Gérard Iooss
email: iooss.gerard@orange.fr

I A U F Dieudonné

Heteroclinic for a 6-dimensional reversible system occuring in orthogonal domain walls in convection

Keywords: Reversible dynamical systems, Invariant manifolds, Bifurcations, Heteroclinic connection, Domain walls in convection

.

Introduction and Results

Let us study the following reversible system in R 6

A (4) = A(1 -A 2 -gB 2 )
(1)

B ′′ = ε 2 B(-1 + gA 2 + B 2 ),
where the coordinates in R 6 are Z = (A 0 , A 1 , A 2 , A 3 , B 0 , B 1 ) = (A, A ′ , A ′′ , A ′′′ , B, B ′ ). This system occurs in the search for domain walls intersecting orthogonally, in a fluid dynamic problem such as the Bénard-Rayleigh convection between parallel horizontal plates (see subsection 1.1 and all details in [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF]). The heteroclinic we are looking for, corresponds to the connection between rolls on one side and rolls oriented orthogonally on the other side. The system [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF] has been also introduced by Manneville and Pomeau in [START_REF] Manneville | A grain boundary in cellular structures near the onset of convection[END_REF], obtained after formal physical considerations and using symmetries for the study of orthogonal walls in the onset of Bénard-Rayleigh convection.

We would like to find analytically a heteroclinic connection (g > 1, ε small) such that A * (x), B * (x) > 0, (A * (x), B * (x)) →

(1, 0) as x → -∞ (0, 1) as x → +∞ .

By a variational argument Boris Buffoni et al [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF] prove the existence of such an heteroclinic orbit, for any g > 1, and ε small enough. This type of elegant proof does not unfortunately allow to prove the persistence of such heteroclinic curve under reversible perturbations of the vector field. This is our motivation for producing analytic arguments, proving such an existence, uniqueness and smoothness in parameters (ε, g) of this orbit, however for limited values 1 < g ≤ 2, fortunately including physical interesting ones. Then we study the linearized operator along the heteroclinic curve, allowing to attack the problem of existence of orthogonal domain walls in convection (forthcoming paper).

After some basic consideration on the system, a first part of the paper (sections [START_REF] Haragus | Grain boundaries in the Swift-Hohenberg equation[END_REF][START_REF] Kapitula | Spectral and Dynamical Stability of Nonlinear Waves[END_REF][START_REF] Kato | Perturbation theory for linear operators[END_REF] is devoted to the proof of Theorem 1. Then section 7, is devoted to the study of the linearized operator along the heteroclinic, some properties of which are necessary for the forthcoming proof of existence of orthogonal domain walls in convection.

We set δ = (g-1) 1/2 . The idea here might be to use the arc of equilibria A 2 + B 2 = 1, which exists for δ = 0, connecting end points M -= (1, 0) and M + = (0, 1), and to prove that for suitable values of δ, the 3-dimensional unstable manifold of M -intersects transversally the 3-dimensional stable manifold of M + , both staying on a 5 dimensional invariant manifold W δ . However, for δ = 0 the situation in M + is very degenerated, with a quadruple 0 eigenvalue for the linearized operator, while it is only a double eigenvalue in M -. Then we are not able to prove, for δ close to 0, that the 3-dimensional unstable manifold of M - exists from B = 0 until B reaches a value close enough to 1.

The strategy here consists to keep in mind that, after changing the coordinate x in x = εx, the limit ε → 0 of the system (1) gives a non C 1 heteroclinic solution such that (i) for x running from -∞ to 0, then (A 0 , B 0 ) varies from (1, 0) to (0, 1 √ g ) on the ellipse A 2 0 + gB 2 0 = 1, while (ii) for x running from 0 to +∞, then (A 0 , B 0 ) varies from (0, 1 √ g ) to (0, 1) in satisfying the differential equation (see the first integral (3)).

B ′ 0 = ε √ 2 (1 -B 2 0 ).
The major difficulty in the proof of Theorem 1 is to prove the existence of the 3-dim unstable manifold of M -until A 0 reaches a neighborhood of 0, and to prove the existence of the 3-dim stable manifold of M + until B 0 reaches a neighborhood of 1/ √ g = 1/ 1 + δ 2 . The usual proofs of existence of such invariant manifolds give only local results, so we need to use here a first integral of the system, expressing that both manifolds lie on a 5-dimensional invariant manifold, and then we are able to extend sufficiently the domain of existence of these manifolds. Indeed we prove the following Theorem 1 Let us choose 0 < δ 0 < 1/3, then for δ 0 ≤ δ ≤ 1, η 0 such that 0 < α = [(1 + δ 2 )η 2 0 -1] 1/2 , and for ε small enough with α = ε 1/3 , the 3-dim unstable manifold of M -intersects transversally the 3-dim stable manifold of M + , except maybe for a finite number of values of δ. The connecting curve which is obtained is locally unique (it is the only curve for this intersection). Moreover its dependency in parameters (ε, δ) is analytic. In addition we have B(x) and B ′ (x) > 0 on (-∞, +∞), the principal part of B(x) being given i) for x ∈ (-∞, 0], by

B 0 (x) = 1 (1 + δ 2 2 ) 1/2 cosh(x 0 -εδx) , cosh x 0 = 1 B 00 (1 + δ 2 2 ) 1/2 , B 00 = B 0 (0) = (1 -η 2 0 δ 2 ) 1/2 ,
ii) for x ∈ [0, +∞), by B 0 (x) = tanh(εx/ √ 2) + B 00 1 + B 00 tanh(εx/ √ 2) .

For x → -∞ we have (A 0 -1, A 1 , A 2 , A 3 , B 0 , B 1 ) → 0 at least as e εδx , while for x → +∞, (A 0 , A 1 , A 2 , A 3 ) → 0 at least as e - √ δ 2 x , and (B 0 -1, B 1 ) → 0 at least as e - √ 2εx .

In section 4 we prove at Lemma 8 the existence of the unstable manifold of M -= (1, 0) until a neighborhood of (A 0 , B 0 ) = (0, 1/ 1 + δ 2 ). Here there is no restriction on the choice of δ, except δ ≥ δ 0 > 0.

In section 5 we prove at Lemma 13 the existence of the stable manifold of M + = (0, 1) until (backward direction) a neighborhood of (0, 1/ 1 + δ 2 ). Here there is a restriction δ ≤ 1, for being able to reach the end point.

In section 6 we prove the transverse intersection of the two manifolds, except maybe for a finite set of values of δ. This ends the proof of Theorem 1.

In section 7 we give in Lemma 19 the properties of the linearized operator along the heteroclinic, which are necessary to prove a persistence result under a reversible perturbation for the heteroclinic in the 8-dimension space (with B ∈ C).

Remark 2 It should be noticed that we show that, in the middle of the heteroclinic, A 0 (0) ∼ αδ which is very close to 0, while B 0 (0) is very close to 1/ √ g. The choice of α = ε 1/3 results from the proof of Lemma 8.

Remark 3

Using symmetries of the system: A → ±A, B → ±B and reversibility symmetry: (A(x), B(x)) → (A(-x), B(-x)), we find 8 heteroclinics. Two are connecting M -to M + with opposite dynamics, two others connect -M -to M + , two connect M -to -M + , and two connect -M -to -M + . The one which interests us is the only one connecting M -to M + with the dynamics running from M -to M + .

Remark 4 It should be noticed that the study made in [START_REF] Manneville | A grain boundary in cellular structures near the onset of convection[END_REF] on the heteroclinic solution for the system (1) uses asymptotic analysis, and catches many properties which are proved rigorously here.

Origin of system (1)

The Bénard-Rayleigh convection problem is a classical problem in fluid mechanics. It concerns the flow of a three-dimensional viscous fluid layer situated between two horizontal parallel plates and heated from below. Upon increasing the difference of temperature between the two plates, the simple conduction state looses stability at a critical value of the temperature difference corresponding to a critical value R c of the Rayleigh number. Beyond the instability threshold, a convective regime develops in which patterns are formed, such as convective rolls, hexagons, or squares. Observed patterns are often accompanied by defects.

Mathematically, the governing equations are the Navier-Stokes equations coupled with an equation for the temperature, and completed by boundary conditions at the two plates. Observed patterns are then found as particular steady solutions of these equations. Very recently, the existence of orthogonal domain walls has been studied by [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF], where the authors handle the full governing equations, showing that the study leads to a small perturbation of the reduced system of amplitude equations [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF].

Starting from a formulation of the steady governing equations as an infinitedimensional dynamical system in which the horizontal coordinate x plays the role of evolutionary variable (spatial dynamics), a center manifold reduction is performed, which leads to a 12-dimensional reduced reversible dynamical system (reducing to 8-dimensional after restricting to solutions with reflection symmetry y → -y). A normal form for this reduced system is obtained, for which, after an appropriate rescaling of the normal form, the principal part is the system (1), with B ∈ C, and B 2 replaced by |B| 2 . The truncation leading to (1) allows to take B real, since its phase does not play any role at this level. Solutions of the system (1) provide leading order approximations of solutions of the full governing equations. In particular, the equilibrium (A 0 , B 0 ) = (0, 1) of the system (1) gives an approximation of convection rolls (in the x direction) bifurcating for Rayleigh numbers R > R c close to R c , whereas the equilibrium (A 0 , B 0 ) = (1, 0) of the system (1) gives the same convection rolls (in the y direction) rotated by an angle π/2 with the phase fixed by the imposed reflection symmetry. A heteroclinic orbit connecting these two equilibria provides then an approximation of orthogonal domain walls (see Figure 1). The parameter ε in

(1) is such that ε 4 is proportional to R 1/2 -R 1/2 c . The parameter g > 1 in (1)
is function of the Prandtl number, while other parameters, which only appear in higher orders, are the wave numbers of the rolls, close to the critical value.

Remark 5

Values of δ such that 0.476 ≤ δ include values obtained for δ in the Bénard-Rayleigh convection problem where g is function of the Prandtl number P (see [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF]). With rigid-rigid, rigid-free, or free-free boundaries the minimum values of g are respectively (g min = 1.227, 1.332, 1.423) corresponding to δ min = 0.476, 0.576, 0.650. The restriction in Theorem 1 corresponds to 1 < g ≤ 2. The eligible values for the Prandtl number are respectively P > 0.5308, > 0.6222, > 0.8078.

Global invariant manifold W δ

The first observation is that we have the first integral

ε 2 (A ′2 ) ′′ -3ε 2 A ′′2 -B ′2 + ε 2 2 (A 2 + B 2 -1) 2 + ε 2 δ 2 A 2 B 2 = 0, (2)
i.e.

2ε 2 A 1 A 3 -ε 2 A 2 2 -B 2 1 + ε 2 2 (A 2 0 + B 2 0 -1) 2 + ε 2 δ 2 A 2 0 B 2 0 = 0 (3)
This defines a 5-dimensional invariant maniford W δ valid for any δ > 0, which contains the heteroclinic curve that we are looking for. The singular points of this manifold are given by

A 1 = A 2 = A 3 = B 1 = 0, 0 = A 0 (A 2 0 + (1 + δ 2 )B 2 0 -1), 0 = B 0 ((1 + δ 2 )A 2 0 + B 2 0 -1)
. For δ > 0, and since (A 0 , B 0 ) = (0, 0) or (±(δ 2 + 2) -1/2 , ±(δ 2 + 2) -1/2 ) do not belong to W δ , we only find the singular points

(A 0 , B 0 ) = (±1, 0), ( 4
) (A 0 , B 0 ) = (0, ±1).
For δ = 0, all singular points belong to a circle of singular points:

A 2 0 + B 2 0 = 1. ( 5
)
3 Linear study of the dynamics 3.1 Case δ > 0 (g > 1)

3.1.1 Neighborhood of M -= (1, 0)
The eigenvalues of the linearized operator at M -are such that λ 4 = -2 or

λ 2 = ε 2 δ 2 , hence ±2 -1/4 (1 ± i),
±εδ.

This gives a 3-dimensional unstable manifold, and a 3-dimensional stable manifold.

Neighborhood of M

+ = (0, 1)
The eigenvalues of the linearized operator at M + are such that λ 4 = -δ 2 or λ 2 = 2ε 2 , hence

±2 -1/2 (1 ± i)δ ′ , ±ε √ 2, δ ′ = √ δ.
This gives again a 3-dimensional unstable manifold and a 3-dimensional stable manifold.

