M Capitaine

C Donati-Martin

Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices

In this paper, we investigate the fluctuations of a unit eigenvector associated to an outlier in the spectrum of a spiked N × N complex Deformed Wigner matrices M N . M N is defined as follows:

where W N is an N × N Hermitian Wigner matrix whose entries have a symmetric law µ satisfying a Poincaré inequality and the matrix A N is a block diagonal Hermitian deterministic matrix A N = diag(θ, A N -1 ), θ ∈ R has multiplicity one and generates an outlier in the spectrum of M N . In the diagonal case, we prove that the fluctuations of a unit eigenvector corresponding to this outlier of M N are not universal: the limiting distribution is the convolution of µ and a Gaussian distribution. In the block diagonal case, the limiting law is replaced by a sequence of approximations with a Gaussian distribution depending on the size N , whose variance may not converge.

Introduction

Wigner matrices are complex Hermitian random matrices whose entries are independent (up to the symmetry condition). They were introduced by Wigner in the fifties, in connection with nuclear physics. Here, we will consider Hermitian Wigner matrices of the following form :

W N = 1 √ N H N
where H N is an Hermitian matrix whose diagonal entries are iid real random variables and those above the diagonal are iid complex random variables, with variance σ 2 . If the entries are independent Gaussian variables, W N =: W G N is a matrix from the Gaussian Unitary Ensemble (G.U.E.).

1

There is currently a quite precise knowledge of the asymptotic spectral properties (i.e. when the dimension of the matrix tends to infinity) of Wigner matrices. This understanding covers both the so-called global regime (asymptotic behavior of the spectral measure) and the local regime (asymptotic behavior of the extreme eigenvalues and eigenvectors, spacings...). Wigner proved that a precise description of the limiting spectrum of these matrices can be achieved.

Theorem 1. [START_REF] Wigner | Characteristic vectors of bordered matrices with infinite dimensions[END_REF][START_REF] Wigner | On the distribution of the roots of certain symmetric matrices[END_REF] µ W N w -→ µ sc a.s. when N → +∞ where

dµ sc dx (x) = 1 2πσ 2 4σ 2 -x 2 1 [-2σ,2σ] (x) (1)
is the so-called semi-circular distribution.

A priori, the convergence of the spectral measure does not prevent an asymptotically negligeable fraction of eigenvalues from going away from the limiting support (called outliers in the following). Actually, it turns out that Wigner matrices do not exhibit outliers. Theorem 2. [START_REF] Bai | Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix[END_REF] Let W N be a Wigner matrix. Denote by λ i (W N ) the eigenvalues of W N ranked in decreasing order. Assume that the entries of H N has finite fourth moment, then almost surely, λ 1 (W N ) → 2σ and λ N (W N ) → -2σ when N → +∞.

In [START_REF] Tracy | Level-spacing distributions and the Airy kernel[END_REF], Tracy and Widom derived the limiting distribution (called the Tracy-Widom law) of the largest eigenvalue of a G.U.E. matrix. Theorem 3. Let W G N be a G.U.E. matrix. Let q : R → R be the unique solution of the differential equation q (x) = xq(x) + 2q(x) 3 such that q(x) ∼ x→+∞ Ai(x) where Ai is the Airy function, unique solution on R of the differential equation f (x) = xf (x) satisfying f (x) ∼ x→+∞ (4π √ x) 1/2 exp(-2/3x 3/2 ). Then lim

N →+∞ P N 2/3 λ 1 (W G N ) √ N -2σ ≤ s = F 2 (s),
where

F 2 (s) = exp - +∞ s (x -s)q 2 (x)dx .
The first main step to prove the universality conjecture for fluctuations of the largest eigenvalue of Wigner matrices has been achieved by Soshnikov [START_REF] Soshnikov | Universality at the edge of the spectrum in Wigner random matrices[END_REF]; in [START_REF] Lee | A necessary and sufficient condition for edge universality of Wigner matrices[END_REF], a necessary and sufficient condition on off-diagonal entries of the Wigner matrix is established for the distribution of the largest eigenvalue to weakly converge to the Tracy-Widom distribution. We also refer to these papers for references on investigations on edge universality.

In regards to eigenvectors, it is well known that the matrix whose columns are the eigenvectors of a G.U.E. matrix can be chosen to be distributed according to the Haar measure on the unitary group. In the non-Gaussian case, the exact distribution of the eigenvectors cannot be computed. However, the eigenvectors of general Wigner matrices have been the object of a growing interest and in several papers, a delocalization and universality property were shown for the eigenvectors of these standard models (see among others [START_REF] Bloemendal | Isotropic local laws for sample covariance and generalized wigner matrices[END_REF][START_REF] Erdös | Local semicicle law and complete delocalization for Wigner random matrices[END_REF][START_REF] Erdös | Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices[END_REF][START_REF] Knowles | Eigenvector distribution of Wigner matrices[END_REF][START_REF] Tao | Random matrices: universal properties of eigenvectors[END_REF] and references therein). Heuristically, delocalization for a random matrix means that its normalized eigenvectors look like the vectors uniformly distributed over the unit sphere. Let us state for instance the following sample result. Theorem 4. (Isotropic delocalization, Theorem 2.16 from [START_REF] Bloemendal | Isotropic local laws for sample covariance and generalized wigner matrices[END_REF]). Let W N be a N × N Wigner matrix satisfying some technical assumptions. Let v(1), . . . , v(N ) denote the normalized eigenvectors of W N . Then, for any C 1 > 0 and 0 < < 1/2, there exists C 2 > 0 such that

sup 1≤i≤N | v(i), u | ≤ N √ N ,
for any fixed unit vector u ∈ C N , with probability at least 1 -

C 2 N -C 1 .
Practical problems (in the theory of statistical learning, signal detection etc.) naturally lead to wonder about the spectrum reaction of a given random matrix after a deterministic perturbation. For example, in the signal theory, the deterministic perturbation is seen as the signal, the perturbed matrix is perceived as a "noise", and the question is to know whether the observation of the spectral properties of "signal plus noise" can give access to significant parameters on the signal. Theoretical results on these "deformed" random models may allow to establish statistical tests on these parameters. A typical illustration is the so-called BBP phenomenon (after Baik, Ben Arous, Péché) which put forward outliers (eigenvalues that move away from the rest of the spectrum) and their Gaussian fluctuations for spiked covariance matrices in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF] and for low rank deformations of G.U.E. in [START_REF] Péché | The largest eigenvalue of small rank perturbations of Hermitian random matrices[END_REF].

3

In this paper, we consider additive perturbations of Wigner matrices. The pionner works on additive deformations go back to Pastur [START_REF] Pastur | On the spectrum of random matrices[END_REF] for the behavior of the limiting spectral distribution and to Füredi and Komlós [START_REF] Füredi | The eigenvalues of random symmetric matrices[END_REF] for the behavior of the largest eigenvalue. We refer to [START_REF] Capitaine | Spectrum of deformed random matrices and free probability[END_REF] and the references therein for a survey on spectral properties of deformed random matrices. The model studied is as follows :

M N := W N √ N + A N ,
where -W N is a complex Wigner matrix, that is a N × N random Hermitian matrix such that {W ii , √ 2RW ij , √ 2IW ij } 1≤i<j are independent identically distributed random variables with law µ. We assume that µ is a symmetric distribution, with variance σ 2 , and satisfies a Poincaré inequality (see Appendix).

-A N is a N × N deterministic Hermitian matrix, whose spectral measure µ A N = 1 N N i=1 λ i (A N ), the λ i (A N )'s denoting the eigenvalues of A N , converges to ν a compactly supported probability whose support has a finite number of connected components. We assume that A N has a fixed number p of eigenvalues, not depending on N , outside the support of ν called spikes, whereas the distance of the other eigenvalues to supp(ν) goes to 0.

