E Tassi

P J Morrison

Mode signature and stability for a Hamiltonian model of electron temperature gradient turbulence

Stability properties and mode signature for equilibria of a model of electron temperature gradient (ETG) driven turbulence are investigated by Hamiltonian techniques. After deriving the infinite families of Casimir invariants, associated with the noncanonical Poisson bracket of the model, a sufficient condition for stability is obtained by means of the Energy-Casimir method. Mode signature is then investigated for linear motions about homogeneous equilibria. Depending on the sign of the equilibrium "translated" pressure gradient, stable equilibria can either be energy stable, i.e. possess definite linearized perturbation energy (Hamiltonian), or spectrally stable with the existence of negative energy modes (NEMs). The ETG instability is then shown to arise through a Kreȋntype bifurcation, due to the merging of a positive and a negative energy mode, corresponding to two modified drift waves admitted by the system. The Hamiltonian of the linearized system is then explicitly transformed into normal form, which unambiguously defines mode signature. In particular, the fast mode turns out to always be a positive energy mode (PEM), whereas the energy of the slow mode can have either positive or negative sign.

I. INTRODUCTION

An important issue for the stability of equilibria of continuous media concerns the existence of negative energy modes (NEMs), spectrally stable modes of oscillation of a medium with negative energy. One reason NEMs are important is because equilibria with them, although linearly or spectrally stable, can be destabilized by arbitrarily small perturbations. For example, if dissipation is added to the dynamics so as to remove energy from a NEM, then it can be proven that the mode becomes spectrally unstable. On the intuitive level, dissipation removes energy from the already negative energy of the mode, which makes it more negative and increases the amplitude of mode. In non-dissipative systems, NEMs can become unstable with the presence of positive energy modes (PEMs) through nonlinear coupling. By this means the system can even develop finite-time singularities while conserving the energy of the nonlinear system (see, e.g. [1] and many original references therein).

In plasma physics, the study of NEMs has a long tradition dating to the early work of Sturrock [10] on streaming instabilities and Greene and Coppi [2] on magnetohydrodynamic (MHD) type dissipative instabilities in confinement systems. NEMs have been studied in many plasma contexts; for example, Vlasov-Maxwell dynamics [3][4][5][6][7][8], Maxwell drift-kinetic [9] theories, wave-wave interaction in the two-stream instability [11,12,19], magnetic reconnection [13],

ideal MHD in the presence of equilibrium flows [14][15][16], magnetorotational instability [17,18], and magnetosonic waves in the solar atmosphere [20].

Because the concept of NEM is intimately related to that of energy, it was proposed in [21] that the Hamiltonian framework is the natural one for defining and investigating this phenomenon, contrary to the usual practice in plasma physics. Indeed, once the Hamiltonian structure of the model under consideration is known, an unambiguous definition of the energy of the system becomes available: the total energy corresponds to the Hamiltonian of the system and the energy of the linear dynamics must come in a natural way from the second variation of this nonlinearly conserved quantity. Moreover, the normal form theory for linear Hamiltonian systems, provides a clear and systematic way for determining the signature of modes in the neighborhood of an equilibrium of the system. Indeed, for systems with discrete degrees of freedom, the Hamiltonian of the linearized system can always be cast, for stable modes, into the sum of Hamiltonians of decoupled harmonic oscillators, each of which possesses a characteristic frequency and a characteristic signature. Namely this signature, which, for each mode, can be positive or negative depending on whether the mode provides a positive or negative contribution to the total energy, provides a systematic way to identify PEMs or NEMs of the system. Finally, given the existence of the Hamiltonian structure, energy-based methods, akin to 'δW ' of MHD, can be used to obtain sufficient conditions for stability of equilibria or to indicate the presence of NEMs. More precisely, an equilibrium has a NEM, if it is spectrally stable but the second variation of its free energy functional, evaluated at that equilibrium, has indefinite sign.

In this paper, we investigate the presence of NEMs and stability properties of a reduced model for electron temperature gradient (ETG) instabilities, in the Hamiltonian framework. ETG turbulence has been considered as one of the mechanisms that enhances anomalous particle and electron thermal fluxes in tokamaks [22,23]. The detection of NEMs, is therefore important in order to see what potentially unstable modes might lie dormant in the absence of dissipation. When destabilized by dissipation, such modes might enhance the anomalous transport across the confining magnetic field. We note, however, that the methods applied for the ETG model, can be applied, in principle, to any ideal plasma model.

The model for ETG turbulence considered here, has been previously investigated in Refs. [24,25], and in Ref. [24] it was described how this model possesses a noncanonical Hamiltonian formulation, with a Poisson bracket that turns out to be essentially identical to that for reduced MHD [26] (see also [START_REF] Morrison | AIP Conf. Proc[END_REF]). In the present paper we first provide further information about the Hamiltonian structure of the model by deriving explicitly its families of Casimir invariants. We then determine sufficient conditions for stability of generic equilibria by making use of the Energy-Casimir method. The investigation of the presence of NEMs is carried out for the case of homogeneous equilibria, for which we derive an explicit condition for the presence of NEMs. This condition warns us that NEMs are present if the value of the equilibrium pressure gradient lies in a given interval whose end points depend on the perpendicular wave vector and the magnetic field curvature. In particular, the length of this interval shrinks to zero as the perpendicular wave number goes to infinity. After obtaining the eigenvalues and eigenvectors of the system, we explicitly construct and carry out the transformation that puts the Hamiltonian for the ETG model into normal form.

The paper is organized as follows. In Sec.II we review the model and its Hamiltonian formulation, and then derive the Casimir invariants. In Sec.III we apply the Energy-Casimir method and obtain conditions for energy stability. In Sec.IV, after reviewing the theory of mode signature for linear Hamiltonian systems we apply it to the ETG model.

Explicit conditions for the existence of NEMs, their relationship to energy stability and spectral stability conditions coming from the dispersion relation are derived, and the normal form transformation is explicitly obtained. Finally, we conclude in Sec.V.

II. HAMILTONIAN STRUCTURE OF THE ETG MODEL

The non-dissipative ETG driven turbulence model of [24,25] is given by

∂ ∂t (1 -∇ 2 )φ = [φ, ∇ 2 φ + x] + p √ r , √ rx , (1)
∂ ∂t p √ r = p √ r , φ + [ √ rx, φ] , (2)
where two-dimensional slab geometry is assumed so that the field variables, φ the stream function and p the pressure, are functions of the Cartesian coordinates (x, y). The quantity [f, g] = ∂f /∂x∂g/∂y -∂f /∂y∂g/∂x is the canonical Poisson bracket. Equations ( 1) and (2) are written in the normalized form described in [24], where the constant parameter r is defined as

r = L 2 n L B L P , (3)
with L n , L B and L P being the characteristic length scales of variation of the background density, magnetic, and electron pressure fields, respectively. The parameter r is thus related to the mechanism providing the drive for the ETG instability. In particular, in the limit r → 0, which is attained if one flattens the electron pressure gradient, the linear dispersion relation indicates that ETG modes degenerate into marginally stable drift waves [25].

In [24], the authors showed that the system (1)-( 2) possesses a Hamiltonian structure in terms of a noncanonical

Poisson bracket. This means (see, e.g. [1,[START_REF] Morrison | AIP Conf. Proc[END_REF]) that the system can be cast in the form

∂χ i ∂t = {χ i , H}, i = 1, • • • , n (4)
with χ i (x, t) indicating a suitable set of n field variables (with n = 2 in our case) and

H[χ 1 , • • • , χ n ] a Hamiltonian
functional that is conserved by the dynamics. The Poisson bracket {, } appearing in ( 4) is an antisymmetric bilinear binary operator satisfying the Leibniz rule and the Jacobi identity. For the model of ( 1) and (2), it was shown [24] that, with the choice χ 1 = φ, χ 2 = p, the Hamiltonian of the system is

H[φ, p] = 1 2 d 2 x φ 2 + |∇φ| 2 - p 2 r ( 5
)
and the Poisson bracket is

{F, G} = -d 2 x(φ -∇ 2 φ -x)[L -1 F φ , L -1 G φ ] + (p + rx) ([L -1 F φ , G p ] + [F p , L -1 G φ ]), (6)
where the operator L, and its inverse L -1 are formally defined so that Lf = f -∇ 2 f , and

L -1 Lf = LL -1 f = f , for
a generic field f . The subscripts on F and G in (6) denote functional derivatives with respect to the fields φ or p.

Noncanonical Poisson brackets such as (6) are characterized by the presence of so called Casimir invariants (see, e.g. Ref. [1]), due to degeneracy in the cosymplectic operator of the bracket. More precisely, a Casimir invariant of a

Poisson bracket is a functional

C(χ 1 , • • • , χ n ) that satisfies {C, F } = 0 (7)
for any functional F of the field variables. Because they commute in particular with any H, Casimir functionals are preserved during the dynamics. In order to derive the Casimir invariants of the ETG model, it is convenient to introduce the variables

P = p √ r + √ rx, λ = φ -∇ 2 φ -x, (8)
which correspond to a "translated" pressure and to a variable analogous to the potential vorticity of the Charney-Hasegawa-Mima equation [START_REF] Weinstein | [END_REF]29], respectively. In terms of these variables the model equations read

∂ ∂t λ = -[L -1 (λ + x), λ] + [P, √ rx], (9)
∂ ∂t P = [P, L -1 (λ + x)], (10)
whereas the Hamiltonian and the bracket become

H(λ, P) = 1 2 d 2 x (λ + x)L -1 (λ + x) -P 2 + 2 √ rPx , (11)
{F, G} = -d 2 xλ[F λ , G λ ] + P([F λ , G P ] + [F P , G λ ]) , (12)
where the bracket is seen to be identical to that for reduced MHD as first given in [26].

Applying (7) we deduce that the equations determining the Casimir invariants for our system are

[C λ , λ] + [C P , P] = 0, [C λ , P] = 0. ( 13
)
By solving (13) we see that the system admits two independent infinite families of Casimirs:

C 1 = d 2 x H(P), C 2 = d 2 x λF (P), (14)
with H and F arbitrary functions. The dynamics described by the inviscid ETG model is then subject to an infinite number of constraints imposed by the conservation of the Casimir invariants (14). For instance, as a consequence of the conservation of C 2 , integrals of the potential vorticity λ over regions bounded by contour lines of P will be conserved during the dynamics (see [30]).

Notice that the constant of motion I found in [24] is given by

I = d 2 x p 2 r + 2(φ -∇ 2 φ)p = 2 √ r d 2 xλP + d 2 xP 2 -2r d 2 xx φ -∇ 2 φ - x 2 , (15)
which is a linear combination of two particular Casimirs of ( 14) with the realization that the time derivative of

-2r d 2 xx φ -∇ 2 φ - x 2 (16)
vanishes if ∂φ/∂x and ∂φ/∂y vanish or are periodic at the boundaries. Therefore, I amounts to a single special case of the general families of (14).

