Abed Bounemoura
email: abedbou@gmail.com

Generic perturbations of linear integrable Hamiltonian systems

In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem which does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is non-resonant is more subtle. Our second result shows that for a generic perturbation, the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of ε -1 ) interval of time, and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation, one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).

Introduction

Let n ≥ 2 be an integer, T n := R n /Z n and B ⊆ R n an open bounded convex subset. Consider a Hamiltonian function H : T n × B → R of the form H(θ, I) = h(I) + εf (θ, I), (θ, I) = (θ 1 , . . . , θ n , I 1 , . . . , I n ) ∈ T n × B, ε ≥ 0 and its associated Hamiltonian system θi (t) = ∂ I i H(θ(t), I(t)) = ∂ I i h(I(t)) + ε∂ I i f (θ(t), I(t)), İi (t) = -∂ θ i H(θ(t), I(t)) = -ε∂ θ i f (θ(t), I(t))

1 ≤ i ≤ n.

For ε = 0, all solutions are stable: the action variables are constant, I(t) = I 0 for all t ∈ R, hence their level sets {I(t) = I 0 } × T n define invariant tori on which the dynamics is quasiperiodic of frequency ∇h(I 0 ). A central question is whether solutions remain stable or become unstable when ε > 0, and fundamental results on this question are due to Kolmogorov, Arnold and Nekhoroshev.

In [START_REF] Kolmogorov | On the preservation of conditionally periodic motions for a small change in Hamilton's function[END_REF], Kolmogorov proved that for a non-degenerate h and for all f , the system defined by H still has many invariant tori, provided it is real-analytic and ε is small enough. What he showed is that among the set of unperturbed invariant tori, there is a subset of positive Lebesgue measure (the complement of which has a measure going to zero when ε goes to zero) who survives any sufficiently small perturbation, the tori being only slighted deformed. This result has been re-proved and extended in many ways, with important contributions of Arnold and Moser, and this is nowadays called the KAM theorem: the literature on the subject is enormous, we simply refer to [START_REF]Smooth ergodic theory and its applications[END_REF] and [START_REF] Sevryuk | The classical KAM theory at the dawn of the Twenty-First Century[END_REF] for recent surveys. A consequence of Kolmogorov's result is that for most solutions, the action variables are almost constant for all times, that is

|I(t) -I 0 | √ ε, t ∈ R.
This last inequality is in fact true for all solutions if n = 2, under a different non-degeneracy condition as proved by Arnold ( [START_REF] Arnol'd | Proof of a theorem of A.N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations[END_REF], [START_REF]Small denominators and problems of stability of motion in classical and celestial mechanics[END_REF]) with a different version of the KAM theorem.

But this stability of the action variables over an infinite time is not true for all solutions if n ≥ 3. In [START_REF]Instability of dynamical systems with several degrees of freedom[END_REF], Arnold constructed an example of h (belonging to the class of nondegenerate functions to which the KAM theorem applies) and an example of f such that the system associated to H has a solution for which

|I(t) -I 0 | ≥ 1, t ≥ τ (ε) > 0,
for all ε > 0 sufficiently small. He then conjectured that this phenomenon holds true for a generic h and a generic f : this is usually called the Arnold diffusion conjecture, and it is one of the main open problem.

Then, in the seventies, Nekhoroshev proved ( [START_REF]An exponential estimate of the time of stability of nearly integrable Hamiltonian systems[END_REF], [START_REF]An exponential estimate of the time of stability of nearly integrable Hamiltonian systems II[END_REF]) that for any n ≥ 2, for a generic h and for all f , along all solutions we have |I(t) -I(0)| ε b , |t| exp ε -a , for some positive exponents a and b, provided that ε is small enough and that the system is real-analytic. So solutions which do not lie on invariant tori are stable not for all time, but during an interval of time which is exponentially long with respect to some power of the inverse of ε. The consequence on Arnold diffusion is that the time of diffusion τ (ε) is exponentially large. The class of generic functions in the Nekhoroshev theorem is different from the class of non-degenerate functions in the KAM theorem of Kolmogorov and Arnold, but their intersection contains, respectively, the class of strictly convex and strictly quasiconvex functions.

It is our purpose in this paper to investigate the fate of these theorems and conjecture in the special case where h is linear, that is

h(I) = α • I := α 1 I 1 + • • • α n I n , α ∈ R n \ {0}.
(1)

But first let us give two reasons why it could be interesting to study perturbations of such linear integrable systems. The first one is obvious: locally around any point the main approximation of a non-linear integrable Hamiltonian is given by its linear part, hence perturbations of linear integrable systems describe the local dynamics of perturbations of non-linear integrable systems. The second reason is that it encompasses several interesting problems. For instance, one encounters such systems (or at least very similar systems) when studying the dynamics in the vicinity of a linearly stable (elliptic) equilibrium point, or a Lagrangian quasi-periodic invariant torus or even more generally a linearly stable isotropic reducible quasi-periodic invariant torus. These problems are of real physical interest (we refer to the work of Chirikov, see [START_REF] Chirikov | How fast is the Arnold diffusion[END_REF] for instance, where these systems are called "weakly" non-linear as any non-linearity has to come from the perturbation). So let us fix h linear, that is h is as in (1) with some vector α ∈ R n , and let us come to the question of whether the KAM theorem, the Nekhoroshev theorem and Arnold diffusion holds true. The first point to observe is that h, being very degenerate, does not belong to the class of integrable Hamiltonians to which the KAM or Nekhoroshev theorem applies, and Arnold's construction of unstable solutions does not directly apply in this setting. But this does not necessarily means than those results or constructions are not applicable. There are two possibilities for the vector α ∈ R n : either it is resonant, that is there exists k ∈ Z n \ {0} such that the Euclidean scalar product k • α = 0, or it is non-resonant, and the results will depend drastically on whether α is resonant or not.

So first assume α is resonant. Examples (that is, for specific perturbations) from [Mos60] and [START_REF] Sevryuk | The classical KAM theory at the dawn of the Twenty-First Century[END_REF] show, respectively, that the Nekhoroshev and the KAM theorem cannot be true for all perturbations. Yet it may be possible that they hold true for a generic perturbation. We will show here that this cannot happen. In fact, for a generic perturbation, we will show that there is a fixed open ball B * in action space such that for ε sufficiently small and for all initial action I 0 ∈ B * , one has

|I(t) -I 0 | ∼ 1, t ∼ ε -1 .
This will be the content of Theorem 1, to which we refer for a precise statement. Hence, in this case, one has diffusion for a generic perturbation and the conjecture of Arnold is true. The fact that the diffusion time is of order ε -1 implies that the Nekhoroshev theorem cannot hold true for a generic perturbation. Moreover, as the result is true for an open ball of initial action whose radius is fixed as ε goes to zero, the KAM theorem is also violated for a generic perturbation (we refer to the comments after Theorem 1 for a more detailed explanation).

Next assume that α is non-resonant. The example of [START_REF] Sevryuk | The classical KAM theory at the dawn of the Twenty-First Century[END_REF] applies as well in this case, and the KAM theorem cannot be true for any perturbation. However, we will prove in Theorem 2 that the KAM theorem holds true for a generic perturbation. As far as the Nekhoroshev theorem is concerned, it is already known that for all perturbations and for all solutions, one has the stability estimates

|I(t) -I(0)| ∆ α (ε -1 ) -1 , |t| exp ∆ α (ε -1 ) ,
where ∆ α is an increasing function encoding some arithmetic properties of the vector α. In the special case where α satisfies a Diophantine condition (that is, |k • α| |k| -τ for some τ ≥ n -1), then ∆ α (x) x 1 1+τ and one recovers exponential estimates as in the Nekhoroshev theorem. This was proved in [START_REF] Bounemoura | Optimal stability and instability for near-linear Hamiltonians[END_REF] in the general case; the Diophantine case was known before, see for instance [START_REF] Fassò | Lie series method for vector fields and Hamiltonian perturbation theory[END_REF]. In [START_REF] Bounemoura | Optimal stability and instability for near-linear Hamiltonians[END_REF] it is also proved that there exists a sequence of ε j -perturbation, j ∈ N, with ε j going to zero as j goes to infinity, such that

|I(t) -I(0)| ∼ ∆ α (ε -1 j ) -1 , |t| ∼ exp ∆ α (ε -1 j ) ,
and therefore the above stability estimates cannot be improved for an arbitrary perturbation. Yet we will prove that for a generic perturbation, one has much better stability estimates: the time of stability is a double exponential, that is

|I(t) -I(0)| ∆ α (ε -1 ) -1 , |t| exp exp ∆ α (ε -1
) .

This will be proved in Theorem 3. The only remaining question is therefore the following.

Question 1. Assume α is non-resonant. Can one prove the existence of unstable solutions (in the sense of Arnold) for a generic perturbation of h(I) = α • I?

Of course, one knows how to construct specific perturbation for which the system has unstable solution: this is done in [START_REF] Bounemoura | Optimal stability and instability for near-linear Hamiltonians[END_REF], but also in [START_REF] Douady | Stabilité ou instabilité des points fixes elliptiques[END_REF] where one can find "less degenerate" perturbations which however cannot be made analytic. For a generic perturbation, according to Theorem 3 the time of instability has to be doubly exponentially large, which is certainly a sign of the difficulty of the problem. As a matter of fact, this question and the Arnold diffusion conjecture are both special case of the following question.

Question 2. Consider an arbitrary integrable system defined by a function h, with n ≥ 3 degrees of freedom. Can one prove the existence of unstable solutions for a generic perturbation of h? Arnold's original conjecture concerns generic integrable Hamiltonians, which certainly do not encompass linear integrable Hamiltonians. Yet the question makes sense for an arbitrary integrable Hamiltonian, and to the best of our knowledge, this is widely open: it is not known if one can construct an example of integrable Hamiltonian (with n ≥ 3) for which all solutions remain forever stable after a generic perturbation (if it possible, this of course would give a negative answer to the question above).

To conclude, let us just add that we believe that Question 1 is actually more difficult to the conjecture of Arnold: as it will be clear from the arguments below, when dealing with a generic perturbation of a linear (non-resonant) integrable system, one is naturally led to look at a generic perturbation of a non-linear but singular (that is, ε-dependent) integrable system, and this singular case turns out to be more complicated than the regular case.

Main results

Let us now state more precisely our results. We will assume, without loss of generality, that our vector α = (α 1 , . . . , α n ) ∈ R n has unit norm, that is

|α| := max 1≤i≤n |α i | = 1.
Throughout the text, we shall use | • | to denote the supremum norm for vectors in R n , but also the induced norm on tensor spaces associated to R n .

Our perturbation f ε will be assumed to depend explicitly on ε ≥ 0, but we will require the following two conditions to hold. First, we will assume that f ε is uniformly bounded (with respect to some suitable norm, as we will define below) with respect to ε, so that ε really describes the magnitude of the perturbation. Then, we will also assume that when ε goes to zero, f ε converges (for the appropriate topology) to f 0 = f . The various generic conditions will be imposed on f , and they will continue to hold for f ε for all ε small enough, uniformly in ε.