The empirical spectral distribution µ M N = 1 N N i=1 λ i (M N ), the λ i (M N )'s denoting the eigenvalues of M N , converges a.s. towards the probability λ := µ sc,σ 2 ν where µ sc,σ 2 is the semicircular distribution with variance σ 2 and denotes the free convolution, see [START_REF] Pastur | On the spectrum of random matrices[END_REF] (in this paper, the limiting distribution is given via a functional equation for its Stieltjes transform), [START_REF] Anderson | An introduction to random matrices[END_REF]Theorem 5.4.5]. We refer to [START_REF] Voiculescu | Free random variables[END_REF][START_REF] Mingo | Free probability and Random matrices[END_REF] for an introduction to free probability theory. Concerning extremal eigenvalues, we proved in [START_REF] Capitaine | Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices[END_REF] that the spikes of A N can generate outliers for the limiting spectrum of M N , i.e. eigenvalues outside the support of the limiting distribution λ. More precisely, we proved the following (see [START_REF] Capitaine | Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices[END_REF]Theorem 8.1] for a more general statement).

Proposition 1. Denote by λ

i (A N ), resp. λ i (M N ), the eigenvalues of A N , resp. M N ranked in decreasing order. Assume that a spike θ = λ i 0 (A N ) of A N satisfies : θ ∈ {u ∈ R\supp(ν), R dν(x) (u -x) 2 < 1 σ 2 }, (2)
then a.s.,

λ i 0 (M N ) -→ N →+∞ ρ θ := θ + σ 2 g ν (θ)
where

g ν : z → 1 z-x dν(x)
is the Stieltjes transform of ν and ρ θ / ∈ supp(λ), i.e. ρ θ is an outlier.

It turns out that we can also describe the angle between the eigenvector associated to the outlier of M N and the corresponding eigenvector associated to the spike θ. Capitaine [START_REF] Capitaine | Additive/multiplicative free subordination property and limiting eigenvectors of spiked additive deformations of Wigner matrices and spiked sample covariance matrices[END_REF] (see also [START_REF] Capitaine | Spectrum of deformed random matrices and free probability[END_REF]) proved Proposition 2. We keep the notation and hypothesis of Proposition 1. Let ξ be a unit eigenvector associated to the eigenvalue λ i 0 (M N ). Then, a.s.

P Ker(A N -θI) (ξ) 2 -→ N →+∞ τ (θ) := 1 -σ 2 1 (θ -x) 2 dν(x). (3)
The aim of this paper is to study the fluctuations associated to the a.s. convergence given above. Note that fluctuations of outliers for deformed non-Gaussian Wigner matrices have been studied only in the case of perturbations A N of fixed rank r. We emphasize that the limiting distribution in the CLT for outliers depends on the localisation/delocalisation of the eigenvector of the spike. Roughly speaking, if the unit eigenvector associated to θ is delocalized, then the limiting distribution of the fluctuations of λ i 0 (M N ) around ρ θ is Gaussian. For localized eigenvector, the limiting distribution depends on the distribution µ of the entries and thus, this uncovers a non universality phenomenon. We refer to [START_REF] Capitaine | Central limit theorems for eigenvalues of deformations of Wigner matrices[END_REF] for these results. We first recall the fluctuations of the largest eigenvalue λ 1 (M N ) when the matrix A N is a diagonal matrix of rank 1 in the localized case, i.e. A N = diag(θ, 0, . . . , 0) with θ > σ.

Proposition 3. [9] The fluctuations of λ 1 (M N ) around ρ θ = θ + σ 2 θ are given by c θ √ N (λ 1 (M N ) -ρ θ ) (law) -→ N →∞ µ N (0, v 2 θ )
where

c θ = (1 -σ 2 θ 2 ) -1 and v 2 θ = 1 2 m 4 -3σ 4 θ 2 + σ 4 θ 2 -σ 2 .
Capitaine and Péché [START_REF] Capitaine | Fluctuations at the edges of the spectrum of the full rank deformed G[END_REF] proved a fluctuation result for any outlier of a full rank deformation of a G.U.E. matrix. Dealing with a diagonal matrix A N , with a spike λ i 0 (A N ) = θ of multiplicity one and limiting spectral distribution ν, their result yields:

Proposition 4. The fluctuations of λ i 0 (M N ) around ρ (N ) θ = θ + σ 2 1 N -1 λ j (A N ) =θ 1 θ -λ j (A N )
1 are given by :

c θ,ν √ N (λ 1 (M N ) -ρ (N ) θ ) (law) -→ N →∞ N (0, σ 2 θ,ν ) (4)
where

c θ,ν = 1 -σ 2 1 (θ-x) 2 dν(x) -1
and

σ 2 θ,ν = σ 2 + σ 4 1 (θ -x) 2 dν(x) 1 -σ 2 1 (θ -x) 2 dν(x) -1 = σ 2 1 -σ 2 1 (θ -x) 2 dν(x) -1 .
We give without proof an extension in the non-Gaussian case of Proposition 3 and Proposition 4. Proposition 5. Let A N be a diagonal matrix with a spike λ i 0 (A N ) = θ of multiplicity one and limiting spectral distribution ν. The fluctuations of

λ i 0 (M N ) around ρ (N ) θ = θ + σ 2 1 N -1 λ j (A N ) =θ 1 θ-λ j (A N )
are given by :

c θ,ν √ N (λ 1 (M N ) -ρ (N ) θ ) (law) -→ N →∞ µ N (0, v 2 θ,ν ) (5)
where

c θ,ν = 1 -σ 2 1 (θ-x) 2 dν(x) -1 and v 2 θ,ν = 1 2 (m 4 -3σ 4 ) 1 (θ -x) 2 dν(x) +σ 4 1 (θ -x) 2 dν(x) 1 -σ 2 1 (θ -x) 2 dν(x) -1 .
The proof of Proposition 5 follows the same lines as the case of a 1-rank matrix in [START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations[END_REF]. Recall that the proof relies on a CLT for quadratic forms. The main changes concern the limiting behavior of some coefficients arising in the CLT (see Appendix).

Remark 1. The expression of the variance v θ,ν is expressed only in terms of integrals in dν. The last term can also be written

1 (θ -x) 2 dν(x) 1 -σ 2 1 (θ -x) 2 dν(x) -1 = 1 (ρ θ -x) 2 d(µ sc ν)(x).
The above formula follows from the fact that ρ θ = H(θ) where H(x) = x + σ 2 g ν (x) and H(ω(x)) = x where w(x) = x -σ 2 g µsc ν (x).

In the following Theorem 5, we consider the simplest case where

A N = diag(θ, A N -1 ), θ ∈ R and A N -1 is a N -1 × N -1 diagonal deterministic matrix such that θ is not an eigenvalue of A N -1 .
In other words, θ is a spike of A N with multiplicity 1. The main result of this paper is the following non universality theorem on fluctuations of eigenvectors associated to the outlier of the deformed Wigner model. Theorem 5. Assume that θ = λ i 0 (A N ) satisfies (2). Let v i 0 be a unit eigenvector associated to the outlier

λ i 0 (M N ). Define τ N (θ) an approximation of τ θ by τ N (θ) = 1 -σ 2 1 (θ -x) 2 dµ A N -1 (x). (6)
Then,

√ N (| e 1 , v i 0 | 2 -τ N (θ)) (law) -→ N →∞ c θ,ν W 11 + Z (7)
in distribution, where Z is a centered Gaussian variable with covariance :

1 2 (m 4 -3σ 4 )A θ,ν + σ 4 B θ,ν (8)
where

   c θ,ν = σ 2 g ν (θ), A θ,ν = -g ν (θ)(1 + σ 2 g ν (θ)) 2 , B θ,ν = 2σ 2 (g ν (θ)) 2 (1 + σ 2 g ν (θ)) -g ν (θ)(1 + σ 2 g ν (θ)) 2 (9)
and g ν is the Stieltjes transform of ν.

