Guillaume James

Dmitry Pelinovsky

BREATHER CONTINUATION FROM INFINITY IN NONLINEAR OSCILLATOR CHAINS

Keywords: 2000 Mathematics Subject Classification. Primary: 37K50; Secondary: 37K60, 37N25 Discrete Klein-Gordon equation, discrete breathers, nonlocal bifurcations, asymptotic methods, fixed-point arguments

Existence of large-amplitude time-periodic breathers localized near a single site is proved for the discrete Klein-Gordon equation, in the case when the derivative of the on-site potential has a compact support. Breathers are obtained at small coupling between oscillators and under nonresonance conditions. Our method is different from the classical anti-continuum limit developed by MacKay and Aubry, and yields in general branches of breather solutions that cannot be captured with this approach. When the coupling constant goes to zero, the amplitude and period of oscillations at the excited site go to infinity. Our method is based on near-identity transformations, analysis of singular limits in nonlinear oscillator equations, and fixed-point arguments.

1.

Introduction. Recent studies of spatially localized and time-periodic oscillations (breathers) in lattice models of DNA [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF][START_REF] Peyrard | Nonlinear analysis of the dynamics of DNA breathing[END_REF] call for systematic analysis of such excitations in the discrete Klein-Gordon equation

ẍn + V ′ (x n ) = γ (x n+1 -2x n + x n-1 ) , n ∈ Z, (1)
where γ > 0 is a coupling constant, V : R → R is a nonlinear potential, and x(t) = {x n (t)} n∈Z is a sequence of real-valued amplitudes at time t ∈ R.

In the classical Peyrard-Bishop model for DNA [START_REF] Peyrard | Nonlinear dynamics and statistical physics of DNA[END_REF], V is a Morse potential having a global minimum at x = 0, confining as x → -∞ and saturating at a constant level as x → ∞. However, recent studies [START_REF] Peyrard | Modelling DNA at the mesoscale: A challenge for nonlinear science?[END_REF][START_REF] Weber | Sharp DNA denaturation due to solvent interaction[END_REF] argued that the Morse potential should be replaced by a potential with a local maximum at x = a 0 > 0, which induces a double-well structure, where one of the wells extends to infinity (both kinds of potentials are depicted in Figure 1). The existence of breathers residing in the potential well near x = 0 can be proved with classical methods such as the center manifold reduction for maps [START_REF] James | Centre manifold reduction for quasilinear discrete systems[END_REF][START_REF] James | Breathers in inhomogeneous nonlinear lattices: An analysis via center manifold reduction[END_REF], variational methods [START_REF] Aubry | Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems[END_REF][START_REF] Pankov | Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices[END_REF], and the continuation from the anticontinuum limit γ → 0 [START_REF] Aubry | Breathers in nonlinear lattices: Existence, linear stability and quantization[END_REF][START_REF] Mackay | Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators[END_REF][START_REF] Sepulchre | Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators[END_REF]. A more delicate problem is the existence of large-amplitude breathers residing in the other potential well that extends to infinity. Large-amplitude stationary solutions bifurcating from infinity as γ → 0 have been obtained in [START_REF] Peyrard | Nonlinear analysis of the dynamics of DNA breathing[END_REF]. These solutions are localized near a single site, say n = 0, and their amplitude diverges as γ → 0. Large-amplitude breathers in a finite-size neighborhood of these stationary solutions have been constructed in [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF] for small coupling γ, using the contraction mapping theorem and scaling techniques. These large-amplitude breathers are confined on the other side of the potential barrier of V , and their amplitude goes to infinity as γ → 0. Existence of large-amplitude breathers oscillating everywhere above the potential barrier of V was left open in [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF].

Our goal is to show the existence of large-amplitude breathers oscillating in several potential wells, setting-up a continuation of these solutions from infinity as γ → 0. To illustrate some key points of our analysis, let us consider the example

V (x) = 1 4 (1 + e -x 2 (x 2 -1)). (2)
Here V has a global minimum at x = 0, a pair of symmetric global maxima at x = ±a 0 with a 0 > 0, and lim x→±∞ V (x) = 1 4 . In the standard anti-continuum limit, one can set γ = 0, x n = 0 for all n ∈ Z\{0}, and consider a time-periodic solution x 0 (t) ≡ x(t) of the nonlinear oscillator equation ẍ + V ′ (x) = 0.

(

) 3
Under a nonresonance condition, this compactly supported time-periodic solution can be continued for γ ≈ 0 into an exponentially localized time-periodic breather solution using the implicit function theorem [START_REF] Mackay | Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators[END_REF]. The phase plane (x, ẋ) and the frequency-amplitude (ω, a) diagram of the nonlinear oscillator equation [START_REF] Aubry | Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems[END_REF] with the potential (2) are shown on Figure 2. In this case, the periodic solution x(t) has a cut-off amplitude at a = a 0 . Only the family of periodic solutions with a ∈ (0, a 0 ) can be continued by the anti-continuum technique developed by MacKay and Aubry [START_REF] Mackay | Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators[END_REF].

In addition, there are two families of unbounded solutions: one corresponds to oscillations on the other side of the potential barrier of V for |x| > a 0 and the other one corresponds to oscillations above the potential barrier. Roughly speaking, the new technique developed in [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF] allows one to obtain large-amplitude breathers "close" to unbounded solutions of the first family for γ ≈ 0.

