices corresponding to 0. 2. Ω s s = (C, 0), the sectorial weak separatrices corresponding to 0. The size of Ω s s goes to 0 as r or r ′ does.

Classification

Orbital necklace

Take s ∈ Σ ∪ {0}. A given point p ∈ V ns s ∩ V sn s corresponds to a point h ns ∈ Ω ns s and a point h sn ∈ Ω sn s . These points must be identified in order to encode the local orbital class of X λ . Because each H ♯ s has connected fibers, the (holomorphic) identifications must be injective.

⊛ If s = 0 and p ∈ V g s , then every function involved in (1.38) for ♯ = ns and ♯ = sn coincide at p except g λ P µ λ/2 λ . The monodromy of g λ P µ λ/2 λ around s n acts as

h sn = h ns exp iπ 1 s + µ λ =: ψ g s (h ns ) . ψ s ψ n 0 0 Ω ns Ω sn ψ g
⊛ If p ∈ V n
s , then every function involved in (1.38) for ♯ = ns and ♯ = sn coincide at p except s ♯ s . Because Ω n s = C the mapping ψ n s must be affine, and it is not hard to check it is a translation:

h sn = h ns + ϕ n s =: ψ n s (h ns ) . (1.39)
⊛ If p ∈ V s s , then every function involved in (1.38) for ♯ = ns and ♯ = sn coincide at p except P µ λ/2 λ exp O ♯ s . Because 0 corresponds to both sectorial weak separatrices, and because those two functions agree on V s s the mapping ψ s s must fix the point 0: 1. Take

h sn = h ns exp 2iπµ λ + o (h ns ) =: ψ s s (h ns ) ∈ Diff (C, 0) . ( 1
ϕ • := (ϕ n • , ϕ s • ) ∈ Holo c (Σ) × Holo c (Σ × (C, 0))
with ϕ s • (0) = 0. We call orbital necklace associated to ϕ • (and µ • implicitly) the complex manifold Ω (ϕ • ) obtained by the analytic atlas consisting in two copies Ω ns and Ω sn of (Σ ∪ {0}) × C, with transitions maps

ψ n • : Ω ns -→ Ω sn (s, h) -→ (s, h + ϕ n s ) ψ s • : (Ω ns , 0) -→ (Ω sn , 0) (s, h) -→ (s, h exp (2iπµ λ + ϕ s s (h))) .
2. A diffeomorphism between two necklaces Ω (ϕ • ) and Ω ( ϕ • ) is the data (Ψ ns • , Ψ sn • ) of s-fibred injective, holomorphic mappings inducing a conjugacy between atlases:

Ψ ns • • ψ ♭ • = ψ ♭ • • Ψ sn • , ♭ ∈ {n , s}
and such that for s = 0

Ψ ns • • ψ g • = ψ g • • Ψ sn • (1.41)
where

ψ g s (h) := h exp iπ 1 s + µ λ .
We say that the necklaces are analytically conjugate. 3. Let Z • be a generic unfolding of formal orbital class µ • as in Theorem 7. We call orbital class of Z • the necklace Orb (Z • ) := Ω (ϕ • ) where ϕ • is built from the mappings (1.39) and (1.40).

Remark 14. We do not take the gate-part identification in account to build the necklace because the construction does not make it at the limit. It is not needed anyway to perform the local classification, because both sectorial normalizing maps always glue on V g s . However the actual space of leaves Ω s on V s is the quotient of the corresponding s-fiber of Orb (Z • ) by ψ g s for s = 0. This is why we must include the condition (1.41) to capture all information relating to orbital conjugacy. Because holomorphic automorphisms of the complex line are rigid there are not many necklaces diffeomorphisms. Hence the orbital necklace of an unfolding is nearly a local invariant, and we prove in the section that it suffices to characterize its local class.

Lemma 10. (Ψ ns

• , Ψ sn • ) is a diffeomorphism of necklaces if and only if there exists c • ∈ Holo c (Σ) × such that Ψ ♯ s (h) = c s h.

Proof. One direction is trivial. Assume then that (Ψ ns • , Ψ sn • ) is a diffeomorphism between necklaces. For each s ∈ Σ ∪{0} the mapping Ψ ♯ s is biholomorphic, thus an invertible linear map h → c ♯ s h. The conjugacy equation Ψ ns s • ψ g s = ψ g s • Ψ sn s implies c ns s = c sn s for s = 0, thus also for s = 0 by continuity.

⊓ ⊔

Integral representation of the saddle orbital invariant

Thanks to Theorem 8 we know the sectorial separatrices s ♯ s glue to a holomorphic function s nsn s on V nsn s , therefore so does R ♯ s appearing in (1.37), yielding a function

R nsn • ∈ Holo c (D nsn ). Proposition 5. Let G • ∈ Holo c (D nsn ) such that (s, x) → Gs(x,0) P λ (x)
is bounded, and let Orb (X • ) be the orbital necklace of X • . The mapping (given in the chart Ω ns )

T • (G • ) : (Orb (X • ) , 0) -→ C (s, h) -→ 1 2iπ ˆγ(p) G s dx P λ ,
where γ (p) is an asymptotic cycle defined in Theorem 9 Item 2. such that H ns s (p) = h, is well-defined and vanishes along {h = 0}. This mapping defines the (linear) period operator

T • : G • -→ T • (G • ) ∈ Holo c (Orb (X • ) , 0) . Proof. Because X λ • F ns s = X λ • F sn s = G λ the difference F sn s -F ns
s is a first integral of X λ , and therefore factors as a map τ s defined on the sectorial space of leaves Ω s s . This can also be seen from Theorem 9. Indeed the value of the integral depends only on the asymptotic tangential homotopy class of γ (p), as an asymptotic tangential homotopy is uniformly continuous. Hence only asymptotic cycles with p ∈ V s s contribute to the period, and the value of the integral only depends on the sectorial leaf F λ | V s s , since there is at most one non-trivial homotopy class of asymptotic cycles per leaf. The same argument shows that τ s (0) = 0, for any asymptotic cycle within the sectorial separatrix is trivial.

⊓ ⊔

The transition map ψ s • of the orbital necklace of X • obey the identity

H sn s (p) = ψ s s (H ns s (p))
while at the same time Recalling how we just defined the period operator, we obtain an integral representation for the saddle-part of the orbital invariant.

H ♯ s (p) = H ∞,♯ s (p) exp O ♯ s (p) , following (1
f • ∈ Holo c (Ω (ϕ • ) , 0) with f • (0) = 0 is called temporal necklace. 2. A diffeomorphism between two temporal necklaces (ϕ • , f • ) and ϕ • , f • is the data (Ψ ns • , Ψ sn • )
of a diffeomorphism between corresponding orbital necklaces and satisfying:

f • = f • • Ψ sn • .
We say that the necklaces are analytically conjugate.

3. Let Z • = U • X • be a generic unfolding under prepared form (1.26) with formal invariants (µ • , u • ) as in Theorem 7. We call local class of Z • the temporal necklace

Class (Z • ) := Orb (X • ) , T • 1 U • - 1 u •
where the period operator T • is defined in Proposition 5.

