Dynamics of Klein-Gordon on a compact surface near an homoclinic orbit

We consider the Klein-Gordon equation on a Riemannian surface which is globally well-posed in the energy space. This equation has an homoclinic orbit to the origin, and in this paper we study the dynamics close to it. Using a strategy from Groves-Schneider, we show that there are many solutions which stay close to this homocline during all times. We point out that the solutions we construct are not small.

are concerned with the following nonlinear Klein-Gordon (KG) equation where p ≥ 1 is an integer, and (u 0 , u 1 ) ∈ H 1 (M ) × L 2 (M ) are real-valued.
In the sequel, we define the scalar product on L 2 (M ) by f, g = 1 Vol M M f g, where Vol M denotes the volume of M , we assume that e n L 2 = 1 and we set e 0 = 1.
-The manifold M and the integer p satisfy either: • M is any compact manifold without boundary of dimension 1 or 2 and p ≥ 1 • M is any compact manifold without boundary of dimension 3 and p = 1.
Moreover, up to a rescaling, we can assume that Vol M = 1.
The stationary solutions of (KG) (solutions which only depend on the space variable) are exactly the constants u = 0, u = m 1/p and u = −m 1/p (see Lemma 2.1). The origin is an equilibrium with an unstable direction. In fact, the eigenvalues of −∆ − m 2 are the (λ 2 k − m 2 ) k∈N . Since 0 < m < λ 1 , the case k = 0 only, gives the hyperbolic directions, corresponding to the solution exp(mt) for t > 0 (resp. exp(−mt) for t < 0). It turns out that (KG) admits a homoclinic orbit to the origin. Indeed, the following x−independent function is a solution to (KG) α(t) = m 1/p (p + 1) 1/(2p) cosh(pmt) 1/p , and in the sequel we will refer to h(t) = (α(t), β(t)), withβ(t) = α(t), as the time homoclinic solution (to the origin) of (KG), see Section 1.5. In this work we aim to study the dynamics of (KG) near this particular trajectory, and we will show the existence of solutions which remain close for all times to this exact temporal solution.
In the case M = S 1 we can precise the dynamics around the equilibrium u = m 1/p (the study near u = −m 1/p is similar since the non-linearity is odd). We linearize the equation near this point (u = m 1/p + v) and we are lead to study the spectrum of (−∂ 2 x + 2pm 2 )v. This operator on S 1 is self adjoint and has pure point spectrum j 2 + 2pm, j ∈ N. Using the Birkhoff normal form theory, Bambusi [1] (see also [3]) has shown that, for a generic choice of m (in order to avoid resonances between the frequencies j 2 + 2pm, j ∈ N), the solutions of (KG) with initial datum of the form m 1/p + v 0 with v 0 small enough, remain close to the equilibrium point m 1/p for very long time. See also the work [8] of J.-M. Delort for quasi-linear equations.
Moreover, using KAM theory, C. E. Wayne [28] and J. Pöschel [26] have proved that, for a generic choice of m, there exist many quasi-periodic solutions near w = m 1/p . See also the recent work of Berti-Biasco-Procesi [4] for derivative wave equations. Observe that quasi-periodic solutions can be constructed even if the equilibrium has a finite number of hyperbolic directions (see [5]).
In higher dimension and for a general manifold, few is known. In the case M is a Zoll manifold, Bambusi-Delort-Grébert-Sjeftel [2] have developed a Birkhoff normal form theory for (KG) near an elliptic equilibrium. Up to now, there is no KAM-type result for (KG) in dimension greater than two.
In this work, we describe some possible behaviours of (KG) near the homoclinic orbit. We state the existence of solutions that travel from a neighbourhood of the origin to turn around the equilibrium m 1/p , close to the homoclinic connection to 0 (see Theorems 1.1 and 1.2). We stress out that these results do not require non resonance conditions. The existence of homoclinic or heteroclinic connections to periodic or quasi-periodic solutions is a question of interest: for example, in the case of parabolic PDE's on a one dimensional bounded domain, homoclinic orbits to equilibrium points can exist but there cannot exist any homoclinic connections to periodic orbits (see [19]). In some other cases some homoclinic connections to small periodic orbits exist while homoclinic orbits to 0 do not exist or remain an open question (see for instance [20,21,18]). More generally, such homoclinic connections to the center manifold often appear when a homoclinic orbit does not persist after a perturbation of the system (see for instance [27]). We refer also to [6], where the existence of homoclinic solutions to a beam equation is studied.