The present paper considers large-amplitude breathers near the second family of unbounded solutions. These two families of breathers are obtained by "continuation from infinity" for arbitrarily small values of γ, but without reaching γ = 0. In this case, the potential V in the nonlinear oscillator equation ( 3) can be simply replaced by

V γ (x) = V (x) + γx 2 . ( 4
)
The potential V γ includes a restoring force originating from the nearest-neighbors coupling in the discrete Klein-Gordon equation [START_REF] Archilla | Demonstration of the stability or instability of multibreathers at low coupling[END_REF]. As γ → 0, the amplitudes and periods of the resulting breathers go to infinity. As a result, we need a careful control of nonresonance conditions in order to prove the existence of such breathers. Although a part of our continuation procedure involving the contraction mapping theorem is close to the one developed in [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF], our mathematical analysis is quite different because our breather solutions scale differently in the different potential wells, which induces some singular perturbation analysis and more delicate estimates than in [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF]. Note also that the contraction mapping theorem has been used by Treschev [START_REF] Treschev | Travelling waves in FPU lattices[END_REF] to prove the existence of other types of localized solutions (solitary waves) in Fermi-Pasta-Ulam lattices, in which nearest-neighbors are coupled by an anharmonic potential having a repulsive singularity at a short distance. In this case, the existence problem yields an advance-delay differential equation with other kinds of mathematical difficulties.

To simplify our analysis, we assume that V is symmetric and bounded, whereas V ′ has a compact support. To be precise, the following properties on V are assumed:

(P1) V ∈ C 2 (R) and V (-x) = V (x) for all x ∈ R; (P2) There is x 0 > 0 such that V ∈ C 7 (-x 0 , x 0 ) and the Taylor expansion of V at x = 0 is V (x) = 1 2 κ 2 x 2 + O(x 6 ) with κ > 0; (P3) 0 ≤ V (x) ≤ V L for all x ∈ R and some V L > 0; (P4) V ′ (x) is compactly supported on [-a 0 , a 0 ] for some a 0 > 0 such that V (x) = V ∞ for |x| ≥ a 0 and V ∞ ∈ (0, V L ].
Assumption (P1) allows us to consider symmetric periodic oscillations, which can be studied on the quarter of the fundamental period. This assumption simplifies the presentation but is not essential, and our analysis could be extended to potentials confining at -∞ (as in Figure 1). Assumption (P2) allows us to develop a contraction mapping argument for the small-amplitude oscillations on the sites n = 0, a procedure which cannot be carried out if a quartic term is present in the expansion of V near the origin. It would be useful to relax this condition, which assumes a very weak anharmonicity of small amplitude oscillations. Note that the quartic term in V (x) near x = 0 is also excluded in the recent analysis of scattering of small initial data to zero equilibrium by Mielke & Patz [START_REF] Mielke | Dispersive stability of infinite-dimensional Hamiltonian systems on lattices[END_REF].

Assumption (P3) allows for large-amplitude oscillations at the central site n = 0. Assumption (P4) allows us to consider linear oscillations of the central site outside the compact support of V ′ . This property is used to solve the singularly perturbed oscillator equation at n = 0. This compact support assumption is quite restrictive, and it would be interesting to relax it in a future work, for example, by considering exponentially decaying potentials (as in example (2)) and treating exponential tails as perturbations of the present case.

We note that Fura & Rybicki [START_REF] Fura | Periodic solutions of second order Hamiltonian systems bifurcating from infinity[END_REF] have proved the existence of periodic solutions bifurcating from infinity for a class of finite-dimensional Hamiltonian systems with asymptotically linear potentials using degree theory. Our analysis is different and consists of two steps. We first reduce the infinite-dimensional Hamiltonian system to a singularly perturbed oscillator equation describing large-amplitude oscillations at the central site, using the contraction mapping theorem. Once this has been achieved, we solve the reduced problem using a topological method (Schauder's fixed point theorem).

Our main result is the existence of the large-amplitude breathers if the potential V satisfies assumptions (P1)-(P4) as well as the technical non-degeneracy condition in equation [START_REF] Mackay | Stability of discrete breathers[END_REF] below. As further problems, it would be interesting to analyze the existence of multibreather solutions bifurcating from infinity, as well as the stability of such solutions, as it was done previously for finite-amplitude breathers near the standard anti-continuum limit (see, e.g., [START_REF] Archilla | Demonstration of the stability or instability of multibreathers at low coupling[END_REF][START_REF] Aubry | Breathers in nonlinear lattices: Existence, linear stability and quantization[END_REF][START_REF] Bambusi | Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators[END_REF][START_REF] Koukouloyannis | On the stability of multibreathers in Klein-Gordon chains[END_REF][START_REF] Mackay | Stability of discrete breathers[END_REF][START_REF] Marín | Finite size effects on instabilities of discrete breathers[END_REF]).

The article is organized as follows. Section 2 describes the main result. Largeamplitude oscillations near n = 0 are analyzed in Section 3. Small-amplitude oscillations for n = 0 are considered in Section 4. The proof of the main theorem is given in Section 5. Section 6 gives a proof that the large-amplitude breather decays exponentially in n ∈ Z.