As we have seen for eigenvalues, the fluctuations depend on the eigenvectors of A N . Without assumption on the eigenvectors of A N , we cannot expect to have convergence of the fluctuations of the eigenvector associated to the outlier. Nevertherless, when A N is a diagonal by block A N = diag(θ, A N -1 ), Theorem 5 can easily be extended by Proposition 6, replacing the convergence result by an approximation result in the spirit of [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF]. Proposition 6. If d LP denotes the Lévy-Prohorov distance, which is a metric for the topology of the convergence in distribution,

d LP ( √ N (| e 1 , v i 0 | 2 -τ N (θ)), Ψ N ) -→ N →∞ 0
where the r.v. Ψ N is given by

Ψ N = c θ,σ W 11 + Z N where c θ,σ = σ 2 g ν (θ) = 2σ 2 1 (θ -x) 3 dν(x),
and Z N is a centered Gaussian variable with variance

V ar(Z N ) = (10) 1 π 2 (C) 2 ∂F k (h)(z 1 ) ∂F k (h)(z 2 ) κ N (z 1 , z 2 ) d 2 z 1 d 2 z 2 (z 1 -σ 2 g(z 1 ) -θ) 2 (z 2 -σ 2 g(z 2 ) -θ) 2 where κ N (z 1 , z 2 ) = 1 2 (m 4 -3σ 4 ) 1 N N -1 i=1 E( Ĝii (z 1 ) Ĝii (z 2 )) + σ 4 1 N E(Tr Ĝ(z 1 ) Ĝ(z 2 )),
Ĝ is the resolvent of the the lower right submatrix of size N -1 of M N , g is the Stieltjes transform of λ = µ sc,σ 2 ν and F k (h) is an almost analytic extension of a smooth function h, equal to 1 in a neighborhood of ρ θ , see [START_REF] Capitaine | Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices[END_REF], [START_REF] Erdös | Local semicicle law and complete delocalization for Wigner random matrices[END_REF].

The proof of Theorem 5 is given in Section 2 and relies upon a representation of the variable | e 1 , v i 0 | 2 in terms of the first entry of the resolvent process {G(z) = (zI -M N ) -1 , z ∈ C\R} through Helffer-Sjöstrand formula. Then, the fluctuations of the process {G 11 (z), z ∈ C\R} are analysed using Schur's formula which enables to express G 11 (z) in terms of random sesquilinear forms. The last Section is an appendix, gathering technical results used in the proof of the main result. Throughout the paper, we will denote by tr the normalized trace :

∀B ∈ M N (C), tr B = 1 N Tr B.
In the following, O(1/N q ) denotes any complex function f N on C + such that there exists a polynomial Q such that for all large N, for any

z in C + , |f N (z)| ≤ Q(| z| -1 ) N q . 8
We denote by λ i (A N ), resp. λ i (M N ), the eigenvalues of A N , resp. M N and u i , resp. v i the normalized associated eigenvectors. Thus u i 0 = e 1 . From the formula

Tr(h(M N )f (A N )) = i,j h(λ i (M N ))f (λ j (A N ))| u j , v i | 2 , we deduce that | e 1 , v i 0 | 2 = h(M N ) 11 (11)
where h (resp. f ) is any smooth function with support in a neighborhood of ρ θ (resp. θ) and is equal to 1 near ρ θ (resp. θ). From Proposition 2, we know that a.s.,

| e 1 , v i 0 | 2 -→ τ (θ). ( 12
)
The proof of Theorem 5 follows the three steps :

1. Representation of h(M N ) 11 = | e 1 , v i 0 | 2 in terms of the resolvent G 11 (z) of M N through Helffer-Sjöstrand formula, where G(z) is the resolvent matrix of M N : G(z) = (zI -M N ) -1 .
2. Fluctuations of the process (G 11 (z), z ∈ C\R).

Conclusion : fluctuations of h(M

N ) 11 .

Representation of h(M N )

We recall Helffer-Sjöstrand's representation formula : let f ∈ C k+1 (R) with compact support and M a Hermitian matrix,

f (M ) = 1 π C ∂F k (f )(z) (M -z) -1 d 2 z ( 13
)
where d 2 z denotes the Lebesgue measure on C.

F k (f )(x + iy) = k l=0 (iy) l l! f (l) (x)χ(y) (14)
where χ : R → R + is a smooth compactly supported function such that χ ≡ 1 in a neighborhood of 0, and ∂ = 1 2 (∂ x + i∂ y ). The function F k (f ) coincides with f on the real axis and is an extension to the complex plane. Note that, in a neighborhood of the real axis,

∂F k (f )(x + iy) = (iy) k k! f (k+1) (x) = O(|y| k ) as y → 0. ( 15
)
We shall apply this formula to h(M N ) 11 for h smooth as above and for a suitably chosen k. Since h, χ are compactly supported, the integral in ( 13) is an integral on a compact set of C. We can thus write h(M N ) 11 as

h(M N ) 11 = - 1 π C ∂F k (h(z)) G 11 (z)d 2 z. (16)
It is well known that (see [START_REF] Capitaine | Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices[END_REF])

G 11 (z) -→ N →∞ 1 z -σ 2 g(z) -θ , ( 17
)
where g is the Stieltjes transform of µ sc ν the limiting spectral distribution of M N . We deduce from [START_REF] Capitaine | Fluctuations at the edges of the spectrum of the full rank deformed G[END_REF] and Helffer-Sjostrand formula (see also Proposition 10) that

τ (θ) = - 1 π C ∂F k (h(z)) 1 z -σ 2 g(z) -θ d 2 z.
If we do not want to add assumptions on the speed of convergence of µ A N to ν, instead of looking at the fluctuations of | e 1 , v i 0 | 2 around τ (θ), we rather look at fluctuations around τ N (θ) defined by [START_REF] Bloemendal | Isotropic local laws for sample covariance and generalized wigner matrices[END_REF]. From Proposition 10,

τ N (θ) = - 1 π C ∂F k (h(z)) 1 z -σ 2 gN-1 (z) -θ d 2 z ( 18
)
where gN-1 is the Stieltjes transform of µ sc µ A N -1 .

We are now interested in the weak convergence of

Φ N := √ N (| e 1 , v i 0 | 2 -τ N (θ)
). Thus, it can be rewritten in terms of the resolvent as follows.

Φ N = - 1 π C ∂F k (h(z)) √ N G 11 (z) - 1 z -σ 2 gN-1 (z) -θ d 2 z. ( 19
)
We first consider the weak convergence of the process

ξ N (z) = √ N (G 11 (z) - 1 z -σ 2 gN-1 (z) -θ ); z ∈ C\R .
2.2 Convergence of the process (ξ N (z))

Dealing with convergence in distribution of processes, the study goes through 2 steps :

1. Convergence of finite dimensional distributions 2. Tightness of (ξ N (z)).

Finite dimensional distributions

We start with Schur's formula

G 11 (z) = 1 z -W 11 √ N -θ -1 N Y * Ĝ(z)Y ( 20
)
where Ĝ is the resolvent of the the lower right submatrix MN-1 of size N -1 of M N and t Y = (W 21 , . . . , W N 1 ). Thus,

MN-1 = ŴN-1 √ N + A N -1
where ŴN-1 is the lower right submatrix of size N -1 of W N . Note that W 11 , Y and Ĝ are independent. Therefore,

ξ N (z) = W 11 + √ N ( 1 N Y * Ĝ(z)Y -σ 2 tr( Ĝ(z))) + √ N σ 2 (tr( Ĝ(z)) -gN-1 (z)) (z -W 11 √ N -θ -1 N Y * Ĝ(z)Y )(z -σ 2 gN-1 (z) -θ) := W 11 + Q N (z) + √ N σ 2 (tr( Ĝ(z)) -gN-1 (z)) D N (z) (21)
We first analyse the two terms D N (z) and √ N (tr( Ĝ(z)) -gN-1 (z)).

Proposition 7.

1.

D N (z) -→ N →∞ (z -σ 2 g(z) -θ) 2 ( 22
)
in probability.

√

N (tr( Ĝ(z)) -gN-1 (z)) -→ N →∞ 0 ( 23
)
almost surely.