In this paper, we get the existence of a large family of homoclinic connections to the center manifold (more precisely heteroclinic connections to some solutions lying in the center manifold). The proof is based on a perturbative method which is classical (see for instance [17,20]) for finite dimensional reversible systems (systems which anticommute with a symmetry S): the key idea is that if a solution hits the reversibility plane {u | u = Su}, then the latter solution is reversible. This method requires more computations in the infinite dimensional cases. It was already performed in a situation close to ours by M. Groves and G. Schneider [12,13,14]: but in their case, they work in the neighbourhood of a bifurcation and get small homoclinic solutions, while in our case the size of the solutions is of order 1 and this requires some additional work on the linearised system, which is one of our main contributions in this paper.
Finally we also mention the recent book of Nakanishi & Schlag [24] on invariant manifolds in the context of dispersive Hamiltonian PDEs.
where f is a non linear function. Let us show that if there exists a nonzero equilibrium, the equation near the smallest equilibrium is of the form (KG) (but with a general non linearity). The equilibrium of (1.1) are the real constants w 0 satisfying f (w 0 ) − w 0 = 0. Observe that if one performs the change of coordinates u = w − w 0 , then there exists a non linear function g such that u satisfies Thus, if f is C 1 and if there exists an equilibrium w 0 = 0, then by the intermediate value theorem, we get that the smallest equilibrium w 0 satisfies f (w 0 ) − 1 ≥ 0. This means that near the smallest non zero equilibrium of (1.1), the equation is of the form For instance in this paper we consider a nonlinearity g(u) = u 2p+1 , u = m 1/p is a (stable) equilibrium and the change of variable u = m 1/p + w leads to 1.3. Another motivation of the problem. -Here we are inspired from the works [12,13,14] of M. Groves and G. Schneider. Consider the non linear wave equation on the real line One can be interested in the construction of "modulated pulse solutions " for (1.2) which are solutions of the form where u(s, y) is a 2π−periodic function in y, and where β ∈ (0, 1) and c β > 0. In the particular case c β = (1 − β 2 ) 1/2 , (1.3) is the Lorentz transform which preserves (1.2) and in general we get Now let m ∈ (0, 1) and β ∈ (0, 1), then we can choose c β > 0 so that c 2 β 1−β 2 = m 2 ∈ (0, 1), which is (KG) up to the change of unknown u −→ u 1/p w. As a consequence, for each m ∈ (0, 1), any solution to (1.4) provides a one-parameter family of solutions to (1.2).
1.4. Hamiltonian structure of (KG). -Denote by v = ∂ t u and introduce Then, the system (KG) is equivalent to We write in such a way that to the continuous phase space X := H 1 × L 2 corresponds the discrete one h 1 × 2 .
We endow this space with the natural norm We define the two dimensional subspace X h ⊂ X spanned by the stable/unstable modes (hyperbolic modes) of X (1.7) X h = X ∈ X | ∀ n ≥ 1, a n = b n = 0 , and we observe that X h is an invariant subspace for (1.6). We also define the infinite dimensional subspace X c ⊂ X spanned by the center modes (elliptic modes) of X We clearly have X = X h ⊕ X c . Denote by P : X −→ X h the projection onto X h and define Q = I − P : X −→ X c the projection onto X c . In the sequel, for X ∈ X , we use the notation X h = P X and X c = QX.
In the coordinates (a n , b n ) n≥0 , the Hamiltonian in (1.5) reads a k e k (x) 2p+1 e n (x)dx, n ≥ 0.