Classification theorem

As prompted by Lemma 10 we define the action of c

• ∈ C {λ} × on functions f • ∈ Holo c (Σ × (C, 0)) by c * • f • : (s, x, y) -→ f s (x, c λ y) .
The action is extended component-wise to tuples of functions.

Theorem 12. (See [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF]) Two generic unfoldings of codimension 1 in the same formal class (µ

• , u • ) ∈ C {λ} × C {λ} [x]
:= O ♯ s • O ♯ s •-1 . Hence Ψ s | V ns s := Y ns s and Ψ s | V sn s := Y sn
s defines a holomorphic conjugacy between X λ and X λ on V s = V ns s ∪ V sn s . This mapping is bounded, thus extends biholomorphically to adh (V s ) by Riemann's removable singularity theorem.

To conclude the proof we only need to check that Ψ s = Ψ -s . Define S s := Ψ •-1 s • Ψ -s which, by construction, is a symmetry of X λ , that is S * s X λ = X λ . A direct computation at a formal level on S s (x, y) = x , y + n+m>1 S n,m (s) x n y m establishes S s = Id. ⊓ ⊔ The classification presented above cannot be complete, as we explain in the upcoming Section 1.9.4.

Dynamical interpretation of the orbital necklace

We describe the relationship between the actual dynamics of X λ (the holonomy of the underlying foliation F λ ) and what could be called the necklace dynamics. We define in the first place what we mean by «the dynamics of X λ » in Section 1.9.1. It encodes the «monodromy» of the canonical first integrals H ♯ s , which really is what the orbital necklace is all about. As X λ does not depend on the choice of s or -s, the splitting of U into squid sectors is artificially superimposed on the dynamics. By rewording this fact as a relationship between orbital necklaces Ω (ϕ s ) and Ω (ϕ -s ), we derive the orbital compatibility condition in Section 1.9.3. The temporal compatibility condition will be derived while performing the temporal realization in Section 1.10.1.

We finally use the compatibility condition to characterize even necklaces, corresponding to invariants ϕ • = (ϕ n • , ϕ s • ) holomorphic as a function of the parameter λ. We show this configuration to be very rare, underlying the lack of completeness of the classification provided by Theorem [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF].

Weak holonomy

Fix once and for all x * ∈ V n s \ρD (by construction of the squid sector in Definition 10 this domain is independent on s ∈ Σ) and a transverse disc

T := {x = x * } ∩ s∈Σ V s s = {x * } × (C, 0) .
Because X λ is in the prepared form (1.26) its integral curves are everywhere transverse to the fibers of the natural projection to the constrained flow-system

γp * = γ P λ • γ X λ • γ p * , γ p * (0) = p * .
Notice that the image of γ p * is included in a single leaf of F λ . Of course if γ is «too long», then γ p * may eventually escape from U. On the contrary if γ p * is defined on the whole

Lemma 11. The nodal holonomy h n

s is an injective holomorphic map on a subdomain T s of T containing both points of intersection of T with the sectorial weak separatrices {y = s ns s (x)} ∪ {y = s sn s (x)}.

Necklace holonomy

While walking along a loop γ : [0, 1] → D λ and lifting it in the foliation F λ to build the holonomy h γ λ , one may follow what happens in the orbital necklace. More precisely, since the image of γ in the (global) leaves space corresponds with just a point, one may wish to understand the «trajectory» induced by γ in the orbital necklace Ω (ϕ • ). We recall that a base-point x * = γ (0) is fixed once and for all now. The restriction of the canonical first integral to the transverse disc T induces an invertible mapping

H s : (x * , y) ∈ T s -→ H ns s (x * , y) ∈ (Ω ns , 0)
whose image contains {0, ϕ n s } (Lemma 11). Starting from h 0 := H s (p * ) we build a sequence of points (h n ) 0≤n≤2d such that h 2m ∈ Ω ns s and h 2m+1 ∈ Ω sn s , the connection between h n-1 and h n being given by the action of ψ ♭n •±1 for ♭ n ∈ {n, g, s} corresponding to the connected component V ♭ s being crossed by γ, the sign being determined by whether the path leaves V ns s («+») or enters it («-»), as long as the partial lift of γ in F λ is defined. We name ∆ γ s the necklace holonomy

∆ γ s : h 0 -→ h 2d .
The maps ∆ γ s and h γ represent the same dynamics since they are conjugate:

H * s ∆ γ s = h γ λ .
(1.42) Let us generalize the construction to abstract orbital necklaces Ω (ϕ • ). Let W be the free group on the letters {n, g, s}. For every s ∈ Σ there exists a group morphism

x * γ Γ n g s H ns H ns ψ s ψ g ψ n h 0 δ h 1 := ψ g (h 0 ) 0 0 δ = (ψ n ) •-1 • ψ g ψ s ψ g ψ n h 0 ∆ h 1 := ψ s (h) 0 0 ∆ = (ψ n ) •-1 • ψ s Ω ns Ω ns Ω sn Ω sn n -1 s n -1 g h 2 h 2 s -s
w s : π 1 (D λ , x * ) -→ W γ -→ ♭ ǫ1 1 • • • • • ♭ ǫ 2d 2d , ♭ j ∈ {n, g, s} , ǫ j ∈ {±1}
defined in such a way that (♭ j , ǫ j ) j≤2d is the sequence obtained as before: ♭ j corresponds to the connected component V ♭j s currently crossed by γ, the sign being determined by whether the path leaves V ns s or enters it. For instance

w s (γ s ) = n -1 g =: γ w s (Γ ) = n -1 s =: Γ .
We omit the proof of the following lemma.

Lemma 12.

1. For every γ ∈ π 1 (D λ , x * ) the word w s (γ) has the form

w s (γ) = d j=1 ♭ -1 j,1 ♭ j,0 .
2. The image W of w s is generated by the words n -1 g and n -1 s:

W = γ , Γ .
3. The mapping w s : π 1 (D λ , x * ) → W is bijective. For any s ∈ Σ we write

p s : W -→ π 1 (D λ , x * ) γ -→ γ s Γ -→ Γ its inverse.
Definition 15. Being given an orbital necklace Ω (ϕ • ) we build a dynamics in the following manner. Take s ∈ Σ and

w = ♭ ǫ1 1 • • • • • ♭ ǫ 2d
2d ∈ W, then denote γ := p s (w). We define the necklace holonomy associated to γ (or w) as the following symbolic expression

∆ γ s := 1≤j≤2d ψ ♭j s •ǫj
.

Depending on γ the expression ∆ γ s may not represent an actual germ of a diffeomorphism, because h γ λ may not be geometrically defined. For that reason the map ∆ • s ranges in the pseudogroup of local diffeomorphisms of Ω ns s . The necklace holonomy representation is the collection ∆ • = (∆ s ) s∈Σ of (pseudo-)group morphisms

∆ s : w ∈ W -→ ∆ s (w) := ∆ ps(w) s .