Proof. Proposition 4.1 in [START_REF] Capitaine | Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices[END_REF] readily yields that

√ N (E tr( Ĝ(z)) -gN-1 (z)) -→ N →∞ 0. ( 24
)
Lemma 2 and Lemma 3 in Appendix yield that

P √ N (tr( Ĝ(z)) -E tr( Ĝ(z)) ) > ≤ K 1 exp - √ N | z| 2 K 2 ,
and then

√ N (tr( Ĝ(z)) -E tr( Ĝ(z)) ) -→ N →∞ 0. (25)
(23) follows from ( 24) and ( 25). Lemma 4 yields that 1 N Y * Ĝ(z)Y -σ 2 tr Ĝ(z) -→ N →∞ 0 in probability and then, using (23

) that 1 N Y * Ĝ(z)Y -σ 2 gN-1 (z) -→ N →∞ 0 in probability. (22) follows.
Let z 1 , . . . , z q in C + . According to [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF] and Proposition 7 , the fluctuations of (ξ N (z 1 ), . . . , ξ N (z q )) rely on the study of the fluctuations of (Q N (z 1 ), . . . , Q N (z q )). Theorem 6. Let (z 1 , . . . , z q ) be in (C \ R) q . Then,

(Q N (z 1 ), . . . , Q N (z q )) (law) -→ N →∞ (G(z 1 ), . . . G(z q ))
where (G(z 1 ), . . . G(z q )) is a C q -valued centered Gaussian random vector whose covariance is given by

E(G(z 1 )G(z 2 )) = 1 2 (m 4 -3σ 4 ) dν(x) (z 1 -σ 2 g(z 1 ) -x)(z 2 -σ 2 g(z 2 ) -x) +σ 4 1 (z 1 -x)(z 2 -x) dλ(x) (26)
and

E(G(z 1 )G(z 2 )) = E(G(z 1 )G(z 2 )) (27)
where m 4 = x 4 dµ(x), λ = µ sc ν and g := g λ is the Stieltjes transform of λ.

Proof. We apply Proposition 9 to the matrices B(z) = Ĝ(z), z ∈ C + . Note that these matrices are random but there are independent of the vector Y .

In order to conclude, we need to show that

I N := 1 N N -1 i=1 ( Ĝii (z 1 ) Ĝii (z 2 ) (P) -→ N →∞ dν(x) (z 1 -σ 2 g(z 1 ) -x)(z 2 -σ 2 g(z 2 ) -x)
and

J N := 1 N Tr( Ĝ(z 1 ) Ĝ(z 2 )) (P) -→ N →∞ 1 (z 1 -x)(z 2 -x) dλ(x).
The second convergence follows from the convergence of µ MN-1 towards λ.

For the first one, since A N is a diagonal matrix, we can use the following estimate which can be easily deduced from [START_REF] Capitaine | Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices[END_REF]) : for k ≤ N -1,

E( Ĝkk (z)) = 1 z -σ 2 g(z) -A N -1 (k, k) + R k,N (z)
with, for any k,

|R k,N (z)| ≤ (1 + |z|) α P (| (z)| -1 )a N (28)
where P is a polynomial with non negative coefficients and a N → N →+∞ 0. We also have from Poincaré inequality that there exists C > 0, for any k,

Var( Ĝkk (z)) ≤ C N (| (z)| -1 ) 4 .
Thus,

E( Ĝkk (z 1 ) Ĝkk (z 2 )) = 1 (z 1 -σ 2 g(z 1 ) -A N -1 (k, k))(z 2 -σ 2 g(z 2 ) -A N -1 (k, k)) +R k,N (z) and 1
N -1 N -1 k=1 E( Ĝkk (z 1 ) Ĝkk (z 2 )) = dµ A N -1 (x) (z 1 -σ 2 g(z 1 ) -x)(z 1 -σ 2 g(z 1 ) -x) + RN (z)
where R k,N (z) and RN (z) satisfy the upper bound [START_REF] Wigner | Characteristic vectors of bordered matrices with infinite dimensions[END_REF]. We thus obtain

1 N -1 N -1 i=1 E( Ĝii (z 1 ) Ĝii (z 2 )) -→ N →∞ dν(x) (z 1 -σ 2 g(z 1 ) -x)(z 2 -σ 2 g(z 2 ) -x)
and from Lemma 3, using that f

N (W ) = 1 N -1 N i=1 [(z 1 -W -A N -1 ) -1 ] ii [(z 2 - W -A N -1 ) -1 ] ii is Lipschitz with constant | (z 1 )| -1 (| (z 2 )| -1 , we can de- duce that 1 N -1 N -1 i=1 Ĝii (z 1 ) Ĝii (z 2 ) (P) -→ N →∞ dν(x) (z 1 -σ 2 g(z 1 ) -x)(z 2 -σ 2 g(z 2 ) -x)
.

We also use to obtain [START_REF] Tao | Random matrices: universal properties of eigenvectors[END_REF] from Theorem 9 that for σ 2 = 1,

E(|y 1 | 4 ) -2 = 1 2 (m 4 -3),
where we recall that µ is the distribution of √ 2 y 1 and √ 2 y 1 . 2

Tightness of the processes {ξ

N (z), z ∈ K}, K compact in {z, | (z)| ≥ ε} Let gN-1 the Stieltjes transform of µ sc µ A N -1 . Define for any z ∈ C ξ N (z) = √ N G 11 (z) - 1 z -σ 2 gN-1 (z) -θ . ξ N is analytic on C + . For any compact set K in C + , define ξ N K = sup z∈K |ξ N (z)| .
First, recall some results from [START_REF] Shirai | Limit theorems for random analytic functions and their zeros. In Functions in number theory and their probabilistic aspects[END_REF]. Let D ⊂ C be a connected (open) domain in the complex plane. Denote by H(D) the space of complex analytic functions in D. Proposition 8. (Proposition 2.5. in [START_REF] Shirai | Limit theorems for random analytic functions and their zeros. In Functions in number theory and their probabilistic aspects[END_REF]) Let X n be a sequence of random analytic functions in D. If X n K is tight for any compact set K, then L(X n ) is tight in P(H(D)). Furthermore, if X n converges to X in the sense of finite dimensional distributions, then L(X n ) converges weakly to a limit L(X).

Since we have for any C > 0 and any p > 0,

P ( X N K > C) ≤ 1 C p E X N p K , (29)
the following lemma turns out to be useful to apply Proposition 8.

Lemma 1. (lemma 2.6 [START_REF] Shirai | Limit theorems for random analytic functions and their zeros. In Functions in number theory and their probabilistic aspects[END_REF]) For any compact set K in D there exists δ > 0 such that

f p K ≤ (πδ 2 ) -1 K δ |f (z)| p m(dz), f ∈ H(D),
for any p > 0, where K δ ⊂ D is the closure of the δ-neighborhood of K. (m denotes the Lebesgue measure).

According to Lemma 1, there exists δ > 0 such that K δ ⊂ C + and for any p > 0,

ξ N p K ≤ (πδ 2 ) -1 K δ |ξ N (z)| p m(dz). Therefore E ξ N p K ≤ (πδ 2 ) -1 K δ E (|ξ N (z)| p ) m(dz) (30)
≤ (πδ 2 ) -1 sup z∈K δ E (|ξ N (z)| p ) m(K δ ). (31)
In order to prove the tightness of ξ N , using Proposition 8, ( 29) and (31), we are going to show that, for any compact set K ∈ C + , there exists a constant C > 0 such that for all large N,

sup z∈K E |ξ N (z)| 2 < C . (32)
We have for any z 1 and z

2 in C + , E (ξ N (z 1 ))ξ N (z 2 )) = N E G 11 (z 1 ) - 1 z 1 -σ 2 gN-1 (z 1 ) -θ G 11 (z 2 ) - 1 z 2 -σ 2 gN-1 (z 2 ) -θ = N E (G 11 (z 1 )G 11 (z 2 )) -N E (G 11 (z 1 )) 1 z 2 -σ 2 gN-1 (z 2 ) -θ -N E (G 11 (z 2 )) 1 z 1 -σ 2 gN-1 (z 1 ) -θ + N 1 z 1 -σ 2 gN-1 (z 1 ) -θ 1 z 2 -σ 2 gN-1 (z 2 ) -θ
We have (see [START_REF] Capitaine | Free convolution with a semicircular distribution and eigenvalues of spiked deformations of Wigner matrices[END_REF] for diagonal perturbation),

E (G 11 (z i )) = 1 z -σ 2 gN-1 (z i ) -θ + O(1/N ), and Var(G 11 (z i ) = O(1/N ).
It readily follows that there exists polynomials P 1 and P 2 with nonnegative coefficients such that

E (ξ N (z 1 )ξ N (z 2 )) ≤ P 1 | z 1 | -1 P 2 | z 2 | -1
and then there exists some polynomial Q with nonnegative coefficients such that for all large N and all z ∈ C + ,

E |ξ N (z)| 2 ≤ Q | z| -1 . (33)
This implies (32).

