Stokes phenomenon and confluence in non-autonomous Hamiltonian systems

This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the sixth Painleve equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system. The problem of analytic classification is also addressed. Key words: Non-autonomous Hamiltonian systems; irregular singularity; non-linear Stokes phenomenon; wild monodromy; confluence; local analytic classification; Painleve equations.


Introduction
We consider a parametric family of non-autonomous Hamiltonian systems of the form shortly written as with a singular Hamiltonian function H(y,x, ) x(x− ) , where H(y, x, ) is an analytic germ such that H(y, 0, 0) has a non-degenerate critical point (Morse point) at y = 0: D y H(0, 0, 0) = 0, det D 2 y H(0, 0, 0) = 0.
The last condition means that the y-linear terms of the right side of (2) are of the form for some λ (0) (0) = 0. For = 0 the system (1) has two regular singular points at x = 0 and x = . At each one of them, the local information about the system is carried by a formal invariant and a monodromy (holonomy) operator. On the other hand, for = 0 the corresponding information about the irregular singularity at x = 0 is carried by a formal invariant and by a pair of non-linear operators. Our main goal is to explain the relation between these two distinct phenomena, and to show how the Stokes operators are related to the monodromy operators. The principal thesis is, that while the monodromy operators diverge when → 0, they each accumulate to a 1-parameter family of "wild monodromy operators" which encode the Stokes phenomenon (Theorem 34). It is expected that this "wild monodromy" should have Galoisian interpretation.
Along the way, we provide a formal normal form and a sectoral normalization theorem for the family (Theorem 14), an analytic classification (Theorem 31), and a decomposition of the monodromy operators (Theorem 32).
In Section 5, we illustrate all this on the example of traceless 2 × 2 linear differential systems x(x − ) dy dx = A(x, )y, A(x, ) ∈ sl 2 (C), det A(0, 0) = 0, for which our description follows from the more general work of Lambert and Rousseau [LR12,HLR13]. Here the relation between the monodromy and the Stokes phenomenon can be summarized as: Theorem. When → 0 the elements of the monodromy group of the system (3) accumulate to generators of the wild monodromy group of the limit system (that is the group generated by the Stokes operators and the exponential torus).
The linear case can be kept in mind as a leading example of which the general non-linear case is a close analogy.
An important example of a confluent family of systems (1), which in fact motivated this study, is the degeneration of the sixth Painlevé equation to the fifth one, presented in Section 6. A more detailed treatment of this confluence will be the subject of an upcoming article [Kli17].

Acknowledgments
This paper was inspired by the works of C. Rousseau and L. Teyssier [RT08], C. Lambert and C. Rousseau [LR12], and A. Bittmann [Bit16a,Bit16b,Bit16c]. It was written during my stay at Centre de Recherches Mathématiques at Université de Montréal. I want to thank Christiane Rousseau for her support and the CRM for its hospitality.

The foliation and its formal invariants
The family of systems (1) defines a family of singular foliations in the (y, x)-space, leaves of which are the solutions. We associate to (1) a family of vector fields tangent to the foliations Z H, (y, x) = x(x − )∂ x + X H,x, (y), where X H,x, (y) = ∂H ∂y2 (y, x, )∂ y1 − ∂H ∂y1 (y, x, )∂ y2 .
The vector field Z H,0 (y, x) has a saddle-node type singularity at (y, x) = 0, i.e. its linearization matrix has one zero eigenvalue, corresponding to the x-direction. It follows from the Implicit Function Theorem that, for small = 0, Z H, has two singular points (y 0 (0, ), 0) and (y 0 ( , ), ) bifurcating from (y 0 (0, 0), 0) = 0 and depending analytically on . The aim of this paper is a study of their confluence when → 0. The two singularities of Z H, have each a strong invariant manifold Y 0 = {(y, x) : x = 0}, resp. Y = {(y, x) : x = }. Away of these invariant manifolds the vector field Z H, is transverse to the fibration with fibers Y c = {(y, x) : x = c}. The (y, x)-space is endowed with a Poisson structure associated to the 2-form the restriction of which on each fiber Y c is symplectic. The vector field Z H, is transversely Hamiltonian with respect to this fibration, the form ω, and the Hamiltonian function H(y, x, ).