Resonant case

Our first result deals with a resonant frequency vector α ∈ R n . It is no restriction to assume that α = (0, α

) ∈ R d × R n-d where α ∈ R n-d is non-resonant, for some 1 ≤ d ≤ n -1.
The perturbation f ε : T n × B → R will be assumed to be only sufficiently smooth. More precisely, given an integer k ≥ 3, we assume that f ε belongs to C k (T n × B), which is the space of k-times differentiable functions on T n × B whose differentials up to order k extend by continuity to T n × B, where B is the closure of B. This is a Banach space with the norm

|f ε | C k (T n × B) := max 0≤l≤k sup (θ,I)∈T n × B |∇ l f ε (θ, I)| .
To introduce our generic condition on the perturbation, let us decompose θ = ( θ, θ) ∈ T d × T n-d and define the (partial) average fε :

T d × B → R by fε ( θ, I) := T n-d f ε ( θ, θ, I)d θ.
Let us further define f = f0 . Our condition is then: (G1) There exists I * ∈ B such that the function f * :

T d → R, defined by f * ( θ) = f ( θ, I * ), is non-constant. Hence there exists θ * ∈ T d such that ∂θ f ( θ * , I * ) = 0. Since the map f ∈ C k (T n × B) → f ∈ C k (T d × B) is linear, bounded, surjective
and hence open, it is easy to check that the subset of functions f satisfying (G1) is both open and dense (in an appropriate topology; for instance it is open in C 0 topology and dense in C k topology, and even in higher regularity when f is more regular). Eventually, let us consider Then there exist positive constants ε 0 , c 1 and c 2 and an open ball B * ⊂ B -c 1 which depend on h and f but not on ε such that if 0 < ε ≤ ε 0 , for any initial action I 0 ∈ B * , there exists a solution (θ(t), I(t)) of the system associated to H with I(0) = I 0 such that

     H(θ, I) = α • I + εf ε (θ, I), ε ≥ 0, f 0 = f, α = (0, α) ∈ R d × R n-d , α ∈ R n-d nonresonant, |α| = 1, |f ε | C k (T n × B) ≤ 1, k ≥ 3.
|I(τ ) -I 0 | ≥ c 1 , τ = c 2 ε -1 .
This is a result of (fast) instability for a generic perturbation, in the case where the frequency is resonant. As the time of instability is linear with respect to ε -1 , this result also implies the failure of Nekhoroshev type estimates for a generic perturbation. Finally, observe that unstable solutions cover, in action space, an open ball which is independent of ε: the set of invariant tori, if any, cannot converge to a set of full measure of invariant unperturbed tori, and therefore the KAM theorem cannot hold true for a generic perturbation.

Note that our condition (G1) is of local nature, and so is our Theorem 1. A global condition can be formulated as below:

(G1 ′ ) For all I ∈ B, there exists θ = θ(I) ∈ T d such that ∂θ f ( θ, I) = 0.
It is not hard to prove (for instance, using a transversality argument) that this condition (G1 ′ ) is still satisfied for an open and dense set of functions f . Here's our global statement.

Theorem 1'. Let H be as in (H res ), with f satisfying (G1 ′ ) and

lim ε→0 | fε -f | C 1 (T d × B) = 0.
Then there exist positive constants ε 0 , c 1 and c 2 which depend on h and f but not on ε such that if 0 < ε ≤ ε 0 , for any initial action I 0 ∈ B -c 1 , there exists a solution (θ(t), I(t)) of the system associated to H with I(0) = I 0 such that

|I(τ ) -I 0 | ≥ c 1 , τ = c 2 ε -1 .
The proof of Theorem 1' will be just a slight modification of the proof of Theorem 1.

Non-resonant case

Our next results deal with a non-resonant frequency α ∈ R n . Let us define the functions

Ψ α by Ψ α (Q) = max |k • α| -1 | k ∈ Z n , 0 < |k| ≤ Q , Q ≥ 1 and ∆ α by ∆ α (x) = sup{Q ≥ 1 | QΨ α (Q) ≤ x}, x ≥ 1.
The vector α is said to be (γ, τ )-Diophantine for some γ > 0 and

τ ≥ n-1 if Ψ α (Q) ≤ γ -1 Q τ : in this case, one has ∆ α (x) ≥ (γx) 1 1+τ
. The perturbation f ε : T n × B → R will be assumed, for simplicity, to be real-analytic but this can be weakened (see the comments after Theorem 2 and Theorem 3). Given two parameters r > 0 and s > 0, let us define the complex neighborhoods

V s T n := {θ ∈ C n /Z n | max 1≤i≤n {|Im(θ i )|} < s}, V r B := {I ∈ C n | inf I ′ ∈B |I -I ′ | < r},
and for simplicity, let us denote

V s,r := V s T n × V r B.
Our assumption is that f ε extends to a bounded holomorphic function on V s,r , and let us define

|f ε | s,r := sup (θ,I)∈Vs,r |f ε (θ, I)|.
Let us now define the (full) average of f ε , that we still denote by fε , by setting fε (I) :=

T n f ε (θ, I)dθ. (2)
As before, let f = f0 and we consider the following condition:

(G2) There exists I * ∈ B such that the Hessian matrix ∇ 2 f (I * ) is non-singular.

The set of functions f which satisfy (G2) is obviously open and dense. We are finally led to consider the following Hamiltonian:

     H(θ, I) = α • I + εf ε (θ, I), ε ≥ 0, f 0 = f, α ∈ R n nonresonant, |α| = 1, |f ε | s,r ≤ 1. (H nonres )
We can now state our second result.

Theorem 2. Let H be as in (H nonres ), with f satisfying (G2) and

lim ε→0 |∇ 2 fε (I * ) -∇ 2 f (I * )| = 0.
Then there exist positive constant ε0 c, c, C and an open ball B * ⊂ B which depend only on h and f but not on ε, such that if 0 < ε ≤ ε0 , there exists a set K ε ⊂ T n × B * , which consists of real-analytic Lagrangian quasi-periodic tori invariant by the Hamiltonian flow of H. Moreover, we have the measure estimate

Leb(T n × B * \ K ε ) ≤ C exp -c∆ α cε -1 Leb(T n × B * )
where Leb denotes Lebesgue measure.

The conclusions of this statement are similar to those of the classical KAM theorem (see for instance [START_REF] Pöschel | Integrability of Hamiltonian systems on Cantor sets[END_REF] or [START_REF]Smooth ergodic theory and its applications[END_REF]): the proof will actually consist of using the condition (G2) on the perturbation to remove the degeneracy on the integrable part.

Observe that the measure estimate of the complement of the set of invariant tori depends on α, as it is exponentially small with respect to ∆ α (ε -1 ); when α is (γ, τ )-Diophantine, the estimate is of order exp -(γ/ε) 1 1+τ . The analyticity assumption is in fact not necessary; one may assume that the system is only Gevrey regular (see [START_REF] Popov | KAM theorem for Gevrey Hamiltonians[END_REF]) or sufficiently smooth. In those cases, the statement is essentially the same, with the obvious modifications (concerning the regularity of each torus and the measure estimate) so we will not pursue this further.

Observe also, as before, that our condition (G2) is local, so Theorem 2 only applies to some (possibly very small) open ball B * . To have a global statement, one can consider the following condition:

(G2 ′ ) ∇ f : B → R n is a diffeomorphism onto its image.
Notice, however, that this condition (G2 ′ ) is no longer generic (one has the same issue for the classical KAM theorem on a fixed domain). Under such a condition, we have the following global statement.

Theorem 2'. Let H be as in (H nonres ), with f satisfying (G2 ′ ) and

lim ε→0 |f ε -f | C 2 ( B) = 0.
Then there exist positive constant ε0 c, c and C which depend only on h and f but not on ε, such that if 0 < ε ≤ ε0 , there exists a set K ε ⊂ T n × B, which consists of real-analytic Lagrangian quasi-periodic tori invariant by the Hamiltonian flow of H. Moreover, we have the measure estimate

Leb(T n × B \ K ε ) ≤ C exp -c∆ α cε -1 Leb(T n × B)
where Leb denotes Lebesgue measure.

As before, the proof of Theorem 2' will be easily obtained from the proof of Theorem 2.

To state our last result, we need to introduce several notations and a definition. Given an integer m ≥ 2, let us denote by P (n, m) the space of polynomials with real coefficients of degree m in n variables, and P 2 (n, m) ⊂ P (n, m) the subspace of polynomials with vanishing homogeneous terms of degree zero and one. The following definition was introduced in [START_REF]Double exponential stability for generic real-analytic elliptic equilibrium points[END_REF], following the work of Nekhoroshev ([Nek73]).

Definition 1. Given positive constants ρ, C and δ, a polynomial

P 0 ∈ P 2 (n, m) is called (ρ, C, δ)-stably steep if for any P ∈ P 2 (n, m) such that |P -P 0 | < ρ, any integer l ∈ [1, n -1],
and any vector subspace Λ ⊆ R n of dimension l, letting P Λ be the restriction of P to Λ, the inequality max

0≤η≤ζ min |x|=η, x∈Λ |∇P Λ (x)| > Cζ m-1
holds true for all 0 < ζ ≤ δ. A polynomial P 0 ∈ P 2 (n, m) is called stably steep if it is (ρ, C, δ)-stably steep for some positive constants ρ, C and δ.

Recall that the average perturbation fε : B → R was defined in (2), and that f = f0 . Given I ∈ B and m ≥ 2, we let T m 2 f (I) ∈ P 2 (n, m) be the polynomial obtained by selecting the homogeneous terms of degree at least 2 and at most m from the Taylor expansion of f at I. Formally, we have

T m 2 f (I)X = m l=2 (l!) -1 ∇ l f (I).(X) l , X = (X 1 , . . . , X n ).
Let us further define m n := [n 2 /2 + 2], where [ • ] denotes the integer part, so that we can finally state the condition we shall require on f :

(G3) There exists I * ∈ B and an integer m ≥ m n such that the polynomial T m 2 f (I * ) ∈ P 2 (n, m) is stably steep.

It was proved in [START_REF]Double exponential stability for generic real-analytic elliptic equilibrium points[END_REF] (using arguments from [START_REF] Nekhoroshev | Stable lower estimates for smooth mappings and the gradients of smooth functions[END_REF]) that the set of stably steep polynomials is, for m ≥ m n , generic in the sense that its complement is a closed semialegebraic subset of positive codimension. The set of functions f which satisfy (G3) is therefore open and dense. We can now state our third result.

|I(t) -I 0 | ≤ C∆ α (cε -1 ) -1 , |t| ≤ ε -1 exp exp -c∆ α (cε -1 ) .
Exactly as in Theorem 2, the real-analyticity assumption is not necessary and can be weakened: if H is only assumed to be α-Gevrey smooth, for α ≥ 1, then one obtains the following estimates:

|I(t) -I 0 | ≤ C∆ α (cε -1 ) -1 , |t| ≤ ε -1 exp exp -c∆ α (cε -1 ) 1 α
, which generalizes the estimates of Theorem 3 (as the case α = 1 corresponds to real-analytic Hamiltonians). Once again, we can have a global statement under the following global condition: (G3 ′ ) There exists an integer m ≥ m n such that for all I ∈ B, the polynomial T m 2 f (I) ∈ P 2 (n, m) is stably steep.

As we already said, the set of stably steep polynomials with m ≥ m n has a complement with positive codimension, but more is true: when m is arbitrary large, the codimension of the complement gets arbitrary large. Using a transversality argument, one can then infer that condition (G3 ′ ) is still satisfied for an open and dense set of functions f . Here's our global statement.

Theorem 3'. Let H be as in (H nonres ), with f satisfying (G3 ′ ) and

lim ε→0 |f ε -f | C m ( B) = 0.
Then there exist positive constant ε0 , c, c and C which depend only on h and f but not on ε, such that if 0 < ε ≤ ε0 , for any solution (θ(t), I(t)) of the system associated to H with

I(0) = I 0 ∈ B -C∆ α (cε -1 ) -1 we have |I(t) -I 0 | ≤ C∆ α (cε -1 ) -1 , |t| ≤ ε -1 exp exp -c∆ α (cε -1 ) .
Again, the proof of Theorem 3' will be the same as the proof of Theorem 3.