All this implies that the 3-dimensional unstable manifold starting at M - is included into the 5-dimensional manifold W δ , as well as the 3-dimensional stable manifold starting at M + is included into the 5-dimensional manifold W δ . This gives a good hope that these two manifolds intersect along a heteroclinic curve...provided that they still exist "far" from the end points M + and M -. The idea is to show that this occurs when δ is not too small.

The limit points M -= (1, 0) and M + = (0, 1) have a degenerate situation for δ = 0, because of the multiple 0 eigenvalue for the linearized operator. For δ = 0, it is possible to build a family of 2-dim unstable invariant manifolds and a family of 2-dim stable manifolds along the arc of equilibria A 2 + B 2 = 1. For δ > 0, the perturbation gives two new 3-dim invariant manifolds, however their transversality is weaker and weaker as B → 1. A "serious" study is then needed, which is the object of our work.

Unstable manifold of M

Change of coordinates

Let us fix 0 < δ 0 ≤ 1/3, and δ 1 > 1, we assume, from now on

0 ≤ B 0 ≤ 1 -η 2 0 δ 2 , η 0 > 1 1 + δ 2 = 1 √ g , α def = (η 2 0 (1 + δ 2 ) -1) 1/2 , ε 2 α 2 ≤ δ 0 ≤ δ ≤ δ 1 , (6)
and let us define new coordinates

Z = ( A * + A 0 , A 1 , A 2 , A 3 , B 0 , B 1 ) t (7)
where

A 0 = A * cancels A ′ 3 with A * 2 def = 1 -(1 + δ 2 )B 2 0 , A * ≥ δα.
In the following α is a "small parameter", the relative size of which, with respect to ε is precized later.

Remark 6

The occurence of A * is linked with a formal computation of an expansion of the heteroclinic in powers of ε, which gives A * as the principal part of A 0 , valid for B 0 < (1 + δ 2 ) -1/2 = 1/ √ g. The hope is to build the unstable manifold until this limit value.

Remark 7

We choose the conditions on δ, δ 0 ≤ δ ≤ δ 1 in the purpose to include known computed values of the coefficient g = 1 + δ 2 , in the convection problems, with different boundary conditions (see [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF]).

We prove below the main result of this section: Lemma 8 For ε small enough, for 0 < δ 0 < 1/3, and δ 1 arbitrary,

δ ∈ [δ 0 , δ 1 ], α 2 = η 2 0 (1 + δ 2 ) -1, ε 2 ≤ δ 0 α 2 , ε = α 3 , the 3-dimensional unstable manifold of M -exists for 0 ≤ B 0 (x) ≤ (1 -η 2 0 δ 2 ) 1/2 , x ∈ (-∞, 0].
It sits in W g , is analytic in (ε, δ), and

A 0 = A * + B 0 O(α 1/2 δ 1/2 e εδx ) A 1 = B 0 O(αδe εδx ) A 2 = B 0 O(αδe εδx ) A 3 = B 0 O(αδe εδx ),
where

0 ≤ 1 -A * ≤ cB 2 0 , A * (0) = 1, A * ≥ δα. Moreover, as x → -∞, A 0 -A * , A 1 , A 2 ,
A 3 are bounded by cεδe 2εδx , and B 0 , (resp. B 1 ) by ce εδx , (resp. cεe εδx ), where c is a constant independent of ε, δ.

Remark 9 We observe that A 0 reaches a value close to 0 since A * reaches δα which is close to 0, while B 0 reaches (1-η 2 0 δ 2 ) 1/2 which is close to 1/(1+δ 2 ) 1/2 = 1/ √ g, not close to 1.

The system (1) becomes

A 0 ′ = A 1 + (1 + δ 2 )B 0 A * B 1 A ′ 1 = A 2 A ′ 2 = A 3 (8) A ′ 3 = -2 A * 2 A 0 -3 A * A 0 2 -A 0 3 B ′ 0 = B 1 B ′ 1 = ε 2 δ 2 B * ( A * 2 -B 2 0 ) + 2ε 2 (1 + δ 2 ) A * B 0 A 0 + ε 2 (1 + δ 2 )B 0 A 0 2 ,
Now, we define the linear operator

L δ =          0 1 0 0 0 (1+δ 2 )B0 A * 0 0 1 0 0 0 0 0 0 1 0 0 -2 A * 2 0 0 0 0 0 0 0 0 0 0 1 2ε 2 (1 + δ 2 ) A * B 0 0 0 0 0 0          , (9)
for which 0 is a double eigenvalue, and such that the non zero eigenvalues satisfy

λ 4 -2ε 2 B 2 0 (1 + δ 2 ) 2 λ 2 + 2 A * 2 = 0. ( 10
)
The discriminant is

∆ ′ = ε 4 B 4 0 (1 + δ 2 ) 4 -2 A * 2 .
Our assumption

B 0 ≤ 1 -η 2 0 δ 2 and ε 2 α 2 ≤ δ ≤ δ 1 , in addition with the con- straint 1 α ≥ (1 + δ 2 ) 2 . ( 11
) implies -∆ ′ ≥ A * 2 .
Then we have two pairs of complex eigenvalues

λ 2 ± = ε 2 B 2 0 (1 + δ 2 ) 2 ± i √ -∆ ′ .
The idea is to find new coordinates able to manage a new linear operator in the form of two independent blocs

±λ r λ i -λ i ±λ r (12)
for which the eigenvalues are ±λ r ± iλ i , where

2λ 2 r = √ 2 A * + ε 2 B 2 0 (1 + δ 2 ) 2 (13) 2λ 2 i = √ 2 A * -ε 2 B 2 0 (1 + δ 2 ) 2 λ 2 r -λ 2 i = ε 2 B 2 0 (1 + δ 2 ) 2 λ 2 r + λ 2 i = √ 2 A * 4λ 2 r λ 2 i = -∆ ′ .
We choose a form of the linear operator as (12) for being able to have good estimates for the monodromy operator associated with the linear operator, the coefficients of which are functions of B 0 ∈ [0, 1 -η 2 0 δ 2 ] (see Appendix A.1).

Estimates for the eigenvalues

First, notice that (13) and

α ≤ (1 + δ 2 ) -2 imply λ r λ i ≥ A * 2 , 2 1/4 A * 1/2 ≥ λ r ≥ A * 1/2 2 1/4 ≥ α 1/2 2 1/4 √ δ, (14)
1 2 3/4 A * 1/2 ≤ λ i ≤ A * 1/2 2 1/4 . (15)

New coordinates

The eigenvector and generalized eigenvector for the eigenvalue 0 are :

Z 0 =         0 0 0 0 A * 0         , Z 1 =         0 -(1 + δ 2 )B 0 0 0 0 A *        
. Now we denote by

V + r ± iλ i V + i , V - r ± iλ i V - i
the eigenvectors belonging respectively to the eigenvalues λ r ± iλ i , -λ r ± iλ i then we define

V + r =            - λr(λ 2 r -3λ 2 i ) 2 A * 2 1 λ r λ 2 r -λ 2 i - λr (λ 2 r -λ 2 i ) (1+δ 2 )B0 A * - (λ 2 r -λ 2 i ) 2 (1+δ 2 )B0 A *            , V + i =            - 3λ 2 r -λ 2 i 2 A * 2 0 1 2λ r - (λ 2 r -λ 2 i ) (1+δ 2 )B0 A * - 2λr (λ 2 r -λ 2 i ) (1+δ 2 )B0 A *           
, and we define new coordinates as

        A 0 A 1 A 2 A 3 0 B 1         = B 0 (x 1 V + r + x 2 λ i V + i + y 1 V - r + y 2 λ i V - i + z 0 Z 0 + z 1 Z 1 ).
We observe that after eliminating z 0 , we still have 6 coordinates, including B 0 as one of the new coordinates.

Remark 10 We notice that we put B 0 in front of the new coordinates, as this results from the analysis, and shorten the computations.

We have now

A 0 = -B 0 λ r (λ 2 r -3λ 2 i ) 2 A * 2 (x 1 -y 1 ) -B 0 λ i (3λ 2 r -λ 2 i ) 2 A * 2 (x 2 + y 2 ) A 1 = B 0 (x 1 + y 1 ) -(1 + δ 2 )B 2 0 z 1 A 2 = λ r B 0 (x 1 -y 1 ) + λ i B 0 (x 2 + y 2 ) (16) A 3 = (λ 2 r -λ 2 i )B 0 (x 1 + y 1 ) + 2λ r λ i B 0 (x 2 -y 2 ) 0 = - (λ 2 r -λ 2 i ) (1 + δ 2 )B 0 A * A 2 + A * B 0 z 0 B 1 = - (λ 2 r -λ 2 i ) (1 + δ 2 )B 0 A * A 3 + A * B 0 z 1 ,
which needs to be inverted. We obtain

B 0 x 1 = (λ 2 r + λ 2 i ) 4λ r A 0 + 3λ 2 r -λ 2 i 4λ r (λ 2 r + λ 2 i ) A 2 (17) + A 1 2 + (1 + δ 2 )B * 2 A * B 1 + (λ 2 r -λ 2 i ) 2 A * 2 A 3 , λ i B 0 x 2 = - (λ 2 r + λ 2 i ) 4 A 0 - λ 2 r -3λ 2 i 4(λ 2 r + λ 2 i ) A 2 (18) - (λ 2 r -λ 2 i ) 4λ r A 1 + (1 + δ 2 )B 0 A * B 1 + 1 4λ r 1 - (λ 2 r -λ 2 i ) 2 A * 2 A 3 , B 0 y 1 = - (λ 2 r + λ 2 i ) 4λ r A 0 - 3λ 2 r -λ 2 i 4λ r (λ 2 r + λ 2 i ) A 2 (19) + A 1 2 + (1 + δ 2 )B 0 2 A * B 1 + (λ 2 r -λ 2 i ) 2 A * 2 A 3 , λ i B 0 y 2 = - (λ 2 r + λ 2 i ) 4 A 0 - λ 2 r -3λ 2 i 4(λ 2 r + λ 2 i ) A 2 (20)
+ (λ 2 r -λ 2 i ) 4λ r A 1 + (1 + δ 2 )B 0 A * B 1 - 1 4λ r 1 - (λ 2 r -λ 2 i ) 2 A * 2 A 3 , B 0 z 1 = (λ 2 r -λ 2 i ) (1 + δ 2 )B 0 A * 2 A 3 + 1 A * B 1 .
Let us now define

X = x 1 x 2 , Y = X = y 1 y 2 ,
then, for ε small enough, we obtain the following useful estimates

A * 1/2 2 3/4 ≤ λ r , λ i < 2 1/4 A * 1/2 , A * ≥ δα ≥ ε 2 α , | A 0 | ≤ 3 B 0 A * 1/2 (|X| + |Y |), |A 1 | ≤ B 0 (|X| + |Y |) + 2B 2 0 |z 1 |, |A 2 | ≤ 2B 0 A * 1/2 (|X| + |Y |), (21)
|A 3 | ≤ 2B 0 A * (|X| + |Y |), |B 1 | ≤ 3ε 2 B 2 0 (|X| + |Y |) + A * B 0 |z 1 |.