Fibered changes of coordinates
We consider the problem of analytic classification of families of systems (1), or orbital analytic classification of vector fields (4), with respect to fiber-preserving (shortly fibered ) changes of coordinates (y, x, ) = (Φ(u, x, ), x, ).
Such a change of coordinate transforms a system (2) to a system using the identity P J t P = det P · J for any 2×2 matrix P .
Definition 1. We call a fibered transformation Φ transversely symplectic if det(D u Φ) ≡ 1, i.e. if it preserves the restriction of ω to each fiber Y x .
Definition 2. Two systems (1) with Hamiltonian functions H(y,x, ) x(x− ) andH (u,x, ) x(x− ) are called analytically equivalent if there exists an analytic germ of a transversely symplectic transformation y = Φ(u, x, ) that is analytic in (u, x, ) and transforms one system to another: Φ * Z H, = ZH , .
Proof. It is enough to show that the system du dx = (D u Φ) −1 ∂Φ ∂x is transversely Hamiltonian, that is, denoting

The formal invariant χ(h, x, )
Theorem 4 (Siegel). Let H : (C 2 , 0) → (C, 0) have a non-degenerate critical point at 0, and let ω be a symplectic volume form. There exists an analytic system of coordinates u = (u 1 , u 2 ) in which The function G H is uniquely determined by the pair (H, ω) up to the involution induced by the symplectic change of variable J : (u 1 , u 2 ) → (u 2 , −u 1 ).  [Vey77] and [FS94]. The uniqueness can be seen by expressing G H (h) in terms of a period map over a vanishing cycle, see Section 2.2.1 below.
-The Theorem 4 provides the existence of an analytic transformation of a Hamiltonian vector fieldẏ = J t (D y H) in dimension 2 to its Birkhoff normal where ±λ = 0 are the eigenvalues of the linear part JD 2 y H(0). The involution (9) corresponds to the freedom of choice of the eigenvalue λ.
-The change of coordinates is far from unique. Indeed, the flow of any vector field ξ = a(u 1 u 2 ) u 1 ∂ u1 − u 2 ∂ u2 preserves the normal form.
Let H = H(y, x, ) be our germ. By the implicit function theorem, for each small (x, ), the function H(·, x, ) has an isolated non-degenerate critical point y 0 (x, ), depending analytically on (x, ). Let y = Φ(u, x, ) be the transformation to the Birkhoff-Siegel normal form for the function y → H(y, x, ) and the form ω = dy 1 ∧dy 2 , depending analytically on (x, ), i.e.
By (7), it brings the system (1) to a prenormal form where h = u 1 u 2 , or equivalently, Definition 6. The function χ(h, x, ) is called a formal invariant of the system (1). For = 0 the formal invariant χ is completely determined by the functions G H (·, 0, ) and G H (·, , ) , which are analytic invariants of the autonomous Hamiltonian systems X H,0, , X H, , (5) on the strong invariant manifolds Y 0 , Y .
Corollary 7. The formal invariant χ(h, x, ) is well-defined up to the involution induced by the symplectic transformation u → Ju. It is uniquely determined by the polar part of the Hamiltonian H(y,x, ) x(x− ) , and it is invariant with respect to fibered transversely symplectic changes of coordinates.
Then ±λ(x, ) are the eigenvalues of the matrix A(x, ) = J t D 2 y H(0, x, ) modulo x(x − ), see Example 8, and the involution (13) corresponds to the freedom of choice of the eigenvalue λ.
Example 8 (Traceless linear systems). A traceless linear system with trA(x, ) = 0 and A(0, 0) ∼ for some λ (0) (0) = 0, is of the form (2) for the quadratic form H(y, x, ) = 1 2 t yJ t A(x, )y. Let ±λ(x, ) be the eigenvalues of A(x, ), and let C(x, ) be a corresponding matrix of eigenvectors of A(x, ), depending analytically on (x, ) and normalized so that det C(x, ) = 1. The change of variable y = C(x, )u, brings the system (14) to