1.5. The homoclinic orbit. -The space-stationary solutions of (KG) exactly correspond to the solutions of (1.10) satisfying a n = b n = 0 for n ≥ 1. In this case, the equation on (a 0 , b 0 ) reads and this system possesses a homoclinic solution to 0 which we will denote in the sequel by h : t → (α(t), β(t)), and we will denote by K 0 the curve which is described (see Figure 1). Indeed we can explicitly compute cosh(pmt) 1/p+1 , and we have the bounds For η > 0 denote by K η the trajectory of (1.11) given by the initial conditions a 0 (0) = η, b 0 (0) = 0 (see Figure 1). In our forthcoming paper [10], we study the long time stability of the trajectory K η for 0 < η 1, by the flow of the system (1.10).
b 0  Our main result describes possible dynamics near the homoclinic orbit: We show that there are many solutions which stay close to h. The proof of this result is inspired from the work [12] of M. Groves and G. Schneider. In our case, the novelty is that we have not to assume m to be small, i.e. we are not dealing with small solutions of (KG).
For any function f ∈ L 2 (M ), denote by f c = f − M f . Then we can state Theorem 1.1. -Let M , p and m satisfy Assumptions 1 and 2. There exist C > 0 and ε 0 > 0 so that the following holds: and so that for all t ≥ 0 Furthermore, as t −→ +∞, u(t, ·) tends to a solution of the local center manifold W c (see Definition 4.6).
Notice that Theorem 1.1 is not a stability result since we only fix the elliptic part of the initial datum, and the statement of the theorem is that there exists a way to define the hyperbolic part such that the solution satsifies (1.15). For example if f c = g c = 0, the corresponding u is the homoclinic orbit h.
The equation (KG) is reversible. More precisely, the equation (KG) is invariant under the transformation u(t, x) −→ u(−t, x). This corresponds to the symmetry S defined by S(a n ) = a n S(b n ) = −b n , for n ≥ 0.
This is a symmetry of reversibility: Namely, denote by V H the vector field (1.10), then Using this symmetry we are able to precise the result of Theorem 1.1 in the case g = 0.
Theorem 1.2. -Let M , p and m satisfy Assumptions 1 and 2. There exist C > 0 and ε 0 > 0 so that the following holds: and so that Furthermore, as t −→ +∞, u(t, ·) tends to a solution of the local center manifold W c .
Since u is reversible, observe that if u tends to w when t −→ +∞, then u tends to Sw when t −→ −∞. Remark 1.3. -Theorem 1.2 proves the existence of heteroclinic orbits from some orbit on the center manifold W c to another one. It would be interesting to precise the dynamics on the center manifold. Actually we can expect the existence of invariant tori on W c which are perturbations of the tori obtained for the linearized equation around the homoclinic orbit (see section 3). Nevertheless, to prove this, we would need a KAM theory for infinite dimensional tori which is an open problem up to now (see however [7]). Notice that even if such theory would exist, it would prove that most of the tori, but not all of them, of the linearized system are preserved. Thus it would be impossible to conclude that our heteroclinic orbits tends to KAM tori of W c when t → ±∞ as we can expect.
1.7. Plan of the paper. -In Section 2 we prove Theorem 2.2. Section 3 is devoted to the study of the linearisation of (KG) around the homoclinic orbit (Theorem 3.2). In Section 4 we construct the center manifold for the Klein-Gordon equation and prove Theorem 1.1. In the appendix, we recall a result which is useful in the proof of Theorem 3.2.
Notations. -In this paper c, C > 0 denote constants the value of which may change from line to line. These constants will always be universal, or depend on the fixed quantities m and p. We denote by N the set of the non negative integers, and N * = N\{0}. We set X = H 1 (M ) × L 2 (M ).

Preliminaries:
Proof of the first results on (KG) 2.1. Stationary solutions of (KG). -We look for the time-independent solutions to (KG), which correspond to equilibrium points for the system (1.10).
We apply ∇ to the equation and get −∆∇g − m 2 ∇g + (2p + 1)g 2p ∇g = 0. We multiply this equation with ∇g and integrate (by parts) on M g k e k . Since (λ k ) k≥0 is non decreasing and 0 < m < λ 1 we have This inequality together with (2.1) implies that ∇g = 0, and we conclude.

2.2.