Orbital compatibility condition

Define

Σ ∩ := Σ ∩ (-Σ) = {s ∈ Σ : -s ∈ Σ} = Σ + ∪ Σ - Σ ± := Σ ∩ ∩ {±ℑ (s) > 0}
the union of two domains on which we can compare the necklace dynamics for s and -s. According to (1.42) and Figure 1.19 we have for s ∈ Σ ∩ the identities between actual diffeomorphisms .43) In order to motivate the definition of compatibility condition, we must explicit the bridge between relations on Σ + and on Σ -. We obtain this connection by rewording algebraically

     H * s ∆ s n -1 g = H * -s ∆ -s g -1 s if s ∈ Σ + H * s ∆ s n -1 g = H * -s ∆ -s n -1 sg -1 n if s ∈ Σ - H * s ∆ s n -1 s = H * -s ∆ -s n -1 s . ( 1
σ : W -→ W γ -→ γ -1 Γ Γ -→ Γ with inverse σ •-1 : W -→ W γ -→ Γ γ -1 Γ -→ Γ satisfies p s = p -s • σ •±1 for s ∈ Σ ± .
The system (1.43) expresses that the necklace holonomy associated to an actual unfolding is compatible with the latter identity. Yet the system explicitly involves the sectorial first integrals H ns • and is therefore not intrinsic to the orbital necklace. The key to resolve this issue is to observe that for s ∈ Σ + the mapping

∆ -s g -1 s ∈ Diff (C, 0)
is hyperbolic and therefore locally analytically linearizable near the fixed-point 0. There exists only one such analytic linearization with prescribed linear part. Because ∆ s n -1 g is an affine map the invertible function

η s := H s • H •-1 -s (1.44)
is a holomorphic linearization of ∆ -s g -1 s . Hence η s can be recovered uniquely, up to its linear part, from the knowledge of the orbital necklace.

Definition 16. For an orbital necklace Ω (ϕ • ) recall the symbolic holonomy representation ∆ • . We say that the orbital necklace Ω (ϕ • ) is a compatible orbital necklace when there exists η

• ∈ Holo c (Σ ∩ × (C, 0)) such that for every s ∈ Σ + (resp. s ∈ Σ -) the mapping η s is a local linearization of ∆ -s g -1 s (resp. ∆ -s n -1 sg -1 n satisfying the next properties. ⊛ For every s ∈ Σ ∩ η ′ s (0) = 1 . ⊛ For every s ∈ Σ ∩ η s • η -s = Id . ⊛ For every s ∈ Σ ∩ η * s ∆ s n -1 s = ∆ -s n -1 s ,
which is equivalent to the conjugacy of the whole dynamics:

η * s ∆ s = ∆ -s • σ •±1 for all s ∈ Σ ± .

Characterization of even purely convergent unfoldings

For the sake of concision we only deal with the case µ • = 0. Proposition 6. Take a purely convergent generic unfolding of codimension 1 with µ = 0, i.e. its orbital necklace ϕ • satisfies ϕ n • = 0. There exist p ∈ N and α

• ∈ C {λ} such that ϕ s • (h) = - 1 p log (1 + α • h p ) if and only if ϕ s • = ϕ s -• .
Remark 15. A generalization of this result for general µ • is performed in [START_REF] Rousseau | Analytic normal forms and inverse problems for unfoldings of 2dimensional saddle-nodes with analytic center manifold[END_REF]. In that case the existence of a non-zero, even ϕ s • forces µ • to be a rational constant belonging to p -1 Z.

Because the local classification for saddle-node vector fields is complete, and since ϕ • extends continuously at s = 0, the configuration presented in the proposition is rather rare. As a consequence the classification presented in Theorem 12 is not complete. We also mention that such unfoldings are locally orbitally conjugate to a normal form (Theorem 5)

X ∞ • + λ κ x p+2 y p+1 ∂ ∂y , κ ∈ N .
As in [START_REF] Teyssier | Examples of non-conjugated holomorphic vector fields and foliations[END_REF] it is indeed possible to show that for the above normal form one has

ϕ s • (h) = - 1 p log (1 -2iπλ κ T • (xy p ) (h)) ,
where the period operator is the one associated to the model X ∞ • . The explicit computations done in Corollary 6 therefore proves our claim, as well as one direction of the proposition since the period T • (xy p ) is actually even.

Let us prove the other direction. If ϕ s • is even then the orbital compatibility condition writes η * s ψ s s = ψ s s for some η s tangent-to-identity. Therefore η s , ψ s s < Diff (C, 0) is Abelian. Since η s and ψ s s are tangent-to-identity there consequently exists [START_REF] Loray | Dynamique des groupes d'automorphismes de C, 0[END_REF] a formal tangent-to-identity change φ s in the variable h, unique d ∈ N, ν ∈ C and t ∈ C\ {0} such that

η s := φ * s η s = Φ 1 Z(d,ν) ψ s s := φ * s ψ s s = Φ t Z(d,ν) Z (p, ν) := h p+1 1 + νh p ∂ ∂h .
Observe that for all t ∈ C the diffeomorphism

Φ t Z(p,0) (h) = h (1 -pth p ) 1/p
is a ramification of homography, therefore we aim at showing ν = 0. This is ultimately done by applying the next lemma.

Lemma 13. [5, Assertions 1.1 to 1.4]In the following ξ is a formal diffeomorphism in the variable h at 0.

1. Let Z, Z be formal vector fields in the variable h at 0. If ξ * Φ 1 Z = Φ 1 Z , then ξ * Z = Z (the converse is trivial). 2. Assume that ξ * Z (p, ν) = aZ (p, ν) with a = 1. Then ν = 0 and ξ is a ramification of homography (in particular analytic).

Show now that ν = 0 and φ s itself is a ramification of homography, forcing ψ s s to be also one. Taking into account the fact that η s linearizes ∆ -s g -1 s = ℓ s ψ s s for ℓ s := exp iπ s , we obtain the relation

L * s η s = ψ s s •-1 • η s = Φ 1-t Z(p,ν)
where

L s := φ •-1 s • (ℓ s φ s ) .
We apply the lemma with ξ := L s and a := 1-t = 1. Hence ν = 0 and L s is a ramification of homography. φ s is a formal linearization of L s which is tangent-to-identity. For values λ of the parameter corresponding to ℓ s / ∈ R (say ℑ (ℓ s ) > 0) the fix-point 0 of L s is hyperbolic: the map φ s is locally holomorphic at 0, unique and therefore given by

φ s := lim n→∞ L •n s ℓ -n s
uniformly on a neighborhood of 0. Since for for every n ∈ N the map

L •n s ℓ -n
s is a ramification of homography the result folds by taking the limit n → ∞.