Geometric interpretation of the invariant χ.
For each small (x, ), the function H(·, x, ) has an isolated non-degenerate critical point y 0 (x, ), depending analytically on (x, ), with a critical value h 0 (x, ). For (x, ) fixed, h ∈ (C, h 0 ), consider the germ of the level set S h (x, ) = {y ∈ (C 2 , y 0 (x, )) : H(y, x, ) = h} ⊂ Y x . As a basic fact of the Picard-Lefschetz theory [AVG12], we know that if h is a non-critical value for H(·, x, ), i.e. h = h 0 , then S h (x, ) has the homotopy type of a circle. Let γ h (x, ) depending continuously on (x, ) be a loop generating the first homology group of S h (x, ), the so called vanishing cycle. And let µ be a 1-form such that ω = dH ∧µ; its restriction to a non-critical level S h (x, ) is called the Gelfand-Leray form of ω and is denoted µ = ω dH .
Its period function over the vanishing cycle is well-defined up to a sign change (orientation of γ h ), and depends analytically on (x, ) [AVG12, chap. 10]. Let G H (·, x, ) be the inverse of the function h → h h0(x, ) p(s, x, ) ds. Then (G H , ω) is the Birkhoff-Siegel normal form of (H, ω).
Indeed, the above formula for G H is invariant with respect to analytic transversely symplectic changes of coordinates: Supposing that H = g(y 1 y 2 , x, ) is in its Birkhoff-Siegel normal form, then the level sets are written as S h = {y 1 = 0, y 2 = g •(−1) (h,x, ) y1 }, and ω dH = dy1 y1· ∂g ∂(y 1 y 2 ) •g •(−1) (h,x, ) , and therefore p(h) = ∂g ∂(y1y2) The above formula for the the Birkhoff-Siegel normal form and hence for the formal invariant χ involves a double inversion which makes it difficult to calculate. The following proposition, which will be proved in Section 7.4, allows to determine it in some special cases. This will be useful in the case of the fifth Painlevé equation (Section 6).
Corollary 10 (Invariant χ for = 0). For = 0, suppose that is the formal invariant of the vector field Z H,0 = x 2 ∂ x + X H associated to H.

Model system (formal normal form)
Definition 11 (Model family). Let χ(h, x, ) be the formal invariant of the system (1). The model family (formal normal form) for the the system (1) is the family of systems which is Hamiltonian with respect to the Hamiltonian function G(u1u2,x, ) x(x− ) , The formal normal form of the family Z H, is the associated family of vector fields The system (16) is integrable with the function h(u) = u 1 u 2 being its first integral, Z G, · h = 0. The general solutions of (16) are of the form where 3 Formal and sectoral normalization theorem Throughout the text we will denote for some δ y , δ u , δ x , δ > 0, and implicitly suppose that δ << δ x so that the singular points x = 0, are both well inside X.
• The u 1 -component of the solution (19) tends to 0 along a negative real trajectory of (23) and to ∞ along a positive real trajectory for |ω ± | < π 2 , and vice-versa for the u 2 -component. Denoting x 1,± ( ) the attractive equilibrium point of (23) and x 2,± ( ) the repulsive one, then , when x → x i,± ( ) along a real trajectory of (23).
Before giving a general theorem on sectoral normalization for the parametric family (1), let us first state it for the limit system with = 0 which has an irregular singularity of Poincaré rank 1 at x = 0.
Theorem 13 (Formal and sectoral normalization at = 0). The system (1) with = 0 can be brought to its formal normal form (16) through a formal transversely symplectic change of coordinates where ψ (k) (u) are analytic in u on a fixed neighborhood U of 0. This formal series is generally divergent, but it is Borel 1-summable, with a pair of Borel sums Ψ (u, x, 0) and Ψ (u, x, 0) defined respectively above the sectors x ∈ X (0), X (0) of Definition 12 (for some 0 < η < π 2 arbitrarily small and some δ x > 0 depending on η), and u ∈ U. The fibered sectoral transformations (y, x) = (Ψ • (u, x, 0), • = , , are transversely symplectic and bring the system (1) with = 0 to its formal normal form.
The Theorem 13 is originally due to Takano [Tak90] for systems (1) whose formal invariant is of the form χ(h, x) = λ (0) + xχ (1) (h). In the case of the irregular singularity of the fifth Painlevé equation it was proved earlier by Takano [Tak83]. Some similar and closely related theorems are due to Shimomura [Shi83], Yoshida [Yos84], and recently by Bittmann [Bit16a,Bit16b], which apply to doubly resonant systems −1 under a condition on positivity of tr ∂ ∂x D y F (0, 0). This condition is not satisfied for Hamiltonian systems (1) but nevertheless allows to treat Painlevé equations.
Theorem 14 (Formal and sectoral normalization). Let Z H, (y, x) be a family of vector fields (4) and let χ(h, x, ) be their formal invariant.
(iii) LetΨ(u, x, ) be an analytic extension of the function given by the convergent series (27).
with S θ ⊂ C denoting the circle through the points 0 and 1 with center on e iθ R + , we can express Ψ ± (u, x, ) through the following Laplace transform ofΨ: In particular, As a consequence, Ψ ± andΨ satisfy the same (∂ u , ∂ x , ∂ )-differential relations with meromorphic coefficients.
The proof will be given in Section 7. The transformations Ψ ± andΨ are unique up to left composition with an analytic symmetry of the model system, see Corollary 27. Definition 16. The solution y = Ψ ± (0, x, ) is called ramified center manifold. It is the unique solution that is bounded on X ± ( ) (cf. [Kli16]).
Remark 17. In the variable x = z, the system (2) takes the form of a singularly perturbed system The domains z ∈ 1 X ± ( ), ∈ E ± , then correspond to the Stokes domains in the sense of exact WKB analysis [KT05], where the Stokes curves would be the real separatrices of the point z = ∞ of the vector field (23) e i(ω±+arg ) z(z−1) λ(0,0) ∂ z with a fixed phase ω ± (24).