Applications

The techniques used to prove Theorem 1, Theorem 2 and Theorem 3 have been already used in the study of the dynamics in a neighborhood of an invariant quasi-periodic torus. In this case, the Hamiltonian is a small perturbation of a linear integrable Hamiltonian system, so our results are in principle applicable; however, the perturbation being quite specific, our generic conditions (G1), (G2) or (G3) cannot be always satisfied. To be more precise, one considers here a Hamiltonian of the form

H(θ, I) = α • I + A(θ)I • I + R(θ, I), (θ, I) ∈ T n × U ( * )
where U is an open neighborhood of the origin in R n , A : T n → Sym(n, R) is a map taking values in the space of real symmetric matrices of size n, and R(θ, I) = O 3 (I) is of order at least 3 in I. The set T α := T n × {I = 0} is invariant by the Hamiltonian flow of H, it is a Lagrangian quasi-periodic torus with frequency α. On the domain T n × B ε , where B ε is the ε-ball around the origin in R n , the Hamiltonian in ( * ) is easily seen to be equivalent to

H ′ (θ, I) = α • I + εA(θ)I • I + ε 2 R(θ, I), (θ, I) ∈ T n × B 1
and this Hamiltonian H ′ is of the form (H res ) or (H nonres ) (depending whether α is resonant or not, and on the regularity of H) with

f ε = A(θ)I • I + εR(θ, I), f = f 0 = A(θ)I • I.
It is not hard to see that Theorem 1 cannot be used in a neighborhood of a resonant invariant quasi-periodic torus to prove topological instability, as (G1) do not hold at I * = 0. Yet under different generic conditions and using techniques similar to those that will be used to prove Theorem 1, one can still prove that generically such a torus possesses some weaker instability properties, as shown in [START_REF]Instability of resonant invariant tori in Hamiltonian systems[END_REF]. Concerning Theorem 2, the situation is much better as it is directly applicable: the condition (G2) amounts to say that the matrix A 0 := T n A(θ)dθ is non-singular. Hence one can show that a non-resonant invariant quasi-periodic torus is always KAM stable, without requiring any Diophantine condition on the frequency vector. This application has already been worked out in [START_REF]Non-degenerate Liouville tori are KAM stable[END_REF].

Finally, Theorem 3 is not sufficient to prove that generically, a non-resonant invariant quasi-periodic torus is doubly exponentially stable. Indeed, in this case f = f 0 is a quadratic function of I and therefore (G3) cannot be checked. The result does apply if (G3) is replaced by the following stronger condition: (G4) There exists I * such that the restriction of ∇ 2 f (I * ) to the orthogonal complement of α ∈ R n is sign-definite.

Theorem 3 do apply if (G3) is replaced by (G4), and its proof can be greatly simplified in this case (this is because, for ε small enough, (G4) implies that fε is quasi-convex in a neighborhood of I * , while (G4) implies that fε is steep in a neighborhood of I * , a much more general condition that will be recalled in §6). However, this condition (G4) is no longer generic. Using Theorem 3 (with the condition (G3) replaced by (G4)), we obtain the following corollary.

Corollary 4. Let H be as in ( * ), assume that H is real-analytic, α is non-resonant and A 0 is sign-definite when restricted to the orthogonal complement of α ∈ R n . Then there exist positive constants c and c such that for ε > 0 small enough and for any solution (θ(t), I(t)) of the system associated to H with |I(0)| ≤ ε, we have

|I(t)| ≤ 2ε, |t| ≤ ε -1 exp exp -c∆ α (cε -1 ) .
This last result gives a slight generalization of a result of Morbidelli and Giorgilli (see [START_REF] Morbidelli | Superexponential stability of KAM tori[END_REF]), in which the above statement is proved assuming α to be Diophantine; note that their proof do not extend to an arbitrary non-resonant vector α as they use Birkhoff normal forms, which in general do not exist unless α is Diophantine. On the other hand, assuming α to be Diophantine (and using Birkhoff normal forms), it is proved in [START_REF] Bounemoura | Double exponential stability for generic invariant tori in Hamiltonian systems[END_REF] that generically, the invariant torus is doubly exponentially stable (an analogous result, in the context of elliptic equilibrium points, is contained in [START_REF]Double exponential stability for generic real-analytic elliptic equilibrium points[END_REF]). We do not know if this last result can be obtained without the assumption that α is Diophantine.

Normal forms

The main ingredients to prove Theorem 1, Theorem 2 and Theorem 3 are normal form results that will be recalled in this section.

First we assume that α ∈ R n is resonant, more precisely α = (0, α

) ∈ R d × R n-d with 1 ≤ d ≤ n -1 and α ∈ R n-d non-resonant. We can also define Ψ α by Ψ α(Q) = max |k • α| -1 | k ∈ Z n-d , 0 < |k| ≤ Q , Q ≥ 1 and ∆ α by ∆ α(x) = sup{Q ≥ 1 | QΨ α(Q) ≤ x}, x ≥ 1.
The proposition below will be used to prove Theorem 1.

Proposition 1. Let H be as in (H res ). Then there exist constants c := c(n), C := C(n, k) and µ 0 := µ 0 (n, k) such that if

µ α(ε) := ∆ α cε -1 -1 ≤ µ 0 , (3)
then, setting B ′ = B -Cµ α(ε), there exist a symplectic embedding Φ :

T n × B′ → T n × B of class C k-1 such that H • Φ(θ, I) = α • I + ε fε ( θ, I) + εf + ε (θ, I) with the estimates |f + ε | C k-1 (T n × B′ ) ≤ Cµ α(ε), |Φ -Id| C k-1 (T n × B′ ) ≤ Cµ α(ε). ( 4
)
This statement is a special case of Theorem 1.1 in [START_REF]Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians[END_REF], to which we refer for a proof.

In the case where α ∈ R n is non-resonant and the Hamiltonian is real-analytic, we have the following proposition, which will be used to prove both Theorem 2 and Theorem 3. Proposition 2. Let H be as in (H nonres ). Then there exist constants c := c(n), c ′ := c ′ (n, s), C := C(n, s, r) and µ 0 := µ 0 (n, s, r) such that if

µ α (ε) := ∆ α cε -1 -1 ≤ µ 0 , (5)
then there exist a real-analytic symplectic embedding Φ : V s/2,r/2 → V s,r such that

H • Φ(θ, I) = α • I + ε fε (I) + εg ε (I) + εf + ε (θ, I)
with the estimates

     |g ε | s/2,r/2 ≤ Cµ α (ε), |f + ε | s/2,r/2 ≤ C exp -c ′ µ α (ε) -1 , |Φ -Id| s/2,r/2 ≤ Cµ α (ε). (6)
As before, this statement is a special case of Theorem 1.1 in [START_REF]Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians[END_REF], to which we refer for a proof.

Proofs of Theorem 1 and Theorem 1'

The proof of Theorem 1 will follow easily from Proposition 1, using arguments similar to those used in [START_REF] Bounemoura | Generic fast diffusion for a class of non-convex Hamiltonians with two degrees of freedom[END_REF] and [START_REF]A note on micro-instability for Hamiltonian systems close to integrable[END_REF].

Proposition 1 provides us with a (symplectic, close to the identity) change of coordinates which reveals that the dominant part of the Hamiltonian is given by h + ε fε , where fε is the (partial) average of f ε . The system associated to this dominant part is no longer integrable, because of the condition (G1) it has unstable solutions and an elementary argument will show that the same is true for the system associated to H. From now on, to simplify notations, when convenient we will suppress all constants depending on n, k, ζ and the function ∆ α by using the symbols , and ∼. We start by assuming ε 1 so that (3) is satisfied and Proposition 1 can be applied. Let H • Φ be the Hamiltonian in formal form, which is defined on T n × B ′ , and has the form

H(θ, I) = α • I + ε fε ( θ, I) + εf + ε (θ, I) = α • Ĩ + ε fε ( θ, I) + εf + ε (θ, I) where I = ( Ī, Ĩ) ∈ R d × R n-d . Consider
           θ(t) = ε∂Ī fε ( θ(t), I(t)) + ε∂Ī f + ε (θ(t), I(t)), θ(t) = α + ε∂ Ĩ fε ( θ(t), I(t)) + ε∂ Ĩ f + ε (θ(t), I(t)), İ(t) = -ε∂θ fε ( θ(t), I(t)) -ε∂θf + ε (θ(t), I(t)), İ(t) = -ε∂ θf + ε (θ(t), I(t)). (8)
Using the fact that fε has unit norm and the estimates (4) on f + ε , we obtain from the first, third and fourth equation of (8) that, for t ≤ τ ∼ ε -1 and ε 1,

| θ(t) -θ(0)| ≤ τ θ(t) 1 + µ(ε) 1 | Ī(t) -Ī(0)| ≤ τ İ(t) 1 + µ(ε) 1 | Ĩ(t) -Ĩ(0)| ≤ τ İ(t) µ(ε) 1.
These inequalities, together with (4) and our choices of θ(0) = θ * and

I(0) = ( Ī(0), Ĩ (0)) ∈ B ζ (I * ), implies that for t ≤ τ , |∂θ fε ( θ * (t), I(t))| 1. ( 9
)
But then using again the third equation of ( 8), one has

İ(t) ε -εµ(ε) ε which eventually gives | Ī(τ ) -Ī(0)| τ ε ∼ 1 and so |I(τ ) -I(0)| 1.
This proves the theorem for H • Φ, but the statement for H is then obvious. Indeed, recall that Φ is symplectic, so that if (θ(t), I(t)) is a solution of the system associated to H • Φ, then ( θ(t), Ĩ(t)) := Φ(θ(t), I(t)) is a solution of the system associated to H. Moreover, Φ is close to the identity as expressed in (4), so that in particular the image of T

But then

| Ĩ(τ ) -Ĩ(0)| ≥ |I(τ ) -I(0)| -|I(τ ) -Ĩ(τ )| -|I(0) -Ĩ(0)|
and using the estimate (4) on Φ, we have, for ε small enough,

| Ĩ(τ ) -Ĩ(0)| ≥ |I(τ ) -I(0)| -2Cµ α(ε) 1.
This concludes the proof.

Proof of Theorem 1'. Now we are assuming that f satisfies (G1 ′ ), that is for all I ∈ B, there exists θ = θ(I) ∈ T d such that ∂θ f ( θ, I) = 0. We are also assuming Once we have obtained a uniform constant, the rest of the proof is now similar to the proof of Theorem 1.

lim ε→0 | fε -f | C 1 (T d × B) = 0. ( 10

Proof of Theorem 2 and Theorem 2'

The proof of Theorem 2 will follow easily from Proposition 2 and a version of the classical KAM theorem that we will recall below. Indeed, as before, Proposition 2 provides us with a (symplectic, close to the identity) change of coordinates which shows that the dominant part of the Hamiltonian is given by h + ε fε + εg ε , where fε is the (full) average of f ε . The system associated to this dominant part is now integrable, and because of the condition (G2), the system associated to h + ε fε is non-degenerate in the sense of Kolmogorov, and so is the system associated to h + ε fε + εg ε as εg ε can be considered as an (integrable) perturbation. One has to be careful because the non-degeneracy is ε-dependent; yet we will see that this causes no problem.

Let us now recall the statement of the classical KAM theorem. Consider a domain D ⊂ R n , two positive constants ̺ and σ and a Hamiltonian H real-analytic on V σ T n × V ̺ D of the form

H(θ, I) = H 0 (I) + H 1 (θ, I), (θ, I) ∈ V σ T n × V ̺ D. (H)
For any I ∈ V ̺ D, the Hessian ∇ 2 H 0 (I) is assumed to be non-singular and moreover, we assume the existence of a constant M > 0 such that

|∇ 2 H 0 (I)| ≤ M, | ∇ 2 H 0 (I) -1 | ≤ M. (11)
We can now state a version of the classical KAM theorem.