III. ENERGY STABILITY

The Hamiltonian formalism provides a systematic procedure for implementing the Energy-Casimir method for investigating stability of equilibria (see, e.g. [1,[START_REF] Marsden | Introduction to Mechanics and Symmetry[END_REF][START_REF] Morrison | [END_REF]), a stability method that originated in plasma physics in [33] that has often been referred to as nonlinear stability. This method has been adopted in many works; for example, in the context of fluid models for plasmas in [START_REF] Hazeltine | Proceedings of the International Conference on Plasma Physics[END_REF][START_REF] Holm | [END_REF][36][37]. It provides sufficient conditions for stability by taking the second variation of the free energy, the Hamiltonian plus Casimir invariants, and extracting conditions that are necessary for definiteness. Because the method is based on nonlinear constants of motion, the stability conditions obtained are stronger than conditions that emerge from dispersion relations, i.e. spectral stability conditions, that follow entirely from the linear equations of motion. Indeed this second variation stability, which we will refer to simply as energy stability, implies linear stability [1,[START_REF] Bokhove | Large-Scale Atmosphere-Ocean Dynamics 2, geometric Methods and Models[END_REF], but the converse is not true. In some works energy stability is called formal stability when an additional convexity estimate is not provided. Usually these estimates are rather trivial and even when they are provided they are only a small part of a mathematically rigorous stability proof -for this reason we eschew this terminology.

For noncanonical Hamiltonian systems, equilibrium solutions can be found by solving the equations that result from extremizing the free energy functional. For our system the free energy functional is given by

F = H + C 1 + C 2
(which is not to be confused with the generic functional F of our Poisson brackets), with H given by ( 12) and C 1,2 by ( 14). For convenience we introduce the new variables according to the transformation

Λ = λ + x, P = P (17)
and then drop the bar on P in the following. In terms of these variables the Hamiltonian and the bracket become

H(Λ, P) = 1 2 d 2 x ΛL -1 Λ -P 2 + 2 √ rPx , {F, G} = d 2 x(x -Λ)[F Λ , G Λ ] -P([F Λ , G P ] + [F P , G Λ ]).
The free energy functional is then explicitly given by

F (Λ, P) = 1 2 d 2 x ΛL -1 Λ -P 2 + 2 √ rPx + d 2 xH(P) + d 2 x(Λ -x)F (P) , (18)
and the equilibrium equations, obtained from setting the first variation δF equal to zero, are

F Λ = L -1 Λ + F (P) = 0, (19)
F P = -P + √ rx + H ′ (P) + (Λ -x)F ′ (P) = 0, ( 20
)
where the prime denotes derivative with respect to the argument of the function. Due to the presence of the arbitrary functions in the Casimirs, such equilibrium equations possess free functions. Specifying these selects from a class of equilibrium solutions. In particular, choosing F corresponds to fixing the relation between the equilibrium stream function φ eq = L -1 Λ eq and the equilibrium translated pressure P eq .

As indicated above, an equilibrium solution of ( 19)-( 20) is energy stable (and therefore linearly) stable, if the second variation of F , evaluated at that equilibrium, has a definite sign. In terms of the variables φ and P, the second variation of F reads

δ 2 F = d 2 x (1 -F ′ (P))|L -1 δΛ| 2 + (1 -2F ′ (P))|L -1 ∇δΛ| 2 + F ′ (P)(δΛ + δP) 2 -F ′ (P)(L -1 ∇ 2 δΛ) 2 + H ′′ (P) -1 + (Λ -x)F ′′ (P) -F ′ (P) |δP| 2 . ( 21
)
From ( 21) one immediately obtains sufficient conditions for positive definiteness of δ 2 F in the case of no flow:

F (P eq ) ≡ 0 and H ′′ (P eq ) > 1 . (22)
From ( 19), F (P eq ) ≡ 0 implies no flow, while H ′′ (P eq ) > 1 gives a condition on the pressure profile. Generally speaking, the situation with flow, when F (P eq ) ≡ 0, is expected to have NEMs [1]. However, this case is more complicated to analyze, with the Poincaré inequality often being of use, but we will not pursue it further here.

IV. NEGATIVE ENERGY MODES

In this section, we first review the theory of NEMs in the finite degree-of-freedom Hamiltonian context [START_REF] Kre Ȋn | Four Papers on Ordinary Differential Equations[END_REF], since it applies directly to finite systems with discrete spectra (e.g. [11]). The situation for continuous spectra is more complicated [5,[START_REF] Morrison | [END_REF][41][42] and will not be considered here. Subsequently, after carrying out a spectral stability analysis of the system linearized around homogeneous equilibria with no flow, we make use of the Hamiltonian formalism in order to detect the presence of NEMs among the stable modes of the linearized system. Finally we carry out the explicit transformation that casts the corresponding Hamiltonian into normal form.

A. Review of mode signature and normal forms for linear Hamiltonian systems

A real canonical Hamiltonian linear system with N degrees of freedom is generated by the canonical Poisson bracket

{f, g} = ∂f ∂q i ∂g ∂p i - ∂f ∂p i ∂g ∂q i , (23)
and a quadratic Hamiltonian

H L = 1 2 A ij z i z j , (24)
where z = (q 1 , ..., q N , p 1 , ..., p N ), A ij are the elements of a 2N × 2N matrix with constant coefficients, and repeated sum notation is assumed with i, j = 1, 2, . . . , N . The resulting equations of motion can then be compactly written as

ż = J c Az, (25)
where

J c = 0 N -I N I N 0 N ( 26
)
is the 2N × 2N canonical symplectic matrix (cosymplectic form). Assuming

z = ze iωt + z * e -iωt , (27)
with * indicating complex conjugate, (25) yields the eigenvalue problem

iω α z α = J c Az α , α = 1, • • • , N, (28)
where we have dropped the tilde on the eigenvectors and have added an eigenvalue label α. In what follows we will assume distinct eigenvalues, precluding the existence of nontrivial Jordan form and possible secular growth in time. We also assume that the eigenvalues ω α are real, which is the case of interest for detecting mode signature.

Because our dynamical variables are real, the remaining N eigenvalues are given by ω -α = -ω α and the corresponding eigenvectors are z α = z * α . Defining Ω := J -1 c , the symplectic two-form, we construct the quantity

h(α, β) := iω α z T β Ωz α = z T β Az α , (29)
where T denotes transpose.

It can be easily shown that the property h(α, β)h(β, α) = 0 holds. Then, from this relation and the antisymmetry of Ω, it follows that

h(α, β) = 0, if β = -α. ( 30
)
On the other hand,

h(-α, α) = z T -α Az α = z * α T Az α = iω α z * α T Ωz α , (31)
is clearly the energy (Hamiltonian H L ) of the mode (z α , ω α ; z * α , -ω α ). Evidently, z * α T Ωz α is a purely imaginary number, and a normalization constant for the eigenvectors can be chosen in such a way that

z * α T Ωz α = ±2i, (32)
with the sign, an invariant, depending on the specific mode under consideration. Note that the left-hand side of [START_REF] Morrison | [END_REF] represents the Lagrange bracket (symplectic two-form) of z * α and z α . If z α is an eigenvector, associated with a positive eigenvalue ω α , and

z * α T Ωz α = -2i, (33)
then (z α , ω α ; z * α , -ω α ) corresponds to a positive energy mode, otherwise it is a negative energy mode. This can be easily seen by observing that, in the case of a PEM, the corresponding energy is given by

h(-α, α) = iω α z * α T Ωz α = 2ω α > 0. ( 34
)
Note that, although here we carried out an analysis with canonical coordinates, Sylvester's theorem guarantees that the signature of a mode (i.e., whether it is a PEM or a NEM), does not depend on the choice of the coordinate system.

The distinction between positive and negative energy modes becomes even more transparent when we are reminded that, for stable modes there exists [3,11,[START_REF] Kre Ȋn | Four Papers on Ordinary Differential Equations[END_REF] a canonical transformation T :

(Q 1 , • • • , Q N , P 1 , • • • , P N ) → (q 1 , • • • q N , p 1 , • • • p N )
, that casts the quadratic Hamiltonian of ( 24) into the following normal form:

H L = 1 2 N α=1 σ α ω α (P 2 α + Q 2 α ) , (35)
where ω α represents the positive eigenvalues of the linearized system, whereas σ i ∈ {-1, 1} is the signature of the mode.

If the system contains unstable modes, then they have a different normal form. However, if the Hamiltonian is restricted to the stable modes, then it can be written as the Hamiltonian for a system of N harmonic oscillators with different frequencies. The modes for which σ = -1, which give a negative contribution to the total energy, correspond to the NEMs, while those corresponding to σ i = 1 are, of course, PEMs.

Once the eigenvalues and eigenvectors of the system are known, the procedure for constructing the map T is algorithmic. First one needs to select, among the 2N eigenvectors of the system, the N eigenvectors z α that satisfy

z * α T Ωz α = -2i, α = 1, • • • , N. (36)
Then the 2N × 2N matrix that defines the transformation is given by

T = col (Re z 1 , Re z 2 ... Re z N , Im z 1 , Im z 2 , . . . , Im z N ) , (37)
which is the matrix with columns given by Re z 1 etc. It can be shown that the transformation constructed in this way is canonical and indeed provides the desired diagonalization.

B. Mode signature and stability for the ETG model Now consider a special case of the no-flow equilibria of Sec. III, the homogeneous equilibria:

Λ eq = L -1 Λ eq = 0, P eq = α P x ,

where α P is a constant. The equilibrium solution [START_REF] Bokhove | Large-Scale Atmosphere-Ocean Dynamics 2, geometric Methods and Models[END_REF] corresponds to the choices

F (P) = 0, H(P) = 1 2 1 - √ r α P P 2 , (39)
for the Casimir functions that appear in equations ( 19) and (20).

Linearizing the model equations around this equilibrium gives the system

Λ = - ∂ ∂y L -1 Λ - √ r ∂ ∂y P, Ṗ = α P ∂ ∂y L -1 Λ .
Expanding the perturbations as Fourier series, as follows:

Λ = Λ = +∞ k=-∞ Λk (t)e -ik•x , P = α P x + P = α P x + +∞ k=-∞ Pk (t)e -ik•x , (40)
yields the amplitude equations

Λk = i k y 1 + k 2 ⊥ Λk + i √ rk y Pk , (41)
Ṗk = -iα P k y 1 + k 2 ⊥ Λk , (42)
whence, the dispersion relation for modes of the form e i(ωt-k•x) is obtained,

ω 2 - k y 1 + k 2 ⊥ ω + α P √ r k 2 y 1 + k 2 ⊥ = 0. ( 43
)
This expression is in agreement with that obtained in Ref. [25]. The eigenvalues correspond to a slow and a fast mode, and are given by

ω k s = k 2(1 + k 2 ⊥ ) (1 -1 -4(1 + k 2 ⊥ )α P √ r), (44)
ω k f = k 2(1 + k 2 ⊥ ) (1 + 1 -4(1 + k 2 ⊥ )α P √ r), (45)
where we have set k y = k. The corresponding eigenvectors are

Λk s,f = -ω k s,f 1 + k 2 ⊥ α P k Pk s,f = - 1 2α P (1 ± 1 -4(1 + k 2 ⊥ )α P √ r) Pk s,f , (46)
for k > 0. The system also possesses the eigenvalues ω k -s,-f = -ω k s,f , whose eigenvectors are the complex conjugates of those of (46).