Stokes operators and accumulation of monodromy
We will define several operators acting as transversely symplectic fibered isotropies on the three following foliations given by three different vector fields: • Foliation in the (u, x)-space given by the model vector field Z G, (18).
• Foliation in the (c, x)-space, c being the constant of initial condition in (19), given by the rectified vector field Z 0, = x(x − )∂ x . Note that a fibered isotropy of Z 0, is necessarily independent of x; it acts on the c-space of initial conditions only.
• Foliation in the (y, x)-space given by the original vector field Z H, (4).

Symmetries of the model system: exponential torus
A vertical infinitesimal symplectic symmetry (shortly infinitesimal symmetry) of the normal form vector field Z G, (18) is a germ of vector field ξ in the (u, x)-space that preserves: Lemma 18. A vector field ξ is an infinitesimal symmetry of Z G, if and only if ξ = X f,x, is a Hamiltonian vector field with respect to ω for a first integral f (u, x, ) of Z G, : The vector field Z G, has the following obvious first integrals (cf. (19)): and (20). Clearly, any function of c = (c 1 , c 2 ) is again a first integral, and since c defines local coordinates on the space of leaves (space of initial conditions), the converse is also true. Note that the map c : u → c(u, x, ) conjugates the vector field Z G, to the "rectified" vector field Z 0, = x(x − )∂ x in the (c, x)-space: It turns out that analytic first integrals are functions of h = c 1 c 2 only.
Proof of Proposition 19. On one hand, c 1 (u, x, ), c 2 (u, x, ) are local coordinate on the space of leaves, hence any first integral is a function of them (depending on ). On the other hand, any analytic germ f (u, x, ) is uniquely decomposed as , we see that for f to be bounded when x → x i,± , we must have f i = 0, i = 1, 2. Therefore f = f 0 which must then be independent of x.
A meromorphic function is a quotient of analytic ones.
Corollary 21. The Lie algebra of analytic infinitesimal symmetries of Z G, consists of Hamiltonian vector fields and is commutative. It is also called the infinitesimal torus.
The time-1 flow map of a vector field (30) is given by Definition 22. A (transversely symplectic fibered) isotropy of the model vector field An isotropy that is analytic in x on a full neighborhood X of both singularities will be called a symmetry.
Definition 23 (Intersection sectors). For ∈ E ± {0} define the left and right intersection sectors and for = 0 let X ∩ i± (0) be their limits. They are the domains of self-intersection of X ± ( ) attached to the points x i,± ( ) (25).
Lemma 24. Let φ i,± (u, x, ) be a sectoral isotropy of the normal form vector field Z G, , analytic and bounded for x ∈ X ∩ i,± ( ), u ∈ U. Then for some analytic germs f i , f j , g j .
Proof. The isotropy φ i,± (u, x, ) is analytic in u on some neighborhood of u = 0 and bounded when x → x i,± . In particular, the restriction of c • φ i,± to any fiber {x = cst = 0, } is analytic in u, and therefore, since c • φ i,± (c, ) is independent of x, it is an analytic function of c on some neighborhood of c = 0.
Writing φ i,± = (φ 1,i,± , φ 2,i,± ), its k-th component is given by and we see that the expansion of Note that the hypersurface {u i = 0} = {c i = 0} consists of all leaves of Z G, that are bounded when x → x i,± inside X ± ( ), and φ i,± must preserve it.
Proposition 25. An isotropy of the normal form vector field Z G, that is bounded and analytic on U × X ± ( ) is a symmetry. It is given by a time-1 flow of some vector field (30).
Proof. An isotropy φ(u, x, ) of the model system bounded and analytic on U × XE ± is in particular bounded and analytic on U × X ∩ i,± ( ) for each ∈ E ± , and therefore by Lemma 24, it is such that for some analytic germs f 1 , f 2 . The transverse symplecticity condition is then rewritten as Corollary 26. The Lie group of (transversely symplectic fibered) symmetries (31) of Z G, is commutative and connected. It is called the exponential torus.
A characterization of the Lie group of symmetries of a general system (1) will be given in Proposition 29.