Theorem 5 (KAM). Let H be as in (H), with H 0 satisfying (11). Assume moreover that ∇H 0 is a diffeomorphism of V ̺ D onto its image. Then there exist constants ν 0 > 0 and E > 0, which depend only on n, σ, ̺ and M , such that if

ν := |H 1 | σ,̺ ≤ ν 0 ,
there exists a set K ν ⊂ T n × D which consists of real-analytic Lagrangian quasi-periodic tori invariant by the Hamiltonian flow of H. Moreover, we have the measure estimate

Leb(T n × D \ K ν ) ≤ E √ νLeb(T n × D)
where Leb denotes Lebesgue measure.

This is exactly the content of Theorem A in [START_REF] Pöschel | Integrability of Hamiltonian systems on Cantor sets[END_REF], to which we refer for a proof. We can now complete the proof of Theorem 2.

Proof of Theorem 2. Recall that we are considering H as in (H nonres ), with f satisfying (G2), which is the existence of

I * ∈ B such that ∇ 2 f (I * ) is non-singular. Recall also that lim ε→0 |∇ 2 fε (I * ) -∇ 2 f (I * )| = 0. ( 12
)
From (G2) we know that there exists a positive constant M > 0 such that

|∇ 2 f (I * )| ≤ M/2, | ∇ 2 f (I * ) -1 | ≤ M/2.
Then, because of (12), for ε sufficiently small with respect to M , we obtain

|∇ 2 f ε (I * )| ≤ M, | ∇ 2 f ε (I * ) -1 | ≤ M. (13)
As in the proof of Theorem 1, we will now suppress all constants depending on n, s, r, M , the distance of I * to ∂B = B \ B and the function ∆ α by using the symbols , and ∼. First, we assume that ε 1 so that µ α (ε) ≤ µ 0 and Proposition 2 applies. Let us first prove the statement for the Hamiltonian H := H • Φ in normal form. To prove such a statement, let us consider the rescaled Hamiltonian Hε := ε -1 H, that is

Hε (θ, I) = ε -1 α • I + fε (I) + g ε (I) + f + ε (θ, I)
that we can write Hε (θ, I) = H 0 (I) + H 1 (θ, I)

with H 0 (I) = ε -1 α • I + fε (I) + g ε (I), H 1 (θ, I) = f + ε (θ, I).

We have

∇ 2 H 0 (I * ) = ∇ 2 fε (I * ) + ∇ 2 g ε (I * ), so that using the estimate (6) on g ε and Cauchy estimates, we obtain

|∇ 2 H 0 (I * ) -∇ 2 fε (I * )| µ α (ε).
For ε 1, using this last estimate, and the inequality (13), we obtain

|∇ 2 H 0 (I * )| 1, | ∇ 2 H 0 (I * ) -1 | 1.
Hence there exists ζ ∼ 1 such that ∇ 2 H 0 (I) is non-singular for all I ∈ B ζ (I * ). Restricting ζ if necessary, we may assume that ∇ 2 H 0 (I) is non-singular for all I ∈ V ̺ B ζ (I * ), for some 0 < ̺ ≤ r/2, with the estimates

|∇ 2 H 0 (I)| 1, | ∇ 2 H 0 (I) -1 | 1.
Moreover, by a further restriction of ζ if necessary, ∇H 0 can be assumed to be a diffeomorphism of V ̺ B ζ (I * ) onto its image. Now we set σ = s/2, and the estimate (6) on f

+ ε yields |H 1 | σ,̺ ≤ ν α (ε) ∼ exp -c ′ µ α (ε) -1 .
Assuming again ε 1 so that ν α (ε) ≤ ν 0 , we can apply we eventually obtain

Leb(T n × B * \ K ε ) exp -cµ α (ε) -1 Leb(T n × B * ).
This concludes the proof.

Proof of Theorem 2'. Now we are assuming that f satisfies (G2 ′ ), that is ∇ f : B → R n is a diffeomorphism onto its image, and

lim ε→0 | fε -f | C 2 ( B) = 0. ( 14
)
Given any I ∈ B, there exist a positive constant M I and an open neighborhood U (I) of I in B such that for any I ′ ∈ U (I), we have

|∇ 2 f (I ′ )| ≤ M I /2, | ∇ 2 f (I ′ ) -1 | ≤ M I /2.
As in the proof of Theorem 1', by a compactness argument one can find a positive constant M such that for all I ∈ B,

|∇ 2 f (I)| ≤ M/2, | ∇ 2 f (I) -1 | ≤ M/2.
Using (14), for ε sufficiently small with respect to M , we get

|∇ 2 fε (I)| ≤ M, | ∇ 2 fε (I) -1 | ≤ M.
Having obtained a uniform constant, the rest of the proof is now similar to the proof of Theorem 2.

6 Proof of Theorem 3 and Theorem 3'

The proof of Theorem 3 will follow from Proposition 2 and a version of the classical Nekhoroshev theorem that we will recall below. Once again, Proposition 2 gives us a (symplectic, close to the identity) change of coordinates which shows that the dominant part of the Hamiltonian, which is given by h + ε fε + εg ε , is integrable. Using the condition (G3), we will show that the system associated to h + ε fε is generic in the sense of Nekhoroshev, and so is the system associated to h + ε fε + εg ε . Here also, we will have to show that the ε-dependence of the integrable part is not an obstacle.

The Nekhoroshev theorem requires a condition of steepness on the integrable part of the Hamiltonian, so we first recall this definition.

Definition 2. A differentiable function h : B → R is steep on a domain B ′ ⊆ B if there exist positive constants C ′ , δ ′ , p l , for any integer l ∈ [1, n -1]
, and κ such that for all I ∈ B ′ , we have |∇h(I)| ≥ κ and, for all integer l ∈ [1, n -1], for all vector space Λ ∈ R n of dimension l, letting λ = I + Λ the associated affine subspace passing through I and h λ the restriction of h to λ, the inequality

max 0≤η≤ξ min |I ′ -I|=η, I ′ ∈λ |∇h λ (I ′ ) -∇h λ (I)| > C ′ ξ p l
holds true for all 0 < ξ ≤ δ ′ . We say that h is (κ, C ′ , δ ′ , (p l ) l=1,...,n-1 )-steep on B ′ and, if all the p i = p, we say that h is (κ, C ′ , δ ′ , p)-steep on B ′ .

Next we need to make a link between this definition of steep function and the definition of stably steep polynomials (Definition 1 in Section 2): a function whose Taylor expansion at some point is stably steep is steep on some neighborhood of this point. Here's a precise statement.

is (κ, C ′ , δ ′ , m -1)-steep on B ζ (I * ) with κ = ̟/2, C ′ = C/2, δ ′ = ζ.
For a proof, we refer to Theorem 2.2 in [START_REF]Double exponential stability for generic real-analytic elliptic equilibrium points[END_REF] or Proposition 3.2 in [START_REF] Bounemoura | Double exponential stability for generic invariant tori in Hamiltonian systems[END_REF]. We can eventually state the theorem of Nekhoroshev ([Nek77], [START_REF]An exponential estimate of the time of stability of nearly integrable Hamiltonian systems II[END_REF]). To do this, we consider again a real-analytic Hamiltonian H as in (H), that is

H(θ, I) = H 0 (I) + H 1 (θ, I), (θ, I) ∈ V σ T n × V ̺ D,
where D is some domain in R n and σ and ̺ are positive constants. The integrable Hamiltonian will be assumed to be a steep function on D, and its Hessian will be assumed to be uniformly bounded on V ̺ D, that is there exists a positive constant M such that for all I ∈ V ̺ D,

|∇ 2 H 0 (I)| ≤ M. (15)
Here's the statement.

Theorem 6 (Nekhoroshev). Let H be as in (H), with H 0 a (κ, C ′ , δ ′ , p)-steep function on D satisfying (15). Then there exists a constant ν 0 > 0, which depends only on n, σ, ̺, κ, C ′ , δ ′ , p and M , and constants a > 0 and b > 0 which depend only on n and p, such that if

ν := |H 1 | σ,̺ ≤ ν 0 ,
then for any solution (θ(t), I(t)) with I(0) ∈ D -ν b , we have

|I(t) -I(0)| ≤ ν b /2, |t| ≤ ν -1 exp ν -a .
This is exactly the content of the main theorem in [START_REF]An exponential estimate of the time of stability of nearly integrable Hamiltonian systems[END_REF], Section §4.4, to which we refer for a proof. We can now complete the proof of Theorem 3.

Proof of Theorem 3. Recall that we are considering H as in (H nonres ), with f satisfying (G3), which is the existence of a point I * ∈ B and an integer m ≥ m n such that the polynomial

T m 2 f (I * ) ∈ P 2 (n, m) is stably steep. Recall also that lim ε→0 max 2≤l≤m |∇ l fε (I * ) -∇ l f (I * )| = 0. (16)
From (G3) we know that there exist positive constant ρ, C and δ such that the polynomial T m 2 f (I * ) ∈ P 2 (n, m) is (2ρ, C, δ)-stably steep. Then, using (16), for ε sufficiently small with respect to δ, the polynomial T m 2 fε (I * ) ∈ P 2 (n, m) is (ρ, C, δ)-stably steep. As in the proofs of Theorem 1 and Theorem 2, we will now suppress all constants depending on n, s, r, m, ρ, C, δ, |∇ 2 f (I * )|, the distance of I * to ∂B = B \ B and the function ∆ α by using the symbols , and ∼. First, we assume that ε 1 so that µ α (ε) ≤ µ 0 and Proposition 2 applies. We will first prove a statement for the Hamiltonian H := H •Φ in normal form. To prove such a statement, again, we consider the rescaled Hamiltonian Hε := ε -1 H, that can be written Hε (θ, I) = H 0 (I) + H 1 (θ, I) with H 0 (I) = ε -1 α • I + fε (I) + g ε (I), H 1 (θ, I) = f + ε (θ, I). Let us first establish the fact that H 0 is steep in a neighborhood of the point I * . Using the estimate (6) on g ε and Cauchy estimates, we obtain that for all 2 ≤ l ≤ m, This concludes the proof.

Proof of Theorem 3'. Now we are assuming that f satisfies (G3 ′ ), that is there exists an integer m ≥ m n such that for all I ∈ B, the polynomial T m 2 f (I) ∈ P 2 (n, m) is stably steep. We are also assuming lim

ε→0 | fε -f | C m ( B) = 0. ( 17
)
As in the proofs of Theorem 1' and Theorem 2', since the set of stably steep polynomials is open, by a compactness argument one can find positive constants ρ, C and δ such that for all I ∈ B, the polynomial T m 2 f (I) ∈ P 2 (n, m) is (2ρ, C, δ)-stably steep. Then, using (17), for ε sufficiently small with respect to δ, for all I ∈ B the polynomial T m 2 fε (I) ∈ P 2 (n, m) is (ρ, C, δ)-stably steep. As before, once we have obtained uniform constants, the rest of the proof is now similar to the proof of Theorem 3.

(

  H res ) Before stating our first result, we need to introduce the following notation: given a positive constant ζ, we define B -ζ := {I ∈ B | B(I, ζ) ⊂ B} where B(I, ζ) is the ball of radius ζ around I. We can now state our first theorem. Theorem 1. Let H be as in (H res ), with f satisfying (G1) and lim ε→0 |∂θ fε ( θ * , I * ) -∂θ f ( θ * , I * )| = 0.