From ( 44)-( 45) we obtain a necessary and sufficient condition for spectral stability, viz.

1 -4(1 + k 2 ⊥ )α P √ r > 0 ⇒ α P < 1 4(1 + k 2 ⊥ ) √ r . ( 47
)
Therefore, if the electron pressure or magnetic field gradients are such that α P < 0, such equilibria are always spectrally stable ∀ k. If α P > 0, on the other hand, the equilibrium will be stable only for k such that 0

< α P < 1/4(1 + k 2 ⊥ ) √ r
is satisfied. In other words, there will always be instability for sufficiently large k ⊥ .

The linearized system of ( 41) and ( 42) inherits a Hamiltonian formulation from the nonlinear system, one that can be written in the framework described in Sec. IV A. We can then take advantage of this fact in order to see whether NEMs are present in the system and to cast the Hamiltonian into its normal form.

By using the relation (see, e.g., [11,43,44])

δF δ Λ = k=+∞ k=-∞ δF δ Λ k e -iky = 1 2π +∞ k=-∞ ∂ F ∂ Λ-k e -iky , (48)
where F ( Λ) = F ( Λk ), it can be shown that the Hamiltonian structure of ( 41) and ( 42), is given by the bracket

{F, G} = +∞ k=1 ik 2π ∂F ∂ Λk ∂G ∂ Λ-k - ∂F ∂ Λ-k ∂G ∂ Λk -α 2 P ∂F ∂ Λk ∂G ∂ P-k + ∂F ∂ Pk ∂G ∂ Λ-k - ∂F ∂ P-k ∂G ∂ Λk - ∂F ∂ Λ-k ∂G ∂ Pk . (49)
and the linear Hamiltonian, which is proportional to δ 2 F ,

H L = +∞ k=1 H k L = 2π +∞ k=1 | Λk | 2 1 + k 2 ⊥ - √ r α P | Pk | 2 , ( 50
)
where we have suppressed the sum on k x (note that k x only appears in the combination k 2 ⊥ = k 2 x + k 2 ). Although not canonical, this formulation, in principle is sufficient in order to detect the presence of NEMs for the equilibrium under consideration. Indeed, as already done for the four-field model of [13], we can make use of the property that NEMs can change their signature only if they become unstable through a "Kreȋn bifurcation" [1], or if the corresponding eigenvalues go through zero frequency. It is then sufficient to identify a NEM in a particular limit, and we are then guaranteed that its signature does not change as long as one of the two above mentioned phenomena does not occur. Sylvester's theorem also guarantees that the signature is independent on the choice of the coordinates we make. If we fix a wave vector k, then we can first evaluate the energy associated with the corresponding mode in the ( Λk , Λ-k , Pk , P-k ) coordinates by inserting eigenvalues and eigenvectors associated to k in the expression for H k L . 2). In this case both modes are PEMs and the system is energy stable. In the plot on the right, αP is positive (αP = 0.5, √ r = 0.2). For 0 < k ⊥ < 1.22, the equilibrium is stable, but the slow mode is a NEM. A Kreȋn bifurcation occurs at k ⊥ = 1.22.

This results in

H L k s,f = 2π 1 -4 √ rα P -2k 2 ⊥ α P √ r ± 1 -4(1 + k 2 ⊥ )α P √ r 2α 2 P | Pk s,f | 2 , (51)
which is the energy of the slow and the fast modes of wave number k (summed over k x ). In order to identify PEMs and NEMs, it is sufficient to consider the limit k ⊥ → 0, which yields

H L k,k ⊥=0 s,f = 2π 1 -4 √ rα P ± 1 -4α P √ r 2α 2 P | Pk s,f | 2 . (52)
We can then see that, for the fast mode (corresponding to the + sign in (52)),

H L k,k ⊥=0 f
is positive, and therefore a PEM. Also, it will remain a PEM as parameters are varied in a continuous way, until the instability threshold is reached. For the slow mode, two cases exist for finite α P . If α P < 0, then H L k,k ⊥=0 s > 0 and, again, we have a PEM.

If 0 < α P < 1/(4 √ r), on the other hand, the slow mode is a NEM.

The above mentioned instability, occurring at large k ⊥ , for α P > 0, indicates a Kreȋn bifurcation, which is one of the possible types of bifurcations occurring in Hamiltonian systems. These situations are illustrated in Fig. 1.

When ω k s is real and negative, that is for α P < 0, both the slow and the fast modes are PEMs and the two branches correspond to two dispersive waves with opposite sign. These waves correspond to inviscid drift waves modified by the presence of the ETG and magnetic field curvature. In the limit r → 0, corresponding to vanishing ETG, the two modes degenerate into a Hasegawa-Mima drift wave. When ω k s is real and positive, i.e. for 0

< α P < 1/4(1 + k 2 ⊥ ) √ r,
both modes are still stable but the slow mode is now a NEM. Comparing the two plots of Fig. 1, one observes that ω k s went from negative to positive, i.e. it crossed through zero frequency, while changing from a PEM to a NEM.

For the parameters chosen for the figure, the instability threshold, due to the presence of ETG, occurs at k ⊥ ≈ 1.22.

For perpendicular wave numbers above this value, the equilibrium is indeed unstable. The transition of the two real eigenvalues into a complex conjugate pair, occurring at k ⊥ ≈ 1.22, is an example of a Kreȋn bifurcation. Note that, as predicted by Kreȋn's theorem (see, e.g., Ref. [1]), if a Kreȋn bifurcation between two eigenvalues occurs, one of them must be a NEM. This is indeed our case.

If we consider now the stability condition (22) for no-flow equilibria, together with [START_REF] Kre Ȋn | Four Papers on Ordinary Differential Equations[END_REF], we find that the homogeneous equilibrium [START_REF] Bokhove | Large-Scale Atmosphere-Ocean Dynamics 2, geometric Methods and Models[END_REF] is energy stable if H ′′ (P eq ) > 1, which is equivalent to α P < 0. Indeed, if this condition is satisfied, the equilibrium is stable, with no NEMs. If we now use the actual pressure p, as a variable, and consider the corresponding homogeneous equilibrium p eq = a p x = √ r(α P -√ r)x, we can reformulate our results in the following way. If the equilibrium pressure gradient is such that a p < -r (i.e., α P < 0), then the system is energy stable for every k. If, on the other hand, -r < a p < -r + 1/4(1 + k 2 ⊥ ), (i.e. 0 < α P < 1/(4(1 + k 2 ⊥ ) √ r) then the system becomes unstable through a Kreȋn bifurcation at a critical k ⊥ , for fixed r. For k ⊥ below this critical value, the equilibrium is spectrally stable but not energy stable. Indeed, the slow mode in this case is a NEM, which makes the equilibrium fragile with respect to the addition of dissipation or nonlinearities.

C. Normal form for the Hamiltonian of the ETG model

The explicit knowledge of the eigenvalues of the linearized system, makes it possible to cast the Hamiltonian into its normal form, as discussed in Sec. IV A. First of all we point out that the transformation ( Λk , Λ-k , Pk , P-k ) → (q 1 , q 2 k , p 1 k , p 2 k ), corresponding to

q 1 k = π kα 2 P ( Pk + α P Λk + P-k + α P Λ-k ) , p 1 k = -i π kα 2 P ( Pk + α P Λk -P-k -α P Λ-k ) , q 2 k = π k ( Λk + Λ-k ), p 2 k = i π k ( Λk -Λ-k ) , (53)
puts the system into canonical Hamiltonian form. Indeed, in terms of the variables (q 1 k , q 2 k , p 1 k , p 2 k ), the bracket of (49) takes the canonical form

{F, G} = ∞ k=1 ∂F ∂q k 1 ∂G ∂p k 1 - ∂F ∂p k 1 ∂G ∂q k 1 + ∂F ∂q k 2 ∂G ∂p k 2 - ∂F ∂p k 2 ∂G ∂q k 2 . ( 54
)
The Hamiltonian (50) in terms of the new variables, on the other hand, reads

H L = 1 2 ∞ k=1 4 i,j=1 A k ij z k i z k j , (55)
where

A k =    a c 0 0 c b 0 0 0 0 a -c 0 0 -c b    , z k = (q k 1 , q k 2 , p k 1 , p k 2 ) (56) with a = - √ rα P k, b = k(1/(1 + k 2 ⊥ ) - √ rα P )
, and c = √ r|α P |k. We emphasize that the transformation (53) is constructed in such a way that the canonical variables q k i and p k i are real. We are thus in the framework depicted in Sec. IV A. For each k, the equations of motion are given by żk

= J c A k z k . ( 57
)
Upon writing the variables as

q k 1,2 = zk 1,2 e iω k t + zk 1,2 * e -iω k t , p k 1,2 = zk 3,4 e iω k t + zk 3,4 * e -iω k t , (58)
and dropping the tilde in the following, we obtain that the eigenvectors of the linearized system are

z k s = q k 1s    1 -B - -i -iB -    , z k -s = q k 1s *    1 -B - i iB -    , z k f = q k 1f    1 -B + -i -iB +    , z k -f = q k 1f *    1 -B + i iB +    . (59)
These are the eigenvectors corresponding to the eigenvalues ω k s , ω k -s , ω k f and ω k -f , respectively. In (59) we introduced the quantities

B ± = b + a ± (b + a) 2 -4c 2 2c , (60)
whereas q 1 k s,f are complex coefficients. Following Sec. IV A, the Lagrange bracket for the slow mode reads

z k -s T Ωz k s = 2i(1 -B 2 -)q k 1s * q k 1s . (61)
Direct calculation shows that

1 -B 2 -= - (b + a) 2 -4c 2 ( (b + a) 2 -4c 2 -(b + a)) 2c 2 > 0. ( 62
)
To obtain the inequality in (62), we made use of the fact that, for stable modes, (b + a) 2 -4c 2 < b + a.

The inequality (62) tells us that, for slow modes, we must pick up the plus sign in the general expression [START_REF] Morrison | [END_REF].

Moreover, if we choose the normalization constants so that

q k 1s = q k 1s * = 1 D - ≡ 1 1 -B 2 - , (63)
we obtain that the Lagrange bracket for slow modes becomes

z k -s T Ωz k s = 2i. (64)
Of course,

z k s T Ωz k -s = -2i, (65)
and consequently, according to the definition, given in Sec. IV A, we have a PEM, when ω k -s = -ω k s > 0, and a NEM when ω k -s = -ω k s < 0. This confirms the results we obtained in Sec. IV B with noncanonical variables.