2. Main results. We shall consider the discrete Klein-Gordon equation (1) for small γ > 0 and assume that the breather is localized near the central site n = 0. We consider oscillations in the potential V γ (x) at the energy level E:

ẍ + V ′ γ (x) = 0 ⇒ E = 1 2 ẋ2 + V γ (x). (5)
Thanks to assumption (P3), the anti-continuum limit γ → 0 is singular for E > V L in the sense that a bounded trajectory of system ( 5) trapped by the quadratic potential γx 2 degenerates into an unbounded trajectory as γ → 0. We would like to select a unique T -periodic solution of ( 5) by fixing its energy E > V L and choosing γ small enough. For a fixed E > V L , we will be working for sufficiently small γ > 0 to ensure that V γ (a) = E admits a unique positive solution a(E, γ). More precisely, thanks to assumptions (P3) and (P4), we obtain a unique solution a = (E -V ∞ ) 1/2 γ -1/2 for γ < (E -V L )/a 2 0 . Fixing ẋ(0) = 0, we can parameterize periodic solutions by x(0) = a(E, γ) > 0, and their period can be written T = T (E, γ). Thanks to assumption (P4), we shall prove (in Section 3) that for any

E > V L T (E, γ) = √ 2π γ 1/2 + λ(E) + O(γ) as γ → 0 (6)
where

λ(E) = 2 √ 2 a0 0 dx (E -V (x)) 1/2 - a 0 (E -V ∞ ) 1/2 .
Thanks to assumption (P1), the T -periodic solution x(t) of the nonlinear oscillator equation ( 5) with x(0) = a > 0 and ẋ(0) = 0 is symmetric with respect to reflections about the points t = 0 and t = T 2 and anti-symmetric with respect to reflection about the points t = T 4 and t = 3T 4 . Therefore, the T -periodic solution satisfies

x(-t) = x(t) = -x T 2 -t , t ∈ R. (7)
The normalized frequency of oscillations is defined by

ω 0 (E, γ) = 2π T (E, γ) γ -1/2 (8)
such that ω 0 (E, γ) → √ 2 as γ → 0 for a fixed E > V L . To avoid resonances of largeamplitude oscillations at the central site n = 0 with small-amplitude oscillations at the other sites n ∈ Z\{0}, we will show (in Section 4) that the following nonresonance conditions

κ 2 -m 2 γω 2 0 (E, γ) + 2γ(1 -cos(q)) = 0, (9)
must be satisfied for all m ∈ Z and all q ∈ [-π, π]. Here κ is the parameter of the expansion of V about x = 0 in assumption (P2). For a fixed E > V L , we shall now consider the non-resonant set of parameters (breather frequency ω = 2π

T and coupling constant γ). We plot on Figure 3 the admissible regions between the boundaries of the non-resonant set, given by the

curves ω = √ κ 2 +4γ m
and ω = κ m , together with the curve ω = √ γ ω 0 (E, γ). Nonresonance conditions (9) are satisfied if γ belongs to the set

C E = ∪ m≥m0 (Γ m , γ m ),
where γ = Γ m and γ = γ m correspond to the intersections of the above curves starting with some m 0 ≥ 1. In this case, Γ m and γ m satisfy implicit equations

√ κ 2 + 4 Γ m m + 1 = Γ m ω 0 (E, Γ m ), κ m = √ γ m ω 0 (E, γ m ), m ≥ m 0 . (10)
Equations ( 10) can be solved for m large enough thanks to expansion [START_REF] James | Centre manifold reduction for quasilinear discrete systems[END_REF] and the implicit function arguments, yielding as m → ∞

Γ m = κ 2 2m 2 1 + κλ(E) πm - 2 m + O(m -2 ) , γ m = κ 2 2m 2 1 + κλ(E) πm + O(m -2 ) .
In particular, we note that Γ m < γ m and |γ m -Γ m | = O(m -3 ) as m → ∞.

We can now state the main result of this article. Note that the existence of breathers is only obtained for a subset CE,ν ⊂ C E of the non-resonant values of the coupling constant γ, because our method breaks down near the boundary of C E . Theorem 1. Assume (P1)-(P4) on V (x) and fix E > V L . Let x(t) be a T (E, γ)periodic solution of the nonlinear oscillator equation [START_REF] Fura | Periodic solutions of second order Hamiltonian systems bifurcating from infinity[END_REF] for small γ > 0 satisfying symmetries [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF] and assume λ ′ (E) = 0, that is,

a0 0 dx (E -V (x)) 3/2 - a 0 (E -V ∞ ) 3/2 = 0. ( 11
)
Fix ν ∈ (0, 1) and consider the set of coupling constants CE,ν = ∪ m≥m0 ( Γm , γm ) ⊂ C E , where Γm , γm are defined by the implicit equations

κ 2 + 4 Γm (m + 1) 2 -ν(m + 1) = Γm ω 0 (E, Γm ), κ √ m 2 + νm = γm ω 0 (E, γm ), ( 12
)
for m ≥ m 0 , and satisfy as m → +∞

Γm = κ 2 2m 2 1 + κλ(E) πm - 2 -ν m + O(m -2 ) , γm = κ 2 2m 2 1 + κλ(E) πm - ν m + O(m -2 ) .
For all sufficiently small γ in CE,ν , there exists a T -periodic spatially localized solution x(t) ∈ H 2 per ((0, T ); l 2 (Z)) of the Klein-Gordon lattice (1) such that

x n (t) = x -n (t), n ∈ Z; x(-t) = x(t) = -x T 2 -t , t ∈ R;
and

∃C > 0 : sup t∈[0,T ] |x 0 (t) -x(t)| ≤ Cγ -1/4 , sup n∈N sup t∈[0,T ] |x n (t)| ≤ Cγ 1/4 . ( 13
) Remark 1. Since x L ∞ = a = O(γ -1/2
) as γ → 0, the first bound in [START_REF] Mielke | Dispersive stability of infinite-dimensional Hamiltonian systems on lattices[END_REF] shows that the relative error

x 0 -x L ∞ / x L ∞ is as small as O(γ 1/4 ). Remark 2. If ν ∈ (0, 1), we still have Γm < γm and |γ m -Γm | = O(m -3 ) as m → ∞.
Therefore, the rate of decrease of the interval widths in the set CE,ν corresponds to the rate of decrease of the widths of the non-resonant intervals in the set C E .