Theorem 3 .

 3 Let H be as in (H nonres ), with f satisfying (G3) andlim ε→0 max 2≤l≤m |∇ l fε (I * ) -∇ l f (I * )| = 0.Then there exist positive constant ε0 , c, c, C and an open ball B * ⊂ B which depend only on h and f but not on ε, such that if 0 < ε ≤ ε0 , for any I 0 ∈ B * and any solution (θ(t), I(t)) of the system associated to H with I(0) = I 0 , we have

Proof of Theorem 1 .

 1 Recall that we are considering H as in (H res ), with f satisfying (G1), which is the existence of I * ∈ B and θ * ∈ T d such that ∂θ f ( θ * , I * ) = 0. Recall also that lim ε→0 |∂θ fε ( θ * , I * ) -∂θ f ( θ * , I * )| = 0. (7) From (G1), we have |∂θ f * ( θ * )| = |∂θ f ( θ * , I * )| := 3ζ > 0. Hence, by (7), for ε small enough with respect to ζ, we have |∂θ fε ( θ * , I * )| ≥ 2ζ > 0. If we let B ζ (I * ) be the ball of radius ζ around I * , for all I ∈ B ζ (I * ) this last inequality gives |∂θ f ( θ * , I)| ≥ ζ > 0. Up to taking ζ smaller with respect to the distance of I * to ∂B = B \ B, we may assume that B ζ (I * ) is contained in B, and then taking ε smaller, we may assume that B ζ (I * ) is contained in B ′ = B -Cµ α(ε).

  the solution (θ(t), I(t)) of the system associated to H • Φ with θ(0) = θ * ∈ T d , θ(0) ∈ T n-d arbitrary and I(0) ∈ B ζ (I * ). It satisfies the following system of equation:

  n × B ζ (I * ) by Φ contains T n × B * , where B * is some open ball in B. Consider an arbitrary solution ( θ(t), Ĩ(t)) with Ĩ(0) ∈ B * , then the corresponding solution (θ(t), I(t)) has I(0) ∈ B ζ (I * ), and therefore by the preceding, |I(τ ) -I(0)| 1.

)

  Given any I ∈ B, there exist a positive constant ζ I and an open neighborhood U (I) of I in B such that for any I ′ ∈ U (I), we have |∂θ f ( θ, I ′ )| ≥ ζ I for some θ = θ(I) ∈ T d . The sets U (I), for I ∈ B, forms an open cover of B from which one can extract a finite open cover. Hence there exists ζ > 0 such that for all I ∈ B, we have |∂θ f ( θ, I)| ≥ ζ for some θ = θ(I) ∈ T d . Using (10), for ε sufficiently small with respect to ζ, we obtain |∂θ fε ( θ, I)| ≥ ζ.

  Theorem 5 with D = B ζ (I * ) and ν = ν α (ε): there exists a set K να(ε) ⊂ T n × B ζ (I * ) which consists of real-analytic Lagrangian quasi-periodic tori invariant by the Hamiltonian flow of Hε . Moreover, we have the measure estimate Leb(T n × B ζ (I * ) \ K να(ε) ) ν α (ε)Leb(T n × B ζ (I * )) ∼ exp -cµ α (ε) -1 Leb(T n × B ζ (I * ))with c = c ′ /2. Now observe that the Hamiltonian flow of Hε differs from the Hamiltonian flow of H by a constant time change: more precisely, (θ(t), I(t)) is a solution of the system associated to Hε if and only if (θ(ε -1 t), I(ε -1 t)) is a solution of the system associated to H.Hence K να(ε) ⊂ T n × B ζ (I * ) isstill invariant by the Hamiltonian flow of H. To conclude, let K ε := Φ -1 (K να(ε) ) and set B * = B ζ/2 (I * ). Using the estimate (6) on Φ, for ε 1 one can ensure that K ε ⊂ T n × B * . Moreover, since Φ is real-analytic and symplectic, then K ε ⊂ T n × B * consists of real-analytic Lagrangian quasi-periodic tori invariant by the Hamiltonian flow of H. Moreover, using again the estimate (6) on Φ, for ε 1 the Jacobian of Φ can be made arbitrarily close to one, and since Leb(T n × B * ) ∼ Leb(T n × B ζ (I * ))

Proposition 3 .

 3 Let h : Bζ(I * ) → R be a function of class C m 0 +1 such that |∇h(I * )| ≥ ̟ > 0 and such that T m I * h is (ρ, C, δ)-stably steep. Then for ζ > 0 sufficiently small with respect to ζ, |h| C m 0 +1 (Bζ ) , ρ, ̟, m 0 , C and δ, the function h

  |∇ l H 0 (I * ) -∇ l fε (I * )| µ α (ε).Therefore for ε 1, the polynomial T m 2 H 0 (I * ) ∈ P 2 (n, m) is stably steep. Moreover, it is easy to see that for ε 1, we have|∇H 0 (I * )| ε -1 1.We can thus apply Proposition 3 to find ζ ∼ 1 such that H 0 is a steep function on B ζ (I * ). Choosing ̺ = r/2 and σ = s/2, we also have supI∈V̺B ζ (I * ) |∇ 2 H 0 (I)| 1 and |H 1 | σ,̺ ≤ ν α (ε) ∼ exp -c ′ µ α (ε) -1 .Assuming again ε 1 so that ν α (ε) ≤ ν 0 , we can eventually apply Theorem 6 with D = B ζ (I * ) and ν = ν α (ε) and we obtain the following statement: for any solution (θ(t), I(t)) of the system associated to Hε withI(0) ∈ B ζ (I * ) -ν α (ε) b , we have |I(t) -I(0)| ≤ ν α (ε) b /2, |t| ≤ ν α (ε) -1 exp ν α (ε) -a .For ε 1, observe that we can assume thatB ζ (I * ) -ν α (ε) b contains B ζ/2 (I * ), hence the estimate above applies for any I(0) ∈ B ζ/2 (I * ). Now let us recall that (θ(t), I(t)) is a solution of the system associated to Hε if and only if (θ(ε -1 t), I(ε -1 t)) is a solution of the system associated to H. Hence for any solution (θ(t), I(t)) of the system associated to H with I(0) ∈ B ζ/2 (I * ), we have|I(t) -I(0)| ≤ ν α (ε) b /2, |t| ≤ ε -1 ν α (ε) -1 exp ν α (ε) -a .Recalling the definition of ν α (ε), for ε sufficiently small this inequality can be arranged to give|I(t) -I(0)| ≤ ν α (ε) b /2, |t| ≤ ε -1 exp exp -cµ α (ε) -1with c ∼ c ′ . Coming back to the original Hamiltonian H, as Φ is symplectic, (θ(t), I(t)) is a solution of the system associated to H • Φ if and only if ( θ(t), Ĩ(t)) := Φ(θ(t), I(t)) is a solution of the system associated to H. Since Φ is closed to the identity as expressed in (6), the image of T n × B ζ/2 (I * ) contains T n × B * for some open ball B * contained in B. Consider an arbitrary solution ( θ(t), Ĩ(t)) with Ĩ(0) ∈ B * , then the corresponding solution (θ(t), I(t)) has I(0) ∈ B ζ/2 (I * ), and therefore |I(t) -I(0)| ≤ ν α (ε) b /2, |t| ≤ ε -1 exp exp -cµ α (ε) -1 . But then, for |t| ≤ ε -1 exp exp -cµ α (ε) -1 , we have | Ĩ(t) -Ĩ(0)| ≤ | Ĩ(t) -I(t)| + |I(t) -I(0)| + |I(0) -Ĩ(0)| and using the estimate (6) on Φ, we have, for |t| ≤ ε -1 exp exp -cµ α (ε) -1 , | Ĩ(τ ) -Ĩ(0)| ≤ ν α (ε) b /2 + 2Cµ α (ε) µ α (ε).


Romain Bachelard
email: bachelard@cpt.univ-mrs.fr

Andrea Antoniazzi

Cristel Chandre

Duccio Fanelli

Michel Vittot

Enhancement of particle trapping in the wave-particle interaction

Keywords: 41.60.Cr, 47.10.Df, 05.45.Gg Free Electron Laser, Hamiltonian dynamics, chaos control

Introduction

A Free Electron Laser (FEL) generates a tunable, coherent, high power radiation. A FEL differs from conventional lasers in using a relativistic electron beam as its lasing medium. The physical mechanism responsible of the light emission and amplification is the interaction between a relativistic electron beam and a magnetostatic periodic field generated in the undulator. Due to the effect of the magnetic field, the electrons are forced to follow sinusoidal trajectories, thus emitting synchrotron radiation. This initial seed, termed spontaneous emission, is then amplified along the undulator until the laser effect is reached.

Among different schemes, Single-Pass high-gain FELs are particularly attractive since they hold the promise of resulting in flexible light sources of broad theoretical and applied interests. The coupled evolution of the radiation field and the N particle beam in a Single-Pass FEL can be successfully modeled within the framework of a simplified Hamiltonian picture [START_REF] Bonifacio | Physics of the high-gain FEL and superradiance[END_REF]. The N + 1 degree of freedom Hamiltonian displays a kinetic contribution, associated with the particles, and a potential term accounting for the self-consistent coupling between the particles and the field. Hence, direct inter-particles interactions are neglected, even though an effective coupling is indirectly provided because of the interaction with the wave.

The theory predicts a linear exponential instability and a late oscillating saturation, for the amplitude of the radiation field. Inspection of the asymptotic phase-space suggests that a bunch of particles gets trapped in the resonance and forms a clump that evolves as a single macro-particle localized in space. The remaining particles are almost uniformly distributed between two oscillating boundaries, and populate the so called chaotic sea.

This observation allowed to derive a simplified Hamiltonian model to characterize the saturated evolution of the laser. Such reduced formulation consists in only four degrees of freedom, namely the wave, the macro-particle and the two boundaries, delimiting the portion of space occupied by the the uniform halo surrounding the inner bulk [START_REF] Tennyson | Self-Consistent Chaos in the Beam-Plasma Instability[END_REF][START_REF] Antoniazzi | Statistical mechanics and Vlasov equation allows for a simplified Hamiltonian description of Single-Pass Free Electron Laser saturated dynamics[END_REF]. Furthermore, the macro-particle rotates around a well defined fixed point and this microscopic dynamics is shown to be responsible for the macroscopic oscillations observed at the intensity level. It can be therefore hypothesized that a significant reduction in the intensity fluctuations can be gained by implementing a dedicated control strategy, aimed at confining the macro-particle in space. In addition, the size of the macro-particle is directly related to the bunching parameter, a quantity of paramount importance in FEL context [START_REF] Antoniazzi | Statistical mechanics and Vlasov equation allows for a simplified Hamiltonian description of Single-Pass Free Electron Laser saturated dynamics[END_REF].

Smith et al [START_REF] Smith | Phase-locked particle motion in a largeamplitude plasma wave[END_REF] showed that a test-wave, being characterized by a frequency close to the one of the wave, can effectively destroy the macro-particle and activate a consequent detrapping process. Dimonte et al [START_REF] Dimonte | Destruction of Trapped-Particle Ocillations[END_REF] implemented this approach on a Travelling Wave Tube and detected a significant reduction of the intensity oscillations, followed, however, by an undesired systematic collapse of its mean-value.