System with new coordinates

The system (8) writen in the new coordinates is computed in Appendix A.2. It takes the following form

x ′ 1 = f 1 + λ r x 1 + λ i x 2 (22) +B 1 a 1 A 0 + c 1 A 2 + d 1 A 3 + e 1 B 1 B 0 - 1 B 0 x 1 -ε 2 (1 + δ 2 )(2 -δ 2 )B 0 2 A * A 0 2 -ε 2 (1 + δ 2 )B 0 2 A * 2 A 0 3 , x ′ 2 = f 2 -λ i x 1 + λ r x 2 + B 1 -a 2 A 0 + b 2 A 1 + c 2 A 2 + d 2 A 3 + e 2 B 1 - 1 B 0 x 2 (23) - 1 4λ r λ i A * B 0 3 A * 2 -2ε 4 B 4 0 (1 + δ 2 ) 4 A 0 2 - 1 4λ r λ i B 0 1 - (λ 2 r -λ 2 i ) 2 A * 2 A 0 3 . y ′ 1 = f 1 -λ r y 1 + λ i y 2 + (24) +B 1 -a 1 A 0 -c 1 A 2 + d 1 A 3 + e 1 B 1 B 0 - 1 B 0 y 1 -ε 2 (1 + δ 2 )(2 -δ 2 )B 0 2 A * A 0 2 -ε 2 (1 + δ 2 )B 0 2 A * 2 A 0 3 , y ′ 2 = -f 2 -λ i y 1 -λ r y 2 + B 1 -a 2 A 0 -b 2 A 1 + c 2 A 2 -d 2 A 3 + e 2 B 1 - 1 B 0 y 2 (25) + 1 4λ r λ i A * B 0 3 A * 2 -2ε 4 B 4 0 (1 + δ 2 ) 4 A 0 2 + 1 4λ r λ i B 0 1 - (λ 2 r -λ 2 i ) 2 A * 2 A 0 3 , B ′ 0 = -ε 2 (1 + δ 2 )B 0 A 3 A * + A * B 0 z 1 , with f 1 = ε 2 δ 2 B 0 (1 + δ 2 )( A * 2 -B 2 0 ) 2 A * , f 2 = - ε 2 δ 2 B 0 (1 + δ 2 )(λ 2 r -λ 2 i )( A * 2 -B 2 0 ) 4λ r λ i A * ,
coefficients a j , b j , c j , d j , e j are defined and estimated in Appendix A.2 in (83,84), (85,86,87), (88,89), (90,91). Here A 0 , A 1 , A 2 , A 3 , B 1 should be replaced by their (linear) expressions (16) in coordinates (x 1 , x 2 , y 1 , y 2 , z 1 ) with coefficients functions of B 0 . The system above should be completed by the differential equation for z 1 . In fact we replace the equation for z ′ 1 by the direct resolution of the first integral (3) with respect to z 1 (see below).

Resolution of (3) with respect of z 1 (X, Y, B 0 )

For extending the validity for the existence of the unstable manifold of M -we need to replace the differential equation for z 1 in using instead the first integral (3). This leads to

B 2 1 = { A * B 0 z 1 -ε 2 B 0 (1 + δ 2 ) A * A 3 } 2 = 2ε 2 A 1 A 3 -ε 2 A 2 2 + ε 2 2 (-δ 2 B 2 0 + 2 A * A 0 + A 0 2 ) 2 + ε 2 δ 2 ( A * + A 0 ) 2 B 2 0 , hence A * 2 z 2 1 = ε 2 δ 2 A * 2 (1 + δ 2 B 2 0 2 A * 2 ) + 2ε 2 B 0 A 3 (x 1 + y 1 ) - ε 4 (1 + δ 2 ) 2 A * 2 A 2 3 - ε 2 B 2 0 A 2 2 + + 2ε 2 A * 2 B 2 0 A 0 2 + 2ε 2 A * B 2 0 A 0 3 + ε 2 2B 2 0 A 0 4 , (26)
where we may observe on the r.h.s., that

δ 2 2 A * 2 < 1 2α 2 , hence ε 2 δ 2 ≤ ε 2 δ 2 (1 + δ 2 B 2 0 2 A * 2 ) ≤ ε 2 δ 2 (1 + 1 2α 2 ),
which is independent of (X, Y ). Moreover there is no linear part in (X, Y ). For further estimates, we make a new scaling

(X, Y ) = αδ(X, Y ), z 1 = εδz 1 . (27)
We notice that (21) implies

| 2ε 2 B 0 A 3 (x 1 + y 1 )| ≤ cε 2 α 2 δ 2 A * (|X| + |Y |) 2 | ε 4 (1 + δ 2 ) 2 A * 2 A 2 3 | ≤ cε 4 α 2 δ 2 (|X| + |Y |) 2 ε 2 B 2 0 A 2 2 ≤ cε 2 α 2 δ 2 A * (|X| + |Y |) 2 ) 2ε 2 A * 2 B 2 0 A 0 2 ≤ cε 2 α 2 δ 2 A * (|X| + |Y |) 2 | 2ε 2 A * B 2 0 A 0 3 | ≤ cε 2 α 3 δ 3 A * 1/2 (|X| + |Y |) 3 ε 2 2B 2 0 A 0 4 ≤ cε 2 α 4 δ 4 A * 2 (|X| + |Y |) 4 ,
so that the factors in the estimates are such that

cε 2 α 2 δ 2 A * ε 2 δ 2 A * 2 ≤ c α 2 A * , cε 2 α 4 δ 4 ε 2 δ 2 A * 4 ≤ c α 2 A * 2 , cε 2 α 3 δ 3 ε 2 δ 2 A * 5/2 ≤ c α 5/2 δ 3/4 A * 2 , c being independent of ε and δ ∈ [δ 0 , δ 1 ]. Now defining z 10 such that 1 ≤ z 10 (B 0 ) def = (1 + δ 2 B 2 0 2 A * 2 ) 1/2 ≤ 1 α , for α ≤ 1/ √ 2, (28)
It results that

z 1 2 = z 10 2 + O α 2 A * (|X| + |Y |) 2 + α 5/2 A * 2 (|X| + |Y |) 3 + α 2 A * 2 (|X| + |Y |) 4
and using

B 2 0 = 1 -A * 2 1 + δ 2 we also have 1 z 10 2 = 2 A * 2 2 A * 2 + δ 2 B 2 0 ≤ 2(1 + δ 2 ) A * 2 δ 2 ≤ c A * 2 δ 2 ,
so that

z 1 = z 10 (B 0 ) 1 + A * 2 δ 2 O α 2 A * (|X| + |Y |) 2 + α 5/2 A * 2 (|X| + |Y |) 3 + α 2 A * 2 (|X| + |Y |) 4 1/2 = z 10 (B 0 ) 1 + O[α 2 (|X| + |Y |) 2 ] 1/2 , for |X| + |Y | ≤ ρ, ρ fixed,
and taking the square root, we obtain

z 1 = z 10 (B 0 ) + Z(X, Y , B 0 ) (29) with Z(X, Y , B 0 ) = O(α(|X| + |Y |) 2 ,
Z(X, Y , B 0 ) being defined in the ball

|X| + |Y | ≤ ρ,
provided that ε is small enough and where ρ is of order 1, not necessarily small with respect to α. Moreover Z is analytic in its arguments and is at least quadratic in (X, Y ). Since z 1 contains z 10 which is independent of (X, Y ), the new system has new "constant terms" and "linear terms", appearing as perturbations of the former ones.

System where z 1 is eliminated

The new system is computed in Appendix A.3. We obtain (notice that B 0 is in factor of the "constant" terms)

X ′ = L 0 X + B 0 F 0 + L 01 (X, Y ) + B 01 (X, Y ), (30)
Y ′ = L 1 Y + B 0 F 1 + L 11 (X, Y ) + B 11 (X, Y ),
where

L 0 = λ r λ i -λ i λ r , L 1 = -λ r λ i -λ i -λ r ,
and with the following estimates, for terms independent of (X, Y )

|F 0 | + |F 1 | ≤ cε 2 α 4 , (31)
for terms which are linear in (X, Y )

|L 01 (X, Y )| + |L 11 (X, Y )| ≤ c ε α 2 (|X| + |Y |), (32)
and for terms at least quadratic in (X, Y ), choosing α small enough and for

|X| + |Y | ≤ ρ, we obtain |B 01 (X, Y )| + |B 11 (X, Y )| ≤ c(α + ε 2 α 2 )(|X| + |Y |) 2 . ( 33
)

Integral formulation for solutions bounded as x → -∞

Let us introduce the monodromy operators associated with the linear operators L 0 , L 1 which have non constant coefficients (functions of B 0 (see [START_REF] Hale | Ordinary differential equations[END_REF]):

∂ ∂x S 0 (x, s) = L 0 S 0 (x, s), S 0 (x, s 1 )S 0 (s 1 , s 2 ) = S 0 (x, s 2 ), S 0 (x, x) = I, ∂ ∂x S 1 (x, s) = L 1 S 1 (x, s), S 1 (x, s 1 )S 1 (s 1 , s 2 ) = S 1 (x, s 2 ), S 1 (x, x) = I.
The coefficients of operators L 0 , L 1 are functions of B 0 , so we need the Lemma 24 in Appendix A.1, with the following estimates, valid for 0

≤ B 0 ≤ 1 -η 2 0 δ 2 , α ≤ (1 + δ 2 ) -2 : ||S 0 (x, s)|| ≤ e σ(x-s) , -∞ < x < s ≤ 0, ( 34
)
||S 1 (x, s)|| ≤ e -σ(x-s) , -∞ < s < x ≤ 0, ( 35
) with σ = α 1/2 δ 1/2 2 1/4 .
We are looking for solutions of (30) which stay bounded for x → -∞. Then, thanks to estimates (34) (35), the system (30) may be formulated as

X(x) = S 0 (x, 0)X 0 + x 0 S 0 (x, s)G 0 (s)ds (36) Y (x) = x -∞ S 1 (x, s)G 1 (s)ds G 0 (s) def = B 0 F 0 + L 01 (X, Y ) + B 01 (X, Y ), G 1 (s) def = B 0 F 1 + L 11 (X, Y ) + B 11 (X, Y )
where X, Y and B 0 are bounded and continuous functions of s, B 0 tending towards 0 as s → -∞.

Strategy

The idea is i) solve (36) with respect to (X, Y ) in function of (X 0 , B 0 ); ii) solve the integro-differential equation for B 0 , with

B 0 | x=0 = B 0 (0).
Then the unstable manifold of M -is given (see [START_REF] Hale | Ordinary differential equations[END_REF]) by

Y | x=0 , z 1 | x=0 in terms of X 0 , B 0 (0)
. The result will be valid for an interval [0, 1 -η 2 0 δ 2 ] for B 0 and it appears that A 0 is then very close to 0 at this end point. The hope is that this should allow to compute its intersection with the 3-dim stable manifold of M + which computation should be valid for B 0 in the interval [ 1 -η 2 0 δ 2 , 1].

Resolution for (X, Y )

Let us define, for κ > 0

C 0 κ = {X ∈ C 0 (-∞, 0]; X(x)e -κx is bounded} equiped with the norm ||X|| κ = sup (-∞,0) |X(x)e -κx |.
We observe that, provided that κ < σ

| x -∞ S 1 (x, s)e κs ds| ≤ e κx κ + σ |S 0 (x, 0)e -κx | ≤ e (σ-κ)x , x ≤ 0, | x 0 S 0 (x, s)e κs ds| ≤ e κx σ -κ , x ≤ 0. Let us choose κ ≤ σ 2 , then | x -∞ S 1 (x, s)e κs ds| ≤ e κx σ = 2 1/4 e κx α 1/2 δ 1/2 , | x 0 S 0 (x, s)e κs ds| ≤ 2 5/4 e κx α 1/2 δ 1/2 , x ≤ 0. Let us assume that ||B 0 || κ ≤ m
holds with m independent of ε, which needs to be proved at next subsection. Hence, the implicit function theorem applies for (X, Y ) in the function space C 0 κ , provided that we can choose κ ≤ σ 2 and ||X|| κ + ||Y || κ ≤ ρ. Using the above estimates for coefficients, we obtain

|X(x)e -κx | ≤ |X 0 | + 2 5/4 α 1/2 δ 1/2 ||B 0 F 0 + L 01 (X, Y ) + B 01 (X, Y )|| κ , hence ||X|| κ ≤ |X 0 | + 2 5/4 α 1/2 δ 1/2 ||B 0 F 0 + L 01 (X, Y ) + B 01 (X, Y )|| κ , (37)
and in the same way

||Y || κ ≤ 2 1/4 α 1/2 δ 1/2 ||B 0 F 1 + L 11 (X, Y ) + B 11 (X, Y )|| κ . ( 38
)
Remark 11 The choice of κ is governed by the behavior of B 0 (x) as x → -∞, which is studied at next subsection.

For ε small enough, estimates on F 1 , B 11 , (38) and

||X|| κ + ||Y || κ ≤ ρ, we obtain, with S def = ||X|| κ + ||Y || κ S ≤ |X 0 | + c[ ε 2 m α 9/2 + Sε α 5/2 + (α 1/2 + ε 2 α 5/2 )Sρ] so that for δ 0 ≤ δ ≤ δ 1 and ε = α 3 , (39)
and ε small enough and ρ such that

(1 + ρ + ρε)ε 1/6 ≤ (1 + 2ρ)ε 1/6 S ≤ (1 + c ′ ε 1/6 )|X 0 | + cε 1/2 m,
which leads finally to

||Y || κ ≤ c(mε 1/2 + ε 1/6 |X 0 |), (40)
||X|| κ ≤ (1 + cε 1/6 )|X 0 | + cε 1/2 m, ( 41
)
where c is a number independent of ε, ε = α 3 small enough, and we assume m bounded by a certain M of order 1, S ≤ ρ, where ρ is fixed arbitrarily, of order 1.