Remark 3. Because the proof of Theorem 1 uses Schauder's fixed point theorem, it does not establish uniqueness of the solution. We anticipate that this is a technical difficulty and the solution is unique for fixed E > V L .

Remark 4. Although we do not attempt here to deal with non-compact potentials, we believe that assumption (P4) can be relaxed if V ′ (x) has a sufficiently fast decay to zero as |x| → ∞. In that case, we conjecture that the non-resonance condition [START_REF] Mackay | Stability of discrete breathers[END_REF] would be replaced by

Q := ∞ 0 1 (E -V (x)) 3/2 - 1 (E -V ∞ ) 3/2 dx = 0,
where V ∞ := lim x→∞ V (x). Figure 4 illustrates that this condition is satisfied for the particular potential (2), for any finite E > V L (we note that the value of Q approaches 0 as E → ∞). Theorem 1 is proved in Section 5, using intermediate results established for the single oscillator equation ( 5) with a forcing term (in Section 3) and for the discrete KG equation (1) linearized at zero equilibrium (in Section 4). We finish the article with Section 6, where we prove that the amplitude of breather oscillations decays exponentially in n on Z in the following sense:

∃D 0 > 0 : sup t∈[0,T ] |x n (t)| ≤ (D 0 γ) (2n-1)/4 , n ∈ N,
for all sufficiently small γ ∈ CE,ν .

3. Large-amplitude oscillations at a central site. We consider here solutions of the nonlinear oscillator equation (5) in the singular limit γ → 0. Assumptions (P1)-(P4) on the potential V are used everywhere, without further notes.

Lemma 1. Fix E > V L . There exists γ 0 = γ 0 (E) > 0 such that for any γ ∈ (0, γ 0 ), there exist exactly two T -periodic solutions of (5) with amplitude

x L ∞ = (E -V ∞ ) 1/2 γ -1/2
satisfying symmetries [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF] and the asymptotic expansion as γ → 0,

T = √ 2π γ 1/2 + 2 √ 2 a0 0 dx (E -V (x)) 1/2 - a 0 (E -V ∞ ) 1/2 + O(γ). (14)
Proof. Thanks to assumptions (P1) and (P3), for a fixed E > V L there exists γ 0 (E) > 0 such that for any γ ∈ (0, γ 0 ) equation V γ (x) = V (x) + γx 2 = E admits only two solutions x = ±a with a > 0 such that V ′ γ (x) > 0 for all x ≥ a. (One can easily compute that γ 0 (E) = (E -V L )/a 2 0 .) The two periodic solutions with symmetries [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF] are constructed from the same bounded trajectory on the phase plane (x, ẋ) departing from either the point (x(0), ẋ(0)) = (a, 0) or the point (x(0), ẋ(0)) = (-a, 0).

Because E = γa 2 + V (a), it follows from assumption (P4) that

a = (E -V ∞ ) 1/2 γ -1/2 , γ ∈ (0, γ 0 ).
Asymptotic expansion of the period T is found from the exact formula

T = √ 2 a -a dx E -V γ (x) = 2 √ 2 a 0 dx E -γx 2 -V (x) . ( 15
)
Thanks to assumption (P4), we know that

V (x) = V ∞ for all x ∈ [a 0 , a], so that T = 2 √ 2 √ γ π 2 -θ 0 + 2 √ 2 a0 0 dx E -γx 2 -V (x) ,
where θ 0 is the smallest positive root of a sin(θ) = a 0 satisfying

θ 0 = arcsin a 0 γ 1/2 (E -V ∞ ) 1/2 = a 0 γ 1/2 (E -V ∞ ) 1/2 + O(γ 3/2 ) as γ → 0. Since E > V L is fixed, there is C(E) > 0 such that E -V (x) ≥ C(E) for all x ∈ [0, a 0 ]. As a result, the asymptotic expansion a0 0 dx E -γx 2 -V (x) = a0 0 dx (E -V (x)) 1/2 + O(γ) as γ → 0,
concludes the proof of the asymptotic expansion [START_REF] Pankov | Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices[END_REF].

Let us represent the solution of ( 5) for E > V L in the form

x(t) = X(τ ) γ 1/2 , τ = γ 1/2 t. (16)
By Lemma 1, we have

X L ∞ = O(1) and T 0 := γ 1/2 T = O(1) as γ → 0 with precise value X L ∞ = (E -V ∞ ) 1/2
and the asymptotic expansion,

T 0 = √ 2π + 2 √ 2γ 1/2 a0 0 dx (E -V (x)) 1/2 - a 0 (E -V ∞ ) 1/2 + O(γ 3/2 ), (17)
as γ → 0. We shall now derive a series of estimates that will be useful for the proof of Theorem 1.