Following the same procedure for the fast mode, we find

z k -f T Ωz k f = 2i(1 -B 2 + )q k 1f * q k 1f . (66)
Given that 1 -B 2 + < 0, the Lagrange bracket for the fast mode becomes

z k -f T Ωz k f = -2i, (67)
after having chosen the following normalization for the eigenvectors:

q k 1f = q k 1f * = 1 D + ≡ 1 B 2 + -1 . (68)
Because ω k f is always positive, according to the definition, (67) tells us that the fast mode, as expected, is always a PEM.

The transformation that casts the Hamiltonian (55) into normal form, following Sec. IV A, will be a real canonical transformation T k that, for each k, maps a new set of coordinates zk

= (Q k 1 , Q k 2 , P k 1 , P k 2 )
, in terms of which the Hamiltonian is diagonal, into z k .

After noticing that

z k -s * T Ωz k -s = -2i, (69)
z k f * T Ωz k f = -2i, (70)
the matrix associated with the application T k is constructed in the following way:

T k =      1 D- 1 D+ 0 0 -B- D--B+ D+ 0 0 0 0 1 D--1 D+ 0 0 B- D--B+ D+      . (71)
Direct calculation shows that

T k T A k T k =     -ω k s 0 0 0 0 ω k f 0 0 0 0 -ω k s 0 0 0 0 ω k f     . (72)
Consequently, the Hamiltonian (55) can be finally written as

H L = 1 2 +∞ k=1 4 i,j=1 (T k zk ) i T A k ij (T k zk ) j = 1 2 +∞ k=1 ω k f Q k 2 2 + P k 2 2 -ω k s Q k 1 2 + P k 1 2 . ( 73
)
The expression (73) corresponds to the normal form for the Hamiltonian of the linearized ETG model for stable modes.

It clearly shows how the corresponding energy can be decomposed into the sum of energies of harmonic oscillators which possess, as characteristic frequencies, those of the fast and slow modes. The harmonic oscillators associated with the fast modes always provide a positive contribution to the total energy. Those associated to the slow modes, on the other hand, give a negative contribution if ω k s > 0, which translates into the condition on the equilibrium pressure gradient discussed in Sec. IV B.

V. CONCLUSIONS

By making use of the Hamiltonian formalism, we have analyzed the mode signature and the stability properties of an ETG fluid model. The families of Casimir invariants of the model were obtained, thereby showing that the dynamics of the model is subject to an infinite number of constraints. A stability condition has been derived, according to which, the absence of equilibrium flow and a restriction on the pressure equilibrium profile imply stability. Subsequently, after reviewing the concept of mode signature in the Hamiltonian framework, we have explicitly determined the energies of stable modes. From the stability viewpoint, the dispersion relation gives us a spectral stability condition which, however, does not give us information about the stronger condition of energy stability. Indeed, our analysis shows that spectrally stable homogeneous equilibria can be of two types, depending on the value of the parameters. If α P < 0, equilibria are spectrally and energy stable (i.e. with no NEMs). If 0 < α P < 1/(4(1 + k 2 ⊥ ) √ r), on the other hand, equilibria are still spectrally stable, but they are not energy stable. Indeed, the sign of the second variation of the free energy functional, in this case, is indefinite because of the presence of NEMs. Equilibria of the second type might then be prone to dissipation-induced or nonlinearity-induced instabilities.

As anticipated in Sec. I, one of the advantages of the Hamiltonian formalism for investigating stability and mode signature, is that it is very general and can be applied to any plasma model with a Hamiltonian structure. The method described and applied in the present paper, nevertheless, refers only to systems with discrete spectrum. A natural and promising future project, would be to analyze mode signature and perform the normal form analysis for more complex equilibria that support continuous spectrum using the techniques developed in [5,[START_REF] Morrison | [END_REF][41][42]. Such equilibria have more spatial dependence, for example.

FIG. 1 :

 1 FIG.1: Depiction of two possible mode signature situations for ETG modes, depending on the sign of αP . The plot on the left refers to the case of negative αP (αP = -0.3, √ r = 0.2). In this case both modes are PEMs and the system is energy stable. In the plot on the right, αP is positive (αP = 0.5, √ r = 0.2). For 0 < k ⊥ < 1.22, the equilibrium is stable, but the slow mode is a NEM. A Kreȋn bifurcation occurs at k ⊥ = 1.22.

Acknowledgments E. Tassi acknowledges fruitful discussions with Ö. D. Gürcan during the Festival de Théorie 2009. It is also a pleasure to acknowledge useful discussions with the Nonlinear Dynamics group at the Centre de Physique Théorique, Luminy. This work was supported by the European Community under the contracts of Association between EURATOM, CEA, and the French Research Federation for fusion studies. The views and opinions expressed herein do not necessarily reflect those of the European Commission. Financial support was also received from the Agence Nationale de la Recherche (ANR EGYPT). In addition, PJM was supported by the US Department of Energy Contract No. DE-FG03-96ER-54346.


Eric Sabourin
email: eric.sabourin@cirad.fr

CERTIFICATION PROCESS AS AN INTERFACE BETWEEN EXCHANGE MARKETING AND RECIPROCITY

Keywords: Agri-food qualification, certification systems, reciprocity, marketing exchange, Brazil qualification des produits, systèmes de certification, réciprocité, échange marchand, Brésil Agri

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

This paper mobilizes the theory of reciprocity in economic anthropology to analyze family farm product qualification process effects in Brazil. All qualification processes guaranteeing the origin, the specificity, the quality process or the standards of a product, may reduce the effects of competition and speculation, which are distinctive of the capitalist exchange. The hypothesis is that a qualification process can contribute to establish a symmetrical relation of reciprocity between producer and buyer. It can also contribute to generate a sharing structure (of quality) among the group of producers. When the producer's name is at stake, reciprocity relationships generate values of trust, reputation, honor and accountability. These ethical values guarantee the legitimacy and the authority of the certification process of quality and origin standards. In fact, it thus creates a territoriality of reciprocity around a specific product. However, apart from this group of qualified producers and also for the marketing of other products which are not quality-labeled, it is the rules of the capitalistic exchange market that apply. Qualification and certification could also induce exclusion. That is why, in this type of process, there has to be some sort of interface between production and the market. The quality certification mechanism reintroduces a dimension of reciprocity (the domestic unit, the peasant community, the co-operative) into the capitalist exchange market system. For this, the certification process should not be, by its cost or its modalities a factor of social or economic exclusion. It does not exclude producer's access to qualification for financial reasons or exclude consumers through its effects increasing the product. This work is based on a comparative study of several experiences of quality certification for family farming agro-ecological products Brazil, around three different mechanisms: External Audit Certification (EAC), Network Participative Certification (NPC) and mutual cocertification.

There are three main chapters: 1) The theoretical frame and context; 2) Case studies and results; 3) Discussion, lessons and perspectives.

COMMERCIALIZATION AND THEORY OF RECIPROCITY

Selling farm products is not just a matter of capitalistic exchange. Although it is now the norm for many of the small and family farms of Brazil [START_REF] Conterato | A mercantilização da agricultura familiar do Alto Uruguai/RS: um estudo de caso no município de Três Palmeiras[END_REF][START_REF] Ploeg | The New peasantries: struggles for autonomy and sustainability in an era of Empire and Globalization[END_REF], one still sees socially controlled markets which also abide by the rules of reciprocity and redistribution to use the different economic services as defined by [START_REF] Polanyi K ; Arensberg C | Trade and Market in the Early empires[END_REF]. In Brazil, as a result of direct relations between producers and consumers, local trade fairs and weekly markets generate social ties calling upon sociability and reciprocity. What is meant by reciprocity here is the dynamic of service reproduction, which leads to the development of social ties, as identified by [START_REF] Mauss | Essai sur le Don[END_REF]. [START_REF] Temple | Teoría de la Reciprocidad[END_REF] defined reciprocity as the repeating of an action or a service in such a way that people may recognize one another and feels they belong to a community. From an anthropological angle, the theory of reciprocity corresponds therefore to a reflective action occurring between individuals, or an inter-subjective relation and not merely the swapping of goods and objects like in the case of an exchange. Two elements of the theory of reciprocity are particularly mobilized for this study:

1. Reciprocity principle is more than a mere gift/counter-gift relationship between peers or symmetrical social groups. The reductionism of this definition, which prevailed for a long time and still prevails sometimes in anthropology, leads to confusing the notions of symmetric exchange and reciprocity. This error persists so long as reciprocity is interpreted following the binary logic of exchange. Exchange is merely a swapping of objects. With [START_REF] Temple | La réciprocité ou la naissance des valeurs humaine[END_REF] the theory of ternary logic (Lupasco) allows for the appearance of a third party in the relationship of reciprocity. This third party can be interpreted as the being of the Agri-food certification process as an interface between exchange & reciprocity Sabourin E relationship or the very structure of inter-subjectivity, which is irreducible for the exchange of goods or services since it does away with the social tie or debt. Economically speaking, reciprocity therefore, is not only an economic category that differs from the capitalist exchange as identified by [START_REF] Polanyi K ; Arensberg C | Trade and Market in the Early empires[END_REF], but an economic principle in contrast and even antagonistic with exchange.

2. The reciprocity relationships can be analyzed in structural terms, and declined according to a few elementary structures (Temple 1998).

In their symmetrical form, these structured relationships of reciprocity give birth to ethical values: a symmetrical bilateral structure produces a feeling of friendship and the symmetrical distribution structure of goods within a group gives rise to justice. This is clearly the most complex point in the theory of reciprocity as put forward by [START_REF] Temple | La réciprocité ou la naissance des valeurs humaine[END_REF]. It is also the most difficult to validate, since it involves not only the production of materialistic goods but also that of feelings and human values. It is, however, a key point that paves the way for analyses and alternative proposals in the field of human economy.

When it comes to marketing farm products, peasants or their organizations have set up several types of interfaces allowing for a coexistence and connection between reciprocity and exchange practices. The first interface is direct selling which involves the meeting of a producer and a consumer. In this case, there is of course an exchange relationship, but the direct contact between the producer and the buyer triggers a binary relationship of reciprocity.

Here we have a one-to-one relationship that leads to emotional values such as friendship, mutual recognition or to ethical values of loyalty and respect. This is also true for the short food-chains socially managed by producers' associations or consumer co-operatives 1 . One could also call to mind the case of the peasant markets and "peasant's baskets" in France that are managed by associations including both farmers and consumers.

All these direct contact situations give rise to human relations, conversations and explanations about products, work, techniques, tricks, etc.. They also give rise to feelings and values of friendship, loyalty and trust between producers and consumers. When a direct relationship is not possible, one possible alternative is to call upon the qualified and interpersonal intermediation of a broker. The broker can establish a ternary relationship of reciprocity through co-knowledge, a network effect and information as to the quality of the product and the producer. A structure of this type produces trust, accountability and reputation (of both the producer and the broker), which in turn produce prestige.

Mechanisms used for product qualification can also enable an articulation between reciprocity and marketing exchange. Product qualification policy or instrument (quality-label or appellation) reduces the effects of competition and speculation following the logic of the capitalistic exchange and helps in the building of territories of reciprocity.