In this paper, we focus on the macro-particle, spontaneously established as a result of the wave-particle interaction process in the saturated regime. In par- ticular we develop a dedicated technique to influence and, possibly, control its evolution, thus opening up the perspective of defining innovative approaches aiming at stabilizing the laser signal. To this end, we consider a test-particle Hamiltonian in a mean-field approach and calculate a small but appropriate control term which acts as a perturbation. The latter is shown to induce an increase in size of the macro-particle. A regularization of the dynamics is also observed, as confirmed by the reconstruction of invariant tori around the massive core.

t I -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3 θ p -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3 θ p -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3 θ p -3 -2 -1 0 1 2 3 -4 -3 -2
The paper is organized as follows. In Sec. 1, we introduce the mean-field model and shortly outline its derivation from the original N-body Hamiltonian. In Sec. 2, a test-particle is "controlled", through the reconstruction of invariant tori of its dynamics. Finally, we draw our conclusion and discuss possible implication of the present analysis.

The mean-field model

As previously anticipated, the dynamics of a Single-Pass FEL is successfully captured by the following Hamiltonian [START_REF] Bonifacio | Physics of the high-gain FEL and superradiance[END_REF] :

H N = N i=1 [ p 2 i 2 -2 I N cos (θ i + φ)], (1)
where (θ i , p i ) are the position and momentum of the i-th particle and (φ, I) stand respectively for the phase and intensity of the radiation. In the meanfield model, the conjugated variables (φ, I) are replaced by two functions of time φ(t) and I(t), obtained from direct simultations of the self-consistent dynamics. In other words, this amounts to formally neglecting the action of the electrons on the field.

The N-body Hamiltonian (1) can therefore be mapped into

HN = N i=1 H 1p (θ i , p i , t),
where

H 1p (θ, p, t) = p 2 2 -2 I(t) N cos (θ + φ(t)). (2)
Hence, the dynamics of a FEL can be adressed by studying the evolution of a test particle, obeying Hamiltonian (2). The functions I(t) and φ(t) act as external fields and are here imposed by assuming their simplified asymptotic behaviour as obtained by a frequency analysis [START_REF] Laskar | Introduction to Frequency Map Analysis[END_REF] :

2 I(t) N e iφ(t) ≈ F -ǫ K k=1 W k e iω k t ,
in the reference frame of the wave.

Hamiltonian control of a test-particle

The Hamiltonian control addresses systems which are close to integrable, i.e. whose Hamiltonian can be written as H = H 0 + V , where H 0 is integrable and V a perturbation of order ǫ (compared to H 0 ). The results we use here have been proven rigorously [START_REF] Vittot | Perturbation theory and control in classical or quantum mechanics by an inversion formula[END_REF][START_REF] Chandre | Control of stochasticity in magnetic field lines[END_REF]. In practice, it can be shown that a suitable control term f of order ǫ 2 exists such that H 0 + V + f has an invariant torus at a given frequency ω 0 . In our case, the perturbation corresponds to the oscillating part of the intensity. The interaction term of Hamiltonian (2) reads 2

I(t) N cos (θ -φ(t)) = F cos θ -ǫRe(e iθ W(t)).
Thus, our integrable Hamiltonian can be cast in the form

H 0 = p 2 2 -F cos θ, (3)
whereas the time-dependent perturbation V is identified as

V (t, θ) = ǫRe(e iθ W(t)). ( 4
)
First, we express Hamiltonian (3) into action-angle variables (ϕ, J) [START_REF] Lichtenberg | Regular and Stochastic Motion[END_REF]. Then, we expand H 0 around J = J 0 , which in turn identifies the region where the invariant torus is reconstructed in :

H 0 (J) = E 0 + ω 0 (J -J 0 ) + δ(J -J 0 ) 2 + O((J -J 0 ) 3 ).
Likewise, the θ-component of perturbation ( 4) is expanded as

e iθ = M m=0 L n=-L α m,n (J -J 0 ) m e inϕ + O((J -J 0 ) M +1 )
The control term reads [START_REF] Chandre | Control of stochasticity in magnetic field lines[END_REF] f

(ϕ, t) = V (J 0 , ϕ, t) -V (J 0 -Γ∂ ϕ V (J 0 , ϕ, t), ϕ, t),
where Γ is a linear operator acting on an element of the Fourier basis as :

Γe i(ωt+nϕ) = e i(ωt+nϕ) i(ω + nω 0 )
.

At the second order in ǫ (which is the main term), the control term reads

f (ϕ, t) = ǫ 2 w(ϕ, t)Γ∂ ϕ v(ϕ, t) -ǫ 2 δ(Γ∂ ϕ v(ϕ, t)) 2 ,
where v and w are the first terms in the expansion of V :

V (J, ϕ, t) = v(ϕ, t) + (J -J 0 )w(ϕ, t) + +O((J -J 0 ) 2 ), which gives Γ∂ ϕ v(ϕ, t) = K k=0 L n=-L nα 0,n W k ω 0 n + ω k e i(nϕ+ω k t) , and
w(ϕ, t) = K k=0 L n=-L α 1,n W k e i(nϕ+ω k t) ,
where W k are the Fourier coefficients of W :

W (t) = K k=0 W k e iω k t .
Numerically, two Fourier modes are considered for W (t) (K = 2), and eleven for the θ-dependent part of the perturbation (4) (L = 5). The perturbation is of amplitude ǫ = 1/5. The expansion is performed around J 0 ≈ 1.33 (which corresponds to the energy E 0 = 0).

In action-angle variables, the regularization of the dynamics (see Fig. 2) is clearly seen. Rather than a single torus, the control term generates a continuous set of invariant tori, thus expanding the regular domain in phase space.

Unfortunately, the exact change of variables from (ϕ, J) to (θ, p) [START_REF] Lichtenberg | Regular and Stochastic Motion[END_REF] has a singularity at the pendulum separatrices. In order to implement our control on the whole space, we use a simplified, but regular, change of variables which mimics the exact one in the region of the invariant torus predicted by the control (the error in this region is less than 4%) (see Fig. 3) :

tan ϕ = θ p .
In terms of the (θ, p) variables, our control term now reads :

f(θ, p, t) = f (arctan θ p , t),
and its regularity is now the one of the function (θ, p) → arctan θ p . Therefore, the controlled dynamics of a test-particle is given by Hamiltonian : In the (θ, p) variables (see Fig. 4), the control is successful in reconstructing some invariant tori around the macro-particle. In other words, it enlarges this regular structure.

H c (θ, p, t) = H 1p (θ, p, t) + f (θ, p, t).

Conclusion

In this paper, we considered a simplified mean-field approach to investigate the saturated dynamics of a Single Pass FEL. In particular, we showed that the size of the macro-particle can be increased by adding a small pertubation to the system, thus resulting in a low cost correction in term of energy. The main idea is to build invariant tori localized at specific positions : this method is utterly general and could be succesfully used to adjust the size of the macroparticle, thus possibly enhancing the bunching factor. Analogously, by limiting the portion of phase-space spanned by the macro-particle, one could aim at stabilizing the laser signal. Two future lines of investigation are foreseen. First, we intend to implement the computed control term in the framework of the original N-body self-consistent picture and explore possible beneficial effects for the evolution of the radiation field. Further, it is planned to apply the above technique to the case of the reduced Hamiltonian of [START_REF] Tennyson | Self-Consistent Chaos in the Beam-Plasma Instability[END_REF][START_REF] Antoniazzi | Statistical mechanics and Vlasov equation allows for a simplified Hamiltonian description of Single-Pass Free Electron Laser saturated dynamics[END_REF]. We point out that an experimental test of the control method on a modified Travelling Wave Tube has been done [START_REF] Chandre | Channeling chaos by building barriers[END_REF] in absence of self-consistency.

Figure 1 .

 1 Figure 1. Left : Normalized intensity of the FEL's radiation simulated from Hamiltonian (1). Right : N = 10000 electrons in phase space, at t = 400, 401.25, 402.5 and 403.75.

Figure 2 .Figure 3 .

 23 Figure 2. Poincaré sections of a test-particle of Hamiltonian H 0 (J)+ V (J, ϕ, t) (left) and H 0 (J) + V (J, ϕ, t) + f (ϕ, t) (right), in the pendulum action-angle variables.

Figure 4 .

 4 Figure 4. Poincaré sections of a test-particle of Hamiltonian H 1p (θ, p, t) (left) and H 1p (θ, p, t) + f (θ, p, t) (right).


Ross Møller

Knuth Nickel Babuska

Malcolm Dekker

Pichat Neumaier Kahan

Kulisch

Bohlender Bohlender

Mosteller / Tukey

Linnaimaa Leuprecht

Oberaigner Jankowski

/ Semoktunowicz

/- Wozniakowski Jankowski

Wozniakowski Kahan

How to manage accuracy and speed? A new "better" algorithm every year since 1999

Precision

u= arithmetic precision u = 2 -53 ≈ 10 -16 for b64 in IEEE-754 (2008) Accuracy for backward stable algorithms Accuracy of the computed sum ≤ (n -1)

× cond × u cond( x i ) = |x i | | x i |
No more significant digit in IEEE-b64 for large cond, i.e. > 10 16

More accuracy . . . More precision: double-double, quad-double, . . . Reliable and significant measure of the time complexity?

The classic way: count the number of flop A usual problem: double the accuracy of a computed result A usual answer for polynomial evaluation (degree n) How to chose the data test? Next step for f.p. summation: reproducibility to improve productivity Web site with common and shared resources: tested + test + make file sources, data files and generators, real and abstract associated measures

) (i) (i) (i+1) r (n-1) (n-2) (n-3) (n-4) (n-5) (0) (1) (2) (3) (4) (
Open and dynamic interaction: load your new algorithm, your new data, run them and let's contribute architectures? compilers?

suggestions and partners are welcome!

  Compensated algorithms: Kahan(72), . . . , Sum2(05), SumK(05) Accuracy of the computed sum u + cond × u K . . . but still depending on the conditioning Distillation: iterate until faithful or exact rounding Error free transformation of [x] → [x (1) ] → • • • → [x * ] such that x i = x * i and [x * ] provides the expected rounded value. Kahan (87), . . . , Zhu-Hayes: iFastSum (SISC-09) More space to keep everything Long accumulator, hardware oriented: Malcolm (71), Kulish (80) Cut the summands: AccSum (SISC-08), FastAccSum (SISC-09) Sum by fixed exponent: HybridSum (SISC-09), OnLineExact (TOMS-10) From faithful to exact rounding -→ Run-time and memory efficiencies are now the discriminant factors Why measure summation algorithm performance? How to measure summation algorithm performance?

  pintool to analyse and visualise the ILP of x86-coded algorithms Pin (Intel) tool (http://www.pintool.org) Outputs: ILP measure (#C, #I), IPC histogram, data-dependency graph Input: x86 64 binary file Developed and maintained by B. Goossens and D. Parello (DALI) Some recent accurate and fast summation algorithms Twice more precision Sum2: Compensated with a VectSum that uses TwoSum DDSum: Recursive sum + double-double arithmetic Faithful or exact rounding iFastSum: SumK with dynamic error control AccSum and FastAccSum: Adaptative computational effort wrt cond. Split the summands by chunk that sums exactly (width depends on n), careful sum of the chunks. Chunk cutting-line fixed in AccSum while more dynamic in FastAccSum HybridSum and OnLineExactSum: Exponent extraction of the summands, careful accumulation in one (HS) or two vectors (OLE) of fixed and short length (2048 in IEEE-b64), and distillate the (very for OLE) short vector with iFastSum.

  The limited Accuracy of Performance Counter MeasurementsWe caution performance analysts to be suspicious of cycle counts . . . gathered with performance counters.