Corollary 1. Let X(τ ) be the T 0 -periodic function defined by the solution of Lemma 1 in parametrization ( 16) for any fixed E > V L . Then, X ∈ C 3 per (0, T 0 ) and for sufficiently small γ > 0, there exists

C(E) > 0 such that X H 1 per ≤ C(E). Moreover, X C 1 ≤ (1 + √ 2) √ E. Proof. We recall that X L ∞ = (E -V ∞ ) 1/2 ≤ √ E.
Let us rewrite the energy conservation (5) in parametrization ( 16):

1 2 Ẋ2 + X 2 + V (γ -1/2 X) = E. ( 18
)
Since V ≥ 0 we have Ẋ L ∞ ≤ √ 2E, which gives the bound on X C 1 . This also gives the uniform bound on X H 1 per since T 0 = O(1) as γ → 0. Let us also rewrite the second-order equation ( 5) in parametrization [START_REF] Peyrard | Nonlinear analysis of the dynamics of DNA breathing[END_REF]:

Ẍ(τ ) + 2X(τ ) + γ -1/2 V ′ (γ -1/2 X(τ )) = 0. ( 19
)
Thanks to assumption (P1), the solution X(τ ) is actually in C 3 per (0, T 0 ).

The potential term of the nonlinear equation ( 19) is a singular contribution to the linear equation as γ → 0. Because of the singular contribution, Ẍ L ∞ grows as γ → 0. Nevertheless, thanks to assumption (P4) of the compact support of V ′ (x), the solution X(τ ) stays in the domain |X| ≥ a 0 γ 1/2 , where V ′ (γ -1/2 X) = 0 for most of the times τ in the period [0, T 0 ]. The following lemma estimates the size of the time interval, for which the solution stays in the domain |X| ≤ a 0 γ 1/2 . Lemma 2. Let X(τ ) be the same as in Corollary 1. Let ∆T 0 be the length of the subset of [0, T 0 ] in which |X(τ )| ≤ a 0 γ 1/2 . Then, ∆T 0 admits the asymptotic expansion

∆T 0 = 2 √ 2γ 1/2 a0 0 dx (E -V (x)) 1/2 + O(γ 3/2 ) as γ → 0. ( 20
)
Proof. Consider the splitting of [0,

T 0 ] into 0, 1 4 (T 0 -∆T 0 ) ∪ 1 4 (T 0 -∆T 0 ), 1 4 (T 0 + ∆T 0 ) ∪ 1 4 (T 0 + ∆T 0 ), 1 4 (3T 0 -∆T 0 ) ∪ 1 4 (3T 0 -∆T 0 ), 1 4 (3T 0 + ∆T 0 ) ∪ 1 4 (3T 0 + ∆T 0 ), T 0 .
Thanks to the symmetries (7), we have

X 1 4 T 0 = X 3 4 T 0 = 0.
In the first, third, and fifth intervals, the second-order equation ( 19) for sufficiently small γ > 0 becomes the linear oscillator Ẍ + 2X = 0.

An explicit solution with X(0) = (E -V ∞ ) 1/2 and Ẋ(0) = 0 has the form

X(τ ) = (E -V ∞ ) 1/2 cos( √ 2τ ), τ ∈ - 1 4 (T 0 -∆T 0 ), 1 4 (T 0 -∆T 0 ) . (21)
The matching condition

X(τ 0 ) = a 0 γ 1/2 at τ 0 = 1 4 (T 0 -∆T 0 ) gives cos T 0 -∆T 0 2 √ 2 = a 0 γ 1/2 (E -V ∞ ) 1/2 .
Using the asymptotic expansion [START_REF] Peyrard | Nonlinear dynamics and statistical physics of DNA[END_REF] for T 0 , we obtain the asymptotic expansion [START_REF] Weber | Sharp DNA denaturation due to solvent interaction[END_REF] for ∆T 0 .

Corollary 2. Let X(τ ) be the same as in Lemma 2 and

Y (τ ) := γ -1/2 V ′ (γ -1/2 X(τ )). ( 22
)
Then, Y ∈ C 1 per (0, T 0 ) and there exists C(E) > 0 such that Y H 1 per ≤ Cγ -3/4 . Proof. Using the bounds on ∆T 0 in Lemma 2, Corollary 1, and assumption (P1) on the potential V (x), we obtain

T0 0 |Y (τ )| 2 dτ = γ -1 T0 0 |V ′ (γ -1/2 X(τ ))| 2 dτ ≤ γ -1 ∆T 0 V ′ 2 L ∞ ≤ C 1 γ -1/2 ,
and

T0 0 | Ẏ (τ )| 2 dτ = γ -2 T0 0 | Ẋ(τ )| 2 |V ′′ (γ -1/2 X(τ ))| 2 dτ ≤ γ -2 ∆T 0 V ′′ 2 L ∞ Ẋ 2 L ∞ ≤ C 2 γ -3/2 , for some constants C 1 , C 2 > 0. The bound on Y H 1
per follows from the above computation.

We shall be working in the space of T 0 -periodic functions in H 2 , H 1 , or L 2 satisfying symmetry [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF]. Therefore, let us denote for all p ≥ 1

H p e := X ∈ H p per (0, T 0 ) : X(-τ ) = X(τ ) = -X T 0 2 -τ , τ ∈ R (23)
and use similar notations for L 2 e and L ∞ e . We are now prepared to deal with the singularly perturbed linear oscillator under the small source term:

Z(τ ) + 2Z(τ ) + γ -1/2 V ′ (γ -1/2 Z(τ )) = γ ε+1/2 F (τ ), (24)
where ε > 0, F ∈ L 2 e , and F L 2 per = O(1) as γ → 0. It is necessary to consider the inhomogeneous problem (24) in order to control the effect of small coupling in the discrete Klein-Gordon equation ( 1) at the central site n = 0. Energy for the perturbed oscillator equation ( 24) can be written in the form