Resolution for B 0

We intend to solve the part of our system for B 0 with B 0 (0

) = B 0 | x=0 .
We notice from ( 16) and ( 21) that

B 1 = εδ A * B 0 z 10 (B 0 ) + Z(X, Y , B 0 ) -ε 2 αδ(1 + δ 2 ) B 0 A * A 3 A 3 = B 0 [ε 2 B 2 0 (1 + δ 2 ) 2 (x 1 + y 1 ) + 2λ r λ i (x 2 -y 2 )], εα(1 + δ 2 ) A 3 A * 2 ≤ 4εα 2 δ A * (|X| + |Y |) ≤ 4εα(|X| + |Y |),
so that it is clear that (see above estimates for Z)

B 1 > 0 for B 0 ∈ (0, 1 -η 2 0 δ 2 ), |X| + |Y | ≤ ρ. ( 42
)
This is coherent with the study of the linearized system near M -: Indeed the principal part of the differential equation for B 0 is

B ′ 0 = εδB 0 A * z 10 (B 0 )
which may be integrated as

B 2 0 = 1 (1 + δ 2 2 ) cosh 2 (x 0 -εδx) , (43)
cosh x 0 = 1 B 0 (0)(1 + δ 2 2 ) 1/2
, which satisfies B 0 = 0 for x = -∞, and B 0 = B 0 (0) for x = 0. More precisely the differential equation for B 0 is now (after replacing (X, Y ) by its expression found at previous subsection)

B ′ 0 = εδ A * B 0 z 10 (B 0 )[1 + f (B 0 )] (44)
where

f (B 0 ) is a non local analytic function of B 0 in C 0 κ , such that ||f (B 0 )|| κ ≤ cα 2 ρ.
Remark 12 We may notice that we might replace cα 2 ρ in the estimate above, by

cα 2 ρe κx → 0 as x → -∞, since X and Y ∈ C 0 κ .
We are looking for the solution such that B 0 = 0 for x = -∞, and B 0 (0) ≤ 1 -η 2 0 δ 2 for x = 0. We can rewrite (44) as

2B 0 B ′ 0 B 2 0 A * z 10 (B 0 ) = 2εδ[1 + f (B 0 )]. ( 45
)
We now introduce the variable v :

v = 1 -1 -(1 + δ 2 2 )B 2 0 1 + 1 -(1 + δ 2 2 )B 2 0 , B 2 0 = 1 1 + δ 2 2 4v (1 + v) 2 , so that (ln v) ′ = 2εδ[1 + f (B 0 )].
We observe that for x running from -∞ to 0,

w = ln v is increasing from -∞ to w 0 = ln v 0 < 0.
Now let us define h continuous in its argument and such that

h(w) = f (B 0 ), B 0 = 1 1 + δ 2 2 1/2 2e w/2 (1 + e w ) ,
and let us find an a priori estimate for the solution B 0 (x), for x ∈ (-∞, 0]. We obtain by simple integration

x 0 w ′ (s) 1 + h(w)(s) ds = 2εδx.
For α small enough we have

1 -cα 2 ρ ≤ 1 1 + h(w) ≤ 1 + cα 2 ρ,
hence (since w < w 0 , and x < 0)

(w 0 -w)(1 -cα 2 ρ) ≤ -2εδx ≤ (w 0 -w)(1 + cα 2 ρ) so that exp( -2εδx 1 + cερ ) ≤ e w0-w ≤ exp( -2εδx 1 -cερ ) and v 0 exp( 2εδ 1 -cα 2 ρ x) ≤ v(x) ≤ v 0 exp( 2εδ 1 + cα 2 ρ x).
It finally results that we obtain an a priori estimate for

B 0 (x) = B 0 (X 0 , B 0 (0))(x) ∈ C 0 κ , (46)
B 0 (X 0 , B 0 (0))(x) = 1 1 + δ 2 2 1/2 2 v(x) (1 + v(x)) , x ∈ (-∞, 0), 2 √ v 0 exp( εδ 1-cα 2 ρ x) 1 + v 0 exp( 2εδ 1-cα 2 ρ x) ≤ 1 + δ 2 2 1/2 B 0 ≤ 2 √ v 0 exp( εδ 1+cα 2 ρ x) 1 + v 0 exp( 2εδ 1+cα 2 ρ x) , (47)
v 0 = 1 -1 -(1 + δ 2 2 )B 2 0 (0) 1 + 1 -(1 + δ 2 2 )B 2 0 (0) < 1.
It remains to notice that we can choose

κ = εδ 1 + cα 2 ρ
in the proof for (X, Y ), which needs to satisfy

κ ≤ σ 2 = α 1/2 √ δ 2 5/4 . ( 48
)
We have already chosen ε = α 3 hence

κ ≤ εδ = δα 3 ≤ α 1/2 √ δ 2 5/4
for α small enough, and (48) is satisfied. The a priori estimate for B 0 allows to prove that there is a unique solution of the integro-differential equation (45) which satisfies the estimate (47) (see [START_REF] Hale | Ordinary differential equations[END_REF]). Since B 0 is in factor in A 0 , A 1 , A 2 , A 3 , B 1 the behavior for x → -∞ of the coordinates of the unstable manifold, is governed by the behavior of B 0 . The estimates indicated in Lemma 8 results from (21), ( 27) and(28). This ends the proof of Lemma 8.

Let us define the hyperplane H 0

B 0 = (1 -η 2 0 δ 2 ) 1/2 .

Intersection of the unstable manifold with H 0

We need to give precisely the intersection of the unstable manifold with the hyperplane B 0 = 1 -η 2 0 δ 2 . This gives a two-dimensional manifold lying in the 4-dimensional manifold W g ∩ H 0 . Taking into account of

A * = δα λ r , λ i ∼ δ 1/2 α 1/2 2 1/4 , ε = α 3 , z 10 ∼ B 00 α √ 2 , |Y (0)| = O(α 1/2 |X 0 | + B 00 α 3/2 ),
we obtain a two-dimensional intersection which is tangent to a plane (parameters x 1 , x 2 ) with principal part given by

A 0 = δα + α 1/2 δ 1/2 2 3/4 B 00 (x 1 -x 2 ) + O(α|X 0 | + α 2 B 00 ) A 1 = αδB 00 x 1 - α 2 δ √ 2 B 00 + O(α 3/2 |X 0 | + α 5/2 B 00 ) A 2 = δ 3/2 2 1/4 B 00 α 3/2 (x 1 + x 2 ) + O(α 2 |X 0 | + α 3 B 00 ) (49)
A 3 = √ 2δ 2 B 00 α 2 x 2 + O(α 5/2 |X 0 | + α 7/2 B 00 ) B 00 = 1 -η 2 0 δ 2 ∼ (1 + δ 2 ) -1/2 , with |x 1 | + |x 2 | ≤ ρ, δ 0 ≤ δ ≤ δ 1 , ε = α 3 , α 2 = η 2 0 (1 + δ 2
) -1 > 0, and where we do not write B 1 since we know that this manifold lies in the 5 dimensional manifold W g .

Stable manifold of M +

We show the following Lemma 13 For ε small enough, , δ 0 ≤ δ ≤ 1, the 3-dimensional stable manifold of M + is included in the 5-dimensional manifold W g , it exists for A 0 , A 1 , A 2 , A 3 in a ball of small radius η (independent of ε), is analytic in parameters (ε, δ),

and reaches B 0 (0) = B 00 def = 1 -η 2 0 δ 2 , with η 2 0 (1 + δ 2 ) = 1 + ε 2/3 . More- over as x → +∞, (A 0 , A 1 , A 2 , A 3 ) → 0 as exp(-δ 2 x), (B 0 -1, B 1 ) → 0 as exp(- √ 2εx), v(x) def = B 0 (x) -1 δ 1/2 ≃ - (1 -B 0 (0))(1 -tanh(εx/ √ 2) 1 + B 0 (0) tanh(εx/ √ 2) , (50)
|v 0 | ≤ 0.293.
Let us define

δ ′ = δ 1/2 ,
and choose a new basis

V - r =          1 -δ ′ √ 2 0 δ ′3 √ 2 0 0          , V - i =          0 -δ ′ √ 2 δ ′2 -δ ′3 √ 2 0 0          , V + r =          1 δ ′ √ 2 0 -δ ′3 √ 2 0 0          , V + i =          0 δ ′ √ 2 δ ′2 δ ′3 √ 2 0 0          , W - 1 =         0 0 0 0 1 -ε √ 2         , W + 1 =         0 0 0 0 1 ε √ 2         ,
for defining new coordinates (x 1 , x 2 , y 1 , y 2 , z 0 , z 1 ) such that Z = (0, 0, 0, 0, 1, 0

) t + δ ′ x 1 V - r + δ ′ x 2 V - i + δ ′ y 1 V + r + δ ′ y 2 V + i + δ ′ z 0 W - 1 + δ ′ z 1 W + 1 A 0 = δ ′ (x 1 + y 1 ) A 1 = - δ ′2 √ 2 (x 1 -y 1 + x 2 -y 2 ) A 2 = δ ′3 (x 2 + y 2 ) ( 51
)
A 3 = δ ′4 √ 2 (x 1 -y 1 -x 2 + y 2 ) B 0 = 1 + δ ′ (z 0 + z 1 ) B 1 = -ε √ 2δ ′ (z 0 -z 1 ).
A simple resolution leads to

x 1 = A 0 2δ ′ - A 1 2 √ 2δ ′2 + A 3 2 √ 2δ ′4 x 2 = - A 1 2 √ 2δ ′2 + A 2 2δ ′3 - A 3 2 √ 2δ ′4 y 1 = A 0 2δ ′ + A 1 2 √ 2δ ′2 - A 3 2 √ 2δ ′4 y 2 = A 1 2 √ 2δ ′2 + A 2 2δ ′3 + A 3 2 √ 2δ ′4 z 0 = B 0 -1 2δ ′ - B 1 2εδ ′ √ 2 z 1 = B 0 -1 2δ ′ + B 1 2εδ ′ √ 2 .
Let us define

u = x 1 + y 1 = δ ′-1 A 0 (52) v = z 0 + z 1 = δ ′-1 (B 0 -1),
then system (1) reads as

A ′ 0 = A 1 , A ′ 1 = A 2 , A ′ 2 = A 3 , A ′ 3 = -A 0 δ 2 + 2δ 2 v + δu 2 + (1 + δ 2 )v 2 , v ′ = 1 δ ′ B 1 , B ′ 1 = ε 2 (1 + δ ′ v) 2δ ′ v + δv 2 + (1 + δ 2 )δu 2 .
With variables (51) this gives

x ′ 1 = - δ ′ √ 2 (x 1 + x 2 ) - δ ′ ug(u, v) 2 √ 2 , x ′ 2 = δ ′ √ 2 (x 1 -x 2 ) + δ ′ ug(u, v) 2 √ 2 , y ′ 1 = δ ′ √ 2 (y 1 + y 2 ) + δ ′ ug(u, v) 2 √ 2 , y ′ 2 = - δ ′ √ 2 (y 1 -y 2 ) - δ ′ ug(u, v) 2 √ 2 , z ′ 0 = -ε √ 2z 0 - εδ ′ 2 √ 2 f (u, v), z ′ 1 = ε √ 2z 1 + εδ ′ 2 √ 2 f (u, v), g(u, v) = u 2 + 2δv + (1 + δ 2 )v 2 f (u, v) = 3v 2 + δ ′ v 3 + (1 + δ 2 )(1 + δ ′ v)u 2 ,
where the linear part is as expected.

For finding the stable manifold of M + we put the system in an integral form, looking for solutions tending to 0 as x → +∞ :

X(x) = e -Lx X 0 - δ ′ 2 √ 2 x 0 e -L(x-s) u(s)G(u, v)(s)ds, Y (x) = - δ ′ 2 √ 2 +∞ x e L(x-s) u(s)G(u, v)(s)ds, (53)
z 0 (x) = e -ε √ 2x z 00 - εδ ′ 2 √ 2 x 0 e -ε √ 2(x-s) f (u, v)(s)ds, z 1 (x) = - εδ ′ 2 √ 2 +∞ x e ε √ 2(x-s) f (u, v)(s)ds, (54)
where

G = g -g , L = δ ′ √ 2 1 1 -1 1 .
We notice that

e Lx = e δ ′ x √ 2 cos δ ′ x √ 2 sin δ ′ x √ 2 -sin δ ′ x √ 2 cos δ ′ x √ 2 , ( 55
) ||e -Lx || ≤ e -δ ′ x √ 2 , x > 0.
The stable manifold is obtained in expressing (Y (0), z 1 (0)) as function of (X 0 , z 00 ).