	How to trust non-reproducible experiment results? How to read the current literature?
	Numerical results in S.M. Rump contributions (for summation) Instruction Level Parallelism (ILP) describes the potential of the instructions of Measures are mostly non-reproducible A synthetic sample: e = (a+b) + (c+d) A synthetic sample: e = (a+b) + (c+d) A synthetic sample: e = (a+b) + (c+d) A synthetic sample: e = (a+b) + (c+d)
	The execution time of a binary program varies, even using the same data 26% for Sum2-SumK (SISC-05) : 9 pages over 34 a program that can be executed simultaneously x86 binary x86 binary Instruction and cycle counting x86 binary Instruction and cycle counting x86 binary Instruction and cycle counting x86 binary Instruction and cycle counting x86 binary
	input and the same execution environment. ... ... ... Cycle 0: i1 ... Cycle 0: i1 ... Cycle 0: i1 ... 20% for AccSum (SISC-08) : 7 pages over 35 i2 i2 i2 Hennessy-Patterson's ideal machine (H-P IM) mov eax,DWP[ebp-16] i1 mov eax,DWP[ebp-16] i1 mov eax,DWP[ebp-16] i1 mov eax,DWP[ebp-16] i1 mov eax,DWP[ebp-16] i1 mov eax,DWP[ebp-16] i1	i4 i4 i4
	Metric Flop count Flop count ratio Measured #cycles ratio Why? Experimental uncertainty of the hardware performance counters Horner CompHorner DDHorner 2n 22n + 5 28n + 4 1 ≈ 11 ≈ 14 1 2.8 -3.2 Spoiling events: background tasks, concurrent jobs, OS interrupts every instruction is executed one cycle after the execution one of the mov edx,DWP[ebp-20] i2 add edx,eax i3 mov edx,DWP[ebp-20] i2 add edx,eax i3 mov edx,DWP[ebp-20] i2 add edx,eax i3 mov edx,DWP[ebp-20] i2 add edx,eax i3 Cycle 1: i3 i5 mov edx,DWP[ebp-20] i2 add edx,eax i3 i5 mov edx,DWP[ebp-20] i2 Cycle 1: i3 add edx,eax i3 producers it depends mov ebx,DWP[ebp-8] i4 mov ebx,DWP[ebp-8] i4 mov ebx,DWP[ebp-8] i4 mov ebx,DWP[ebp-8] i4 mov ebx,DWP[ebp-8] i4 mov ebx,DWP[ebp-8] i4 Non deterministic issues: instruction scheduler, branch predictor no other constraint than the true instruction dependency (RAW) add ebx,DWP[ebp-12] i5 add ebx,DWP[ebp-12] i5 add ebx,DWP[ebp-12] i5 add ebx,DWP[ebp-12] i5 add ebx,DWP[ebp-12] i5 Cycle 2: i6 add ebx,DWP[ebp-12] i5 8.7 -9.7 External conditions: temperature of the room add edx,ebx i6 add edx,ebx i6 add edx,ebx i6 add edx,ebx i6 add edx,ebx i6 add edx,ebx i6
	Flop count vs. run-time measures Timing accuracy: no constant cycle period on modern processors (i7. . . ) Measure the #cycles and the #IPC running the code with the H-P IM ... ... ... ... ... ...
	Flop counts and measured run-times are not proportional Uncertainty increases as computer system complexity does The picture is blurred: the computing chain is wobbling around maximal exploitation of the program ILP
	Run-time measure is a very difficult experimental process Which one trust? If we combine all the published speedups (accelerations) on the well processor and ILP in practice: superscalar and out-of-order execution Architecture and micro-architecture issues: multicore, hybrid, speculation execution times approaching to zero? S. Touati (2009) Compiler options and its effects known public benchmarks since four decades, why don't we observe ILP measures the potential of the algorithm performance

20% for AccSumK-NearSum (SISC-08b) : 6 pages over 30 less that 3% for FastAccSum (SISC-09) : 1 page over 37 Lack of proof, or at least of reproducibility Measuring the computing time of summation algorithms in a high-level language on today's architectures is more of a hazard than scientific research. S.M. Rump (SISC, 2009) . . . in the paper entitled Ultimately Fast Accurate Summation Software and System Performance experts' point of view D. Zaparanuks, M. Jovic, M. Hauswirth (2009) Can Hardware Performance Counters Produces Expected, Deterministic Results?

In practice counters that should be deterministic show variation from run to run on the x86 64 architecture. . . . it is difficult to determine known "good" reference counts for comparison. V.M.

[START_REF] Weaver | Can hardware performance counters produce expected, deterministic results?[END_REF]

  ∈ [10 3 , 10 7 ] and cond ∈ [10 8 , 10 40 ] = [ 1/u, 1/u 2.5 ] Not abstract enough: instruction set dependence, compiler choice Good abstraction level? Assembler program or high level programming language?

	Time complexity parameters of the summation algorithms Only n for Sum2, SumK: constant accuracy improvement n and cond for AccSum, iFastSum: adaptive accuracy improvement Exponent range of the summands: Z-H exhibit no influence for HybridSum and OnlineExactSum for large n Rump's generator of arbitrary ill-conditioned dot product (SISC-05), modified for summation and to cover an arbitrary exponent range. Low level choices are crucial n 2sum Fast2sum 2sum Fast2sum 10 3 13 11 13.7 9 10 4 6.5 6.5 5.8 4 10 5 5 5.5 3.8 3.3 10 6 4.7 5.6 3.8 3.3 Cycle ratios (vs. Sum) vary for different EFT and compilers Conclusion How to measure summation algorithm performance? ILP and the PerPI Tool Experiments with recent summation algorithms Conclusion Highly accurate algorithm -→ reliable performance evaluation Hardware counter based measure: uncertainty and no reproducibility PerPI: a software platform to analyze and visualise ILP Reliable: reproducibility both in time and location Realistic: correlation with measured ones Useful: a detailed picture of the intrinsic behavior of the algorithm Optimisation tool: analyse the effect of some hardware constraints [GLPP-Para10] Computing time: More science? Less hazard? No definitive answer Length: n Focus inside OnLineExact cond = 10 16 gcc icc Why measure summation algorithm performance? Flop count: not significant PerPI result is far from perfect
	Conclusion Exploratory tool: gives us the taste of the behavior of our algorithms
	running on "tomorrow" processors
	25 / 30


Siyamak Sarabi
email: siyamak.sarabi@hei.fr

Arnaud Davigny
email: arnaud.davigny@hei.fryann.riffonneau@sncf.frvcourtecuisse@geredis.fr

Yann Riffonneau

Vincent Courtecuisse

Benoit Robyns
email: benoit.robyns@hei.fr

Benoit

CONTRIBUTION AND IMPACTS OF GRID INTEGRATED ELECTRIC VEHICLES TO THE DISTRIBUTION NETWORKS AND RAILWAY STATION PARKING LOTS

HAL is

INTRODUCTION

Nowadays, the concept of smart grid and its definition, application, impacts and contribution to the conventional electrical grid are widely discussed in academic and industrial societies. The problem of traditional energy sources restrictions, their environmental impacts and technical criteria improvement of electrical grids lead to introducing new concept of grids which have bidirectional contribution between production and consumption. In such idea, the presence of renewable energy sources such as solar and wind and electric vehicle (EV) with vehicle-to-grid (V2G) technology become highlighted. These distributed generation sources can be stored in peak hours and reused during off-peak hours. On the contrary, it is important also to take into consideration the unwanted impacts of growing electrified transportation on the electrical grid, due to the coincidence of daily load peak and charging time of EVs [START_REF] Shafiee | Investigating the Impacts of Plug-in Hybrid Electric Vehicles on Distribution Congestion[END_REF]. In France, a perspective planning estimates 2 million electric vehicles up to 2020 [START_REF] Negre | Green Book on charging infrastructure of electric vehicle[END_REF]. ERDF, an electricity distribution network company in France for a fleet of 1 million EV has estimated 2.5TWh annual energy consumption during charging process, which counts 0.5% of total consumption in 2013. While, in sense of instantaneous power demand for case of rapid charging with 43 kW charging power, 47 % of available power of the national grid can be solicited. This can be reached up to 100% at MV and LV distribution network level [START_REF]ERDF website[END_REF]. To cope with upcoming issues due to this large demand, the implementation of charging infrastructures and supervised intelligent charging/discharging energy management strategies are in hand projects for future. In order to define potentials of electric vehicles to respond to the grid services, the impacts of different scenarios, containing coordinated and uncontrolled charging at different operational modes, should be taken into consideration.

In this paper, the impacts of uncontrolled charging of EV fleets and coordinated charging planning with optimization will be analysed on two case studies. The first case is a railway station which is serving the EV charging infrastructure in its parking (20/0.4 kV) and the second case is a distribution grid at a distribution substation level (90/20-15 kV). Both cases will be presented precisely thereafter. The remainder of this paper is organised as follow. First of all, the charging profile of electric vehicle fleet considering their arrival SOC, driving distance and battery capacity are calculated. Secondly, the uncontrolled charging scenarios for each case study with their specifications will be presented. Afterward, the contribution of EV fleet by charging coordination and V2G capacity offer will be explained for each case study. Then the results and comparison will be addressed and finally the paper will be concluded.

ELECTRIC VEHICLE CHARGING PROFILE

Plug-in electric vehicles charging process depends on different criteria: battery capacity and autonomy of the vehicle, arrival state of charge (SOC) and desired departure SOC and charging power rate of the charging system. In France, three charging power rates based on the location and occasion of charging have been considered. These configurations are based on IEC 62196 which is an international standard for set of electrical connectors and charging modes for electric vehicles maintained by International Electrotechnical Commission [4]. Normal Charging (NC), a single phase AC charging with 3 to 3.7 kW charging power rate, Accelerated Charging (A.C.), a three phase AC charging with 23 kW charging power rate and Rapid Charging (RC), a three phase AC charging with 43 kW charging power rate. In this study, we assumed to have basic information for calculation of charging profile of EV fleet for each case study. To calculate the charging profile following equations should be considered [START_REF] Sarabi | Traffic-based Modeling of Electric Vehicle Charging Load and its Impact on the Distribution Network and Railway Station Parking Lots[END_REF].

ᵃᵃᵃ = 1 - ᵃ ᵃᵅ × 100% (1)
ᵃ = (1 -ᵃᵃᵃ ) × ᵃ × (2) ᵃ = ᵃ ᵃ (3)
ᵃᵃ (ᵅ) = ᵃ 0 , ᵃ < ᵅ < ᵃ + ᵃ , ᵃ"ᵅᵅᵃ"ᵅℎᵃ"ᵅᵃ" (4) Where in (5) arrival SOC is calculated based on ᵃ driving distance of EV i and ᵃᵅ the autonomy of the same EV. ᵃ is the energy needed by the same EV to be fully charged with its battery capacity of ᵃ in kWh and charging efficiency of . The charging time ᵃ of this EV can be obtained from its energy need and charging rate ᵃ of the charging station (ᵃ can be 3, 23 or 43 kW based on chosen type of charging).