Remark 16. In the coordinate induced by φ s the subgroup η s , ψ s s is given by

G := h (1 + ξ • h p ) 1 /p , h (1 + α • h p ) 1 /p =: η • , ψ s • for α • , ξ • ∈ C {λ}. Hence L s must be of the form L s (h) = ℓ • h (1 + δ s h p ) 1 /p ,
and we deduce

ξ • = α • 1 -ℓ p • .
(1.45)

In this specific configuration we observe explicitly that the transition mapping η s cannot be defined for all values of s ∈ Σ save when α • = 0, because of (1.45).

Instances of complete classifications

Complete temporal classification

We first describe the range of the period operator (Proposition 5). The next theorem is showed in Section 1.10.1.1.

Theorem 13. For a generic unfolding X • of codimension 1 we recall its orbital necklace Orb (X • ) = Ω (ϕ • ) and associated transition map η • as in (1.44).

1. Let f • ∈ Holo c (Orb (X • ) , 0) such that f • (0) = 0. There exists G • ∈ Holo c Σ × C 2 , 0 with G • = O (P • ) + O (y) such that T • (G • ) = f • . 2. We can find such a function G • satisfying G • = G -• on Σ ∩ if, and only if ∀s ∈ Σ + ∞ n=0 f s • ∆ s g -1 s n • η s = ∞ n=0 f -s • ∆ -s s -1 g n . (1.46)
Notice that because ∆ s g -1 s and ∆ -s s -1 g are tangent to exp iπ -1 s + µ λ Id and exp iπ -1 s -µ λ Id respectively, both sums converge geometrically.

Definition 17. Let (ϕ • , f • ) be a temporal necklace (Definition 13). We say that it is a compatible temporal necklace if Ω (ϕ • ) is a compatible orbital necklace and if f • fulfills the condition (1.46). This is equivalent to the following statement: for s ∈ Σ + let φ ±s denote the unique solution vanishing at 0 of the discrete cohomological equations

φ s -φ s • ∆ s g -1 s = f s φ -s -φ -s • ∆ -s s -1 g = f -s , then η * s φ s = φ -s .
Remark 17. It is sufficient to ensure the relation holds on Σ + because the orbital necklace is compatible. We refer to Remark 18 for more details.

Corollary 5. Being given X The proof relies on solving two Cousin problems, one in (x, y)-space and the other one in s-space.

• define Fol (X • ) := U • X • : U • ∈ Holo C 3 , 0 × and Compat (X • ) := {f • ∈ Holo c (Σ × (C, 0)) : f • satisfies (1.46)} . 1. We have a complete classification Mod loc (Fol (X • )) ≃ Compat (X • ) / C {λ} × . 2. If X • = X ∞ • we have normal forms NF loc (Fol (X ∞ • )) := u • 1 + u • Q • : u • ∈ C {λ} [x] × ≤1 , Q • ∈ Section (1, τ ) X ∞ • where τ := 0 if µ 0 / ∈ R ≤0 1 + ⌊-µ 0 ⌋ otherwise Section (1, τ ) := xP τ λ yC {λ, P τ λ y} .
V ns

V sn V s V n s n s s V g Γ sn Γ ns (a) Spatial contours -Σ Σ Σ + Γ - Γ + Σ - (b) Parametric contours
F sn s -F ns s = δ s • H ns s on V s s 0 on V g s ∪ V n s .
Proof. This is a slight variation on the classical Cauchy-Heine transform. See Figure 1.20 (a). For (x, y) ∈ V ♯ s we set

F ♯ s (x, y) := 1 2iπ ˆΓ ♯ s δ s • H ns s (z, y) z -x dz ,
which by hypothesis defines an element of Holo c D ♯ if Γ ♯ s is deformed slightly to lie outside adh V ♯ s . The rest follows from Cauchy's formula.

⊓ ⊔

The same construction applies for the parametric Cousin problem, using the corresponding contours given in Figure 1.20 (b).

Lemma 15. Given d • ∈ Holo c (Σ + × U) such that s → ds s is bounded, there exists a function D • ∈ Holo c (Σ × U) such that for all s ∈ Σ + we have D s -D -s = d s . Proof. For s ∈ Σ simply set D s := 1 2iπ ˆΓ + d z z -s dz + 1 2iπ ˆΓ - d z z + s dz .

⊓ ⊔

Let us proceed now with the proof of Theorem 13. Lemma 14, applied to the temporal invariant δ s := f s , yields two sectorial functions F ♯ s with prescribed difference over the saddle sector. Because F sn s -F ns s is a first integral of X λ the function defined by

G s := X λ • F ns s on V ns s X λ • F sn s on V sn
s is holomorphic and bounded on V s , therefore extends holomorphically to U. By construction its period coincides with f s . Because F ♯ s has bounded derivatives [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector fields[END_REF],

G • ∈ Holo c (Σ × U) is of the desired form: G s (x, y) = P s (x) ∂F ♯ s ∂x + R λ (x, y) ∂F ♯ s ∂y + y (1 + µ λ x) ∂F ♯ s ∂y .
This proves Item 1.

In general there is no reason for G s = G -s to hold when s ∈ Σ ∩ . We must therefore modify G • by adding to it a function of the form X

• • D • with D • ∈ Holo c (Σ × U), in such a way that G • := G • + X • • D • be even. The equation G -s = G s reads G s -G -s = X λ • (D -s -D s ) .
According to Theorem 10 Item 4. we need the identity

f s = T s (G s ) = T s (G -s )
to hold for s ∈ Σ ∩ . We prove below the hypothesis of Theorem 13 Item 2. guarantees that very property. Taking this fact for granted, we deduce the existence of d • such that G s -G -s = X λ • d s for all s ∈ Σ + using Theorem 10 Item 4. again. This function can be so chosen that lim s→0 d s = 0, in which case s → ds s is bounded. Then Lemma 15 yields the expected D • , completing the proof of Item 2.

Remark 18. From Theorem 10 Item 4. we know that T s (G s -G -s ) = 0 for all s ∈ Σ + if and only if T -s (G s -G -s ) = 0 for all s ∈ Σ + . Therefore f s = T s (G -s ) for all s ∈ Σ ∩ if and only if the equality holds merely on Σ + . Proposition 7.