Canonical general solutions
The model system has a canonical general solution u(x, ; c) (19), depending on an "initial condition" parameter c ∈ C 2 , uniquely determined by a choice of a branch of the function E χ (h, x, ) (20). Correspondingly, y(x, ; c) = Ψ ± (u(x, ; c), x, ) is a germ of general solution of the original system on Y × X ± ( ). In order for this solution to have a continuous limit when → 0, one has to split the domain X ± ( ) in two parts, corresponding to the two parts of X ± (0), by making a cut in between the singular points x 1,± , x 2,± along a trajectory of (23) through the mid-point 2 (see Figure 2). Let us denote X ± ( ) the upper and X ± ( ) the lower part (with respect to the oriented line λ (0) (0) R) of the cut domain The two parts of X ± ( ) intersect in the left and right intersection sectors X ∩ i,± ( ) (Definition 23) attached to {x 1,± , i = 1, 2, and for = 0 also in a central part along the cut. Now take two branches E χ (h, x, ) and E χ (h, x, ) of E χ (h, x, ) on the two parts of the domain, that agree on the right intersection sector X ∩ 2,± , and have a limit when → 0. Correspondingly they determine a pair of general solutions of the model system and a pair of canonical general solutions of the original system Since the transformation Ψ ± is unique only modulo right composition with an exponential torus action T a (u, ) (31), which acts on u • (x, ; c) as

Formal monodromy
The formal monodromy operators are induced by monodromy acting on the solutions u • (x, ; c), • = , , of the model system. For = 0 the induced action of formal monodromies along simple counterclockwise loops around each singular point x i,± = 0, on the 3 foliations is given by: • Monodromy operators of the model system acting on the foliation of the normal form vector field Z G, commutatively by The total monodromy of the model system is given by • Formal monodromy operators acting on the space of initial conditions c commutatively by and a formal total monodromy • Formal monodromy operators N i,± (y, x, ) acting on the foliation of the original vector field Z H, : and The canonical solutions u ± , u ± of the model system on the domains X ± , X ± , are defined such that they agree on the right intersection sector X ∩ 2,± . Therefore on the left intersection sector they are connected by the total formal monodromy operator x ∈ X ∩ 1,± , and by the formal monodromy N xi,± on the central cut between the two domains for = 0 (cf. Figure 2).

Stokes operators and sectoral isotropies
Let y = Ψ ± (u, x, ) be the normalizing transformation on X ± ( ). We call Stokes operators the operators that change the determination of Ψ ± over the left or right intersection sectors. If x ∈ X ∩ i,± ( ), then for = 0 we denotē the corresponding point in X ± ( ) on the other sheet, and extend this notation by limit to = 0. Namely, Then the Stokes operators are the operators which for = 0 are the Stokes operators in the usual sense that send the Borel sum of the formal x-seriesΨ(u, x, 0) in one non-singular direction to the Borel sum in a following non-singular direction.
To each of these Stokes operators we associate sectoral isotropies of the 3 foliations.
The term σ j,i,± (0, 0) is responsible for the ramification of the ramified center manifold y = Ψ ± (0, x, ) of the original vector field Z at the sector X ∩ i,± ( ).
Proof. The isotropy S i,± (u, x, ) is analytic in u on some neighborhood of u = 0 and bounded in x with lim x→xi,± S i,± (u, x, ) = u. By Lemma 24, h,ci, ) , and c j • S i,± = c j · e fj (h, ) + g j (h, c i , ) where f i , f j , g j are some analytic functions of (h, c i , ).