Global well-posedness of (KG). -The aim of this section is to prove the following result. b k e k we set We also define a k e k the spectral projection away the mode 0. Observe that under Assumption 1 there exist In the sequel we will need • When M has dimension 1 or 2, for all 2 ≤ q < +∞ Proof. -By Sobolev, in each of the previous cases, there exists C q > 0 so that for all We then can prove the following a priori estimate be a solution of (KG) and denote by H 0 the constant value of H(u(t, ·)) along this trajectory. For all t for which the solution is defined, we have Proof. -We assume that M has dimension 1 or 2. By the conservation of the energy for (KG), we have We apply the Young inequality with c 1 = u 2 , c 2 = 1, q = p + 1 and ε = m −2/(p+1) and deduce (2.5).
Recall the notations (2.2) and (2.3). Then we have By (2.5), b 0 and J are bounded and moreover by Lemma 2.3 and (2.5) we deduce that M |U | q dx ≤ CJ q/2 for all q ≥ 2. Thus (2.6) gives H 0 as a polynomial in a 0 with bounded coefficients and thus a 0 has to be bounded by a constant depending only on H 0 . The argument is similar when M has dimension 3 and p = 1.
Remark 2.5. -It is easy to check that the points u = ±m 1/p correspond to the minimum of H, and that for this choice H 0 = − p 2(p + 1) m 2+2/p . In particular, the r.h.s of (2.5) is nonnegative.
Proof of Theorem 2.2. -First we show that (KG) is locally well-posed. Let (u 0 , u 1 ) ∈ X and for T > 0 define the space For f : R −→ R, a continuous and bounded function, we can define a bounded operator With this definition, denote by K(t) the free wave propagator Then we show that the mapping Φ defined by is a contraction of E T for T small enough. Namely, by the Sobolev embeddings and the fact that In order to prove the global well-posedness, we iterate the previous argument. We then obtain a sequence of times T n with T n+1 = c 0 min m −2 , (u(T n ), ∂ t u(T n ) −2p X such that the solution is defined on (− ∞ n=0 T n , ∞ n=0 T n ). By Lemma 2.4, for any t > 0, (u(t), ∂ t u(t)) −2p X ≥ C(H 0 ), hence n≥0 T n = +∞ which gives the result. Remark 2.6. -The result of Theorem 2.2 indeed holds for any m ∈ R.

The linearized equation around the homoclinic orbit
In this section, we study the system , and for n ≥ 1 )a n − (2p + 1)a 2p 0 a n . Indeed, if one considers that a 0 is given by (3.1), the system (3.2) is the linearisation of (KG) around the solution u = a 0 . In particular, denoting V (t) = (2p + 1)a 2p 0 (t) and w = k≥1 a k e k , (3.2) is equivalent to the following linear wave equation with time-dependent potential Notice that the system (3.1), (3.2) is not Hamiltonian for the canonical structure, but it is reversible. On the other hand, for a 0 given, the system (3.2) is Hamiltonian for the canonical structure.
We now study the linear evolution of (3.2), which will be useful in the proof of Theorem 1.1. Introduce the notations In other words, every solution to ∂ t Z c = L t c Z c is bounded.
Proposition 3.1 is a direct consequence of the next result. Define the tori by Then the corresponding solution of (3.1), (3.2) is homoclinic to T c for some c ∈ 2 (N, R).
This result can also be interpreted as a linear scattering result for the system (3.2). The infinite dimensional system (3.1), (3.2) has homoclinic orbits to invariant tori of arbitrary large dimension (finite or infinite). These tori correspond to the case a 0 = b 0 = 0 which is stable by (3.1). In that case (3.2) becomes the standard harmonic oscillator in infinite dimension with frequency vector ω(m) = (ω n ) n≥1 and ω n = λ 2 n − m 2 .
Proof. -Take the homoclinic (α, β) in the plane (a 0 , b 0 ), and observe that R α 2p (t)dt < ∞. Now, for n ≥ 1 we introduce the norms on R 2 , |x, y| n = (λ 2 n − m 2 )x 2 + y 2 1/2 and the norm on h 1 × 2 (a n , b n ) n≥1 = n≥1 |a n , b n | 2 n 1/2 . Then for all n ≥ 1, we deduce from Lemma A.1 that the trajectory (a n , b n ) satisfies |a n , b n | n ≤ C|a n (0), b n (0)| n . Next, we denote by S the flow of the linear part of (3.2), i.e. with a 0 = 0. Then the solution of (3.2) reads By construction S preserves the norm . n and thus where we used the boundedness of a n and the integrability of α 2p . Denote by Finally we remark that the trajectory t → S(t) is non resonant (for example when M = S 1 , this happens for a generic choice of the mass m, see [3]), S(t) a + n , b + n t≥0 densely covers the torus T c .