When the name of a producer, his locality or social or professional group is at stake, one is indeed confronted with reciprocity relationships which give rise to values in connection with reputation, honor and accountability. These values trigger the production of quality and a commercial cycle for these so-called "qualified" products. It is therefore these ethical values that guarantee the legitimacy and authority of the certification process establishing the standards of quality and origin. Capitalist businesses have, for that matter, identified the importance of these values (reputation, trust, social accountability) and converted them into marketing objects via the privatization of standard quality control services. Above all, however, and especially in the case of farm food products,

1 Or the two combined, as is the case in Brazil in the region of Porto Alegre between the MST and consumers' co-operatives

Agri-food certification process as an interface between exchange & reciprocity Sabourin E the real economic sanction comes from the consumer himself. Moreover, the guarantee of quality (organic for example) is often inherent to the production itself. There is absolutely no reason a priori why any farmer who has engaged in the process of converting to agroecological farming methods, be it for ethical reasons (produce healthy products, land preservation) or materialistic reasons (added value), should attempt to fraud or decide to go back to the way he was farming before.

The key to the interface is a matter of trust between the consumer and the producer, between producers themselves, or between producers and middlemen.

It is this question as to the credibility of a product's quality that has to be validated and go through a checking process known as certification.

My hypothesis is that the certification mechanism itself is no neutral technological datum. On the contrary, it is a social construction based, before any other consideration, much more on human trust than on the fact that the product is in compliance with production processes or technological standards. These facts have already been identified by the Conventions economy [START_REF] Dupuy | Convention et Common Knowledge[END_REF][START_REF] Eymard-Duvernay F | La Négociation de la Qualité[END_REF]Valeschini, 1995) and confirmed by the presence of participative certification procedures based more on the validation of social compliance than on technological compliance. These approaches, however, do not push the logic of producing social and ethical values right through. Indeed, it would be logical, in the case of farm and health food products, if it led to co-certification or to putting an end to the external marketing certification process.

In Brazil, various co-certification and participative certification systems have been set up between producers or between producers and consumers (Medaets & [START_REF] Medeiros | A Ação coletiva no controle da qualidade da produção orgânica familiar: Análise comparativa entre a certificação por auditoria externa e a certificação participativa em rede[END_REF]. These alternatives are due to the development of local agro-ecological markets and to the effects of reputation. The second part of this paper is a comparative analysis of three certification systems (External Audit Certification-EAC, Network Participative Certification -NPC and co-certification) in which the advantages and limitations of centripetal and decentralized certification are highlighted.

THREE CASE STUDIES IN BRAZIL

In the three cases, we are talking about family farmers producing and marketing their own agro-ecological production. In Amazonia (Mato Grosso State), the system is a co-operative of agro-ecological products aimed for the export and domestic market. In the South (Santa Catarina State), the aim is for the produce to be marketed through the standard marketing channels of the capitalistic exchange (supermarkets, specialty stores), or even exported. In the Nordeste, the produce is marketed via direct selling on local markets in several towns in the state of Paraiba.

Group certification by way of external audit (Cooperagrepa MT & Ecocert)

The Cooperagrepa organic farmers' co-operative (300 member families) at Terra Nova do Norte (Mato Grosso State), has been producing guarana, Brazilian nuts, brown sugar, coffee and honey since 2003. The certification process at this co-operative is carried out by the private firm Ecocert Brazil [START_REF] Whyte | Produits de Qualité, Territoires et développement durable, le cas de l'Agreco-SC[END_REF]. The group contract has been signed for 50 farms by Ecocert who requires at least 30 farmers for this modality.

Ecocert performs the external audit on the basis of samples taken from the farms (10 on average each year). Internal inspections are completed by an agronomist from Cooperagrepa who is responsible for all recordings and documents for each farm between two visits of the Ecocert inspectors. The costs of the external audit are too high for the smaller members of Cooperagrepa. The co-operative is receiving funds from SEBRAE (Brazilian Support for Small Businesses Unit)

Agri-food certification process as an interface between exchange & reciprocity Sabourin E for 5 years. The entire cost of the audit is covered by the funds received from SEBRAE for the first three years, and 50% for the other two years. In 2009, the cost of the audits will amount to 700 reais (R$) per farm (350R$ charged to the producer). The money is deposited by SEBRAE and the farmer prior to the Ecocert's visit and the certificate is supplied following payment. The larger the group, the lower the individual cost. The cost of an individual Ecocert certification system is 1,000R$/year. Checking that the farm is in compliance is largely a technical process. The group of producers goes through a transition period during which it must perform the required recordings in relation the requirements established by the production model. At the end of this period, the certification organization checks that the group is in compliance, a procedure which involves inspecting all the farms within the group and performing laboratory tests on a series of randomly collected samples.

The decision or not to award certification is not taken by the individuals who carried out the inspections, but by other members of the certification organization. Thus, in accordance with ISO specifications, there is a separation between the inspection itself and the certification process. All recordings are retained by the certification organization. Technicians are sent out to check up on the farms and specially trained for inspection procedures. Recruitment of these inspectors is subject to a number of criteria that reduce the possibility of there being any other type of relationship between the inspector and the person being inspected.

ECOVIDA agro-ecological network

ECOVIDA Agroecologia network was founded in 1998 in the state of Santa Catarina to promote agro-ecology, convey information between its members, create legitimate methods for the production of credibility and provide a guarantee of techniques used by its members" [START_REF] Agroecologia | A Legalização da Agricultura Orgânica e da Certificação de Produtos Orgânicos no Brasil: o posicionamento da Rede Ecovida de Agroecologia[END_REF]. The network covers the states of Santa Catarina, Paraná, Rio Grande do Sul and São Paulo and includes over 120 organizations and collective farmer systems (formal and informal groups, associations and producer and consumer co-operatives) and 23 support entities directly affecting approximately 1,500 families.

Several international standards, such as IFOAM/IOAS grower groups (Ifoam, 2009), the EEC Rule nº 2.092/91 or the MAPA nº 6 standard of the Brazilian Ministry of Agriculture dating from 2002 pave the way for certification among organized groups of producers (Brazil, Mapa, 2002). These standards help in establishing group profiles and the conditions of their certification, the idea being to increase the opportunities for access to this service in so far as the mechanism considerably reduces certification costs.

The ECOVIDA network has adopted Network Participative Certification (NPC) in several of these groups.

According to Trujilo Ferrari (1983), "compliance" represents the "alteration or change in the behavior and beliefs of an individual or a group, in a direction established by a larger group". Any alteration involves a process of positive, negative and neutral sanctions. Thus, in the case of NPC, a serious investment is required on behalf of the individuals involved in the system if they want to attain "social compliance". Evaluation of social compliance differs from technological compliance in three ways: it is carried out on the basis of the outcome of meetings and committees where systemic observation and the pooling of information are devoid of a technological profile; the compliance requirements refer to social conditions and the implementing social factors thereof but do not refer to production techniques. Certification is granted (or not) by technicians who have trained in agro-ecology and not by experts who have been trained to carry out inspection/audits. The ethical committee's "visits" constitute the second quality control check. Technicians, farmers and sometimes consumers who pay these visits to the production units come from an external environment. The third quality control check is carried out after completion of peers' reviewing within the frame of the core of the network or group. It is at this point in the process that the certification decisions are made.

Agri-food certification process as an interface between exchange & reciprocity Sabourin E

Agro-ecological markets in the Paraiba

In the state of Paraiba in the Nordeste of Brazil, markets selling agro-ecological products2 of Lagoa Seca, Massaranduba & Campina Grande, fit into the strategy of the Polo Sindical of Borborema (PSB a regional peasant trade union federation), which is to increase the autonomy of family farms in relation to conventional channels over which they have no control. The target consumer has a low income and the prices on these markets are no different from the prices of the same standard products on the other markets of the region. The prices are also lower than the same conventional products in the supermarkets.

The PSB initiative has led a group of farmers to come together via the founding of an association and adopt product control guaranteeing both quality and origin. Farmers from the agro-ecological groups are commissioned to pay regular visits to each producer and check his culture processes, technical patterns and the nature of inputs used. Each farmer may find himself switching from evaluating to being evaluated. The risk of fraud is negligible since the main goal for all concerned is to patronize customers, and this means not cheating on the quality of the products on sale. This system of mutual (or reciprocal) certification has the great advantage of being free of charge, not increasing production costs and not engaging farmers with certification firms.

In the co-certification process, quality control becomes an inherent part of the production structure so long as it finds itself repositioned in a system of reciprocity. These markets work well thanks to the involvement of a few families who ensure that the rules are being respected. The other participants, who are ideologically less involved, are simply opportunistic about the system. Their interest in agro-ecology is essentially related to the emergence of consumer demand and access to a marketing channel that pays more for their work than selling wholesale. Agro-ecological markets are not simply due to the changing of farming techniques or public organization incentives. The reason of their success lies in their qualified and established (and therefore collective and institutionalized) response to consumer demand for quality products devoid of pesticides.

DISCUSSION, LESSONS AND PERSPECTIVES

Comparative analysis of the three experiences

Analysis of the three certification systems presented focuses on practices, the type of relationships they correspond to, results and the nature of mobilized or engendered values. Each of the studied systems can be treated as a logic as well as a specific language.

EAC is based on a high and (perhaps excessively) stringent registration level and external inspections. The aim of these external inspections is to reduce the risk of opportunism, by centralizing the certification decision which is taken on the basis of the inspection documentation forms supplied by the certifier. It has been pointed out by the EEC ( 2002) that these EAC certification modalities do not distinguish between big agro-industrial businesses and small family or peasant type farms. Thus, the fact that inspection standards and the amount of effort to conform are the same for all, regardless of the size of the enterprise, often leads to operations that are unnecessary

Agri-food certification process as an interface between exchange & reciprocity Sabourin E and unsuitable for the small farmers. This criticism also applies to the sampling criteria, tests and analyses, recording procedures of inputs and sales, etc…. These procedures, which have all been "globalized" as a result of ISO specifications and guides, are not in keeping with the specificities of local, rural environments or the potential of local co-operative relationships and reciprocity in terms of developing credibility. Moreover, these certification systems, which are undertaken by a private enterprise, increase costs and limit access to certification for small, family and peasant farms. Lastly, and amongst other things, because of the financial requirements and the amount of documents that has to be supplied, the privately managed certification system is not very efficient at qualifying family farms. This matter, however, cannot be ignored since in Brazil agro-ecological and organic food product markets must comply with a procedure that involves being granted certification for the production of organic foods (Brazil, Mapa 2002).

NPC and mutual co-certification are based on "social compliance" to a mutual objective of a quality culture system and transformation/packaging process of the product and both formal and informal agreements made to reach this objective. The decentralized (or participative) certification decision gives rise to an exclusion mechanism when it is established that the agreements that were made within the group have not been respected; consumers (a minima one consultation) are also enlisted in the certification decision.

Group Certifications aims at a degree of excellence as regards inspections and recommendations in connection with standards for the export market. The decision depends on a collection of data and documents revealing the traceability of the inspected product. The social compliance system advocated by NPC is dependent on visits and reviewing undertaken by peers. These two distinctive operations vary according to a number of criteria or objectives that are less numerous and stringent than enforced in the case of certification by an external audit (table 1).