Fluctuations of the eigenvector

From the previous subsections, we have that, for any bounded continuous function

F from C(K ; C) to C, E(F (ξ N )) -→ N →∞ E(F (H)) (34)
where

H(z) = 1 (z -σ 2 g(z) -θ) 2 (W 11 + G(z))
with W 11 independent of the Gaussian process (G(z))) and

K ε ⊂ {z, | (z)| ≥ ε}. Recall from (19) that Φ N defined as √ N (| e 1 , v i 0 | 2 -τ N (θ))
has the representation

Φ N = - 1 π C ∂F k (h)(z)ξ N (z)d 2 z. (35)
Theorem 7. Φ N converge in distribution to Φ where Φ is given by

Φ = - 1 π C ∂F k (h)(z)H(z)d 2 z = c θ,ν W 11 - 1 π C ∂F k (h)(z) 1 (z -σ 2 g(z) -θ) 2 G(z)d 2 z where c θ,ν = 2σ 2 1 (θ -x) 3 dν(x), and
Z := - 1 π C ∂F k (h)(z) 1 (z -σ 2 g(z) -θ) 2 G(z)d 2 z
is a centered Gaussian variable with variance given by (8) and (9) .

Proof. The proof follows from the proof of Lemma 6.3 in [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF].

i) From Theorem 6, Subsection 2.2.2, the process {ξ

N (z), | (z)| > } converges to {H(z), | (z)| > } and thus we have convergence in distribution of C ∂F k (h)(z)ξ N (z)d 2 z to C ∂F k (h)(z)H(z)d 2 z where C = {z, | (z)| > }.
ii) It remains to control the remaining terms where the integral is restricted to {z, | (z)| ≤ }. The proof follows the same line as in [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF] based upon the following estimate, from (33) : The convergence of Φ n to Φ then follows, using the same lines of [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF].

sup N E(|ξ N (z)|) ≤ Q(| (z) -1 |). Recall that | ∂F k (h)(x + iy)| ≤ C(h)
The computation of c θ,ν follows from Proposition 10 :

c θ,ν = - 1 π C
∂F k (h)(z) 1 (z -σ2 g(z) -θ) 2 dz = Res( 1 (z -σ 2 g(z) -θ) 2 , ρ θ ).

A straightforward computation gives

Res( 1 (z -σ 2 g(z) -θ) 2 , ρ θ ) = -

ϕ 1 (ρ θ ) (ϕ 1 (ρ θ )) 3 = φ 1 (θ)
where ϕ(z) = z -σ 2 g(z) (g := g λ ) and φ(z) = z + σ 2 g ν (z). Let us now prove [START_REF] Capitaine | Spectrum of deformed random matrices and free probability[END_REF]. From [START_REF] Tao | Random matrices: universal properties of eigenvectors[END_REF] Proof. The theorem is an extension to the multi-dimensional case of the CLT for quadratic forms of Bai-Silverstein given in the appendix of [START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations[END_REF]. This extension can be proved along the same lines as in [START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations[END_REF]. See also [START_REF] Bai | Central limit theorem for eigenvalues in a spiked population model[END_REF]. F k (h)(z) φ(z)dz = 2iπRes(φ, ρ θ ).

Computation of Helffer-Sjöstrand's integral

  |y| k near the real axis (see[START_REF] Füredi | The eigenvalues of random symmetric matrices[END_REF]).Taking k large enough to cancel the singularity of Q(|y -1 |), we can show thatC ∂F k (h)(z)ξ N (z)d 2 zN is a tight sequence and, for any δ, we can find such that, for all N ,{z,| (z)|≤ } | ∂F k (h)(z)|E|ξ N (z)|d 2 z ≤ δ.

2 z 2 dν

 2 σ 2 g(z) -θ) 2 (z -σ 2 g(z) -x) d σ 2 g(z) -θ) 2 (z -x) d 2 z and E(GG * ) = (E|y 1 | 4 -2)a(z k , zl ) + b(z k , zl ) k,l

Proposition 10 .

 10 Let h be a smooth function with compact support in (ρ θ -2δ, ρ θ + 2δ) and satisfying h ≡ 1 sur [ρ θ -δ, ρ θ + δ]. Let χ be a compactly supported function on (-L, L), and χ = 1 around 0. We denote by D = (ρ θ -2δ, ρ θ + 2δ) × (-L, L). Let φ a meromorphic function in D, with a pôle in ρ θ . Then,I(φ) := 1 π C ∂F k (h)(z) φ(z)d 2 z = -Res(φ, ρ θ ) (36)where F k (h) is defined in[START_REF] Erdös | Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices[END_REF]. Res(φ, ρ θ ) denotes the residue of the function φ at the point ρ θ .Proof. Let small enough such thatF k (h)(z) = 1 for z ∈ B(ρ θ , ). Set D = D\B(ρ θ , ). φ is holomorphic on D . Since F k (h) has compact support in D, we have, 0 = ∂D F k (h)(z) φ(z)dz = ∂D F k (h)(z)φ(z)dz + ∂B(ρ θ , ) F k (h)(z) φ(z)dz = 2i D ∂F k (h)(z) φ(z)d 2 z + ∂B(ρ θ , ) φ(z)dzwhere the first term is obtained by Green's formula using that ∂φ(z) = 0 on D .D ∂F k (h)(z) φ(z)d 2 z -→ →0 πI(φ)and ∂B(ρ θ , )

They consider fluctuations around this point depending on N in order to not prescribe speed of convergence of µA N to ν.

dλ(x)

The functions φ x (z) = 1 (z-σ 2 g(z)-θ) 2 (z-σ 2 g(z)-x) for x ∈ supp(ν) and φ x (z) = 1 (z-σ 2 g(z)-θ) 2 (z-x) for x ∈ supp(λ) satisfies the hypothesis of Proposition 10. Straightforward computations lead to :

Appendix

Poincaré inequality and concentration phenomenon

A probabilty µ satisfies a Poincaré inequality if for any C 1 function f : R → C such that f and f are in L 2 (µ),

If the law of a random variable X satisfies the Poincaré inequality with constant C P I then, for any fixed α = 0, the law of αX satisfies the Poincaré inequality with constant α 2 C P I . Assume that probability measures µ 1 , . . . , µ M on R satisfy the Poincaré inequality with constant C P I (1), . . . , C P I (M ) respectively. Then the product measure

C P I (i) in the sense that for any differentiable function f such that f and its gradient gradf are in

Lemma 3. Lemma 4.4.3 and Exercise 4.4.5 in [START_REF] Anderson | An introduction to random matrices[END_REF] or Chapter 3 in [START_REF] Ledoux | The Concentration of Measure Phenomenon[END_REF]. Let P be a probability measure on R M which satisfies a Poincaré inequality with constant C P I . Then there exists K 1 > 0 and K 2 > 0 such that, for any Lipschitz function

Quadratic forms of random vectors

Lemma 4 (cf Lemma 2.7 [START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices[END_REF]).

Assume that (B(z)) * = B(z) and that there exists a constant a > 0 (not depending on N ) such that for any z in C \ R ||B(z)|| ≤ a. We assume that B satisfies, for any z, z ∈ C \ R,

Let Y N = (y 1 , . . . , y N ) be an independent vector of size N which contains i.i.d complex standardized entries with bounded fourth moment and such that E(y 2 1 ) = 0. Let (z 1 , . . . , z q ) be in (C \ R) q . Set The sequence (V N ) N converges to a C q -valued centered Gaussian random vector G with parameters E(GG T ) = (E|y 1 | 4 -2)a(z k , z l ) + b(z k , z l ) k,l


D Benmerzouk

J-P P Barbot

Grazing Analysis for Synchronization of chaotic hybrid systems

In this paper, a Grazing bifurcation analysis is proposed and a way to chaos is presented. Moreover, based on this analysis an observer design for the synchronization of chaotic hybrid system is given.