Symmetry group of the system
Proposition 29. The group of (analytic transversely symplectic fibered) symmetries of a system (1) is either 1. isomorphic to the exponential torus: this happens if and only if the system is analytically equivalent to the model (16), or 2. isomorphic to a finite cyclic group.
If the symmetry group is non-trivial, then the system has an analytic center manifold (bounded analytic solution on a neighborhood of both singular points).
This can be satisfied only if • either σ ij,± (h, c j ) = 0 for all i, j, i.e. if S 1,± = id, S 2,± = id and the system is analytically equivalent to its formal normal form, • or there is k ∈ N such that c j σ ij,± (h, c j ) = 0 contains only powers of c k j for all i, j, and e ka = 1, i.e. a ∈ 2πi k Z.

Analytic classification
Definition 30 (Analytic invariants). The collection (χ, {S 1,+ , S 2,+ , S 1,− , S 2,− }) is called an analytic invariant of a system (1). Two analytic invariants (χ, • either χ =χ and there is an element φ(u, ) of the exponential torus, analytic in , such that: • or χ(h, x, ) = −χ(−h, x, ) and there is an element φ(u, ) of the exponential torus, analytic in , such that: Note that the definition of E ± , X ± and x i,± depends on λ(x, ) = χ(0, x, ), therefore the relationλ = −λ entails the renaming By the construction, an analytic invariant of a system (1) is uniquely defined up to the equivalence. Proof. If y = Φ(ỹ, x, ) is an analytic transformation from one system to another, then the sectoral normalizations y = Ψ ± (u, x, ) andỹ =Ψ ± (u, x, ) = Φ • Ψ ± provide the same analytic invariant. Conversely, if the analytic invariants are equivalent, then up to modifying one of the normalizing transformation, one can suppose that they are in fact equal, in which case Φ ± =Ψ ± • Ψ •(−1) ± are analytic transformations between the systems on E + and E − . In fact Φ + = Φ − is an analytic on the whole -neighborhood E. Indeed, the composition Φ + • Φ •−1 − is a symmetry of the second system on the intersection E + ∩ E − , and as such it is determined by its value at x = 0; but sincẽ

Decomposition of monodromy operators
For = 0, let x 0 ∈ X ± ( ) {0, } be a base-point, and let two counterclockwise simple loops around the singular points x i,± , i = 1, 2, be as in Figure 3. Correspondingly, we have two monodromy operators M xi,± acting on the foliation by the solutions of the original system (1) by analytic continuation along the loops. Since the monodromy operators M xi,± act on the foliation, they are independent of the choice of the twoparameter general solution on which they act on the left (a different general solution is where S i,± are the Stokes operators (41) and N i,± are the formal monodromy operators (37). Hence Their right action on analytic extension of the canonical general solutions y • ± (33) to the whole X ± ( ) is given by cf. Figure 2.
Note that in general, a composition of the two monodromies may not be defined if the image of the first does not intersect the domain of definition the second.
1. For = 0, the pseudogroup generated by the monodromy operators is called the (local) monodromy pseudogroup. The pseudogroup generated by the corresponding action on the initial condition c is its representation with respect to the general solution y • ± (x, ; c).
Note that the pseudogroup (44) is independent of the freedom of choice of the sectoral normalizations Ψ • ± of Theorem 13. One of the main goals of this paper is to understand the relation between the monodromy pseudogroup for = 0 and the wild monodromy pseudogroup for = 0.
Suppose that the formal invariant χ(h, x, ) is such that and therefore along which the exponential factor e Then the formal monodromy operators N 0 (u, x, ), N (u, x, ), resp. N 0 (u, x, ), N (u, x, ), converge along each such sequence to a symmetry of the model system (element of the exponential torus) κ ∈ C * . This implies that also the monodromy operators M · i,± (c, ), resp. M xi,± (y, x, ), converge along such sequences Theorem 34. Suppose that the formal invariant of the form (45). Then the monodromy operators of the system (1) for = 0 accumulate along the sequences { n } n∈±N (46) to a 1-parameter family of wild monodromy operators In particular, if we replace κ by e −2πi ∂χ (0) becomes an identity, we obtain the Stokes operators The vector fieldẏ equals to the push-forward Ψ • ± (·, x, 0) * X h of the vector field X h = u 1 ∂ u1 − u 2 ∂ u2 , where • = if i = 1 and • = if i = 2, which "generates" the commutative Lie algebra of bounded infinitesimal symmetries on the sector X • (0).