Remark 3.4. -Notice that in the proof of Theorem 3.2 we crucially use that a 0 = α, i.e. that we linearize around the homoclinic orbit. A natural question is to ask whether this result still holds if (a 0 , b 0 ) is any periodic solution of (3.1). In that case R a 2p 0 (t)dt = +∞ and Lemma A.1 does not apply. In particular it is not clear at all that the trajectories remain bounded. This seems to be a difficult and interesting problem related to the reducibility of (3.3) when V is not a priori small (the case V small and quasi-periodic in time can be solved by a KAM approach, see [9,11] for the last results in the Schrödinger case).

Denote by L
the restriction of L t to X h . We can identify the space X h = R 2 and we denote by , the Euclidian scalar product. Then Then in view of (1.12) we deduce that this solution corresponds to σ and we obtain the first bound in (3.7). With a suitable choice of µ we get (3.6).
To find ρ, we write ρ = γ γ , and look for γ of the form γ =αz, when t ≥ 1. Then z satisfies the equationαz + 2αż = 0 and therefore It is then straightforward to check that γ =αz can be extended to a C ∞ function on R. Then the bound on σ gives |z(t)| ≤ Ce 2mt and |ż(t)| ≤ Ce 2mt for all t ≥ 0, which in turn implies that |γ(t)| ≤ Ce mt and |γ(t)| ≤ Ce mt for all t ≥ 1. Hence the second bound in (3.7). We can moreover choose the initial value ρ of the form claimed in the statement of the lemma.
Then we can explicitly compute where βγ − γβ is a non vanishing constant, since it is the Wronskian of the equation The bound (3.8) then follows from (3.7).

Homoclinic orbits to the center manifold
In this section we prove Theorem 1.1. As we mentioned in the introduction, this part closely follows the argument of Groves-Schneider [14]. Nevertheless in contrast to their work, we are not dealing with m small and thus with small solutions. Even if some results already appear in [14], we reproduce here all the proofs for the convenience of the reader. First we recall the general strategy, illustrated by 1. To begin with, we consider a spatially truncated system for which the non-linearity sees only the elliptic (or central) modes of small amplitude. We construct global solutions to this system that are close to the homoclinic orbit, namely the hyperbolic part is close to h = (α, β) for all times t ≥ 0, and the elliptic part is small only for t ≤ ε −1 where ε is a small parameter. (Subsection 4.2) In the next steps we will prove that the elliptic part of these solutions actually stay small for all t, i.e. that the set of all these solutions is the global center-stable manifold W cs of the truncated system, and thus a local center-stable manifold for (KG) at time t = 0.
2. We construct a global center manifold W c for the truncated system by constructing the solutions that are close to the origin for all times. (Subsection 4.3) 3. The crucial point consists in linking the two previous steps: we prove that the solutions of the truncated system constructed in step 1 actually tend to some solutions contained in the local center manifold as t → +∞ and thus remain small for all times t ≥ 0. As the central part remains unconditionally small, we conclude that these solutions are actually solutions of the original system (without truncation). This gives us a global center stable manifoldW cs which is parametrized by the initial value V s of the stable part that we add to h and by V c the initial value of the elliptic part, both being small. (Subsection 4.4) 1 , b 1 , ..., a n , b n , ...) (schematic) here all is δ-close to 0 ≤ C ε e −rt ≤ Cε 2 Figure 3. Schematic overview of the strategy and the notations. We will see later that we take δ = ε 3/2 with ε 1.

Preliminary results. -
To begin with, we write an equivalent formulation of (KG). Denote by The equation (KG) is equivalent to By the result of Theorem 2.2, the equation (4.2) admits, for a given initial condition X(0) ∈ X a unique global solution X ∈ X so that sup t∈R X(t) ≤ C( X(0) X ). We fix C 0 > 0 large enough, so that all the initial conditions X(0) we consider in the sequel satisfy X(0) ≤ C 0 . Therefore all the solutions X we are going to consider satisfy We denote D c the projection on X c of such initial conditions: where δ is a parameter much smaller than 1. For any function G : X −→ R, denote by Lemma 4.1.