Introduction

The aim of this paper is to generate chaos for piecewise smooth systems which undergo a specific relied bifurcation named grazing bifurcation. The grazing phenomena and non-smooth bifurcations in dynamic systems are well studied at least since thirty years [START_REF] Feigin | On the structure of C-bifurcation boundaries of piecewise continuous systems[END_REF]. Mario di Bernardo and collaborators have contributed in the theory development regarding to the sliding bifurcations (di Bernardo & al., 2000) and they have introduced successfully the relied Poincaré map (di Bernardo & al., 2001). Independently from this, some authors have studied the existence of dynamic models's solutions near the approximate linear one [START_REF] Benmerzouk | Observability analysis using Lyapunov-Schmidt method[END_REF]. In this paper, we mixed the two approaches in order to analyze the same problem but around a periodic solution with grazing behavior. From this, a way to chaos is highlighted by perioddoubling solution. This analysis is based on the topological degree theory and will be apllied to synchronization of chaotic systems, note that from the well known paper of Nijmeijer and Mareels (Nijmeijer & al., 1997), a synchronization of chaotic systems may be studied as an observer design problem. Thus our proposed analysis is finally used in order to design an observer for chaotic hybrid system. These illustrate the efficiency of the proposed approach and it's practical interest. The paper is organized as follow: In section II, the problem statement is established. The problem analysis is presented in section III and a way to chaos is proposed in section IV. Thank to results of previous sections, an observer analysis and design, with simulation results, are given in section V.

Some recalls and problem statement

Let us consider the following piecewise smooth system: . n being a norm defined on R n .

ẋ = F 1 (x) if H(x) ≥ 0 F 2 (x) if H(x) < 0 ( 2
Remark 1 Throughout the paper, for all definition, proposition and theorem, D is considered as an open bounded and connex domain even if it is not a necessary assumption for the considered purpose but this assumption is necessary for the global result.

C abs (I, D) is the set of absolutely continuous functions defined on I and having values in D provided with the same norm as C(I, R n ). According to (Bresis, 1999), (C(I, D), ) and (C abs (I, D), ) are Banach spaces. H is a phase space boundary between regions of smooth dynamics, H defines the set: S = {x(t) ∈ D : H(x(t)) = 0} witch is termed the switching manifold. S divide the phase space into two regions:

S + = {x(t) ∈ D : H(x(t)) ≥ 0} S -= {x(t) ∈ D : H(x(t)) < 0}
Both vector fields F 1 and F 2 are defined on both sides of S, the flows Φ i , i = 1, 2 generated by each vector field are defined as the operators that satisfy: Bernardo & al., 2001), di Bernardo and coauthors chow that a grazing (denoted grazing point) occurs at x = 0 if the following conditions are satisfied: A-1)

∂Φ i (x,t) ∂t = F i (Φ i (x, t)) and Φ i (x, 0) = x, i = 1, 2. In (di
H(0) = 0. A-2) ∇H(0) = 0. A-3) for i = 1, 2. < ∇H(0), ∂Φi ∂t (0, 0) >=< ∇H(0), F 0 i > = 0, A-4) for i = 1, 2. ∂ 2 H(Φ(0,0)) ∂t 2 = (< ∇H(0), ∂F 0 i ∂x F 0 i > + < ∂ 2 H(Φ(0,0)) ∂x 2 F 0 i , F 0 i >) > 0, A-5) (< L, F 0 1 >< L, F 0 2 >) > 0.
where

F 0 i = F i (Φ i (0, 0)), i = 1
, 2, L is the unit vector perpendicular to ∇H(0) and < ., . > is a usual scalar product on R n . The first two conditions state that H well defines the switching manifold. Condition A-3) states that grazing takes place at the origin of x = 0 i.e. the vector field is tangent to S. Condition A-4) ensures that the curvature of the trajectories in S + and S - has the same sign with respect to H (and without loss of generality, this sign is assumed to be positive). Condition A-5) ensures that in a sufficiently small neighborhood of the grazing point, there is no sliding behavior. Now, the system (2.1) is assumed to depend smoothly explicitly or implicitly on a parameter µ. Moreover, at µ = 0 and x(0) = 0, there is a periodic orbit x(t) that grazes at the point x(0). The solution is hyperbolic and hence isolated such that there is no points of grazing along the orbit other than x(.) = 0. As the previous conditions are defined on an open set then there exist two sufficiently small neighborhoods of µ = 0 and x(t) such that A-i) (1,2,3,4,5) are verified. If the vector field is continuous at grazing i.e. F 0 1 = F 0 2 = F but has discontinuous first derivatives, then the Poincaré map is given by (di Bernardo & al., 2001):

P (x, µ) = P 1 (x, µ) if < ∇H, x > > 0 P 2 (x, µ) if < ∇H, x > < 0
Where:

P 1 (x, µ) = N x + M µ + o( x n-1 , µ) P 2 (x, µ) = N (x + v 1 | < ∇H, x > |) 3 2 + v 3 (< ∇H, ∂F 2 ∂x x >)(| < ∇H, x > |) 1 2 + v 2 x(| < ∇H, x > |) 1 2 + M µ + o( x 2 n-1 , µ))
N is a nonsingular matrix (n -1) × (n -1), M is a nonzero n -1 dimensional vector.

v 1 = 2 √ 2 (< ∇H, ∂F 1 ∂x F >) 3 2 ( 1 3 ( ∂ 2 F 2 ∂x 2 - ∂ 2 F 1 ∂x 2 )F 2 + ∂F 2 ∂x ∂F 1 ∂x F - 1 3 (( ∂F 1 ∂x ) 2 + 2( ∂F 2 ∂x ) 2 )F - 1 < ∇H, ∂F 2 ∂x F > ( ∂F 2 ∂x - ∂F 1 ∂x ) F ( 1 3 < ∇H, ( ∂ 2 F 2 ∂x 2 )F 2 > + < ∇H, ( ∂F 2 ∂x ∂F 1 ∂x )F > - 2 3 < ∇H, ( ∂ 2 F 2 ∂x 2 ) 2 F >))) v 2 = 2 √ 2 < ∇H, ∂F 1 ∂x F > ( ∂F 2 ∂x - ∂F 1 ∂x ) v 3 = 2 √ 2 < ∇H, ∂F 1 ∂x F ><∇H, ∂F 2 ∂x F > ( ∂F 2 ∂x F - ∂F 1 ∂x F )
and o(α, µ) -→ 0 when (α, µ) -→ (0, 0) .

For the sake of compactness, ∂F 1 ∂x , ∂F 2 ∂x and ∇H stand respectively for ∂F 1 ∂x (0), ∂F 2 ∂x (0) and ∇H(0).

In order to avoid the 3 2 type singularity in the Poincaré map, the topological degree theory is considered [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]. Therefore, the main definitions and results are recalled in this context: Considering C 1 (I, D) the set of C 1 functions defined on a domain I having values on a domain D of R n , the norm for C 1 (I, D) is defined as follows: (Bresis, 1999).

x ∈ C 1 (I, D) : x 1 = sup t∈I x(t) n + sup t∈I ẋ(t) n (C 1 (I, D), . C 1 ) is a Banach space
Definition 1 (Katok & al., 1997) Let β ∈ C 1 (D, R n ), D the domain of definition of β, D the corresponding closed domain, Z β = x ∈ D : ∂β(x) ∂x = 0 and con- sider a point p ∈ R n -β(∂D ∪ Z β ),
where ∂D is the borderline of D. The topological degree of β at point p defined on D is given by:

deg(β, p, D) := x∈D&β(x)=p (sign ∂β(x) ∂x )
The following result permits to construct a topological degree for a continuous function:

Theorem 1 (Katok & al., 1997) Let h ∈ C(D), p ∈ R n -h(∂D) and the distance r = 1 2 inf x∈D ( p -h(x) n ) > 0,
then there exists a function g ∈ C 1 (D): gh < r such that

deg(β, p, D) = deg(g, p, D).
The following theorem allows to compute the topological degree of some function instead of another one (generally more complicated).

Theorem 2 (Poincaré-Bohl) [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF] Let β and g two continuous functions defined on D such that the following assumption is satisfied: A-6) For every x ∈ ∂D : the line joining β(x) to g(x) (i.e. y = (1λ)β(x) + λg(x), λ ∈ (0, 1)) does not contain the origin, then:

deg(β, p, D) = deg(g, p, D) Remark 2 Assumption A-6) is verified if β T (x)g(x) > 0 for every x ∈ ∂D.
Now, the importance of this notion is highlighted in the following proposition:

Proposition 1 [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF] Let D be open, bounded and connex domain,

β is C 1 (D, R n ), Z β = ∅ and p ∈ R n -β(∂D), then the number of equation's solutions β(x) = p in D is equal to |deg(β, p, D)|.