For the EAC Ecocert type, the evaluation (inspection) is carried out be an inspector and a centralized certification decision is taken by a council held within the certifier organization.

The Associação de Certificação Participativa of the Ecovida network does this job thanks to the evaluations and ethical advice given by the local Groups (called nucleos). In the case of Paraiba, it is the agro-ecological producers' association that ensures standards are met, via meetings and technical visits to the farms. The role of consumer pressure and mutual knowledge have a considerable impact in the Paraiba region and sometimes also within the Ecovida network. In both cases, the certification decision is decentralized and taken by actors who are involved in the production process. There is no separation between the roles of inspection and certification. In the case of the nucleos, on the contrary, the two roles are separated.

In the case of NPC, there is no external inspector, but the visits are ensured by farmers and technicians who have been trained in agro-ecology and are qualified to check compliance to the standards as specified by the Ecovida network (table 1)

The fact that the documentation required by NPC is less complicated than that of EAC doesn't necessarily mean that the quality of the inspection should be questioned. Moreover, the decentralized and participative nature of NPC ensures the producers' competence, not only in terms of their knowledge of standards and quality control, but also in terms of the management and accounting of their operations (Medaets & [START_REF] Medeiros | A Ação coletiva no controle da qualidade da produção orgânica familiar: Análise comparativa entre a certificação por auditoria externa e a certificação participativa em rede[END_REF] In the NPC system, meetings gathering the Groups' families, the intensity of the mutual exchange of information, the establishing of reputation among peers and the exclusion of all those who do not abide by the signed agreements are all factors contributing in the construction of a "social compliance". This social compliance is the fruit of a social pressure that finds its source within the group. It is motivated by shared values and shared social and materialistic interests. In the case of the markets of Paraiba, where the documentation Agri-food certification process as an interface between exchange & reciprocity Sabourin E required for auto-certification process is even simpler, one may argue the intensity of oral exchanges. These exchanges are a source of mutual information and social pressure, factors that contribute in the construction of the producer and product's reputation. As regards the inspection mechanisms, the main expectation of a certification system is by and large that it should lead to reducing the asymmetry of information and fraud and/or opportunism. The three systems all fulfill this goal with no significant differences in terms of results, but with a big difference in terms of costs and processes (table 1).

Table 1 Comparing of quality control modalities

Lessons and perspectives

Qualification policies (quality-label or appellation) restrict the effects of competition and speculation in the capitalistic exchange. It can be noted that this interface can be managed

Agri-food certification process as an interface between exchange & reciprocity Sabourin E and regulated by a community of producers. This is the case with the mutual and gratuitous auto-certification process of agro-ecological markets in the Paraiba region, which is carried out according to a principle of reciprocity. A system of this type not only avoids extra costs which add to the price of the product, but helps in maintaining the producer's competence and autonomy. Indeed, the producer's credibility as regards the quality of his products is based on other elements such as mutual knowledge, direct physical relationships between producers and consumers and, of course, reputation, which supplement and validate the auto-certification or centrifugal qualification process.

An external firm can be commissioned to undertake the qualification process but firms of this type are run according to a contractual and commercial logic. This solution considerably increases production costs and places the farmers in the position of being dependent and subordinate, and even lacking the competence to validate and control the quality of their own products. On the other hand, as the external certifier firms are recognized by the rules of international trade, they may be demanded when the producers are targeting the export market.

For this reason, Brazilian farmers have adopted the intermediate NPC system, which maintains the autonomy and the development of farmers' skills, at a low added cost whilst ensuring external and international recognition.

In fact, the NPC system works as an interface between the logics of reciprocity and exchange on two counts. Firstly, it creates a link between domestic production and market exchange by protecting the qualified production from the effects of competition. Furthermore, the guarantee and control of product quality in the NPC system is based on reciprocity and mutual aid between peers. However, this virtually gratuitous certification mechanism (3 days of work/family/year) defies the capitalistic mercantile logic but at the same time it enables producers to enter into dialogue with it, especially concerning international standards required for the export market. In this sense, NPC is a hybrid certification mechanism suitable for systems that combine the logics of exchange and reciprocity. The theory of reciprocity allows interpreting the social meaning and the economic logic constructed by human communities during the course of history, behind services which, sometimes imperceptibly, have been have been categorized as capitalist exchanges. Capitalism, however, is not particularly concerned about ethical values or a more equitable project for the entire humanity.

However, all the possibilities of qualification systems for rural products have not still been explored. The case of mutual co-certification by the Association of agro-ecological producers of the Agreste and Paraiba, shows that a centripetal qualification validation is feasible. Centripetal qualification (or qualification from within) and the personalized quality-label constitute one of the keys for an endogenous or auto-centered rural development that satisfies first and above all the needs of local communities.

CONCLUSIONS

This paper proposes an original questioning of social equity in terms of innovation for agroecological products qualification and certification processes. It proposes an alternative with participative certification mechanisms which are quite cheaper and sustainable than EAC.

In terms of global reflexion on innovation, it remembers us not to forget social and economical innovation. We are not condemned to depend on technological innovations, always absorbed and transformed in some more costs for consumers by capitalism.

The capitalist exchange market sets the homogeneity of firms' standardized certification standards against heterogeneity, varied procedures or Network Participative Certification registration modalities. According to the logic of the capitalist market, these variations could lead to reducing the credibility of the NPC system. However, when it comes to the point, it is more the case of certification resulting from external audit, for which it cannot be claimed it

Agri-food certification process as an interface between exchange & reciprocity Sabourin E translates the diversity of production systems and conditions, the diverse realities of different countries, terroirs and producers, and particularly social and cultural realities. This is one of the assets of certification by social compliance. It should not, however, become a complete substitute to technical compliance. This is why the producer's opinion as peers, and that of the consumers as judges, is absolutely essential. Notions of reputation, trust and knowledge sharing are at the heart of the qualification process of products, especially agro-food products [START_REF] Ostrom E | A behavioural approach to the rational-choice theory of collective action[END_REF]2003). Social pressure has been recognized for its efficiency in the process of guaranteeing quality credibility. Such values and relationships are typical elements of reciprocity structures. In Brazil, several authors have put forward the idea of developing the debate about social compliance as an efficient alternative mechanism in terms of quality control in the frame of family peasant farming.

At least, what has changed in the way of dealing with this issue over the past 15 years? In fact, the participative and group qualification and certification initiatives emerged in developing countries, where the theoretical analysis in terms of reciprocity or solidarity and plural economy is more diversified and active than in the North.

The debate, which is still timid in Brazil, will pave the way for applying the theory of reciprocity and the notion of system interface.

The example of the agro-ecological markets in the Paraiba region and of the Ecovida groups shows that a personalized and centripetal qualification, via the process of co-certification by peers, is equally efficient and considerably more economical than certification through private external firms. Indeed, consumer interest and their ability to acquire information and enter in more symmetrical relationships of reciprocity with producers is still the basis to the principle of product qualification. This approach opens new perspectives for research and action: on the one hand, around learning and values in the context of relations of reciprocity and, secondly, on how to control the interface between practices of exchange and reciprocity.

Family farmers direct selling markets of agro-ecological products (type of basically organic ecological farming with no chemical inputs).


Olivier Bournez
email: olivier.bournez@lix.polytechnique.fr

Daniel S Graça
email: dgraca@ualg.pt

Emmanuel Hainry
email: emmanuel.hainry@loria.fr

Robust computations with dynamical systems

Keywords: Verification, Model-checking, Computable Analysis, Analog Computations

 was done only for not-so-natural systems from a classical mathematical point of view (discontinuous differential equation systems, discontinuous piecewise affine maps, or perturbed Turing machines), we prove that the results presented there can be generalized to Lipschitz and computable dynamical systems. In other words, we prove that the perturbed reachability problem (i.e. the reachability problem for systems which are subjected to infinitesimal perturbations) is co-recursively enumerable for this kind of systems. Using this result we show that if robustness to infinitesimal perturbations is also required, the reachability problem becomes decidable. This result can be interpreted in the following manner: undecidability of verification doesn't hold for Lipschitz, computable and robust systems. We also show that the perturbed reachability problem is co-r.e. complete even for C ∞ -systems.

Introduction

The investigations on the relationships between dynamics and computations attracted the attention of several research communities. One of them is highly motivated by the question of computer aided verification, and in particular by the question of computer aided verification of hybrid systems (see e.g. [START_REF]Hybrid Systems: Computation and Control: 7th International Workshop[END_REF]).

One main motivation of this community is to get some "as automatic as possible" computer systems, that would take as input the description of a continuous or discrete (or hybrid) system, and the description of some property, call it "safety", and that would tell whether the system satisfies it or not.

The point is, by undecidability of the halting problem of Turing machines, there is no hope to get a fully decidable procedure if the provided formalism for describing hybrid systems allows the description of Turing machines, and if the formalism for describing the property allows to talk about their halting. Some classes of models, such as Timed Automata [START_REF] Alur | Automata for modeling real-time systems[END_REF], have been shown to provide subclasses of systems for which verification of the reachability property is decidable.

However, unfortunately, very simple classes of linear hybrid automata [START_REF] Henzinger | What's decidable about hybrid automata?[END_REF] or piecewise constant derivative systems [START_REF] Asarin | Reachability analysis of dynamical systems having piecewise-constant derivatives[END_REF] have been shown of being able to simulate arbitrary Turing machines. As a consequence, verification procedures are semi-decision procedures and not decision procedures. A more general result can be found in [START_REF] Collins | Continuity and computability of reachable sets[END_REF], where semidecidability is shown for non-linear systems in general, and decidability is proved for systems satisfying some particular condition.

Since the proofs of undecidability or, more generally, of simulation of Turing machines, often involve to encode the configuration of a Turing machine (or of a two counter automata) into some real numbers, and since this require infinite precision, in the hybrid system verification community a folklore conjecture appeared saying that this undecidability is due to non-stability, non-robustness, sensitivity to initial values of the systems, and that it never occurs in "real systems" [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF].

For example, Martin Fränzle writes in [START_REF] Fränzle | Analysis of hybrid systems: An ounce of realism can save an infinity of states[END_REF] "Hence, on simple informationtheoretic grounds, the undecidability results thus obtained can be said to be artifacts of an overly idealized formalization. However, while this implies that the particular proof pattern sketched above lacks physical interpretation, it does not yield any insight as to whether the state reachability problem for hybrid systems featuring noise is decidable or not. We conjecture that there is a variety of realistic noise models for which the problem is indeed decidable".