1. For all s ∈ Σ + we have

η * s ∞ n=0 T s (G -s ) • ∆ s g -1 s n = ∞ n=0 f -s • ∆ -s s -1 g n . 2. f • = T • (G -• ) if and only if condition (1.46) holds. Proof. Set φ s := ∞ n=0 f s • ∆ s g -1 s n , φ -s := ∞ n=0 f -s • ∆ -s s -1 g n and φ s := ∞ n=0 T s (G -s ) • ∆ s g -1 s n .
1. Take p * ∈ T . Any asymptotic cycle γ (p * ), used to compute the period in Proposition 5, is tangentially asymptotically homotopic to the lift in F λ of the limit of nested cycles

lim m→∞ γ m γ m := p -s s -1 g -m n -1 s s -1 g m : [-m, m] -→ D λ (1.47) γ m+1 | [-m,m] = γ m .
We let γ m be the lift of γ m in F λ with γ m (0) = p * . The quantity F ns -s ( γ m (-m)) + ´ γm G -s dx P λ represents the analytic continuation of F ns -s along γ m . By construction the additive monodromy of the continuation of F ns -s is given by the period f -s when turning around the saddle-like singularity of V ns -s . Hence

ˆ γm G -s dx P λ + F ns -s ( γ m (-m)) -F ns -s ( γ m (m)) = m n=0 f -s • ∆ -s s -1 g -n - m n=1 f -s • ∆ -s s -1 g -n n -1 s • H ns -s (p * ) .
Because F ns -s extends continuously to {x = s} and lim n→∞ γ n (±n) = (s, 0), taking the limit we obtain in the chart Ω ns

-s T s (G -s ) • η s = φ -s -φ -s • ∆ -s s -1 gn -1 s .
According to Definition 16, for s ∈ Σ + the identity η * s ∆ s g -1 s = ∆ -s s -1 gn -1 s holds, so that summing over all terms η * s T s (G -s ) • ∆ s g -1 s n for n ∈ Z ≥0 yields the expected result:

η * s φ s = φ -s . 2. The direct implication is trivial. Assume conversely that η * s φ s = ∞ n=0 f -s •∆ -s s -1 g n , i.e. φ s = φ s . Because φ s -φ s • ∆ s g -1 s = f s and φ s -φ s • ∆ s g -1 s = T s (G -s ) we recover f s = T s (G -s ). ⊓ ⊔ 1
T s (G n,m ) (h) = h m × (-m) n+mµ λ Γ (n + mµ λ ) × t λ,n,m × T s,m t λ,n,m := 1 2 n p+q=n n p p-1 j=0 1 -s µ λ + 2j m q-1 j=0 1 + s µ λ + 2j m T s,m := -2s m mµ λ 1 + sµ λ × Γ -m 2s + mµ λ 2 Γ -m 2s -mµ λ 2 .
For given s small enough, the period is zero if and only if n + mµ λ ∈ Z ≤0 . The period is a holomorphic function of λ if and only if mµ • ∈ Z (in which case µ is a rational constant).

Remark 19.

1. Notice that taking the limit s → 0 in Σ leads to

lim s→0 T s,m = lim λ→0 t λ,n,m = 1
recovering classical computations [START_REF] Elizarov | Tangents to moduli maps[END_REF][START_REF] Teyssier | Équation homologique et cycles asymptotiques d'une singularité noeud-col[END_REF][START_REF] Teyssier | Examples of non-conjugated holomorphic vector fields and foliations[END_REF] performed for λ = 0. 2. The eventual lack of evenness of the period comes from the term T s,m . Since it is independent on n, any period T • (y m g (x)), g ∈ C {λ, x}, is even in s if and only if mµ ∈ Z.

Proof. We perform the computation over

V nsn s = D λ \ (V g s ∪ V n s ). Because H s (x, y) := y (x -s) -1 2s - µ l 2 (x + s) 1 2s - µ l 2
is constant on the leaves of F λ we can parameterize an asymptotic path as

x ∈ γ ∞ -→ x, h (x -s)

1 2s + µ l 2 (x + s) -1 2s + µ l 2
where h = H ns s (p * ) and γ ∞ = lim p→∞ γ p (as in (1.47)) is the projection on {y = 0} of the asymptotic cycle γ (p * ). This projection does not depend on the choice of p * ∈ T .

Remembering the computations performed in Proposition 3, we introduce the Pochhammer contour P s ∈ π 1 (D λ , x * ) whose encoding in the dynamics necklace is given by w s (P s ) = n -1 gs -1 ng -1 s .

For h ∈ (Ω ns s , 0) let P s (h) be the lift in F λ of P s starting from p * ∈ T with h = H s (p * ). Both necklace holonomies ∆ s ( γ) and ∆ s Γ are linear in the same coordinate H s | T , and therefore commute. Because the Pochhammer contour is a commutator we have ∆ s n -1 gs -1 ng -1 s = Id .

Hence P s (h) is a (non-trivial) element of the fundamental group of the corresponding leaf of F λ . As a matter of consequence This quantity equals 1 if and only if mµ λ ∈ Z. ⊓ ⊔ Remark 20. The fact that the complete invariant ϕ n • introduced in Section 1.1.3 for affine unfoldings is obtained from the above proposition for µ • := 0 and m := -1 is not fortuitous and can be explained very much like in [START_REF] Teyssier | Examples of non-conjugated holomorphic vector fields and foliations[END_REF]. The best heuristics is the relation

X λ = X ∞ λ -a λ P λ ∂ ∂y = X ∞ λ -
a λ P λ y × y ∂ ∂y so that locally conjugating X ∞ • to X • is somehow equivalent to solving analytically

X ∞ λ • F λ = a λ P λ y .
A more precise approach would require to study the generic saddle-node unfolding near (0, 0, ∞) whose node-and saddle-parts corresponds to saddle-and node-parts near (0, 0, 0).

We deduce from this proposition the following result, concluding the proof of Corollary 5.

f • (0) = 0} Q • -→ T • (Q • )
is bijective.

In the next paragraph we generalize this result for all purely convergent unfoldings.

Proof. This is a consequence of the following two facts:

⊛ the period operator sends y m x n to some monomial h m T s,n,m , ⊛ T s,n,m = 0 for s ∈ (Σ, 0) if and only if T 0,n,m = 0, that is n + mµ λ ∈ Z ≤0 .

The choice of τ prevents T s,mτ (k+1),m to vanish so that the operator is formally invertible. The fact that the inverse operator maps convergent power series f • to convergent power series is a consequence of lim s→0 T s,1+τ (k+1)m,m = T 0,1+λm,m = 0 and of estimates established in [START_REF] Teyssier | Examples of non-conjugated holomorphic vector fields and foliations[END_REF] for T 0,1+mλ,m . ⊓ ⊔

Normal forms for pure convergence

Here we assume that Z • is a generic unfolding of codimension 1 whose orbital invariant has null node-part:

ϕ n • = 0 .
We say that the unfolding is purely convergent.

= u • 1 + u • Q • X ∞ • + yR • ∂ ∂y : Q • , R • ∈ Section (1, τ )
is a family of normal forms for Convergent (1, τ ). Two vector fields in normal forms are locally analytically conjugate if and only if there exists c • ∈ C {λ} × such that R λ (x, y) = R λ (x, c λ y) Q λ (x, y) = Q λ (x, c λ y) .

Remark 21.

1. Notice that the normal forms are not in prepared form (1.26).

2. This collection of normal forms depends analytically on the parameter, hence we have a presentation where the invariants (the functions Q • and R • ) depends analytically on the parameter. This is very much the opposite case of Theorem 12 (compare with Section 1.9.4).

We show this result in two steps, following the strategy presented in [START_REF] Schäfke | Analytic normal forms for convergent saddle-node vector fields[END_REF][START_REF] Rousseau | Analytic normal forms and inverse problems for unfoldings of 2dimensional saddle-nodes with analytic center manifold[END_REF] for λ = 0.