H(τ ) = 1 2 Ż2 + Z 2 + V (γ -1/2 Z). ( 25
)
Because the homogeneous equation with F ≡ 0 admits a T 0 -periodic solution 24) is a singular perturbation of a linear oscillator), a source term of order one would generate a large output as γ → 0, i.e. the output Z -X H 1 per is going to be γ -1/2 larger than the source term γ ε+1/2 F L 2 per . This can be intuitively understood by linearizing equation (24) around X. Indeed, the linearized operator

X ∈ H 2 e with ∂ E T 0 (E, γ) = O(γ 1/2 ) (equation (
L 0 := d 2 dτ 2 + 2 + γ -1 V ′′ γ -1/2 X(τ ) (26)
admits a nontrivial kernel Ker(L 0 ) = span{ Ẋ} in H 2 per (0, T 0 ). However, Ker(L 0 ) = {0} in the subspace H 2 e of even T 0 -periodic functions, under the condition

∂ E T 0 (E, γ) = 0. In this subspace, we have L -1 0 L(L 2 per ,H 1 per ) = O(γ -1/2 ), due to the fact that ∂ E T 0 (E, γ) = O(γ 1/2
) (this follows from standard computations based on the variation of constants method). Thanks to the scaling of the source term considered in (24), we will get Z -

X H 1 per = O(γ ε ) if F L 2 per = O(1) and ε ≤ 1 4 (see Lemma 3 below). Since X H 1 per = O(1) as γ → 0 by Corollary 1, the perturbation Z -X H 1 per is smaller than the unperturbed solution X H 1 per if ε > 0.
There exists an obstacle on the direct application of the Implicit Function Theorem to obtain T 0 -periodic solutions of the singularly perturbed oscillator equation (24). The obstacle comes from the power series expansion

V ′ (γ -1/2 Z) = V ′ (γ -1/2 X) + γ -1/2 V ′′ (γ -1/2 X)(Z -X) +γ -1 V ′′′ (γ -1/2 X)(Z -X) 2 + ...,
which generate large terms in the limit of small γ > 0 because of the singular perturbation in the nonlinear potential. To avoid this difficulty, we use the fact that V ′ (γ -1/2 Z) has a compact support, and transform the search of periodic solutions of equation ( 24) to a root-finding problem to which the Implicit Function Theorem can be applied. We shall prove the following.

Lemma 3. Let X ∈ H 2
e be the solution of Lemma 1 in parametrization ( 16) for any fixed E > V L . Let B δ be a ball of radius δ > 0 in L 2 e centered at 0.

Fix ε ∈ 0, 1 2 . If a0 0 dx (E -V (x)) 3/2 - a 0 (E -V ∞ ) 3/2 = 0, ( 27
)
there exist γ 0 (ε, δ, E) > 0, η(ε, δ, E) > 0, and C(ε, δ, E) > 0 such that the inhomogeneous equation ( 24) with γ ∈ (0, γ 0 ) and F ∈ B δ admits a unique solution

Z = G γ,ε (F ) ∈ H 2 e satisfying Z(τ 0 ) = a 0 γ 1/2 , |H(τ 0 ) -E| < η(ε, δ, E)γ ε for some τ 0 ∈ (0, T0 4 ). Moreover, Z is close to X in H 1 e with Z -X H 1 per ≤ C(ε, δ, E)γ ε, ε = min 1 4 , ε (28)
and satisfies the estimate

Z C 1 ≤ 3 √ E. (29)
In addition, Z(τ

) ≥ a 0 γ 1/2 for τ ∈ [0, τ 0 ], Z(τ ) ∈ [0, a 0 γ 1/2 ] for τ ∈ [τ 0 , T0 4
], and there exists θ(ε, δ, E) > 0 such that

T 0 4 -τ 0 ≤ θ(ε, δ, E)γ 1/2 . ( 30
)
Proof. We shall use a kind of shooting method to transform the differential equation (24) to a root-finding problem. Let us consider an initial-value problem for the second-order differential equation ( 24) starting with the initial data Z(0) = Z 0 and Ż(0) = 0, where Z 0 is a positive γ-independent parameter. Thanks to the compact support in assumption (P4), we solve the inhomogeneous linear equation

Z(τ ) + 2Z(τ ) = γ ε+1/2 F (τ ), τ ∈ [0, τ 0 ], (31)
where τ 0 will be determined from the condition Z(τ 0 ) = a 0 γ 1/2 . We shall prove that for small γ > 0, a unique

τ 0 ∈ 0, π 2 √ 2 exists.
In what follows we denote

W 0 := Ż(τ 0 ) and τ 0 := π 2 √
2 -∆ 0 . We shall consider W 0 ∈ R -as a free parameter and express (Z 0 , τ 0 ) (or equivalently (Z 0 , ∆ 0 )) as a function of W 0 .