Problem analysis

Recall that a fixed point of P (., µ) represents an mT periodic orbit, where m is a natural number and T is the period. Therefore searching a solution of (2.1) is equivalent to resolve the following equation:

P (x, µ) = x (3.2)
Two possibilities appear: First case: <∇H, x > > 0 As, the flow evolves entirely in S + , the solution's analyze only considers the first equation of the system (2.1) and the problem becomes to analyze the equation given by:

γ(x, µ) = N x + M µ + o(||x|| n-1 , µ) -x = 0 (3.3)
If the following assumption is satisfied: A-7) (N -I) is nonsingular, where I is the identity matrix.

The Implicit functions theorem gives:

Proposition 2 Under assumptions A-i) (i=1,2,3,4,5,7) , there exist a neighborhood

V µ=0 in R, a neighborhood V x=0 in R n-1 and an unique application x * :V µ=0 -→ V x=0 solution of γ(x * (µ), µ) = 0 such that x * (0) = 0.
Furthermore, the periodic solution of (2.1) x * depends continuously on µ.

Second case: < ∇H, x > < 0.

The problem of finding a periodic solution of problem (2.1) is equivalent to analyze the following equation:

β(x, µ) = P 2 (x, µ) -x = 0 (3.4) where P 2 (x, µ) = N (x + v 1 (δ(x)) 3 2 + v 2 x(δ(x)) 1 2 +v 3 (< ∇H, ∂F 2 ∂x x >)(δ(x)) 1 2 ) +M µ + o( x 2 n-1 , µ) (3.5) And δ(x) = -< ∇H, x >.
For continuity reason β(x, µ) is considered equal to 0 for all x such that < ∇H, x > ≥ 0.

In order to approach β by a quadratic function having the same solutions number as β, the following "alternative" function β is considered:

β(x, µ) = N x + M µ -x + (δ(x)) 1 2 (N (v 1 (δ(x)) 3 2 +v 2 x(δ(x)) 1 2 + v 3 (< ∇H, ∂F 2 ∂x x >)(δ(x)) 1 2 )) +o( x 2 n-1 , µ) (3.6)
The linear part of β is: v(x, µ) = N x + M µx and the quadratic part is:

w(x, µ) = N (v 1 (δ(x)) 2 + v 2 x(δ(x)) + v 3 (< ∇H, ∂F 2 ∂x x >)(δ(x))) + o( x 2 , µ)
So, from the theorem 2, if the line joining β(x, µ) to β(x, µ) does not contain 0 i.e. if the following assumption is satisfied: A-8) v

T (x, µ)w(x, µ) > 0 for all x ∈ ∂D and any real µ in the neighborhood of zero, then:

deg(β, 0, D) = deg( β, 0, D)
where deg(β, 0, D) and deg( β, 0, D) are respectively the degree of the application β and β defined on D at point 0.

Thus the problem to analyze becomes:

β(x, µ) = 0 (3.7)
The Implicit functions theorem gives:

Proposition 3 Under assumptions A-i) for i = 1, 2, 3, 4, 5, 7, 8), there exist a neighborhood v µ=0 in R, a neighborhood v x=0 in R n-1 and an unique application

x * * : V µ=0 -→ V x=0 solution of β(x * * (µ), µ) = 0 such that x * * (0) = 0.

Way to Chaos

Roughly speaking, a chaotic system is characterize by two properties, the first one is a great sensitivity with respect to the initial conditions. This implies that a long term predictions are almost impossible despite the deterministic nature of the system. The second property is "the strange" structure of its attractor. Moreover, according to (Glendinning, 1994) or (Wiggins, 1990), one way to characterized the chaotic behavior of the system (2.1) (and so for the corresponding equation (3.7)) is to determine three distinct points x, y and z such that: β(x, µ) = y, β(y, µ) = z and β(z, µ) = x. This well be done in three steeps.

First step: analyze of the equation:

β(x, µ) = y = x + η 1 (4.8)
where for a sake of simplicity, η 1 stands for a vector defined in R n-1 , having only one component equal to some fixed value(noted also η 1 ) and the others components are nulls. Thus, the equation (4.8) is equivalent to:

Ψ 1 (x, µ, η 1 ) = β(x, µ) -x -η 1 = 0 (4.9)
And the next proposition is obtained with the same arguments that the previous one:

Proposition 4 Under assumptions A-i)for i = 1, 2, 3, 4, 5, 7, 8), there exist a neighborhood υ µ=0 in R, a neighborhood υ η 1=0 in R , a neighborhood υ x=0 in R n-1 and an unique application x * * * : υ µ=0 × υ η 1=0 -→ υ x=0 solution of Ψ 1 (x * * * (µ, η 1 ), µ, η 1 ) = 0 such that x * * * (0, 0) = 0.

Second step: analyze of the equation:

β( β(x, µ), µ) = z = y + η 2 = x * * * (µ, η 1 ) + η 1 + η 2 (4.10)
where η 2 stands for a vector defined on R n-1 , having only one component equal to some fixed value(noted also η 2 ) and the others are nulls. Thus, equation (4.10) is equivalent to:

Ψ 2 (µ, η 1 , η 2 ) = β(x * * * (µ, η 1 ) + η 1 , µ) -x * * * (µ, η 1 ) -η 1 -η 2 Ψ 2 (µ, η 1 , η 2 ) = 0 (4.11)
The following assumption: A-9) ∂Ψ 2 ∂η 1 (0, 0, 0) = 0. is necessary to obtain: Proposition 5 Under assumptions A-i) for i = (1, 2, 3, 4, 5, 7, 8, 9), there exist a neighborhood ν µ=0 in υ µ=0 , a neighborhood ν η 1=0 in υ η 1=0 , a neighborhood ν η 2=0 in R and an unique application η *

1 :ν µ=0 × v η 2 =0 -→ ν η 1 =0 solution of Ψ 2 (η * 1 (µ, η 2 ), µ, η 2 ) = 0 such that η * 1 (0, 0) = 0.
Third step: analyze of the equation:

β( β( β(x, µ), µ), µ) = x (4.12)
The equation (4.12) is equivalent to:

Ψ 3 (µ, η 2 ) = β(x * * * (µ, η * 1 (µ, η 2 )) + η * 1 (µ, η 2 ) + η 2 ), µ) -x * * * (µ, η * 1 (µ, η 2 )) = 0 (4.13)
and the following assumption : A-10) ∂Ψ 3 ∂η 2 (0, 0) = 0. is necessary to have: Proposition 6 Under assumptions A-i) for (i = 1, 2, 3, 4, 5, 7, 8, 9, 10), there exist a neighborhood θ µ=0 in ν µ=0 , a neighborhood θ η 2=0 in ν η 2=0 and an unique application η * 2 :

θ µ=0 -→ θ η 2=0 solution of Ψ 3 (µ, η * 2 (µ)) = 0 such that η * 2 (0) = 0.
The next corollary sums up the previous results:

Corollary 1 Under assumptions A-i) for (i= 1, 2, 3, 4, 5, 7, 8, 9, 10), the system (2.1) admits a chaotic solution.

The aim of the proposed approach is to generate a chaotic system in order to propose a type of emitter which is chaotic and hybrid. The number of assumptions may appear too important but the first five ones are the standard assumptions for the grazing. Now, in order to highlight the interest of the proposed method, an analyze of the way to chaos and approximated observer for synchronization goal are realized on an example.

Observer analysis and design:

Let us consider the following system defined in

D ⊂ R 3 : ẋ = F 1 (x, α, ε) if x 2 ≤ 0 F 2 (x, α, ε) if x 2 > 0 (5.14)
where x = (x 1 , x 2 , x 3 ) T and α, ε are real parameters. Moreover, F 1 (x, α, ε) is the vector field such that:

           F 11 (x, α, ε) = -(x 2 + 1) + ε((x 1 + 3 sin x 3 ) 2 +(x 2 + 1) 2 -(1 + α))(-x 1 -3 sin x 3 ) F 12 (x, α, ε) = (x 1 + 3 sin x 3 ) + ε((x 1 + 3 sin x 3 ) 2 +(x 2 + 1) 2 -(1 + α))(-x 2 -1)) F 13 (x, α, ε) = 0 (5.15)
And F 2 (x, α, ε) is the vector field such that:

                   F 21 (x, α, ε) = -(x 2 + 1) + ε((x 1 + 3 sin x 3 ) 2 +(x 2 + 1) 2 -(1 + α))(-x 1 -3 sin x 3 ) F 22 (x, α, ε) = (x 1 + 3 sin x 3 ) + ε((x 1 + 3 sin x 3 ) 2 +(x 2 + 1) 2 -(1 + α))(-x 2 -1)) F 23 (x, α, ε) = -9x 2 + ε((x 1 + 3 sin x 3 ) 2 +(x 2 + 1) 2 -(1 + α))(-x 1 -3 sin x 3 ) -ε(x 3 -9 cos ωt -x 1 ) (5.16)
where S = x ∈ R 3 : x 2 = 0 , S + = x ∈ R 3 : x 2 ≥ 0 and S -= x ∈ R 3 : x 2 < 0 .