For the sake of clarity we only deal here with the case ℜ (µ 0 ) > 0 , particularly implying τ = 0.

The initial data is a formal class (µ • , u • ) and a compatible temporal necklace (ϕ • , f • ), as in Definition 17, with ϕ n • = 0. Proposition 9. One can find r ′ > 0 and a covering of (C\ {±s}) × r ′ D into two modified, infinite canonical sectors V ♯ s , ♯ ∈ {ns, sn}, such that the following properties hold. 1. There exists a unique collection of holomorphic vector fields In other words, two vector fields as above are locally orbitally conjugate for all λ ∈ Λ if and only if R

• (y) = R • (c • y) for some c • ∈ C {λ} × .
It might happen that R • = R -• in Item 1., preventing X • to be a generic unfolding. The compatibility condition precisely guarantees that R • = R -• so that X ∞ • + yxR • is the expected normal form in that case. The following lemma hence completes orbital realization with normal forms. Lemma 16. Let X • be a collection of vector fields in the form (1.48), with associated orbital necklace Ω (ϕ • ). Then R • = R -• if and only if Ω (ϕ • ) is compatible.

The temporal realization is a straightforward consequence of the next concluding result, as was done to prove Corollary 5. This is a generalization of Corollary 6 to generic unfoldings under normal forms. Proposition 10. Let X • be a generic unfolding of codimension 1 in normal form of Theorem [START_REF] Klimeš | Confluence of Singularities of Nonlinear Differential Equations via Borel-Laplace Transformations[END_REF]. The operator

Section (1, τ ) -→ {f • ∈ Holo c (Σ × C, 0) : f • (0) = 0} Q • -→ T • (Q • )
is bijective.

Orbital realization on Σ (proof of Proposition 9)

The proof is achieved by iterating a Cauchy-Heine integral transformation solving a certain sectorial Cousin problem, like in Lemma 14, to obtain H ♯

• . In fact we seek two functions O ♯

• such that 1. The modified squid sector V ns s is obtained from the union of a squid sector of radius r as in Definition 10 and the half-rays exp -iπ 8 R ≥r and exp 9iπ 8 R ≥r . The construction is analogous for V sn s , as well as their saddle-, gate-and node-parts. 2. Let r ′ > 0 be given. We call modified canonical sector V ♯ s the product V ♯ s × r ′ D. 3. We say that a triple (r ′ , O sn

H ♯ • := H ∞,♯ • exp O ♯ • as in (1.38). s s s n Γ ns Γ sn V ns V sn V s V n V g
• , O ns • ), with O ♯ • ∈ Holo c D ♯ , is adapted to a domain Ω = (C, 0) if H ns s (V s s ) ⊂ Ω for all s ∈ Σ.

The next result is the basis of the construction. We omit the proof, which is a straightforward generalization of its counterpart in [START_REF] Schäfke | Analytic normal forms for convergent saddle-node vector fields[END_REF] for s = 0 and can be found in [START_REF] Rousseau | Analytic normal forms and inverse problems for unfoldings of 2dimensional saddle-nodes with analytic center manifold[END_REF]. Instead we focus on the constructions involved, stressing the few steps where the case s = 0 does not extend straightforwardly. with some constant K > 0 depending only on µ 0 , ρ and r.

Remark 22. We mention the fact that the assumption ℜ (µ λ ) > 0 guarantees the convergence of the integrals near ∞, since f s H ♯ s (z, y) ∼ ∞ Cz -µ λ for fixed y ∈ r ′ D. allows to get rid of the term R s (0, y), so we may as well assume that R s only depends on y, concluding the proof. ⊓ ⊔ So far we have proven Proposition 9 Item 1. The second item can be shown in exactly the same way as in [START_REF] Schäfke | Analytic normal forms for convergent saddle-node vector fields[END_REF][START_REF] Rousseau | Analytic normal forms and inverse problems for unfoldings of 2dimensional saddle-nodes with analytic center manifold[END_REF], so we shall skip additional details.

Gluing antipodal realizations (proof of Lemma 16)

Assume that R λ is holomorphic with respect to λ. The orbital necklace Ω (ϕ • ) is compatible: η

• := H • • H -• conjugates the necklace dynamics ∆ • and ∆ -• , while η •-1 • = η -• . Conversely, if Ω (ϕ • ) is compatible then ψ • := H •-1 • • η • • H -• is
a self-map of T conjugating the holonomy representation of R • and R -• on T . This map is tangentto-identity in the y-variable. Because the union of leaves L p of F s for p ∈ T contains a uniform, connected neighborhood U of {(±s, 0)}, there exists a family of paths γ (x) linking x to x * so that h γ(x) s (x, y) ∈ T for every (x, y) ∈ U. The map built à la Mattei-Moussu [START_REF] Mattei | Holonomie et intégrales premières[END_REF] Ψ s : (x, y) -→ x , h -γ(x) s • ψ s • h γ(x) s (x, y) is therefore well-defined, biholomorphic and locally conjugates X • and X -• . The conclusion follows from the uniqueness clause of Proposition 9 Item 1, as the linear part of Ψ s in the y-variable is 1. More details can be found in [START_REF] Rousseau | Analytic normal forms and inverse problems for unfoldings of 2dimensional saddle-nodes with analytic center manifold[END_REF]. Item 2., we can give a polynomial bound on the growth of x → Q s (x, y), so that it must be of the expected form after a final correction in the y-variable to normalize Q s (0, y) = 0. The claim follows from the same reasoning as in Theorem 13 in order to perturb Q • so that the resulting function is even in s and has same period.

Fig. 1 . 1 :

 11 Fig. 1.1: Typical members of the simplest saddle-node bifurcation

Fig. 1 . 2 :

 12 Fig. 1.2: Solutions of x 2 y ′ = y

Lemma 2 .

 2 For every λ ∈ Λ we have ϕ (s) sin π 2s = π .

exp 1 u

  du is well-defined and holomorphic on D\ [0, 1]. It is the only solution of the equation which tends to 0 at 0 over R <0 . It must therefore coincide with ŝ when one of the two objects represents a holomorphic function on D. The conclusion follows from the fact that ϕ n 0 embodies the monodromy of the multivalued continuation of s - 0 on D\ {0}. ⊓ ⊔ Let us present now the classification theorem.

⊛⊛

  The set of biholomorphic mappings U → Ũ from an open set U ⊂ C n onto another one Ũ is written Diff U → Ũ . As before this construction can be germified near any A, à ⊂ C n , yielding the set Diff A → à whose elements Ψ belong to some Diff U → Ũ with A ⊂ U, à ⊂ Ũ and Ψ (A) = Ã. In the special case A = {p} and à = {p} we conventionally write Diff ((C n , p) → (C n , p)) instead. If moreover p = p we name Diff (C n , p) the (pseudo)group of germs of a diffeomorphism fixing p. ⊛ A tuple of power series Ψ

Fig. 1 . 4 :

 14 Fig. 1.4: Some leaves of the foliation induced by X ∞ 0 on R 2 . Mixed saddle (left) and node (right) behaviors are apparent.