The unique solution of the linear equation (31) with the initial data Z(0) = Z 0 and Ż(0) = 0 is given by

Z(τ ) = Z 0 cos( √ 2τ ) + 1 √ 2 γ ε+1/2 τ 0 F (τ ′ ) sin( √ 2(τ -τ ′ ))dτ ′ . ( 32
)
At τ = τ 0 , we have the system of nonlinear equations

Z 0 sin( √ 2∆ 0 ) = a 0 γ 1/2 -1 √ 2 γ ε+1/2 τ0 0 F (τ ′ ) sin( √ 2(τ 0 -τ ′ ))dτ ′ , Z 0 cos( √ 2∆ 0 ) = -1 √ 2 W 0 + 1 √ 2 γ ε+1/2 τ0 0 F (τ ′ ) cos( √ 2(τ 0 -τ ′ ))dτ ′ ,
which can be rewritten in the equivalent form

Z 0 = -1 √ 2 W 0 cos( √ 2∆ 0 ) + a 0 γ 1/2 sin( √ 2∆ 0 ) + 1 √ 2 γ ε+1/2 τ0 0 F (τ ) sin( √ 2τ )dτ, 1 √ 2 W 0 sin( √ 2∆ 0 ) = -a 0 γ 1/2 cos( √ 2∆ 0 ) + 1 √ 2 γ ε+1/2 τ0 0 F (τ ) cos( √ 2τ )dτ
(multiply each equation by sin( √ 2∆ 0 ) and cos( √ 2∆ 0 ) and sum the resulting equations). This problem can be rewritten

             Z 0 = -1 √ 2 W 0 cos( √ 2∆ 0 ) + a 0 γ 1/2 sin( √ 2∆ 0 ) + 1 √ 2 γ ε+1/2 π 2 √ 2 -∆0 0 F (τ ) sin( √ 2τ )dτ, ∆ 0 = - a0 W0 sinc( √ 2∆0) γ 1/2 cos( √ 2∆ 0 ) + 1 √ 2W0 sinc( √ 2∆0) γ ε+1/2 π 2 √ 2 -∆0 0 F (τ ) cos( √ 2τ )dτ, (33)
where sinc(x) = sin (x)/x. Assuming that W 0 ∈ (-∞, W max ) is bounded away from 0 and γ ∈ (0, γ max ) is small enough, the second equation can be solved by the contraction mapping theorem for ∆ 0 ∈ 0, π 2 √

2 , and then the first equation determines Z 0 . This yields finally

Z 0 = - 1 √ 2 W 0 + 1 √ 2 γ ε+1/2 π 2 √ 2 0 F (τ ) sin( √ 2τ )dτ + O(γ) (34)
and

∆ 0 = - a 0 γ 1/2 W 0 + 1 √ 2W 0 γ ε+1/2 π 2 √ 2 0 F (τ ) cos( √ 2τ )dτ + O(γ 3/2 ). (35)
Then Z 0 = O(1) and ∆ 0 = O(γ 1/2 ) as γ → 0, so that existence of a unique

τ 0 ∈ 0, π 2 √ 2
follows. We can now continue solution Z(τ ) to τ > τ 0 starting with the initial conditions Z(τ 0 ) = a 0 γ 1/2 and Ż(τ 0 ) = W 0 < 0. Our aim is to solve the inhomogeneous differential equation

Z(τ ) + 2Z(τ ) + γ -1/2 V ′ (γ -1/2 Z(τ )) = γ ε+1/2 F (τ ), τ ∈ [τ 0 , τ * ], (36)
where τ * > τ 0 is the first time where Z(τ * ) = 0. For this purpose the first step is to show that τ * > τ 0 exists. Taking the derivative of the energy H in (25) with respect to τ and using equation (36), we infer that Ḣ = γ ε+1/2 ŻF. (37

) Let E * := H(τ 0 ) = 1 2 W 2 0 + a 2 0 γ + V ∞ and assume that E * > V L ≥ V ∞ .
Note that E * becomes now the parameter of the solution family in place of W 0 .

Let us denote by τ 1 the maximal time such that |Z(τ )| < a 0 γ 1/2 and Ż(τ ) < 0 for all τ ∈ (τ 0 , τ 1 ). We have

Ż(τ ) = -2(H(τ ) -V (γ -1/2 Z(τ )) -Z 2 (τ )), τ ∈ [τ 0 , τ 1 ]. ( 38
)
Using formula (38) and integrating equation (37), we obtain for all τ ∈ [τ 0 , τ 1 ],

|H(τ ) -E * | ≤ γ ε+1/2 τ τ 0 2(H(τ ′ ) -E * + E * -V (γ -1/2 Z(τ ′ )) -Z 2 (τ ′ ))|F (τ ′ )|dτ ′ .
Using the triangle inequality, we find that there exist γ 0 > 0 and a γ-independent constant C(E * , ε) > 0 such that for all γ ∈ (0, γ 0 ),

|H(τ ) -E * | ≤ C(E * , ε)γ ε+1/2 F L 2 per , τ ∈ [τ 0 , τ 1 ]. ( 39
) Now it follows that 1 2 Ż2 (τ 1 ) = H(τ 1 ) -Z 2 (τ 1 ) -V (γ -1/2 Z(τ 1 )) ≥ E * -V L + O(γ ε+1/2 ),
hence Ż(τ 1 ) = 0 for γ small enough since E * > V L . Consequently one has Z(τ 1 ) = -a 0 γ 1/2 and Ż(τ ) < 0 for all τ ∈ [τ 0 , τ 1 ]. By the intermediate value theorem, this yields the existence of τ * ∈ (τ 0 , τ 1 ) such that Z(τ * ) = 0 and Z(τ ) ∈ (0, a 0 γ 1/2 ) for all τ ∈ (τ 0 , τ * ).