Let us define for this section

C 0 κ = {X ∈ C 0 [0, +∞); X(x)e κx is bounded}
equiped with the norm

||X|| κ = sup (0,+∞) |X(x)e κx |.
Using (55), the system (53,54) gives two scalar equations with unknown functions (u, v). We obtain:

u(x) = e -δ ′ x √ 2 u 0 (x) - δ ′ 2 ∞ 0 e - δ ′ |x-s| √ 2 cos[ δ ′ |x -s| √ 2 - π 4 ]u(s)g(u, v)(s)ds, v(x) = e -ε √ 2x z 00 - εδ ′ 2 √ 2 ∞ 0 e -ε √ 2|x-s| f (u, v)(s)ds (56)
with

u 0 (x) = x 10 cos δ ′ x √ 2 -x 20 sin δ ′ x √ 2 .
We may observe that we have the explicit solution of the second equation of (56) for u ≡ 0. Indeed

v(x) = e -ε √ 2x z 00 - εδ ′ 2 √ 2 ∞ 0 e -ε √ 2|x-s| f (0, v)(s)ds corresponds to look for v such that v ′ = (z ′ 0 + z ′ 1 ) = -ε √ 2(z 0 -z 1 ) z ′ 0 -z ′ 1 = -ε √ 2v - εδ ′ √ 2 (3v 2 + δ ′ v 3 ), hence v ′′ = 2ε 2 v + ε 2 δ ′ (3v 2 + δ ′ v 3 ), v, v ′ → x→+∞ 0, which gives v ′2 = ε 2 2 v 2 (2 + δ ′ v) 2 .

It results that

v ′ = - ε √ 2 v(2 + δ ′ v) (57)
since v grows for x > 0 and v < 0 and

|δ ′ v| = 1 -B 0 < 1 implies 2 + δ ′ v > 1.
Finally we obtain

v(x) = v 0 e -ε √ 2x 1 + v 0 δ ′ 2 (1 -e -ε √ 2x
) .

Using the first integral (3)

For extending the domain of validity for the stable manifold of M + , instead of using the differential equations for z 0 and z 1 (or v) we use the first integral (3):

B 2 1 = ε 2 2 (B 2 0 -1 + A 2 0 ) 2 + ε 2 δ 2 A 2 0 B 2 0 + 2ε 2 A 1 A 3 -ε 2 A 2 2 , hence B 2 1 = ε 2 2 [(B 2 0 -1) 2 + 2δ(1 + δ 2 )u 2 (2δ ′ v + δv 2 ) +δ 2 u 4 + 8δ 3 (x 1 y 1 -x 2 y 2 )].
Taking the square root gives the traces of the stable and the unstable manifolds on W g . The stable manifold needs satisfy B 1 = B ′ 0 > 0 , since B 0 < 1 for x = 0, and B 0 = 1 for x = ∞. Assuming that the sign of B 1 does not change in the interval, we obtain

B 1 = ε √ 2 (1 -B 2 0 ) 1 + 2δ ′ (1 + δ 2 )u 2 (2v + δ ′ v 2 ) + δu 4 + 8δ 2 (x 1 y 1 -x 2 y 2 ) (2v + δ ′ v 2 ) 2 1/2 (58)
Remark 14 We notice that this implies that v < 0, v ′ > 0, and |v(x

)| max = |v 0 | = |z 00 + z 1 (0)| is then O(α 2 ) close to h(δ), where
h(δ) = 1 √ δ 1 - 1 1 + δ 2 .
We observe that

B 1 = B ′ 0 = ε √ 2 [1 -B 2 0 ]
may be easily integrated on (0, +∞) with B 0 (∞) = 1), and moreover leads to

z 0 -z 1 = 1 2 v(2 + δ ′ v) (59)
i.e.

z 1 = - δ ′ 4 (z 0 + z 1 ) 2 < 0 (60)
which is the solution of (54) for u = 0. We notice that (59) is exactly (57). It results that (58) may be written as

v ′ = -εv √ 2(1 + δ ′ 2 v) 1 + 2δ ′ (1 + δ 2 )u 2 (2v + δ ′ v 2 ) + δu 4 + 8δ 2 (x 1 y 1 -x 2 y 2 ) (2v + δ ′ v 2 ) 2 1/2 .
Let us assume that

|X(x)|, |Y (x)|, |u(x)| ≤ γ|v(x)|, x ∈ [0, +∞) (61) with 0 < γ < 1 6δ . Using 2 + δ ′ v > 1
this implies that for |X|, |Y | satisfying (61) we have

2δ ′ (1 + δ 2 )u 2 (2v + δ ′ v 2 ) ≤ 2γδ ′ (1 + δ 2 )|u|, δu 4 + 8δ 2 (x 1 y 1 -x 2 y 2 ) (2v + δ ′ v 2 ) 2 ≤ δγ 2 u 2 + 4 9 .
It results that for γ|u| such that

γ|u| ≤ 1 3δ ′ [12(1 + δ 2 ) + √ 2] -1 (62) then 2δ ′ (1 + δ 2 )u 2 (2v + δ ′ v 2 ) + δu 4 + 8δ 2 (x 1 y 1 -x 2 y 2 ) (2v + δ ′ v 2 ) 2 < 1 2 ,
and the square root is analytic in v with

v ′ = -ε √ 2v(1 + δ ′ 2 v)[1 + Z(X, Y, v)], |Z| ≤ 1/4. (63)
Then we can integrate the integro-differential equation, as in section 4.10. We introduce the new variable w as

w ′ = v ′ v(1 + (δ ′ /2)v) , w = ln -v 1 + (δ ′ /2)v , v = - e w 1 + δ ′ 2 e w ;
w decreases from w 0 to -∞ for x ∈ (0, ∞), while v grows from v 0 < 0 to 0.

Z(X, Y, v) = h(X, Y, w), |h| ≤ 1/4.
We then obtain, by simple integration

ε √ 2x(1 -1/4) ≤ w 0 -w(x) ≤ ε √ 2x(1 + 1/4).
Remark 15 The constant 1/4 above may later be replaced by

ce -δ ′ x/ √ 2
since we show later that |X| and |Y | lie in C 0

δ ′ / √ 2 .
We deduce the estimate

v 0 1 -tanh( εx(1-1/4) √ 2 ) 1 + B 0 (0) tanh( εx(1-1/4) √ 2 ) ≤ v(x) ≤ v 0 1 -tanh( εx(1+1/4) √ 2 ) 1 + B 0 (0) tanh( εx(1+1/4) √ 2 ) ( 64
)
where

v 0 = B 0 (0) -1 δ ′ < 0.
The a priori estimate for v obtained in (64) allows to prove (see [START_REF] Hale | Ordinary differential equations[END_REF]) the existence and uniqueness of a solution for (63), provided that ( 61) is satisfied on the whole interval x ∈ [0, ∞).

Estimate for u

Let us first show that for δ ∈ (0, 5), x ∈ [0, +∞) and for γ|u| ≤ c 1 (δ) where c 1 is defined in (68), then

g(u, v) = u 2 + v[2δ + (1 + δ 2 )v] < 0 |g(u, v)| ≤ 2δ|v|. (65)
Using (61), we observe that ( 65) is valid as soon as

γ|u| + (1 + δ 2 )|v 0 | ≤ 2δ. ( 66
)
We wish to reach v 0 = δ ′-1 (B 00 -1) where B 00 is the value of B 0 we have reached with the unstable manifold of M -. So next computations should be valid for v 0 such that |v 0 | ≤ |δ ′-1 (1 -B 00 )|. Using

η 2 0 ∼ 1 1 + δ 2 ,
since α is as close to 0 as we wish, we obtain

1 -B 00 = 1 -(1 -η 2 0 δ 2 ) 1/2 = 1 - 1 1 + δ 2 + O(α 2 ).
Then conditions (66) and (62) lead to

γ|u| ≤ c 1 (δ), (67)
with

c 1 (δ) = min{2δ - 1 √ δ [(1 + δ 2 ) -1 + δ 2 ], 1 3 √ δ [12(1 + δ 2 ) + √ 2] -1 }. ( 68
)
We observe that c 1 (δ) > 0 for δ ∈ (0, 5), c 1 (δ) ∼ 2δ for δ close to 0. Now the estimates (64) of v for x ∈ [0, +∞) lead to

|v 0 | e -5ε √ 2x 4 1 -|v0δ ′ | 2 (1 -e -5ε √ 2x 4 ) ≤ |v(x| ≤ |v 0 | e -3ε √ 2x 4 1 -|v0δ ′ | 2 (1 -e -3ε √ 2x 4
) , so that

|v 0 |e -5ε √ 2x 4 ≤ |v(x)| ≤ |v 0 | 1 -|v0δ ′ | 2 e -3ε √ 2x 4 (69)
and to reach B 00 we need to satisfy

h(δ)e -5ε √ 2x 4 ≤ |v(x)| ≤ c 0 (δ)e -3ε √ 2x 4 (70) with c 0 (δ) = 2δ 3/2 ( 1 + δ 2 + 1) 2 .
Let us consider (56), then ( 65) and ( 69) lead to

|u(x)e 5ε √ 2x 4 | ≤ |X 0 |e ( -δ ′ √ 2 + 5ε √ 2 4 )x + δδ ′ |v 0 | 1 -|v0δ ′ | 2 ∞ 0 e -δ ′ |x-s| √ 2 + 5ε √ 2(x-s) 4 |u(s)e 5ε √ 2s 4 |e -3ε √ 2s 4 ds.
We have

∞ 0 e -δ ′ |x-s| √ 2 + 5ε √ 2(x-s) 4 e -3ε √ 2s 4 ds ≤ 2 √ 2e -3ε √ 2x 4 δ ′ (1 -16ε 2 /δ ′ ) , so that ||u|| 5ε √ 2 4 ≤ |X 0 | + 2 √ 2δ|v 0 | (1 -|v0δ ′ | 2 )(1 -16ε 2 /δ ′ ) ||u|| 5ε √ 2 4
.

We notice that, for

|v 0 | ≤ 1 δ ′ (1 -B 00 ) 2 √ 2δ|v 0 | (1 -|v0δ ′ | 2 )(1 -16ε 2 /δ ′ ) ≤ c 2 (δ) with c 2 (δ) = 2 √ 2δc 0 (δ) (1 -16ε 2 /δ ′ ) ,
and we observe that

2 √ 2δc 0 (δ) = 4 √ 2δ 5/2 ( 1 + δ 2 + 1) 2 < k < 1 for δ ∈ [δ 0 , 1],
so that, since δ 0 is arbitrarily fixed > 0, and ε small enough, for any v satisfying (69) with |v 0 | ≤ 1 δ ′ (1 -B 00 ), by a fixed point argument, we obtain a unique

u ∈ C 0 5ε √ 2 4 such that ||u|| 5ε √ 2 4 ≤ 1 1 -k |X 0 |.
Remark 16 Notice that we neglected terms of order α 2 in the estimates leading to the calculus of k which is stricly <1. However, because of the flexibility of choice for δ, we may check that for ε (i.e. α) small enough the choice δ ∈ [δ 0 , 1] is valid.