-Consider the function F given by (4.1) and let X, X ∈ X . Assume moreover that |X h |, |X h | ≤ δ < 1. Then there exists a constant C > 0 such that Proof.
-Writing X = u v and u = a + U with a ∈ X h and U ∈ X c we have by the fact that |a| ≤ δ and the Sobolev embedding To get (4.4), just remark that for |u|, |u | ≤ Cδ and then apply this inequality to We write the linearisation of (4.2) around the homoclinic orbit h = α β . Set X = Z + h and define (4.5) As a result, Z satisfies the equation Similarly as in Lemma 4.1 we have . We will be interested in solutions to (4.9) for which the center part is small, that's why we introduce the following truncated system (4.10) . We now look for solutions to (4.10) with small hyperbolic part. Fix 0 < r < m and denote by We introduce a new small parameter ε which will quantify the size of Z, namely Z ≤ Cε 2 . We will choose ε = δ 2/3 in such way the truncation θ can be removed.
Then there exists a unique solution Z ≡ Z Vc,Vs to (4.10) in and so that Moreover, this solution satisfies This result means, that for any fixed V c ≤ ε 2 , |V s | ≤ ε 2 , there exists a unique choice of V u = Z h (0), ρ (0) ∈ R so that the corresponding solution has a small hyperbolic part for all times and a central part small for relatively long time (comparing with the time necessary in order that the homoclinic orbit h reaches a neighbourhood of radius ε 2 of the origin). Let us call local center-stable manifold the set of all the solutions verifying this property: We define W cs by W cs = Vc ≤ε 2 ,|Vs|≤ε 2 X Vc,Vs (0) . Theorem 4.10 below will ensure that for ε sufficiently small, this set is actually the standard centerstable manifold of the equilibrium 0 for the initial equation (4.2).
Proof. -We define the map F : and show that for δ > 0 small enough, F is a contraction in B + δ . • To begin with, we show that F : Let |Z h | ≤ δ, then by (4.7), for all t ≥ 0 On the other hand by Proposition 3.1 and (4.7) which in turn implies that F(Z) c r < ∞.
• We now show that F is a contraction in B + δ . By Lemma 4.2 and therefore and then Thus, (4.16) and (4.17) show that F is a contraction whenever 0 < δ < 1 is small enough, and we can deduce that there exists a unique fixed point Z ∈ B + δ . By definition of F, it is clear that this solution satisfies (4.11).

4.3.
The local center manifold. -In this subsection we want to construct solutions that remain close to the origin for all time. Actually this will be achieved by constructing a local center manifold. Again we consider the approximation of the initial problem (4.2), in which we truncate the non-linearity The next result shows that there exist solutions to (4.18) with small hyperbolic components. For 0 < r < m we define and so that Moreover, this solution satisfies The set of these solutions is the center manifold of 0 for equation ( It is clear that (σ 0 e −mt , ρ 0 e mt ) form a basis of the solutions of ∂ t X h = Λ h X h . Denote by K the propagator of the equation ∂ t X c = Λ c X c . Then for X ∈ E r we define which in turn implies that G(X) c r < ∞. Hence, with (4.22) we get G(X) ∈ B δ . • In a second time, by (4.4) we can write and therefore (4.23) and then Thus, (4.23) and (4.24) show that G is a contraction in B δ whenever 0 < δ < 1 is small enough. The bound (4.20) is given by (4.22).
We are now able to give a parametrisation of the local center manifold given in Definition 4.6. Let and, in view of (4.21), Ψ is quadratic at the origin, . This is a particular case of a result of Mielke [23].
As a consequence we can prove that the origin is Lyapunov stable within the center manifold: Proof. -It is here convenient to work in the coordinates (a n , b n ) n≥1 . Recall that from (2.6) the Hamiltonian of (4.2) reads and observe that J/C ≤ X c 2 ≤ CJ. We first show that |H 0 | ≤ Cε 4 . By assumption, J(t ) ≤ Cε 4 , while by (4.20) we obtain (4.27) |a 0 (t)|, |b 0 (t)| ≤ Cδ 6 = Cε 9 ≤ Cε 2 for all t ∈ R.