There were several attempts to formalize and prove (or to disprove) this conjecture: it has been proved that small perturbations of the trajectory still yields undecidability [START_REF] Henzinger | Robust undecidability of timed and hybrid systems[END_REF]. Infinitesimal perturbations of the dynamics for a certain model of hybrid systems has shown to rise to decidability [START_REF] Fränzle | Analysis of hybrid systems: An ounce of realism can save an infinity of states[END_REF]. This has been extended to several models by [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF]. In [START_REF] Asarin | Noisy Turing machines[END_REF] it is shown that Turing machines exposed to small stochastic noise can decide the Halting problem, since its computational power when the error converges to 0 is ≈ Π 0 2 . Let us look at the result presented in [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF]: they consider several classes of widely used models of dynamical systems: Turing machines, piecewise affine maps, linear hybrid automata, and piecewise constant derivative systems. For each of them a notion of "perturbed" dynamics is introduced and the computational power of the corresponding perturbed systems is studied. Perturbations are defined for each model using a notion of metrics on the state space. For a given model, with reachability relation R, the idea is to perturb the dynamic by a small ε, and then take (as the perturbed dynamics of the system) the limit (intersection) R ω of the perturbed reachability relations as this ε tends to 0. In that setting, a system is said "robust" if its reachability relation does not change under small perturbations of the dynamics, i.e. R ω is equal to R [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF]. This has a close resemblance with the notion of "structural stability" for dynamical systems: a system A is structurally stable if, roughly, ε-perturbed systems converge to A as ε → 0, a concept widely studied in the dynamical system theory see e.g. [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields[END_REF], [START_REF] Hirsch | Differential Equations, Dynamical Systems, and an Introduction to Chaos[END_REF].

In [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF], the authors show that for Turing machines, piecewise affine maps, linear hybrid automata, and piecewise constant derivative systems, the relation R ω belongs to the class Π 1 0 (it is co-recursively enumerable), and moreover, any Π 1 0 relation can be reduced to a relation R ω of a perturbed system: any complement of a recursively enumerable set, can be semi-decided by an infinitesimally perturbed system. This means that, for any robust system, its reachability problem is decidable. Indeed, as any system, its reachability problem is semi-decidable (recursively enumerable), and since it is robust, the complement of its reachability problem must be recursively enumerable, from which it follows that the reachability problem must be recursive for robust systems.

In other words, this gives a (partial) answer to the above mentioned conjecture: verification is decidable for robust systems, if the notion of robustness is the one considered here. If one prefers, undecidability of verification arises only when non-robust systems are considered.

In this paper we extend the result of [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF] for the case of Lipschitz and computable (in the sense of recursive analysis [START_REF] Weihrauch | Computable Analysis: an Introduction[END_REF]) systems defined on a compact set, considered as a model of computation. We present both continuous-time and discrete-time versions of our results.

Our aim is to reinforce in some sense the previous result: it follows that verification is decidable for robust systems considered in classical mathematics and computer science, that is to say for robust, Lipschitz, and computable dynamics. In other words, undecidability of verification is really a by-product of non-robustness, even if the system does not rely on trivial (piecewise) dynamics as in [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF].

In a more provocative way, undecidability of verification for safety properties over a compact domain is indeed an artifact of modelization for very general and natural classes of systems.

Formal setting: continuous-time dynamical systems

We begin with some definitions.

Definition 1. A function f : R m → R k is said Lipschitz over a set X if there is some K > 0 such that for all x, y ∈ X one has f (x) -f (y) ≤ K x -y . (1)
In particular it is well known that C 1 functions are Lipschitz over a compact set X ⊆ R m and that an initial-value problem

x ′ = f (t, x) x(t 0 ) = x 0
where f is Lipschitz, have an unique solution (see e.g. [START_REF] Birkhoff | Ordinary Differential Equations[END_REF]).

Definition 2 (Dynamical systems). Let X ⊂ R d , and consider some function f : X → X. Then we can define a (homogeneous inputless) discrete or continuous-time dynamical system associated to (X, f ) as follows:

-In the discrete-time case, a trajectory is a sequence of points {x 0 , x 1 , ...} ∈ X N , satisfying f (x i+1 ) = x i for all i ∈ N. -In the continuous-time case, a trajectory is a solution of the differential equation ẋ = f (x), x(0) = x 0 ∈ X , i.e. a derivable function φ : R + 0 → X, satisfying φ(0) = x 0 , and φ ′ (t) = f (φ(t)) for all t.

Note that dynamical systems are deterministic: there is only one trajectory starting in a given initial point.

In this paper we consider dynamical systems as recognizers of languages: Σ denotes the alphabet Σ = {0, 1} and Σ * denotes words over this alphabet. Therefore we need to encode words over Σ as points in X. This is done using the encoding ν :

Σ * → [0, 1] defined by: if w = w 1 . . . w n ∈ {0, 1} * , where w 1 , . . . , w n ∈ {0, 1}, then ν(w) = n i=1 (2wi+1) 4 i
. This classical encoding is rather arbitrary. Similar encodings would still yield the results proved in this paper.

We want to avoid not-so-interesting ways to get uncomputability:

-First, we restrict ourselves to dynamics over a compact domain. It is known that smooth systems can robustly simulate Turing machines [START_REF] Graça | Robust simulations of Turing machines with analytic maps and flows[END_REF], [START_REF] Graça | Computability with polynomial differential equations[END_REF] (configurations are coded as integers), if the perturbations are ≤ ǫ, for some fixed ǫ > 0. If we do some mapping from an unbounded domain to a bounded domain, these fixed ǫ-perturbations correspond to infinitesimal perturbations in the compact space. Further allowing infinitesimal errors in an unbounded space seems to originate a degree of modelization which is exceedingly artificial to be considered.

There is no loss in generality in assuming the compact domain to be [-1, 1] d . -Second, we want to avoid undecidability due to the impossibility of distinguishing two reals in the recursive analysis setting: this is why we assume that when some computation is accepted this can be stated by considering a value that is clearly below some threshold (1/4), and that this quantity is clearly above a bigger threshold (1/2) when the computation is not terminated.

This leads to the following definition:

Definition 3 (Considering a dynamical system as a language recognizer). Let H be a discrete/continuous-time dynamical system over space X = [-1, 1] d . Let V accept be the set of x ∈ X with x ≤ 1/4 and V compute be the set of x ∈ X with x ≥ 1/2. We say that H computes a language L ⊂ Σ * (or that L is the language of H), over alphabet Σ = {0, 1}, if the following holds: for all w ∈ Σ * , w ∈ L iff the trajectory of H starting from (ν(w), 0, • • • , 0, 1) reaches V accept . For robustness reasons, we assume that for any w ∈ L, the corresponding trajectory always stays in V compute .

Formal setting: recursive analysis

Recursive analysis or computable analysis, was introduced by Turing [START_REF] Turing | On computable numbers, with an application to the Entscheidungsproblem[END_REF], Grzegorczyk [START_REF] Grzegorczyk | Computable functionals[END_REF], Lacombe [START_REF] Lacombe | Extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles III[END_REF]: see [START_REF] Weihrauch | Computable Analysis: an Introduction[END_REF] for an up-to-date monograph presentation of recursive analysis using a computability point of view, or [START_REF] Ko | Computational Complexity of Real Functions[END_REF] for a presentation using a complexity theory point of view. Following Ker-I Ko [START_REF] Ko | Computational Complexity of Real Functions[END_REF], let ν Q : N → Q be the following representation 6 of dyadic rational numbers by integers: ν Q ( p, q, r ) → p-q 2 r , where ., ., . : N 3 → N is an elementarily polynomial time computable bijection.

A sequence of integers (x i ) i∈N ∈ N N converges quickly toward x (denoted by (x i ) i∈N

x) if the following holds for all i:

|ν Q (x i ) -x| < 2 -i . A point x = (x 1 , . . . , x d ) ∈ R d is said computable if for all j, there is a computable sequence (x i ) i∈N ∈ N N (i.e. a computable function a : N → N such that x i = a(i) for all i ∈ N) satisfying (x i ) i∈N x j . A function f : X ⊂ R d → R
, where X is compact, is said computable if there exists some d-oracle Turing machine M such that, for all x = (x 1 , . . . , x d ) ∈ X, for all sequences (x j i ) i∈N x j , M taking as oracles these d sequences computes a sequence (

x ′ i ) i∈N with (x ′ i ) i∈N f (x). A function f : X ⊂ R d → R d
, where X is compact, is said computable if all its projections are.

Formal setting: robustness

We now introduce the settings of [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF], based on an idea of [START_REF] Puri | Dynamical properties of timed automata[END_REF].

Definition 4 (ε-perturbation). Consider a discrete/continuous-time dynamical system H = (X, f ). Given ε > 0, its ε-perturbation H ε is the discrete/continuous-time system H ε defined over the same space X, where:

1. (x 0 , x 1 , ...) is a trajectory of H ε (the trajectory may not be unique), in the case where H is discrete-time, if x i+1 -f (x i ) ≤ ε for all i ∈ N; 2. φ : R + 0 → X is a trajectory of H ε (the trajectory may not be unique), in the case where H is continuous-time, if φ(0) ∈ X and φ ′ (t) -f (φ(t)) ≤ ε for all t ∈ R + 0 .

Note that the ε-perturbation H ε of a dynamical system H is not, in general, a dynamical system since it is not deterministic (several trajectories may start with a given initial point).

Similarly to what is done in Definition 3, we can define a language computed by H ε , which we denote as L ε : w ∈ L ε iff there is some trajectory of H ε which starts from (ν(w), 0, • • • , 0, 1) and reaches V accept .

The following result is immediate.

Lemma 1. For 0 < ε < ε ′ , L ε ⊂ L ε ′ .
Following the idea given in [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF], we consider L ω = ∩ ε>0 L ε .

Definition 5 (Robustness). A dynamical system H is said robust iff its language L is equal to L ω .

Main results

The following results are extensions from those obtained in [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF].

Theorem 1 (L ω is co-r.e. for Lipschitz and computable Systems I).

Assume that language L is computed by a discrete-time system H defined over [-1, 1] d which transition function is Lipschitz and computable. Then L ω is corecursively enumerable.