(Fig. 1 . 5 :

 15 Fig. 1.5: Real foliation of the vector field Z when l ∈ R ×

Example 9 .

 9 is indeed bijective: any first integral factors uniquely and holomorphically through H. The function H (x, y) := xy 2 is a first integral of the resonant linear saddle X (x, y) := 2x ∂ ∂x -y ∂ ∂y . Its fibers are the connected Riemann surfaces xy 2 = c , c ∈ C. Notice that the two branches of {xy = 0} are disconnected when the singularity 0 is removed from them.

Example 11 .

 11 Theorem 2 asserts Mod loc (Affine (1)) ≃ N with normal forms NF loc (Affine (1)) := X κ • : κ ∈ N .

  Families Z λ λ∈ Λ having singularities either generically elementary degenerate (e.g., coalescing saddle-nodes) or non-degenerate but reached multiple times, can be studied through (extra-parametric) generic unfoldings (Z λ ) λ∈Λ by specializing values λ = φ λ .

Fig. 1 . 6 :

 16 Fig. 1.6: The sectorial decomposition of a saddle-node of codimension 1. The white invariant curve in the saddle-part (sectorial separatrix) is mapped to 0 in leaves space

Fig. 1 . 7 :

 17 Fig. 1.7: The Glutsyuk connection

  the orbital model, is formally orbitally equivalent to Z • , the notation (• • • ) ∂ ∂y denotes a transverse holomorphic perturbation and U • ∈ C {λ, x, y} × . Moreover the singularities of Z λ are located at points in {y = 0} corresponding to the roots of the polynomial P λ (x) := x 2 + λ . (1.23)

Fig. 1 . 10 :

 110 Fig. 1.10: Modulus of the ycomponent of a leaf over a sector V ♯ in the x-variable represented as a height map in real 3-space

Fig. 1 . 11 :

 111 Fig. 1.11: An asymptotic cycle and its projection on the x-variable. (Modulus of the y-coordinate of the leaf as a height-map)

Proposition 4 .

 4 Let X and Y be two germs of a holomorphic vector field on a domain U such that [X, Y ] = 0. If f is holomorphic on U ( resp. a formal power series at some point p ∈ U), then Ψ (x, y) := Φ f (x,y) Y (x, y)

  This family of solution is unique up to the addition with an arbitrary formal power series belonging to C [[λ]], corresponding to the choice of the value F λ (0).Discounting additional straightforward computations, we just established a formal classification with normal forms. Theorem 7.[START_REF] Rousseau | Analytic normal forms and inverse problems for unfoldings of 2dimensional saddle-nodes with analytic center manifold[END_REF]We have complete classificationsMod for (1) ≃ C {λ} [x] × ≤1 × Mod orb for(1)Mod orb for (1) ≃ C {λ} with normal forms NF for (1) := C {λ} [x] × ≤1 NF orb for (1) NF orb for (1) := P • (x)

Definition 5 .

 5 Let us call Param (1) the complex line C viewed as the s-space. The branched covering λ : Param (1) -→ C s -→ λ (s) := -s 2 , satisfying the identity

Fig. 1 . 3 (

 13 Fig. 1.12: Two non-equivalent decompositions by squid sectors with the same value of the parameter λ = -s 2 . Notice that the nature of the singular points (node-or saddle-like) swaps from one configuration to the other.

2 .

 2 The critical set ∆ of λ is the origin. 3. A (germ of a) holomorphic function f : Param (1) → C factors as f • λ with f holomorphic if and only if f is even ( i.e. Z /2Z-invariant).

Fig. 1 .

 1 Fig. 1.13: A leaf of a nodelike foliation over a slit disc, cut out from a hyperbolic singularity. (Modulus represented as a heightmap, argument as colors in the base)

  starting from some (x * , y * ) ∈ C × × C. We obtain the path tangent toW t ∈ R -→ (x (t) , y (t)) , y(t) = y * exp (ϑt) satisfying the identity H (x (•) , y (•)) = cst. Notice that lim t→±∞ x (t) = 0 ⇐⇒ ±ℜ (ϑl) < 0 lim t→±∞ y (t) = 0 ⇐⇒ ±ℜ (ϑ) < 0 or y * = 0 .

Fig. 1 . 14 :

 114 Fig. 1.14: The typical leaf above a slit disc. (Modulus represented as a height-map, argument as colors in the base)

Fig. 1 . 15 ⊛

 115 Fig. 1.15

Fig. 1 .

 1 Fig. 1.16: Orbital sectorial identifications

.40) Definition 12 .

 12 Let µ • ∈ C {λ} be given.

Corollary 4 .

 4 Let Ω (ϕ n • , ϕ s• ) be the orbital necklace of X • . Then for all s ∈ Σ × {0}ϕ s s = -2iπT s (P • R nsn • ) . 1.8.5.3 Temporal invariant Definition 13. Let µ • ∈ C {λ} and u • ∈ C {λ} [x]× ≤1 be given. 1. The data of an orbital necklace Ω (ϕ • ) and

×

  ≤1 are locally ( resp. orbitally) equivalent if and only if their temporal ( resp. orbital) necklaces are analytically conjugate. In other words we have local classifications Orb : Mod orb loc (1) -→ Holo c (Σ) × Holo c (Σ × (C, 0)) / C {λ} × Class : Mod loc (1) -→ Holo c (Σ) × Holo c (Σ × (C, 0)) × Holo c (Σ × (C, 0)) / C {λ} × . Proof. Let us present only the orbital part of the proof, the temporal part being easier according to Theorem 10 Item 4. and Proposition 4 Item 1. One way is clear: if Z • and Z • are locally orbitally conjugate by Ψ then the conjugacy factors as a diffeomorphism (Ψ ns • , Ψ sn • ) between the orbital necklaces Orb (Z • ) and Orb Z • . Because the conjugacy is analytic in λ the induced linear change of coordinate h → c • h in leaves space is also analytic in λ. Conversely, assume the existence of a diffeomorphism (Ψ ns • , Ψ sn • ) conjugating the necklaces Orb (Z • ) and Orb Z • . Invoking Lemma 10 we can apply the rescaling (s, x, y) → (s, x, c s y) to Z. Without changing notations, in the new coordinates we must have Ψ ♯ s = Id for each s ∈ Σ ∪ {0}. This particularly implies the identities ϕ ♭ s = ϕ ♭ s for ♭ ∈ {n, s}. But these quantities measure the obstruction to glue over V ns s ∩ V sn s the transitions between corresponding sectorial normalization mappings Y ♯ s

Π

  : (x, y) -→ x outside P -1 λ (0). As a consequence we can lift (smooth) paths in the punctured base γ : [0, 1] → D λ , D λ := rD\P -1 λ (0) , through Π into leaves of F λ and starting from points in T . More precisely, being given p * := (x * , y * ) ∈ T there exists a unique (germ at 0 of a) solution t -→ γ p * (t) = (γ (t) , y (t))

[0, 1 ]Fig. 1 .