Next, using Lemma 2.3 we obtain, for t = t which proves the claim. Next, with (4.27) we can write for all t ∈ R which implies (4.26).  By (1.13), with this choice we have |h(t)| ≤ Cε 2 for all t ≥ t ε /2. Next, we define Θ ∈ C ∞ (R, R) so that and with |Θ (t)| ≤ C/t ε for all t ∈ R. We set Proposition 4.8.
-Consider the function Y defined in (4.29). Then there exists X solution of (4.18) within W c so that for all t ∈ R Proof. -The function Y satisfies the equation Consider now a solution X of the equation ∂ t X = Λ X + F ( X) and set ∆ = Y − X. Then ∆ satisfies (4.32) The proof of the proposition consists in finding a solution of (4.32) so that (4.30) holds and so that where again we choose δ 2 = ε 3 . We claim that we can define a contraction J : It is clear that the fixed point will satisfy (4.32). In a second time we will check that (4.30) holds.
• We now define X = Y − ∆ and it remains to show that X ∈ W c . By definition, it is sufficient to prove that | X h (t)| ≤ δ for all t. Write Y = Θ(Z + h). By (4.12) and the choice of t ε we get that |Y h (t)| ≤ Cε 2 for all t ∈ R and then which implies the result, for ε > 0 small enough. Proposition 4.9. -Let |V s |, V c ≤ ε 2 and ε small enough. The solution Z of (4.10) with Z c (0) = V c and Z s (0) = V s constructed in Proposition 4.3 satisfies Z c (t) ≤ δ for all t ≥ 0. In particular X = Z + h is a solution of (4.2) for t ≥ 0.
This bound together with (4.13) conclude the proof.
Gathering the results of the previous propositions, we get the following Theorem, which ensures that W cs is the standard center-stable manifold of 0 for equation (4.2). (where Z Vc,Vs is given by Proposition 4.3) defined for t ≥ 0 are solutions of (4.2). Moreover, Z Vc,Vs belongs to E + r and satisfy Z c (t) ≤ Cε 2 ≤ δ, Z h (t) ≤ Cε 2 , for all t ∈ R + ; and there exists X ∈ W c such that This completes the proof of Theorem 1.1.  And since ρ(0) = 1 0 (see Lemma 3.5), we have Sρ(0) = ρ(0). So, if V c = SV c , then Z Vc,0 (0) = SZ Vc,0 (0).
As a consequence of the previous lemma, X Vc,0 is a reversible solution. From Theorem 4.10, we then get that on the one hand, Z Vc,0 = X Vc,0 − h belongs to E r and satisfy (Z Vc,0 ) c (t) ≤ δ, (Z Vc,0 ) h (t) ≤ Cε 2 , for all t ∈ R; and on the other hand, there exists X ∈ W c such that (X Vc,0 − X)(t) ≤ Cε −2 e −rt for all t ≥ 0, and (X Vc,0 − SX)(t) ≤ Cε −2 e rt for all t ≤ 0.
This means that X Vc,0 is an heteroclinic connection between the two solutions X and SX of the center manifold W c . Then Theorem 1.2 follows since the condition V c = SV c and V s = 0 is equivalent to (1.16).

Appendix A
In this appendix we recall a result concerning the long time behaviour of the solution of the ordinary differential equationẍ = (−α 2 + q(t))x, t ∈ R where α is a real constant and t → q(t) is a continuous function. Of course the solution of such linear equation are globally defined, but we would like to know whether the solution are bounded or not. It turns out that even if q ∞ < α 2 and q(t) −→ 0 when t −→ ±∞, the solutions may grow indefinitely (1) . The good condition concerns the integrability of |q|: are bounded on R. More precisely, there exists C > 0 which only depends on q so that α 2 |x(t)| 2 + |y(t)| 2 ≤ C α 2 |x(0)| 2 + |y(0)| 2 for all t ∈ R.
The proof is classical (see for instance [16, page 212]) but does not precise the bound on the solution in term of the initial datum. The following argument is more explicit.