Confluence in 2×2 traceless linear systems and their differential Galois group
To illustrate the matter of the previous section, let us consider a confluence of two regular singular points to a non-resonant irregular singular point in a family of linear systems where A is a 2×2 traceless complex matrix depending analytically on (x, ) ∈ (C×C, 0), such that A(0, 0) = 0 has two distinct eigenvalues ±λ (0) (0). The Theorem 14 in this case can be found in the thesis of Parise [Par01] and in the work of Lambert and Rousseau [LR12] (see also [HLR13]). It provides us with a canonical fundamental solution matrices where the transformation matrix Ψ ± (x, ) is bounded on XE ± , and is a solution to the diagonal model system The solution basis Y • ± (x, ) is also called a mixed basis: the first (resp. second) column spans the subspace of solutions that asymptotically vanish when x → x 1,± ( ) (resp. when x → x 2,± ( )), and it is an eigensolution with respect to the corresponding monodromy operator M x1,± (resp. M x2,± ) associated to its eigenvalue e ±2πi λ(x 1,± , ) (resp. e ±2πi λ(x 2,± , ) ). A general solution is a linear combination Let K be the field of meromorphic functions of the variable x on a fixed small neighborhood of 0, equipped with the differentiation d dx . For a fixed small , the local differential Galois group (also called the Picard-Vessiot group) of the system (51) is the group of K-automorphisms of the differential field K Y (·, ) , generated by the components of any fundamental matrix solution Y (x, ). The differential Galois group acts on the foliation associated to the system by left multiplication. Fixing a fundamental solution matrix Y = Y • ± , then each automorphism is represented by a right multiplication of Y • ± by a constant invertible matrix, hence the differential Galois group is represented by an (algebraic) subgroup of SL 2 (C) acting on the right.
It is well known [MR91,SP03] that the differential Galois group is the Zariski closure of = 0: the monodromy group generated by the two monodromy operators around the singular points 0 and , = 0: the wild monodromy group 1 generated by the Stokes operators and the linear exponential torus 2 which acts on the fundamental solutions Y • ± as The question is how are these two different descriptions related?

Confluent degeneration of the sixth Painlevé equation to the fifth
The sixth Painlevé equation is where ϑ = (ϑ 0 , ϑ t , ϑ 1 , ϑ ∞ ) ∈ C 4 are complex constants. It is a reduction to the q-variable of a time dependent Hamiltonian system [Oka80] with a polynomial Hamiltonian function It has three simple (regular) singular points on the Riemann sphere CP 1 at t = 0, 1, ∞.
The fifth Painlevé equation P V 4 P V : q = 1 2q is obtained from P V I as a limit → 0 after the change of the independent variable t = 1 + t , ϑ t = 1 , ϑ 1 = − 1 +θ 1 + 1, which sends the three singularities tot = − 1 , 0, ∞. At the limit, the two simple singular points − 1 and ∞ merge into a double (irregular) singularity at the infinity. The change of variables (56), changes the function · H V I to , and the Hamiltonian system to dq dt = ∂H V I, (q, p,t) ∂p , dp dt = − ∂H V I, (q, p,t) ∂q , whose limit → 0 is a Hamiltonian system of P V . In the coordinate x = 1 t + , the above system is written as with H(q, p, x, ) = −(1 + t )H V I, (q, p,t) and Theorem 14 can be applied.
Theorem 37. The formal invariant χ of the system (57) is Proof. Letq hence by Proposition 9 the Birkhoff-Siegel invariant of H(q, p, , ) is hence by Proposition 9 the Birkhoff-Siegel invariant of H(q, p, 0, ) is The Theorem 13 for the limit system = 0 is in this case due to Takano [Tak83], see also [Shi83,Yos85]. A separate paper [Kli17] will be devoted to a more detailed study of the confluence P V I → P V and of the non-linear Stokes phenomenon in P V through the Riemann-Hilbert correspondence.