 11 Fig.1.17: Generators of π 1 (D λ , x * ) for s ∈ Σ

Fig. 1 . 18 :

 118 Fig. 1.18: The dynamics induced by the weak holonomies in the orbital necklace

Fig. 1 . 19 :

 119 Fig. 1.19: Comparing the necklace dynamics

Item 1 .

 1 is a direct restatement of Theorem 13, Theorem 12 and Definition 13 Item 3. We give the proof of Item 2. in Section 1.10.1.2, based on the explicit computation of the period operator for the formal model in terms of the Gamma function. The dominant terms are Γ (1 + m (τ (k + 1) + µ • )) for m ∈ Z ≥0 : the presence of the monomial x τ helps keeping far away from poles for small |s|.1.10.1.1 Range of the period operator (proof of Theorem 13)

Fig. 1 . 20 :

 120 Fig. 1.20: The contour used for the Cauchy-Heine transforms

.10. 1 . 2 3 . 8 .

 1238 Computation of the period of the model X ∞ • Below we compute periods explicitly, extending the calculation of Section 1.1.Proposition Let G n,m (x, y) := x n y m , m ∈ N, and X • := X ∞• . Then for all s ∈ Σ we have (in the chart Ω ns s )

˛˛ 2 - 1 2 sin π m 2s -π mµ λ 2 =

 2122 Ps(h) G n,m dx P λ = T s (G n,m ) -T s (G n,m ) • ∆ s g -1 s .Summing over the forward orbit of ∆ s g -1 s we obtainT s (G n,m ) (h) = ∞ ℓ=0 Ps•∆s((g -1 s) ℓ )(h) dx .We recognize the integral representation of the Beta function. Up to the presence of µ λ and m, the remaining computations are done identically to those in Proposition 3.The function is even if and only if T s,m = T -s,m . Using the reflection formula we findT s,m T -s,m = (-1) mµ λ sin π m 2s + π mµ λ exp iπm 1 2s + µ λ -exp -iπm 2sexp iπm 1 2s -µ λ -exp -iπm 2s .

Corollary 6 .

 6 Recall the notations of Corollary 5. The operator Section (1, τ ) -→ {f • ∈ Holo c (Σ × C, 0) :

Theorem 14 .

 14 For τ ∈ Z ≥0 define Convergent (1, τ ) := {Z • : ϕ n • = 0 , µ 0 / ∈ R ≤τ }Recall the definition of Section (1, τ ) given in Corollary 5 Item 2. The collection NF loc (Convergent (1, τ )) :

  Mod loc (Convergent (1)) ≃ Section (1, τ ) × Section (1, τ ) / C {λ} ×

  X • ∈ X (Σ × C × r ′ D) of the form X s = X ∞ λ + yxR s ∂ ∂y , R • ∈ Holo c (Σ × r ′ D) ,(1.48)and such that the associated canonical first integrals H ♯ • ∈ Holo D ♯ have connected fibers and satisfy H sn s = ϕ ♭ s • H ns s for ♭ ∈ {n, g, s} and s ∈ Σ.2. The action of holomorphic automorphisms of the orbital necklaceΩ (ϕ • ) by h → c • h induces an action c * • : R • -→ R • • (c • Id) .

Fig. 1 . 21 :

 121 Fig. 1.21: The modified (unbounded) squid sectors and associated Cauchy-Heine contours

Theorem 15 .

 15 Consider some (r ′ , O sn • , O ns • ) adapted to Ω. Take any f • ∈ Holo c (Σ × Ω) vanishing along {h = 0} and with bounded derivative f ′ • = ∂f• ∂h on Σ ∪ {0}, then define F ♯ • by F ♯ s (x, y) := 1 2iπ ˆΓ ♯

For r ′ > 0 g

 0 and O := (O sn • , O ns • ) adapted to Ω, we write E (O) the pair (F ns • , F sn • ) built in the previous theorem for f • := ϕ s • . Define the recursive sequence (O n ) n∈∈Z ≥0 starting with O 0 := (0, 0) andO n+1 := E (O n ) , n ≥ 0 .(1.50) Show it converges in the Banach space H := Holo c (D ns ) × Holo c (D sn ) (equipped with the sup-norm). For the sake of clarity we write H n the canonical first integral H ns s built from O ns s . We can assume that all ϕ s s are holomorphic and have bounded derivatives on some disc ηD = (C, 0). Then we choose s exp 2π |µ λ | , and K is the constant appearing in Theorem 15. We wish to ensure that(∀s ∈ Σ) (∀n ∈ N) (∀y ∈ r ′ D) (∀z ∈ V s s ) |H n (z, y)| ≤ η . (1.51) By construction of H n we have for (z, y) ∈ V ns s |H n (z, y)| ≤ r ′ M exp sup V ns s |O ns s | . Therefore if for some n ∈ N we have sup V ns s |O ns n | ≤ 2π M ηK sup ηDdϕ s s dh , then we first find that |H n (z, y)| < r ′ M exp η . (r ′ , O n ) is adapted to ηD and then, using Theorem 15 Item 4 a., we obtain sup These estimates show by induction on n that, with the above choice of r ′ , the relation (1.50) defines a bounded sequence (O n ) n ⊂ H. It so happens that the components of the sequence (O n ) n converge for the Krull topology in Holo c D ♯ [[y]], almost by construction, hence we can ensure it converges in the space Holo c D ♯ by borrowing the argument of [30]. Lemma 17. [30]Let D be a domain in C m and consider a bounded sequence (f p ) p∈N of Holo c (D) satisfying the additional property that there exists some point z 0 ∈ D such that the corresponding sequence of Taylor series (T p ) p∈N at z 0 is convergent in C [[zz 0 ]] (for the projective topology). Then (f p ) p converges uniformly on compact sets of D towards some f ∞ ∈ Holo c (D). The cited reference likewise provides the remaining claims of the upcoming proposition. for some constant C > 0 whose existence is guaranteed by Proposition 11 Item 2. Therefore for any fixed y ∈ r ′ D the partial function x → R s (x, y) is affine. A change of coordinates of the form (λ, x, y) -→ (λ, x, y exp N λ (y))

1. 10 . 2 . 3 A

 1023 section to the period operator (proof ofProposition 10) This is really Theorem 15. Being givenX • ∈ Holo c (Σ × C × r ′ D) in normal form, the functions F ♯ • of (1.49) induces a function Q • := X • • F ♯ • ∈ Holo c (Σ × C × r ′ D)with prescribed period as in Section 1.10.1.1. Applying again the arguments of Corollary 7

  .38). For p ∈ V s s we have H ∞,sn

	s	(p) = H ∞,ns s	(p) exp 2iπµ λ so that
	H sn s (p) = H ns s (p) exp (2iπµ λ + O sn s (p) -O ns s (p)) .

Coalescing complex planar stationary points