End of the proof of Lemma 13

We may now estimate X and Y given by ( 53), and we need to check that (61) is satisfied. Now, from (53) we obtain

||X|| 5ε √ 2 4 ≤ |X 0 | + δδ ′ |v 0 | (δ ′ -ε)(1 -|v0δ ′ | 2 ) ||u|| 5ε √ 2 4 ≤ |X 0 | 1 + δδ ′ |v 0 | (1 -k)(δ ′ -ε)(1 -|v0δ ′ | 2 ) ≤ |X 0 | 1 + δ ′ k 2 √ 2(1 -k)(δ ′ -ε) , ||Y || 5ε √ 2 4 ≤ δδ ′ |v 0 | (1 -k)(δ + 4ε)(1 -|v0δ ′ | 2 ) |X 0 | ≤ δ ′ k 2 √ 2(1 -k)(δ ′ + 4ε) |X 0 |. We observe that 1 + δ ′ k 2 √ 2(1 -k)(δ ′ -ε) < 1 1 -k ,
hence, using (69) for a lower bound for v(x) we see that for X 0 such that

|X 0 | ≤ (1 -k)γ|v 0 |,
then conditions (61) are realized, and (67) is satisfied as soon as

|X 0 | ≤ (1 -k) γ c 1 (δ).
This ends the proof of the existence, uniqueness and analyticity in parameters of the solution of ( 53) and (56). Then (54) allows to find z 0 , z 1 with

||z 0 || ε √ 2 ≤ |z 00 | + εδ ′ [3 + (1 + δ 2 )γ 2 ] 2 √ 2 x 0 v 2 (s)e ε √ 2s ds ≤ |z 00 | + δ ′ [3 + (1 + δ 2 )γ 2 ] 2 v 0 1 -|v0|δ ′ 2 2 , ||z 1 || ε √ 2 ≤ δ ′ [3 + (1 + δ 2 )γ 2 ] 10 v 0 1 -|v0|δ ′ 2 2
.

We notice that

|z 1 (0)| ≤ δ ′ [3 + (1 + δ 2 )γ 2 ] 10 c 0 (δ) 1 -|v0|δ ′ 2 |v 0 |,
and for δ ∈ [δ 0 , 1] δ ′ [3 + (1 + δ 2 )γ 2 ] 10 c 0 (δ) 1 -|v0|δ ′ 2 ≤ 0.0402[3 + 2γ 2 ]
which is < 1/4 for a good choice of γ (still free choice). Hence

|z 1 (0)| ≤ 1 4 (|z 00 | + |z 1 (0)|) leads to |z 1 (0)| ≤ 1 3 |z 00 |, |v 0 | = |z 00 + z 1 (0)| ≤ 4 3 |z 00 |.
This ends the proof of existence, uniqueness and analyticity in parameters of the stable manifold of M + . The exponential estimates declared in Lemma 13 follow from the linear study of section 3 as x → +∞. The asymptotic expression of v(x) follows from (64) after replacing 1/4 by the better estimate Ce - √ δ 2 x . The bound for v 0 comes from h(δ) with δ = 1. This ends the proof of Lemma 13.

Intersection of the stable manifold with H 0

We need to compute the intersection of the 3-dimensional stable manifold of M + with the hyperplane H 0 defined by

B 0 = 1 -η 2 0 δ 2 . ( 71
)
We then obtain a 2-dimensional sub-manifold living in the 4-dimensional manifold W g ∩ H 0 . We have by construction

A 0 = δ 1/2 (x 10 + y 10 ), A 1 = - δ √ 2 (x 10 + x 20 -y 10 -y 20 ) ( 72
)
A 2 = δ 3/2 (x 20 + y 20 ) A 3 = δ 2 √ 2 (x 10 -x 20 -y 10 + y 20 ),
where y 10 and y 20 are expressed in function of X 0 = (x 10 , x 20 ), with the restriction

|x 10 | + |x 20 | ≤ η.
Below, we need to express the tangent plane to the intersection of the stable manifold with the hyperplane H 0 . This is given by

Y 0 = - δδ ′ √ 2 ∞ 0 e -Ls 1 -1 u(s)v(s)ds, with u(x) = e -δ ′ x √ 2 u 0 (x) -δδ ′ ∞ 0 e -δ ′ |x-s| √ 2 cos[ δ ′ |x -s| √ 2 - π 4 ]u(s)v(s)ds, u 0 (x) = x 10 cos δ ′ x √ 2 -x 20 sin δ ′ x √ 2 , v(x) = v 0 e -ε √ 2x 1 + v 0 δ ′ 2 (1 -e -ε √ 2x ) , v 0 = 1 δ ′ (B 00 -1), so that u(x) = l(v 0 , δ, x)X 0 ∈ C 0 5ε √ 2 4 is linear in X 0 , hence Y 0 = L 1 (v 0 , δ)X 0 , (73)
with a 2x2 matrix L 1 depending analytically of δ ∈ [δ 0 , 1].

Intersection of the two manifolds

We need to see the intersection of the plane (49) tangent to the unstable manifold of M -, with the plane tangent to the stable manifold of M + given by (72), satisfying (73).

We then find a linear system with 4 unknowns (x 1 (u) , x 2 (u) , x

20 ), with the restrictions |x

(s) 10 | + |x (s) 20 | ≤ η, |x 1 (u) | + |x 2 (u) | ≤ ρ.
We then have (x

(s) 10 + y (s) 10 ) = δ ′ α + α 1/2 2 3/4 B 00 (x 1 (u) -x 2 (u) ) -(x (s) 10 + x (s) 20 -y (s) 10 -y (s) 20 ) = √ 2αB 00 x 1 (u) -α 2 B 00 (74) (x (s) 20 + y
(s) 20 ) = α 3/2 2 1/4 B 00 (x 1 (u) + x 2 (u) ) (x (s) 10 -x (s) 20 -y (s) 10 + y (s) 20 ) = 2α 2 B 00 x 2 (u) ,
where we need to express (y

(s) 10 , y (s)
20 ) as a linear function of (x

(s) 10 , x (s) 20 ) (see (73). Let us define X (s) 0 = x (s) 10 x (s) 20 , Y (s) 0 = y (s) 10 y (s) 20 , X (u) = x 1 (u) x 2 (u)
, then we have

X (s) 0 = δ ′ 2 α + α 2 B00 4 α 2 B00 4 + M 1 X (u) , Y (s) 0 = δ ′ 2 α -α 2 B00 4 -α 2 B00 4 + M 2 X (u) ,
with

M 1 = α 1/2 B 00 2 1/4 4 1 -2 1/4 α 1/2 -1 + 2 3/4 α 3/2 -2 1/4 α 1/2 + √ 2α √ 2α -2 3/4 α 3/2 , M 2 = α 1/2 B 00 2 1/4 4 1 + 2 1/4 α 1/2 -1 -2 3/4 α 3/2 2 1/4 α 1/2 + √ 2α √ 2α + 2 3/4 α 3/2 .
The matrix M 2 is invertible with

M -1 2 = 4 α 1/2 B 00 2 1/4 det(M ′ 2 ) √ 2α + 2 3/4 α 3/2 1 + 2 3/4 α 3/2 -2 1/4 α 1/2 - √ 2α 1 + 2 1/4 α 1/2 det(M ′ 2 ) = [ √ 2α(1 + 2 1/4 α 1/2 ) 2 + (1 -2 3/4 α 3/2 )(2 1/4 α 1/2 + √ 2α)] = 2 1/4 α 1/2 + 2 √ 2α + O(α 3/2 ).
It results that

M 1 M -1 2 ∼ 1 + O(α 1/2 ) -2 + O(α 1/2 ) -2 3/2 α + O(α 3/2 ) -1 + O(α 1/2 ) X (s) 0 = 1 + O(α 1/2 ) -2 + O(α 1/2 ) -2 3/2 α + O(α 3/2 ) -1 + O(α 1/2 ) Y (s) 0 + O(α 3/2 ) √ 2δ ′ α 2 + O(α 5/2
) .

(75) Equation (75) represents a 2-dim affine plane resulting from the 4-dim linear system expressing the intersection of the two manifolds. This is in fact the 2 compatibility conditions of the system (74) while solving with respect to (x 1 (u) , x 2 (u) ), and gives a condition on coordinates of the stable manifold. This affine plane needs to intersect the tangent plane to the stable manifold given by (73) with Y (s) 0 expressed as a linear function of X (s) 0 . We deduce that (73) combined with

X (s) 0 = L 2 (v 0 , δ)Y (s) 0 + O(α 3/2 ) leads to Y (s) 0 = L 1 (v 0 , δ)L 2 (v 0 , δ)Y (s) 0 + O(α 3/2 ).
We notice that

v 0 = 1 √ δ (B 00 -1), B 00 = 1 -η 2 0 δ 2 , η 2 0 (1 + δ 2 ) = 1 + α 2 ,
so that the 2x2 matrix L 1 (v 0 , δ ′ )L 2 (v 0 , δ) is a function of α (which is as small as we wish), and depends analytically of δ ∈ [δ 0 , 1]. It is then clear that 1 might be an eigenvalue of this matrix operator only for isolated values of δ. Indeed for δ small enough, the norm of L 1 (v 0 , δ) is small, which does not allow an eigenvalue 1 for L 1 (v 0 , δ)L 2 (v 0 , δ). It then results that for any δ ∈ [δ 0 , 1] except maybe for a finite set of values, and for α small enough, we obtain a unique solution

Y (s) 0 = O(α 3/2 ), leading to X (s) 0 = O(α 3/2 ) which is coherent with the condition |x (s) 10 | + |x (s) 20 | ≤ η.
Moreover adding the 3 first equation of (74) gives

0 = δ ′ α -α 2 B 00 + α 1/2 2 3/4 B 00 (x 1 (u) -x 2 (u) ) + √ 2αB 00 x 1 (u) + α 3/2 2 1/4 B 00 (x 1 (u) + x 2 (u) ) hence x 2 (u) = x 1 (u) - 2 7/4 α 1/2 + 2 3/2 α 1 -α √ 2 x 1 (u) + 2 3/4 (α 1/2 δ ′ -α 3/2 B 00 ) B 00 (1 -α √ 2) , so that δ ′ α + α 1/2 2 3/4 B 00 (x 1 (u) -x 2 (u) ) ≃ -2αB 00 x 1 (u) = O(α 3/2 ). It results that (x 1 (u) , x 2 (u) ) = O(α 1/2 ), which then satisfies the condition |x 1 (u) | + |x 2 (u) | ≤ ρ.
Finally, from (49) we obtain

A 0 (0) = O(B 00 α 3/2 ) A 1 (0) = O(B 00 α 3/2 ) (76) A 2 (0) = O(B 00 α 2 ) A 3 (0) = O(B 00 α 5/2 ).
The uniqueness of the intersection of the tangent planes between the unstable manifold of M -and the stable manifold of M + proves that it is transverse while they both sit on W g and cross the hyperplane (71). Since it is the transverse intersection of two manifolds, depending analytically on parameters (ε, δ), the resulting curve depends analytically on these parameters. We observe that, along this intersection, and by construction, B 1 (x) = B ′ 0 (x) > 0. Its principal part on (-∞, 0] is given by (43) with B 0 (0) = B 00 = 1 -η 2 0 δ 2 , and on [0, +∞)by (50).

The Theorem 1 is then proved. Moreover, for the heteroclinic solution, we can improve the a priori estimates given at Lemma 8. Taking into account the size of variables for x = 0, we have now Corollary 17 For x ∈ (-∞, 0] there exists c > 0 independent of ε such that for the heteroclinic curve

| A 0 (x)| ≤ cαB 0 (x)e εδx |A 1 (x)|, |A 2 (x)|, |A 3 (x)| ≤ cα 3/2 B 0 (x)e εδx
We also give estimates for x > 0. Using (56) and playing on the flexibility of choice for δ, we can find χ < 1 independent of ε, such that for δ ∈ [δ 0 , 1] we have

||u|| δ ′ √ 2 (1-χ) ≤ |X 0 | + 2 √ 2δc 0 (δ) χ(2 -χ) ||u|| δ ′ √ 2 (1-χ) , with 2 √ 2δc 0 (δ) χ(2 -χ) < k ′ < 1. Hence ||u|| δ ′ √ 2 (1-χ) ≤ 1 1 -k ′ |X 0 |
and as above we see that there exists C > 0 such that

||X|| δ ′ √ 2 (1-χ) + ||Y || δ ′ √ 2 (1-χ) ≤ C|X 0 |
, so that, using (51) and (76) we have the following Corollary 18 For x ∈ [0, +∞) there exists c > 0 independent of ε such that for the heteroclinic curve

|A (m) 0 (x)| ≤ cα 3/2 e -δ ′ x √ 2 (1-χ) , m = 0, 1, 2, 3.