Proof of Theorem 14 and of Proposition 9
The proof of Theorem 14 is loosely based on the ideas of Siegel's proof of Theorem 4 [SM71, chap. 16 and 17]. We construct the normalizing transformation y = Φ ± (u, x, ) in a couple of steps as a formal power series in the u-variable with coefficients depending analytically on (x, ) ∈ XE ± , and then show that the series is convergent. The main tool to prove the convergence is the Lemma 38 below.
be a power series in the u-variable with coefficients bounded and analytic on (x, ) ∈ XE ± . We will write {φ ± } m := φ ±,m .
Denoting φ ±,m := sup (x, )∈XE± |φ ±,m (x, )| the supremum norm over XE ± , let be a majorant power series to φ. We will write The following lemma is the essential technique in Siegel's proof.

7.1
Step 1: Ramified straightening of center manifold and diagonalization of the linear part Suppose that the system is in a pre-normal form, (59) We will show that there exists a ramified transversely symplectic change of variable bounded and analytic on the domain XE ± of Definition 12, that brings the system to a form with f ± (w, x, ) = O(|w| 2 ), ∂f1,± ∂w1 + ∂f2,± ∂w2 = 0. The solution w = 0 of the transformed system (61), corresponds to a bounded ramified solution y = φ ±,0 (x, ) of the system (59). The paper [Kli16], see Theorem 39 below, shows that there is a unique such solution on the domain XE ± ; this it is the "ramified center manifold" of the corresponding foliation.
The variableỹ = y − φ ±,0 (x, ) then satisfies . The transformation matrix T ± (60) must then satisfy The existence of such a transformation T ± bounded on XE ± is known [LR12,HLR13] when A ± is analytic. In our case the matrix A ± is ramified, but their proof works anyway. We will obtain T ± directly using Theorem 39. Writing R = r ij i,j and then the terms t i,± , i = 1, 2, are solutions to Riccati equations and the terms b i,± are solution to Combining the equations (59) for φ ±,0 and (62) for t ± , in which r ij = r i,j (φ ±,0 , x, ), we get an analytic system for which the existence of a unique bounded solution on XE ± is assured by the following theorem.
(i) The system (63) possesses a unique solution in terms of a formal power series in (x, ):φ This series is divisible by x(x − ), and its coefficients satisfy φ kj ≤ L k+j (k + j)! for some L > 0.
(iii) Let Φ(x, ) be analytic extension of the function given by the convergent series For each point (x, ), for which there is θ ∈ ]− π 2 , π 2 [ such that S θ · (x, ) ⊆ XE ± , with S θ ⊂ C denoting the circle through the points 0 and 1 with center on e iθ R + , we can express φ ± (x, ) as the following Laplace transform of Φ: In particular, φ + (x, 0) = φ − (x, 0) is the functional cochain consisting of the pair of Borel sums of the formal seriesφ(x, 0) in directions on either side of λ (0) R.
Since the trace of the linear part of both systems (59) and (61) is null, then by the Liouville-Ostrogradskii formula det T (x, ) is constant in x and equal to det T (0, ) = 1. Therefore the transformation (60) is transversely symplectic, and by Lemma 3 the transformed system (61) is transversely Hamiltonian.
The equation (71) has a unique bounded solution {Ψ j,± } m given by the integral where {F j,± } m is the right side of (71), t λ (x, ) =    − λ (0) log x + ( λ (0) + λ (1) ) log(x − ), for = 0, − λ (0) x + λ (1) log x, for = 0, is a branch of the rectifying coordinate for the vector field on X ± ( ), and the integration follows a real trajectory of the vector field (23) in X ± ( ) from a point x 2,± , if m 1 − m 2 + (−1) j < 0, x i,± is as in (25), to x, along which the integral is well defined. Note that the convergence of the constructed formal transformation Φ ± is equivalent to the convergence of Ψ ± (68). We prove the convergence of the latter series using Lemma 38. For this we need to estimate the norms of (72) and (73).
Let us show that the transformation y = Ψ ± (u, x, ) obtained as a composition of the transformations of Steps 1-3 is transversely symplectic and thereforeχ ± = χ.