Proof. Let n ∈ N\{0} and decompose X in d-dimensional hypercubes V 1 , ..., V s of size 1 n . We build a finite automaton A n , which states are V 1 , ..., V s , that roughly recognizes L 1 n . To complete the description of this automaton we need to define two things: (i) the set of accepting states and (ii) the transition rule δ n . The set of accepting states consists of those hypercubes which overlap V accept , i.e. hypercubes that have vertices within distance ≤ 1/4 of the origin. These hypercubes can easily be identified. Now we have to present the transition rule of A n . The following construction is depicted in Fig. 1. Let V j be some hypercube. Then pick its central point x j (this is an easily computable rational) and compute a rational approximation f (x j ) of f (x j ) with precision 1 n . Because f is Lipschitz, there will be some Lipschitz constant K > 0 satisfying condition (1) for all x, y ∈ X. Then, if x ∈ V j is another point of the same hypercube, we have

f (x) -y ≤ 1 n ⇒ (2) f (x j ) -y ≤ f (x j ) -f (x j ) + f (x j ) -f (x) + f (x) -y ≤ K + 2 n .
By other words, if y is an ε-perturbed image of a point of V j , then this point will be within distance K+2 n of f (x j ). We use this fact to proceed as follows. After computing f (x j ), determine all the hypercubes which are within distance ≤ (K +2)/n of this point (in Fig. 1 this corresponds to all hypercubes covered by the ball of center f (x j ) and radius (K + 2)/n). This can be done algorithmically, in finite time, since it is only necessary to check which are the hypercubes (which are finitely many) that have vertices within distance ≤ (K + 2)/n of f (x j ). Let W 1 , ..., W i be these hypercubes. Then we define the transition rule over the hypercubes as follows: δ(V j ) = {W 1 , ..., W i }. This defines the automaton A n . Now we say that a point x ∈ X is accepted by A n if it lies in an accepted hypercube. Let L * 1 n be the language accepted by A n . From (2), the dynamics of A n includes those of the 1 n -perturbed system

H 1 n . Hence L 1 n ⊆ L * 1 n
. On the other side, it is not difficult to see that the dynamics of A n are included in those of H K+3 n . 7 Therefore

L 1 n ⊆ L * 1 n ⊆ L K+3 n ⇒ ∩ ∞ n=1 L * 1 n = ∩ ε>0 L ε = L ω .
Let us now show that

L ω = ∩ ∞ n=1 L * 1 n
is co-r.e., as required. Let w ∈ Σ * . Then build a Turing machine which performs the following steps: i=0 Repeat i++ Simulate A i with input ν X (w) Until ν X (w) is rejected Reject w This shows that the complement of L ω is r.e., as required.

We now introduce a (classical) tool to discretize a continuous-time system (see e.g. [START_REF] Ott | Chaos in Dynamical Systems[END_REF]).

Definition 6 (Stroboscopic map). Consider the continuous-time system H = (X, f ). Then its corresponding stroboscopic map g : X → X is given by g(x 0 ) = φ(1), where φ :

R + 0 → R satisfies φ ′ (t) = f (φ(t)), φ(0) = x 0 .
In other words, g(x 0 ) gives the point where the trajectory of H, starting on x 0 , would reach after one time unit. Lemma 2. Let H = (X, f ) be a continuous-time system, and let H = (X, g) be its discrete-time counterpart, where g is the stroboscopic map of H. Then, if L, L are the languages computed by H and H, respectively, under the same input encoding, we have:

1. L = L; 2. L ω = Lω and L ε = Lε , for all ε > 0.

Proof. Note that for all x 0 ∈ X, if φ : R + 0 → R, φ(0) = x 0 is a trajectory of H and (x 0 , x 1 , ...) is a trajectory of H, we must have φ(k) = x k for all k ∈ N. Therefore, because only the accepting trajectories reach V accept and then stay there, we must have L = L. Similar arguments can be used to show the identity L ε = Lε that, by its turn, implies L ω = Lω .

Theorem 2 (L ω is co-r.e. for Lipschitz and computable Systems II). Assume that language L is computed by a continuous-time system H defined over [-1, 1] d with a Lipschitz and computable transition function. Then L ω is co-recursively enumerable.

Proof. Using the same assumptions of the previous lemma, to show that the language L computed by the continuous-time system H = (X, f ), where X = [-1, 1] d , is co-r.e., it is enough to show that L is a co-r.e. language. If we show that g is Lipschitz and computable, then this can be done as indicated in the proof of Theorem 1. It is well-known (see e.g. [START_REF] Birkhoff | Ordinary Differential Equations[END_REF]) that if f satisfies equation (1) over X, and if φ 0 , φ 1 are solutions of the differential equation

x ′ = f (x) over X, then φ 0 (t) -φ 1 (t) ≤ φ 0 (0) -φ 1 (0) e Kt .
By other words, the stroboscopic map g is Lipschitz on X, with Lipschitz constant e K . Moreover, g is computable by using an algorithmic version of Euler's method on the time interval [0, 1] for the differential equation ẋ = f (x).(cf. [START_REF] Ko | Computational Complexity of Real Functions[END_REF]).

The following lemma is a corollary from the results of [START_REF] Collins | Continuity and computability of reachable sets[END_REF], [START_REF] Collins | Effective computability of solutions of differential inclusions -the ten thousand monkeys approach[END_REF]: one can semi-compute any trajectory of H. Therefore one can semi-decide if a trajectory finishes in V accept , and thus semi-decide L. Lemma 3. Let L be the language computed by a system H = (X, f ), where X = [-1, 1] d and f is Lipschitz and computable. Then L is r.e.

Since a system is robust iff L = L ω , the following corollary is immediate.

Corollary 1 (Robust =>Recursive for Lipschitz and Computable Systems). In the conditions of the lemma above, if H is robust, then L is recursive.

We now prove that Turing machines can be simulated robustly with C ∞systems. Recall that C ∞ -systems are also Lipschitz. This is a generalization of a similar result from [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF], which holds for piecewise systems (which are not of class C 1 ).

Theorem 3 (Perturbed reachability is complete in Π 0 1 ). Let A be a recursively enumerable language. Then there is a computable and C ∞ dynamical system H such that L ω = Ā.

Proof. For any Turing machine M , Theorems 4.4 and 4.5 from [START_REF] Bournez | On the computational power of dynamical systems and hybrid systems[END_REF] give an explicit way to build a continuous-time dynamical system (X, f ) where X is compact and f M is C ∞ such that y ′ = f M (y) simulates M . It is easy to observe that the function f M built there is computable. Changing coordinates, and reasoning through homothety, we can assume without loss of generality X to be [-T, T ] 3 and that the accepting configuration of M correspond to the origin c = (0, 0, 0) ∈ [-T, T ] 3 , and that the initial starting point corresponding to some input w is of the form c = (ν(w), 0, 1) as in Definition 3. This yields a dynamical system such that a word w is accepted by Turing machine M iff the trajectory starting from (ν(w), 0, 1) reaches the origin c. Let C be the cubic neighborhood of c defined by [-0.1, 0.1] 3 , and C ′ be the cubic neighborhood of c defined by [-0.2, 0.2] 3 . The resulting system is also such that a non-accepted word corresponds to a trajectory that stays forever outside of neighborhood C ′ .

We construct from this 3-dimensional smooth dynamical system f M a smooth 4-dimensional dynamical system g M whose perturbed version semi-recognizes the complement of the language recognized by M .

Actually, robustness will only be shown for one component, the extra component added to the original 3-dimensional system, which is the most critical for the simulation, . The other 3 components can be made robust using the construction presented in [START_REF] Graça | Robust simulations of Turing machines with analytic maps and flows[END_REF], [START_REF] Graça | Computability with polynomial differential equations[END_REF] which, although originally done for unbounded spaces, can be brought to the compact space [-T, T ] 3 via a transformation using a function like, e.g. arctan, and can still be used here. However, giving all the details would span the contents of this paper outside allowable size, so we only show robustness for the crucial component.

We will use the notation x, y for 3-dimensional vectors, and h for the fourth dimension. This 4-dimensional system is mainly the original smooth dynamical system embedded in the hyperplane h = 3 of R 4 , however with two changes. First, the neighborhood C of the original dynamical system becomes rejecting in the new system. Second, the zone h ≤ 1 becomes accepting in the new system.

The idea is that for any word w accepted by M , the original system f M will eventually come to C, and hence the perturbed dynamical system g M will eventually go up and reject. For any word w not accepted by M , the system g M will slowly drift "down" until it reaches the accepting zone h ≤ 1.

In order to get a smooth dynamics, we start by the following technicalities. This is an easy and classical mathematical exercise to prove that for any computable reals a < b, function where ẋ = C(x) is a smooth and computable vector field with all the trajectories going to the origin.

In other words, -if h ≥ 4, anything that arrives in the layer h ≥ 4 goes "up" and is rejected.

if 2 < h < 4:

• if x ∈ C, then it goes "up" and it is ultimately rejected.

• if x ∈ C, then it basically simulates the original system f M . -if 1 < h < 2, then it goes "down" until it reaches h ≤ 1, -if -1 ≤ h < 1, then it goes to the origin.

Consider the trajectory starting from (ν(w), 0, 1, 3): suppose that word w is accepted by M : the corresponding trajectory of f M will eventually go to the origin in some finite time t, and then the trajectory of g M will reach neighborhood C, and hence will go up for ever. Taking ǫ small enough (depending on t) any ǫ-perturbed trajectory of g M will be close at time t to this trajectory, and hence also in C, and hence going for ever up afterwards.

Suppose that word w is not accepted by M : the corresponding trajectory of f M will run for ever outside of C ′ . For any ǫ > 0, we can easily construct a trajectory of g M that drifts down slowly in the fourth coordinate until reaching h ≤ 1, and then going to the origin.

Through a change of coordinates, the obtained system fulfills all constraints of Definition 3 and of the statement of the Theorem.

In other words, "perturbed reachability is complete in Π 0 1 " as this is termed in [START_REF] Asarin | Perturbed Turing machines and hybrid systems[END_REF].

Conclusion

In this paper we showed that, on compact sets, the perturbed reachability problem is co-r.e. for Lipschitz and computable systems. We also proved that for Lipschitz and computable systems which are robust to infinitesimal perturbations, the reachability problem is decidable. The results are both for the discrete and continuous-time case. It would be interesting to know what happens at a more refined level, i.e. if from a complexity point of view.

Fig. 1 .

 1 Fig. 1. A figure depicting various elements used in the demonstration of Theorem 1.

  h a,b (x) = exp(-1/(x -a) 2 -1/(b -x) 2 ) for a < x < b and 0 for x ≥ b or x ≤ a is C ∞ and computable. If we denote by γ the constant γ = b a h a,b (x)dx, and by χ a,b (x) the function 1/γ x a h a,b (x), one gets a computable and C ∞ function that values 0 for x ≤ a, and 1 for x ≥ b, and a value in interval [0, 1] in between. Given a < b < c, the function χ a,b,c (x) = χ a,b (x) -χ b,c (x) values 0 for x ≤ a, 1 in b, and 0 for x ≥ c, with a value in interval [0, 1] in between. In the same spirit, let χ C : R 3 → R be some function that values 1 over neighborhood C, and 0 outside neighborhood C ′ .Formally the new system g M is then defined on [-T, T ] 3 × [-1, 5] ⊂ R 4 as follows: g M (x, h) = (y, h ′ ) where h ′ = χ 4,5 (h) + χ C (x)χ 2,3,4 (h) -χ 0,1,2 (h) + χ -1,-0.5,0 (h) and y = f M (x)(1 -χ C (x))χ 2,3,4 (h) + C(x)χ 0,0.5,1 (h),

Many other natural representations of rational numbers can be chosen: they still yield the same class of computable functions -see[START_REF] Weihrauch | Computable Analysis: an Introduction[END_REF][START_REF] Ko | Computational Complexity of Real Functions[END_REF].

Here we suppose that x = x ∞ = max 1≤i≤n |xi|. However, a similar result holds for other norms since all norms are equivalent in a finite-dimensional space.

Acknowledgments. D. Graça was partially supported by Fundação para a Ciência e a Tecnologia and EU FEDER POCTI/POCI via SQIG -Instituto de Telecomunicações.