Study of the linearized operator

Let us redefine the heteroclinic connection we found at Theorem 1 as

(A * (x), B * (x)) ⊂ R 2 with 1 < 1 + δ 2 0 ≤ g = 1 + δ 2 ≤ 1 + (0.825) 2
, and where we know that, for ε small enough

B * (x) > 0, B ′ * (x) > 0 (A * (x), B * (x)) → (1, 0) as x → -∞ (0, 1) as x → +∞ , 1 ε 2 C ′′ + (1 -gA 2 * -3B 2 * )C -2gA * B * A , L g D = 1 ε 2 D ′′ + (1 -gA 2 * -B 2 * )D.
Let us define the Hilbert spaces

L 2 η = {u; u(x)e η|x| ∈ L 2 (R)}, D 0 = {(A, C) ∈ H 4 η × H 2 η ; A ∈ H 4 η , C ∈ D 1 } D 1 = {C ∈ H 2 η ; ε -2 ||C ′′ || L 2 η + ε -1 ||C ′ || L 2 η + ||C|| L 2 η def = ||C|| D1 < ∞}
equiped with natural scalar products. Below, we prove the following Lemma 19 Except maybe for a set of isolated values of g, the kernel of M g in L 2 η is one dimensional, span by (A ′ * , B ′ * ), and its range has codimension 1, L 2orthogonal to (A ′ * , B ′ * ). M g has a pseudo-inverse acting from L 2 η to D 0 for any η > 0 small enough, with bound independent of ε.

The operator L g has a trivial kernel, and its range which has codimension 1, is L 2 -orthogonal to B * (B * / ∈ L 2 ). L g has a pseudo-inverse acting respectively from L 2 η to D 1 for η > 0 small enough, with bound independent of ε. Remark 20 The above Lemma is useful for proving the persistence under reversible perturbations of our heteroclinic. This is done in a forthcoming paper and appears to be more difficult than the symmetric case solved in [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF].

Asymptotic operators

Let us define the operators obtained when x = ±∞ :

M - ∞ A C = -A (4) -2A ε -2 C ′′ -(g -1)C , M + ∞ A C = -A (4) -(g -1)A ε -2 C ′′ -2C , L - ∞ D = ε -2 D ′′ -(g -1)D, L + ∞ D = ε -2 D ′′ .
Notice that all these operators are negative. Furthermore, their spectra in L 2 (R) are such that

σ(M - ∞ ) = (-∞, -c -], c -= max{2, (g -1)} > 0, σ(M + ∞ ) = (-∞, -c + ], c + = c -, σ(L - ∞ ) = (-∞, -(g -1)], σ(L + ∞ ) = (-∞, 0].
Operators M g and L g are respectively relatively compact perturbations of the corresponding asymptotic operators M ∞ and L ∞ defined as

M ∞ = M - ∞ , x < 0 M + ∞ , x > 0 , L ∞ = L - ∞ , x < 0 L + ∞ , x > 0 ,
Their essential spectrum, i.e. the set of λ ∈ C for which λ -M g (resp. λ -L g ) is not Fredholm with index 0, is equal to the essential spectrum of M ∞ (resp. L ∞ ) (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]). The latter spectra are found from the spectra of M ± ∞ and L ± ∞ :

σ ess (M ∞ ) = (-∞, -c + ], σ ess (L ∞ ) = (-∞, 0].
In particular, this implies that 0 does not belong to the essential spectrum of M g , so that the operator M g is Fredholm with index 0. Moreover operators M ∞ and L ∞ are self adjoint negative operators in L 2 , and M ∞ has a bounded inverse [START_REF] Kato | Perturbation theory for linear operators[END_REF].

||M -1 ∞ || L 2 ≤ 1 c + .
This last property remains valid in exponentially weighted spaces, with weights e η|x| , and η sufficiently small, since this acts as a small perturbation of the differential operator (see [START_REF] Kapitula | Spectral and Dynamical Stability of Nonlinear Waves[END_REF]

M g u = f in decomposing u = zU * + v, v = Q 0 u, (M ∞ + A g )v = Q 0 f
and we need to satisfy the compatibility condition f, U * = 0, while z is arbitrary and we obtain for v :

(I + M -1 ∞ A g )v = M -1 ∞ Q 0 f,
where the operator M -1 ∞ A g is now a compact operator for which -1 is not an eigenvalue, since v ∈ U ⊥ * . It results that there is a number c independent of ε such that

||v|| L 2 η ≤ c||f || L 2 η .
From the form of operator M g and using interpolation properties, we obtain for

v = (A, C) ||(A, C)|| D0 ≤ c||f || L 2 η
with a certain c independent of ε.

Properties of L g

Notice that L g is self adjoint in L 2 (R) and that

L g B * = 0, but B * / ∈ L 2 (R).
This property allows to solve explicitely the equation L g u = f ∈ L 2 η with respect to u ∈ L 2 η (using variation of constants method), and shows that it has a unique solution, provided that

R f B * dx = 0. We obtain u(x) = ∞ x ε 2 B * (x) B 2 * (s) F (s)ds with F (s) = ∞ s f (τ )B * (τ )dτ for s ≥ 0 = - s -∞ f (τ )B * (τ )dτ for s ≤ 0.
By Fubini's theorem we can write for x ≥ 0

u(x) = ε 2 B * (x) ∞ x f (τ )B * (τ ) τ x ds B 2 * (s) dτ and, for x ≤ 0 u(x) = -ε 2 B * (x) x -∞ f (τ )B * (τ ) 0 x ds B 2 * (s) dτ -ε 2 B * (x) 0 x f (τ )B * (τ ) 0 τ ds B 2 * (s)
dτ .

The asymptotic properties of B * (x) at ±∞ imply, for x ≥ 0 Then, we choose η = 1 2 εδ, so that the pseudo-inverse of L g has a bounded inverse in L 2 η :

|u(x)|e ηx ≤ Cε 2 ∞ x |f (
|| L g -1 || ≤ c 2 ,
where c 2 is independent of ε. Using the form of L g we obtain easily

||u|| D1 ≤ c 3 ||f || L 2 η with c 3 independent of ε.
Remark 21 The choice made for η is such that

η < εδ, η < ε √ 2,
for values of δ for which Theorem 1 is valid. This means that as x → -∞ (A * -1, B * ), and, as x → +∞ (A * , B * -1) tend exponentially to 0 faster than e -η|x| .

In fact, L g has the same properties as the operator M i in the proof of Lemma 7.3 in [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF], see also [START_REF] Haragus | Grain boundaries in the Swift-Hohenberg equation[END_REF]: L g is Fredholm with index -1, when acting in L 2 η , for η small enough. L g has a trivial kernel, and its range is orthogonal to B * , with the scalar product of L 2 (R).

Properties of M g

We saw that M g is Fredholm with index 0. Furthermore the derivative of the heteroclinic solution belongs to its kernel:

M g A ′ * B ′ * = -A (5)
the system (1) where we look for an heteroclinic U for g = g 0 . Then, we have

F (U (g0) * , g 0 ) = 0, D U F (U (g0) * , g 0 ) = M g0 ,
and since M g0 ζ j = 0, j = 0, 1, using the equivariance under operator T a , we obtain (denoting F 0 = F (U (g0) * , g 0 ) and [..] (2) the argument of a quadratic operator)

0 = M g0 Y + (g -g 0 )∂ g F 0 + 1 2 D 2 UU F 0 [a 1 ζ 1 + Y ] (2) + +O(|g -g 0 |[|g -g 0 | + |a 1 | + ||Y ||] + ||Y || 3 ).
The projection Q 0 of this equation allows to use the implicit function theorem to solve with respect to Y and then obtain a unique solution

Y = Y(a 1 , g), with Y = -(g -g 0 ) M g0 -1 Q 0 ∂ g F 0 - 1 2 M g0 -1 Q 0 D 2 UU F 0 [a 1 ζ 1 ] (2) + +O(|g -g 0 |(|g -g 0 | + |a 1 |) + |a 1 | 3 )).
Then projecting on the complementary space, (only one equation since we work in the subspace orthogonal to ζ * 0 ), we may observe (see the proof below) that P 0 ∂ g0 F 0 = 0 and then obtain the "bifurcation" equation as

q(a 1 , g -g 0 ) = O((|g -g 0 | + |a 1 |) 3 ),
where the function q is quadratic in its arguments and

q| g=g0 ζ 1 = 1 2 P 0 D 2 UU F 0 [a 1 ζ 1 ] (2) .
This equation is just at main order a second degree equation in a 1 depending on g -g 0 . Provided that the discriminant is not 0, the generic number of solutions is 2 or 0. If the discriminant is 0 for g = g 0 , we just go a little farther in g, and obtain a non zero discriminant, since the discriminant cannot stay = 0, because of the analyticity in g of the heteroclinic. This is true except for a set of isolated values of g. We can then use the implicit function theorem for finding corresponding solutions for the system with higher order terms. In fact we already know a solution, corresponding to U

(g) * = U (g0) * + (g -g 0 )∂ g U (g0) * + h.o.t.
which corresponds to specific values for a 1 and Y, of order O(g -g 0 ). It then results that there is at least another solution of order O(g -g 0 ), so that there exists another heteroclinic, in the neighborhood of the known one (then in contradiction with Theorem 1).

Remark 22 The above proof with only 1 dimension in the Kernel

, provides Y = -(g -g 0 ) M g0 -1 ∂ g F 0 + O((g -g 0 ) 2
), which gives a unique heteroclinic. Since we found only one heteroclinic, this shows that the kernel is of dimension 1.

Proof of P

0 ∂ g F 0 = 0 Lemma 23 Any (u, v) in the kernel of M g satisfies R A * B * (B * u + A * v)dx = 0, and ∂ g F 0 (U * , g) = (A * B 2 * , A 2 * B * ) belongs to the range of M g , hence P 0 ∂ g F 0 = 0.

Proof.

Differentiating with respect to g the system (1) verified by the heteroclinic, we obtain

M g ∂ g A * ∂ g B * = A * B 2 * A 2 * B * = ∂ g F 0 (U * , g), hence (A * B 2 * , A 2 * B * ) belongs to the range of M g . When (u, v) ∈ ker M g , then (u, v) ∈ ker M * g where M g = M * g , when the adjoint is computed with the scalar product of L 2 , hence R A * B * (B * u + A * v)dx = 0. ( 79
)
Hence, the eigenvectors ζ * 0 , ζ * 1 of the adjoint M * g (the orthogonal of this 2dimensional eigenspace is the range of M g ), are orthogonal to

∂ g F 0 = (A * B 2 * , A 2 * B * )| g0 in L 2 .

A Appendix

A.1 Monodromy operator

Let us prove the estimate for the monodromy operators. We prove the following Lemma 24 For η 0 δ ≤ A * ≤ 1, and α -1 ≥ (1 + δ 2 ) 2 and the following estimates hold

||S 0 (x, s)|| ≤ e σ(x-s) , -∞ < x < s ||S 1 (x, s)|| ≤ e -σ(x-s) , -∞ < s < x with σ = α 1/2 δ 1/2 2 1/4 .
We start with the system

x ′ 1 = λ r x 1 + λ i x 2 x ′ 2 = -λ i x 1 + λ r x 2
where λ r and λ i are functions of x. When η 0 δ ≤ A * ≤ 1, α -1 ≥ (1 + δ 2 ) 2 , we have, for ε small enough (see ( 14))

λ r ≥ α 1/2 δ 1/2 2 1/4 = σ. Now we have (x 2 1 + x 2 2 ) ′ = 2λ r (x 2 1 + x 2 2 ) hence (x 2 1 + x 2 2 )(x) = e x s 2λr(τ )dτ (x 2 1 + x 2 2 )(s), which, for x < s, leads to (x 2 1 + x 2 2 )(x) ≤ e σ(x-s) (x 2 1 + x 2 2 )(s).
The proof is then done for the operator S 0 . The estimate for S 1 is obtained in the same way.