In the next α = 0.1 and ω1.0001, moreover sinwt came from a bounded oscillatory and thus all states are bounded.

Remark 3 In order to guaranty that the behavior of (5.14) stays on bounded domain the case ε < 0 is not considered.

The associated Poincaré map, defined in D R 2 (i.e. corresponding to x 1 = 0), is:

P (x 2 , x 3 , µ, ε) =                            (P 2 , P 3 ) T = ((e -4επ x 2 , x 3 ) T -( 1 2 (1 -e -4επ )µ, 0) T +o( x , µ)) if (-x 2 ≥ 0) (P 2 , P 3 ) T = ((e -4επ x 2 , x 3 ) T -( 1 2 (1 -e -4επ )µ, 0) T +(4εx 3 2 2 , (-8 9 ε + 8ε 2 )x 3 2 2 -2x 3 εx 1 2 2 -2εx 1 2 2 ) T +o( x 2 , µ)) if (-x 2 < 0)
So, finding a periodic solution of (5.14) is equivalent to analyze:

P (x, µ, ε) = (x 2 , x 3 ) T
(5.17)

The equation (5.17) is equivalent to:

β(x, µ, ε) := P (x, µ, ε) -(X 2 , x 3 ) T = 0 (5.18)
The corresponding "alternative" function is:

β(x, µ, ε) = β1 (x, µ, ε), β2 (x, µ, ε) ⊺ (5.19)
Where, the approximations at order o( x 2 , µ) are

β1 (x, µ, ε) = e -4επ x 2 - 1 2 (1 -e -4επ )µ + 4εx 2 2 -x 2 β2 (x, µ, ε) = (- 8 9 ε + 8ε 2 )x 2 2 -2x 3 εx 2 -2εx 2 -x 3
Following the proposed way to chaos, the assumptions A-i) (i=1,2,3,4,5,7,8,9,10) are satisfied for x 2 > 2πεµ 1-4επ , any x 3 in the neighborhood of 0 and µ in the neighborhood of 0 (for the simulation the staring state are x 1 = x 3 = 0, x 2 = 0.5, and ε = 0.01), then a chaotic behavior appears in the 3 dimension representation (see figure 1). Now, in order to design an observer we firstly consider the unperturbed subsystems associated to (5.14) (i.e. those corresponding to ε = 0 in (5.15) and (5.16)) and given by:

ẋ = f 1 (x) if x 2 ≤ 0
(5.20) where x = (x 1 , x 2 , x 3 ) T and f 1 (x) = (f 11 (x), f 12 (x), f 13 (x)) T is the vector field such that:

   f 11 (x) = -(x 2 + 1) f 12 (x) = x 1 + 3 sin x 3 f 13 (x) = 0 (5.21) ẋ = f 2 (x) if x 2 > 0 (5.22)
And f 2 (x) = (f 21 (x), f 22 (x), f 23 (x)) T is the vector field such that:

   f 21 (x) = -(x 2 + 1) f 22 (x) = (x 1 + 3 sin x 3 ) f 23 (x) = -9 sin ωt -(x 2 + 1)
(5.23)

Both the subsystems (5.20) and (5.22) are supposed to have the same output h(x) = x 1 . Simple computations show that dim(span(dh(0), dL f i h(0), dL 2 f i h(0)) is equal to 3, for i = 1, 2, thus from (Isidori, 1999), the observer linearization problem for these subsystems is solvable in the neighborhood v 0 of x = 0, furthermore, it is possible to define in v 0 , vectors g i (x), for i = 1, 2, witch satisfy:

L g i h(x) = L g i L f i h(x) = 0 and L g i L 2 f i h(x) = 1
, for all x in v 0 and i = 1, 2. The choice of the subsystems and also the output permits to obtain the same vector solution g(x) = 0, 0, -1 cos x 3 ⊺ Consequently, both subsystems (5.20) and (5.22) have relative degree r at point 0 equal to 3 and thus there exist a diffeomorphism φ in the neighborhood v 0 of 0 and new coordinates z j = φ j (x) for j = 1, 2, 3 such that they are equivalent to the following system:

   ż1 = z 2 ż2 = z 3 ż3 = b(z) + a(z)u (5.24) Where: a(z) = L g L 2 f 1 h(x(t)), b(z) = L 3 f 1 h(x(t)
), u is some real control and y = z 1 is the output.

Remark 4 In the neighborhood v 0 of 0, the continuous function a(z) is nonzero and thus considering the following feedback control law: u = 1 a(z) (-b(z) + v), the system (5.24) is equivalent to:

   ż1 = z 2 ż2 = z 3 ż3 = v (5.25)
It is important here to point out that thank to the condition on the relative degree r = 3 for all subsystems, the control law which linearizes the system in a canonical form does not depend on the switching sequence. This may be very useful for stabilization purpose and find a Common Control Lyapunov function CCLF (Moulay & al., 2007) which is an extension to Hybrid system of the well-known CLF [START_REF] Lin | A universal formula for stabilization with bounded controls[END_REF].

As the system (5.24) is observable, now, from the work of (Boutat & al., 2004), it is possible to give conditions in order to recover also the discrete states and the unknown input. This analysis and observers design will be presented in more details in forcoming paper and this on the basis of the work presented in (Barbot & al., 2006). Nevertheless, hereafter, in order to highlight the interest of our approach and in the same time the robustness of high gain observer design, such kind of observer is designed on the basis of system (5.24), where the last row is supposed bounded but unknown:

     . ẑ1 = ẑ2 + 1 ε (ẑ 1 -z 1 ) . ẑ2 = ẑ3 + 100 ε (ẑ 1 -z 1 ) . ẑ3 = 10000 ε (ẑ 1 -z 1 )
(5.26)

The simulation results are correct. In figure 2

CONCLUSION

The proposed method reclaim the resolution of general Poincaré map, this problem may be very difficult without grazing and more tedious with grazing, but as it is shown in the example, this computations may be simplified using some symmetric matrices and usual methods as polar transformations. On another hand, we think that our grazing analysis may be also applied to other type of borderline collisions. Nevertheless, we start our study by grazing phenomena behavior because it is a natural prolongation of the classical smooth ODE and as it is developed in the numerical example, this approach will be very interesting in chaotic systems synchronization using piecewise smooth systems, it allows particularly to double the crypts security of the concerned circuit, a more detailed work will be proposed in a future work.

  .1) where x : I -→ D, D is an open bounded connex domain of R n with n ≥ 2 ,(generally, as I is the time interval I ⊂ R + ). And F 1 , F 2 : C abs (I, D) -→ C(I, D) where C(I, D) is the set of continuous functions defined on I and having values in R n , the norm for C(I, D) is defined as follows: x ∈ C(I, D) : x = sup t∈I x(t) n

Figure 1 :

 1 Figure 1: Strange attractor in 3D

  -a and 3-a the state x 2 is shown and so the switching condition clearly appears. After that the states in the space of the canonical form are shown z 1 and ẑ1 in figure2-b, z 2 and ẑ2 in figure 2-c, z 3 and ẑ3 in figure 2-d in the same way in figure 3 the observation errors are given e 1 = z 1 -ẑ1 in figure 3-b, e 2 = z 2 -ẑ2 in figure 3-c and finally e 3 = z 3 -ẑ3 in figure 3-d.

Figure 2 :

 2 Figure 2: a-x 2 , b-z 1 ẑ1 , c-z 2 ẑ2 , d-z 3 ẑ3

Figure 3

 3 Figure 3: a-x 2 , b-z 1 -ẑ1 , c-z 2 -ẑ2 , d-z 3 -ẑ3