ᵃ (ᵅ) = ᵃᵃ (ᵅ) = , 1 < ᵅ < 144 (

UNCONTROLLED CHARGING SCENARIOS

In the concept of uncontrolled charging of EV fleet, there is no control signal from supervision system through charging infrastructure to the EV plugged-in to the grid. In other words, like most of the electric devices the users can charge their device's battery just by plugging the device in to the socket and charging process will start instantaneously. This can be the same for EV arriving to home, office or public charging stations and simply start to be charged just by being plugged-in. In this study, we have considered that all the EVs just after their arrival time will be plugged-in and charged. The types of electric vehicle considering in this study are based on current EV market contribution of different EV producers in France [START_REF] Sarabi | Supervision of Plug-in Electric Vehicles Connected to the Electric Distribution Grids[END_REF]. In addition the daily driving distance based on statistical data is normally distributed between 10 and 50 km per trip. The two case studies with their characteristics and charging scenarios are as follow;

Case A: Railway Station Parking Lots

The railway stations in France are the subject of smart grid developments with possible energy interactions between the grid and the users. Their connection to the distribution grid is through a MV/LV substation with range of 20-15/0.4 kV. These stations are supposed to act as an energy hub with different components such as local Renewable Energy Sources (RES) and electric vehicle charging stations. To analyse the impact of daily arrived EV to the parking, different scenarios have been considered. In this study the possibility and contribution of coordinated charging/discharging is also taken into account. These scenarios are based on arrival time and departure time of the EVs to the parking. The arrival time of the vehicles are distributed normally between 7:00 to 9:00 whereas the departure time is distributed normally between 16:00 to 20:00. The average availability interval is around 10 hours during which EV can be used for charging coordination. The total number of 10 charging station has been considered as base case. The capacity of the railway station is also evaluated from 1 to 50 charging stations. Charging stations are divided into several types of charging process, different cases have been studied and are described as follow; Normal Charging (NC) Normal charging case is the principle charging option for majority of EVs with 3 to 3.7kW charging rate. For an EV with 20 kWh battery capacity, it takes 6 to 8 hours to be fully charged. In this scenario all the vehicles arriving to the parking will choose NC charging mode. Accelerated Charging (A.C) Accelerated charging case is the charging mode with three phase circuit and 23 kW charging rate power. For an EV with 20 kWh battery, 1 hour is enough to be fully charged. In this scenario all the EVs are considered as choosing AC charging mode.

Rapid Charging (RC) Rapid charging is the case for charging station in highways and freeways where the users need to have their battery fully charged very short. Charging rate of 43 kW lead to charging a 20 kWh battery in less than half an hour. This scenario is also considered that all EV in the parking can be charged at this rate.

Mixed Charging (MC)

In this option 40% of the station are RC charging mode, 20% of them are AC charging mode and 40% of the rest are NC charging mode.

Case B: Distribution Network Substation

For the distribution system operators (DSO) managing the load is a major issue specially when there is local RES production at their network. The presence of EV load is also important as their simultaneous charging load can cause major problem for the grid. In this study we have focused on a distribution substation in the range of 90/20-15 kV with 36 MVA transformers. This grid has also distributed wind productions, which motivates to use its production locally. With rate of EV production in France, it is supposed for this grid to serve near 1300 EVs for the year 2020. The uncontrolled charging profile of this fleet will be analysed where all the EVs have been considered as Normal charging mode arrived around 8:00 at office parking and ask for charging instantaneously. Their departure time is also around 17:00.

CONTRIBUTION OF ELECTRIC VEHICLES

Statistical studies show that in majority of the world cities, the vehicles which are used for daily home/work trips are parked and unused near to 95% of the daily time. This brings high potential for EVs to provide ancillary services for the electrical grid [START_REF] Sarabi | The Feasibility of the Ancillary Services for Vehicle-to-Grid Technology[END_REF]. Reference [START_REF] Sarabi | The Feasibility of the Ancillary Services for Vehicle-to-Grid Technology[END_REF] concludes that the peak power shaving service can be competitive for EV fleet where the interests cover the majority of stakeholders (e.g. TSO, DSO, energy provider and customers). This action can be implemented either by shifting the charging time of the vehicle to the off-peak hours or using V2G technology for discharging the batteries during peak hours and charging them again at off-peak periods. The impacts of this contribution strategy in contrary with uncontrolled charging scenarios are discussed in the rest of the paper for each case study. For both cases the coordination will be analysed with NC Annual energy bill is based on different tariffs of subscribed power. For under studying railway station a tariff fixed by energy supplier company in France (EDF), for users with more than 240 kVA subscribed power is considered. The annual invoice is calculated based on the following equation.

Where the first part of second side is related to consumed energy, ᵃ , in kWh by its price, ᵃ! , in €/kWh for 5 periodical tariffs j. The second part is related to penalty for subscribed power violation (SPV) with ᵃ the reduction coefficient related to each tariff interval and the price of kW violation defined by ᵃ in €/kW. ∆ᵃ is the amplitude of each violation in each 10 minutes average time. Finally, the third part which is a fixed value defined by the subscribed power ᵃ and base rate value in €/kW/year. We can now rewrite the Cost function based on load profile to define the best load profile minimizing the cost function:

ᵃᵅᵅᵅ = (ᵃ! ᵃᵃ (ᵅ) ᵃ!(ᵅ) ) = + ᵃ. ᵃ (ᵃᵃ (ᵅ) -ᵃ (ᵅ)) + . ᵃ (7)
The optimum load profile (ᵃᵃ ) obtained from this problem will be used in second step as an objective for charging/discharging coordination block. This block aims to minimize the difference between optimum LP and LP which is hosting the EVs by coordinating the appropriate charging/discharging time intervals. The problem is formulated in specific form of Mixed Integer Linear Programming (MILP) so called as Binary Linear Programming (BLP) where the acceptable variable can be just zeros and ones [START_REF] Jw | Practical optimization: a gentle introduction[END_REF].

ᵃᵃᵃᵃ = ⎩ ⎨ ⎧ ᵅᵅᵅ ᵃ ᵅ ᵃ ᵅ = ᵃ ᵃ. ᵅ ≤ ᵃ ᵅ ∈ ℤ (8) ᵃᵃ (ᵅ) = [ᵃ , ᵃ , ᵃ , … , ᵃ ] × ᵃ ᵅ = [1, ᵅ] ∈ ℕ, ᵃᵅᵅᵃᵃ"ᵅ ᵅᵃ# ᵃᵃᵅ (9) ᵃᵃᵅ (ᵅ) = ᵃᵃᵃ (ᵅ) -ᵃᵃ (ᵅ) (10) ᵯ (ᵅ) = -|ᵃᵃᵅ (ᵅ)| (11) ⎩ ⎨ ⎧ ᵃ (ᵅ) = ᵯ (ᵅ) × ᵃ ᵅ = [1,144] ∈ ℕ, 10 ᵅᵅᵅ ᵅᵅᵃ"ᵅ ᵅᵅ 24 ℎᵅᵅᵅᵅ ᵃ = [0,1], ᵃᵅᵅᵃᵅᵅ (12)
ᵆᵆ#ᵆ+ᵆ&ᵆ$ᵆ5ᵆ*ᵆ7ᵆ&:

ᵅᵅᵅ … ᵃ (ᵅ) = s.t. ( 13
) ᵃ . = ᵃ = ᵃᵃᵃ (14) ᵃᵃᵃ ≤ ᵃ. = ᵃ ≤ ᵃᵃᵃ (15) ᵃ = ᵃᵃᵅ (ᵅ) |ᵃᵃᵅ (ᵅ)| , ᵃ = [-1,1] ∈ ℤ ( 16
)
As it is shown in the equation the objective is to find the best charging/discharging schedule (ᵃᵃ (ᵅ)) which minimizes the cost function. Equation [START_REF] Sarabi | Supervision of Plug-in Electric Vehicles Connected to the Electric Distribution Grids[END_REF] gives the standard formulation of MILP problems. ᵃᵃᵅ (ᵅ), is the difference between current daily load profile (DLP) and optimum LP that should be fulfilled by EV charging schedule.

To prevent trivial answer of all zeros for ᵃ , the coefficient of objective function will be introduced in form of ᵯ (ᵅ) function. The vector ᵃ also defines at which time step the vehicles can be charged [START_REF] Shafiee | Investigating the Impacts of Plug-in Hybrid Electric Vehicles on Distribution Congestion[END_REF] and at which one it can be discharged (-1). The constraints of the optimization problem are the availability of the vehicles and their desire SOC at departure time. To prevent excessive negative impacts of battery cycle life in V2G mode the minimum SOC of 20% is also considered [START_REF] Sarabi | The Feasibility of the Ancillary Services for Vehicle-to-Grid Technology[END_REF]. An example of algorithm output for 10 EVs is depicted in figure below. The consumption load profile without EVs (CON) is compared with uncontrolled NC scenario (EVN) and coordinated algorithm (EVC). It shows that the algorithm is able to allocate the EVs load in somehow that the Subscribe Power (SP) violation is avoided, load variation is reduced and energy is consumed at lower cost periods.

The process for set of different number of EVs has been evaluated where the results are brought in Table I and II. The results in benefit rows are the percentage of each invoice reduction thanks to using coordinated algorithm.

The point is that these benefits in sense of SPV are very important. Also for energy invoice, increased number of EVs leads to more benefit percentage. The results of different uncontrolled charging scenarios explained in previous section (NC, MC, A.C and RC) and coordinated strategy are compared using SPV indicator in function of different number of EVs. These results are depicted in Fig. 3. As it is shown, in all of the uncontrolled scenarios the SPV is increased linearly as the number of EVs is increased. While in coordinated scenario (EVC), the SPV factor has small rate of increment and this leads to serving higher number of EVs without excessive impact on energy invoice by minimizing the range of SPV. To have it minimized, the same problematic as previous case study is taken into account with constraints of EVs availability, battery SOC limitation, RES intermittency, energy tariff and subscribed power limit. This optimization problem is solved using interior point method for its nonlinear objective function. The results for some sample days are depicted in following figures. In Fig. 4, three scenarios have been presented at upper subplot. The "PS without EV" is the base scenario where it comes from difference between consumption and local RES. This is the power measured at distribution substation entrance so called as PS point and shows how much power is absorbed or injected to the grid. PS-NOEV is the scenario of uncontrolled charging abbreviated from Non Optimized EV and finally PS-OEV which is the Optimized EV scenario. As it is shown in PS-NOEV the vehicles are charged at their arrival time to the office parking. While in PS-OEV the EVs are not charged to avoid extra SPV and additionally they are discharged because of high cost energy period between 9:00 and 11:00. Instead, they are charged during lower III shows the contribution of EV fleet thanks to V2G technology for SPV and EA components. In fact by using V2G mode, the EVs can reduce the possible SPV caused by conventional load and bring 37.98% benefit over SPV component. Also for EA components thanks to the V2G some part of load profile in high cost energy intervals can be transferred to lower cost interval. This brings 0.44% reduction for EA component.

CONCLUSION

The impacts and contribution of electric vehicles on the distribution grid and railway station parking lots have been assessed. Different uncontrolled charging modes with coordinated charging/discharging strategy have been compared. For railway stations serving EV charging infrastructure, the presence of energy management system leads to increase number of EVs that can be hosted by parking. In addition the annual energy invoice paid by station can be reduced up to 6.32% for case of 10 EVs. Also the more EVs are served the more the benefit is important. For distribution grid with RES production coordination strategy can provide ETC minimization. In addition V2G storage capacity can provide subscribed power optimization and absorbed energy price reduction. As a perspective of the work the concentration will be on V2G constraints related to battery discharging constraints to find an appropriate point between both grid utilities and EV owners' benefits. In addition the efforts will be on designing a real-time based supervision strategy that can bring benefits near to the obtained optimized results.

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Fig. 1 :

 1 Fig. 1: Optimization process for energy invoice minimization.

Fig. 2 :

 2 Fig. 2: Upper subplot: Example of a winter day showing charging/discharging coordination, Lower subplot: energy tariff.

Fig. 3 :

 3 Fig. 3: Different scenario impacts on SPV indicator.Case B: Energy Transmission Cost MinimizationThe distribution grid system operators have to pay the transmission system operators (TSO) a yearly bill related to energy transmission which is called energy

Fig. 4 :

 4 Fig. 4: Example of SPV avoidance and low price energy consumption.

Fig. 5 :

 5 Fig. 5: Example of RES surplus consumption by EVs.