A.2 Computation of the system with new coordinates

Let us look for the system (8) writen in the new coordinates, first in forgetting quadratic and higher orders terms

B 0 x ′ 1 = (λ 2 r + λ 2 i ) 4λ r A 1 + (1 + δ 2 )B 0 B 1 A * + 3λ 2 r -λ 2 i 4λ r (λ 2 r + λ 2 i ) A 3 + A 2 2 + (1 + δ 2 ) 2 A * B 2 0 ε 2 δ 2 ( A * 2 -B 2 0 ) + 2(1 + δ 2 ) A * A 0 -(λ 2 r -λ 2 i ) A 0 = B 0 f 1 + (λ 2 r + λ 2 i ) 4λ r B 0 (x 1 + y 1 ) + A 2 2 + 1 4λ r A 3 , λ i B 0 x ′ 2 = - (λ 2 r + λ 2 i ) 4 A 1 + (1 + δ 2 )B 0 B 1 A * - λ 2 r -3λ 2 i 4(λ 2 r -α) A 3 - (λ 2 r -λ 2 i ) 4λ r A 2 + (1 + δ 2 )B 2 0 ε 2 A * δ 2 ( A * 2 -B 2 0 ) - 1 4λ r (λ 2 r + λ 2 i ) 2 A 0 = λ i B 0 f 2 - (λ 2 r + λ 2 i ) 4 B 0 (x 1 + y 1 ) - (λ 2 r -λ 2 i ) 4λ r A 2 + 1 4 A 3 - 1 4λ r (λ 2 r + λ 2 i ) 2 A 0 ,
with

f 1 = ε 2 δ 2 B 0 (1 + δ 2 )( A * 2 -B 2 0 ) 2 A * , f 2 = - ε 2 δ 2 B 0 (1 + δ 2 )(λ 2 r -λ 2 i )( A * 2 -B 2 * ) 4λ r λ i A * , hence x ′ 1 = f 1 + λ r x 1 + λ i x 2 , (80)
x

′ 2 = f 2 -λ i x 1 + λ r x 2 ,
and in the same way

y ′ 1 = f 1 -λ r y 1 + λ i y 2 , y ′ 2 = -f 2 -λ i y 1 -λ r y 2 , (81)
z ′ 1 = 2ε 2 δ 2 ( A * 2 -B 2 0 ) A * = 2f 1 (1 + δ 2 )B 0 , B ′ * = - (λ 2 r -λ 2 i ) (1 + δ 2 )B 0 A * A 3 + A * B 0 z 1 .
We notice that the following estimates hold

|f 1 | ≤ B 0 ε 2 δ 2 A * ≤ B 0 ε 2 δ α , ( 82
)
|f 2 | ≤ B 0 ε 4 δ 2 A * 2 ≤ B 0 ε 2 δ.

A.2.1 Full system in new coordinates

We intend to derive the full system (1) with coordinates (x 1 , x 2 , y 1 , y 2 , B 0 , z 1 ). Differentiating (17) and (18) we see that we respectively need to add to the previous expressions (80) for x ′ 1 and x ′

2 1 B 0 A * 2 √ 2λ r ′ A 0 + (3λ 2 r -λ 2 i ) 4 √ 2λ r A * ′ A 2 + ε 2 (1 + δ 2 ) 2 B 2 0 2 A * 2 ′ A 3 + (1 + δ 2 )B 0 2 A * ′ B 1 -ε 2 (1 + δ 2 ) 2 B 0 2 A * 2 [3 A * A 0 2 + A 0 3 ] + B 0 ε 2 (1 + δ 2 ) 2 A 0 2 2 A * - B 1 B 0 x 1 . and 1 B 0 - A * 2 √ 2λ i ′ A 0 - (λ 2 r -λ 2 i ) 4λ r λ i ′ A 1 - (λ 2 r -3λ 2 i ) 4 √ 2λ i A * ′ A 2 + ε 2 (1 + δ 2 ) 3 B 3 0 4λ r λ i A * ′ B 1 + 1 B 0 1 4λ r λ i 1 - (λ 2 r -λ 2 i ) 2 A * 2 ′ A 3 - 1 4λ r λ i B 0 1 - (λ 2 r -λ 2 i ) 2 A * 2 [3 A * A 0 2 + A 0 3 ] - ε 4 B 3 0 (1 + δ 2 ) 4 4λ r λ i A * A 0 2 - B 1 B 0 x 2 .
We then arrive to the system (22,23,24,25). We observe that (using (11))

A * ′ = - (1 + δ 2 )B 0 A * B 1 (λ 2 r ) ′ = - (1 + δ 2 )B 0 B 1 √ 2 A * (1 -ε 2 √ 2(1 + δ 2 ) A * ) (λ 2 i ) ′ = - (1 + δ 2 )B 0 B 1 √ 2 A * (1 + ε 2 √ 2(1 + δ 2 ) A * ) A * 2 √ 2λ r ′ = a 1 B 0 B 1 , |a 1 | ≤ c A * 3/2 , (83)
A * 2 √ 2λ i ′ = a 2 B 0 B 1 , |a 2 | ≤ c A * 3/2 , (84)
- (λ 2 r -λ 2 i ) 4λ r λ i ′ = b 2 B 0 B 1 , |b 2 | ≤ cε 2 A * 3 , (85)
(3λ 2 r -λ 2 i ) 4 √ 2λ r A * ′ = c 1 B 0 B 1 , |c 1 | ≤ c A * 5/2 , (86)
- (λ 2 r -3λ 2 i ) 4 √ 2λ i A * ′ = c 2 B 0 B 1 , |c 2 | ≤ c A * 5/2 , (87)
ε 2 (1 + δ 2 ) 2 B 2 0 2 A * 2 ′ = d 1 B 0 B 1 , |d 1 | ≤ c A * 3 , (88)
1 4λ r λ i 1 - (λ 2 r -λ 2 i ) 2 A * 2 ′ = d 2 B 0 B 1 , |d 2 | ≤ c A * 3 , (89)
-ε 2 (1 + δ 2 )B 0 A * A 3 [a 1 A 0 + c 1 A 2 + d 1 A 3 - x 1 B 0 ], b 02 (X, Y ) = - 1 4λ r λ i A * B 0 3 A * 2 -2ε 4 B 4 0 (1 + δ 2 ) 4 A 0 2 + e 2 ε 4 (1 + δ 2 )B 2 0 A * 2 A 3 2 -ε 2 (1 + δ 2 )B 0 A * A 3 [-a 2 A 0 + b 2 B 0 (x 1 + y 1 ) + c 2 A 2 + d 2 A 3 - x 2 B 0 ], C 01 (X, Y ) = α 2 δ 2 A 0 3   -ε 2 (1+δ 2 )B0

Figure 1 :

 1 Figure 1: Orthogonal domain wall

Cε 2 2εδ 0 x 2 η

 02 τ )e ητ |(τ -x)e -η(τ -x) dτ , and for x ≤ 0|u(x)|e -ηx ≤ Cε 2 2εδ x -∞ |f (τ )e -ητ |e -(η+εδ)(x-τ ) dτ + |f (τ )e -ητ |e (η-εδ)(τ -x) dτ . The bound ||u|| L 2 η ≤ c 2 ||f || Lfollows from classical convolution results between functions in L 2 and functions in L 1 ,

Remark 25

 25 We have S 0 (x, s) = e x s λr(τ )dτ cos( x s λ i (τ )dτ ) sin( x s λ i (τ )dτ ) -sin( x s λ i (τ )dτ ) cos( x s λ i (τ )dτ ) .

( 1 + δ 2 = e 1 B 1 , |e 1 ε 2 ( 1 + δ 2 ) 3 B 2 0 1 A. 3 . 1 2 e 2 A * 2 B 0 -b 2 ( 1 + δ 2 ) A * B 2 0 ,M+ A * B 2 0 b 2 (x 1 + y 1 ) + ( 1 + δ 2 ) 2 ε 2 B 3 0 A * b 2 A 3 , 2 + e 1 ε 4 ( 1 + δ 2 )

 12111212321312220212021112032412 4λ r λ i A * ′ = e 2 B 0 B 1 , |e 2 | ≤ c A * 3 ,(91)with c independent of ε and δ ∈ [δ 0 , δ 1 ].A.3 Elimination of z System after scalingAfter the scaling (27) our system (22,23,24,25) takes the formX ′ = L 0 X + B 0 F 0 + B 01 (X, Y ) + z 1 M 01 (X, Y ) +z 1 2 B 0 n 0 + C 01 (X, Y ), Y ′ = L 1 Y + B 0 F 1 + B 11 (X, Y ) + z 1 M 11 (X, Y ) +z 1 2 B 0 n 1 + C 11 (X, Y ),where F 0 , F 1 , n 0 , n 1 are two-dimensional vectors M 01 , M 11 are linear operators in (X, Y ), B 01 , B 11 are quadratic and C 01 , C 11 are cubic in (X, Y ), all functions of B 0 . More precisely we have 01(X, Y ) = εδ m 01 (X, Y ) m 02 (X, Y ) , m 01 (X, Y ) = A * B 0 a 1 A 0 + c 1 A 2 + (d 1 -2e 1 (1 + δ 2 )ε 2 B 0 A * )A 3 -x 1 B 0 , m 02 (X, Y ) = A * B 0 -a 2 A 0 + c 2 A 2 + (d 2 -2e 2 (1 + δ 2 )ε 2 B B 01 (X, Y ) = αδ b 01 (X, Y ) b 02 (X, Y ) , b 01 (X, Y ) = -ε 2 (1 + δ 2 )(2 -δ 2 )B 0 2 A * A 0

n 1 ,α 4 ,

 14 M 11 , B 11 , C 11 are deduced respectively from n 0 , M 01 , B 01 , C 01 in changing (a 1 , c 1 , b 2 , d 2 , e 2 ) into their opposite.A.3.2 System after elimination of z 1Let us replace z 1 by z 10 + Z(X, Y , B 0 ) in the differential system for (X, Y ). The new system becomes (notice that B 0 is in factor of the "constant" terms)X ′ = L 0 X + B 0 F 0 + L 01 (X, Y ) + B 01 (X, Y ), Y ′ = L 1 Y + B 0 F 1 + L 11 (X, Y ) + B 11 (X, Y ),which is (30) withF 0 = F 0 + z 10 2 n 0 , L 01 (X, Y ) = z 10 M 01 (X, Y ), B 01 (X, Y ) = B 01 (X, Y ) + Z(X, Y )M 01 (X, Y ) + C 01 (X, Y ) +2z 10 Z(X, Y )B 0 n 0 + Z(X, Y ) 2 B 0 n 0 .In using estimates (21), (83) to (91), it is straightforward to check that|F 0 | + |F 1 | ≤ cε 2 |M 01 (X, Y )| ≤ c εδ A * (|X| + |Y |),hence |L 01 (X, Y )| + |L 11 (X, Y )| ≤ c ε α 2 (|X| + |Y |).

For higher order terms

  we have|B 01 (X, Y )| ≤ cα(|X| + |Y |) 2 , |2z 10 Z(X, Y )n 0 | ≤ c ε 2 α 2 (|X| + |Y |) 2 , |Z(X, Y )M 01 (X, Y )| ≤ cε(|X| + |Y |) 3 , |Z(X, Y ) 2 n 0 | ≤ cε 2 (|X| + |Y |) 4 , |C 01 (X, Y )| ≤ cα(|X| + |Y |) 3 ,hence, choosing α small enough and for|X| + |Y | ≤ ρ,(92)we obtain|B 01 (X, Y )| + |B 11 (X, Y )| ≤ c(α + ε 2 α 2 )(|X| + |Y |) 2 .

  with a range orthogonal to U * in L 2 . Let us define the projections Q 0 on U ⊥ * and P 0 on U * , which are orthogonal projections in L 2 , then we need to solve in L 2

	η
	section 3.1).
	We show at section 7.3.1 that the kernel of M g is one-dimensional (except for a finite set of values of g), spanned by (A ′ * , B ′ * )

def = U *

at least as e εδx for x → -∞, and at least as e - √ 2εx for x → +∞. The system (1) is now considered with B 0 complex valued, so in (1) B 2 is replaced by |B| 2 .

For being able to prove any persistence result under reversible perturbations of system (1) in R 4 × C 2 we need to study the linearized operator at the above heteroclinic solution. We follow the lines of [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF].

The linearized operator is given by

We show below (see section 7.3.1) that the kernel of M g , is one dimensional, then this implies that the range of M g needs satisfy the orthogonality with only one element. In fact, because of selfadjointness in L 2 , the range of

where there are only 2 possible good dimensions (on each side). This gives a bound = 2 to the dimension of the kernel of M g . Let us show that dimension 2 of ker M g implies non uniqueness of the heteroclinic, which contradicts Theorem 1, hence the only possibility is that the dimension is one. Let us choose arbitrarily g 0 and assume that the kernel of M g0 consists in

where ζ 0 = (A ′ * , B ′ * )| g0 and let us decompose a solution of (1) in the neighborhood of g 0 as U = T a (U

where T a represents the shift x → x + a, where a, a 1 ∈ R, and Y belongs to a subspace transverse to ker M g0 . Let us denote by Q 0 and P 0 = I -Q 0 , projections, respectively on the range of M g0 , and on a complementary subspace (Q 0 may be built in using the eigenvectors ζ * 0 , ζ * 1 of the adjoint operator M * g0 ). Let us denote by F (U, g) = 0