We now express the distance |τ * -τ 0 | from the energy (25) controlled by bound (39). From equations (37) and (38), we obtain

τ * -τ 0 = τ * τ0 Ż(τ ) Ż(τ ) dτ = - τ * τ0 Ż(τ )dτ 2(H(τ ) -V (γ -1/2 Z(τ )) -Z 2 (τ )) .
Using bound (39), we have for all τ ∈ [τ 0 , τ * ] as γ → 0,

1 2(H(τ ) -V (γ -1/2 Z(τ )) -Z 2 (τ )) = 1 2(E * -V (γ -1/2 Z(τ )) -Z 2 (τ )) + O(γ ε+1/2 ),
so that the change of variables x = γ -1/2 Z(τ ) gives

τ * -τ 0 = γ 1/2 a0 0 dx 2(E * -V (x) -γx 2 ) + O(γ ε+1/2 ) as γ → 0. (40)
Combining this expansion of τ * -τ 0 with the expansion of τ 0 = π 2 √

2 -∆ 0 obtained in (35), and using the fact that W 0 = -2(E * -V ∞ -a 2 0 γ), we end up with the expansion

τ * (E * , γ, F ) = π 2 √ 2 - a 0 γ 1/2 2(E * -V ∞ ) + γ 1/2 a0 0 dx 2(E * -V (x)) + O(γ ε+1/2 ), (41)
as γ → 0, uniformly in F ∈ B δ for fixed ε > 0 and δ > 0.

Let us now examine the regularity of τ * with respect to the variable (E * , γ, F ) including the functional parameter F . From the standard fixed-point reformulation of the Cauchy problem for differential equations, it follows that the map (W 0 , γ, F

) → Z is C 1 from (-∞, W max ) × (0, γ max ) × L 2 e into C 1 ([0, π √ 2 ]). Consequently, the map (E * , γ, F, τ ) → Z(τ ) is C 1 from (E min , +∞) × (0, γ max ) × L 2 e × 0, π √ 2 into
R, where we have denoted

E min = 1 2 W 2 max + V ∞ . Since Z(τ * ) = 0 and Ż(τ * ) = 0, the implicit function theorem ensures that τ * is C 1 with respect to (E * , γ, F ) ∈ (E min , +∞) × (0, γ max ) × B δ .
Now we come back to equation ( 24). If we manage to find a value of E * such that

τ * (E * , γ, F ) = 1 4 T 0 (E, γ), (42)
then the symmetry constraints satisfied by F ∈ L 2 e in (23) imply that Z ∈ H 2 (0, T 0 ) also satisfies the symmetry conditions in (23). Indeed, Z(τ ) and Z(-τ ) are solutions of the same Cauchy problem, hence Z(τ ) = Z(-τ ) by Cauchy's theorem. Using the same argument one obtains Z(τ + T 0 /4) = -Z(-τ + T 0 /4) (this requires in addition the evenness of V ). These two equalities imply Z(0) = -Z(T 0 /2) = -Z(-T 0 /2) = Z(T 0 ) and in the same way Ż(0) = Ż(T 0 ) = 0. Consequently, Z(τ ) and Z(τ +T 0 ) are solutions of the same Cauchy problem, hence Z is T 0 -periodic. Therefore, Z ∈ H 2 e if there is E * solving equation (42).

Using expansions ( 17) and (41), equation ( 42) can be rewritten

R(E * , γ ε ) = 0, (43)
where

R(E * , γ ε ) = a0 0 dx E -V (x) - a 0 √ E -V ∞ - a0 0 dx E * -V (x) + a 0 √ E * -V ∞ + O(γ ε ) (44)
as γ → 0, uniformly in F ∈ B δ (we omit the dependency of R with respect to E and F in notations). Now let us consider the non-degeneracy condition (27), which is equivalent to ∂R ∂E * (E, 0) = 0. Due to the non-differentiability of R at γ = 0, we cannot solve (43) directly by the implicit function theorem, but rewrite the problem in the form

E * -E = O(γ ε + (E * -E) 2 ) ( 45
)
using the fact that ∂R ∂E * (E, 0) = 0. By the contraction mapping theorem, if E > V L , ε ∈ 0, 1 2 , δ > 0, F ∈ B δ are fixed, there exist constants γ 0 (ε, δ, E) > 0 and C 0 (ε, δ, E) such that equation ( 45) admits a unique solution E * near E for all γ ∈ (0, γ 0 ), with

|E * -E| ≤ C 0 (ε, δ, E)γ ε . (46)
Moreover, the map (γ, F ) → E * is C 1 on (0, γ 0 ) × B δ by the uniform contraction principle. This completes the proof of existence and local uniqueness of solution Z. Estimate (30) follows directly from expansion (40).

It remains to prove bounds (28) and (29). For this purpose, we first consider a time interval in which Z and X are both given by the explicit solution of a linear equation. Recall that

X(τ ) = E -V ∞ cos ( √ 2 τ ), 0 ≤ τ ≤ τ0 , (47)
where

τ0 = 1 √ 2 arccos a 0 γ 1/2 (E -V ∞ ) 1/2 . Now consider τ0 = min(τ 0 , τ0 ) = T0 4 + O(γ 1/2
). By combining the explicit expressions of X and Z (see equation ( 32)), estimates (34) and ( 46), and observing that

W 0 = -2(E * -V ∞ ) + O(γ), one obtains Z -X L ∞ (0,τ0) ≤ C(ε, δ, E)γ ε , Ż -Ẋ L ∞ (0,τ0) ≤ C(ε, δ, E)γ ε , (48)
for some C(ε, δ, E) > 0 for all γ ∈ (0, γ 0 ). Since τ0 = T0 4 + O(γ 1/2 ), it follows from equation ( 32) and (47) that

X L ∞ (τ0,T0/4) + Z L ∞ (τ0,T0/4) = O(γ 1/2 ) as γ → 0. ( 49
)
As a result, we extend the first bound in (48) to

Z -X L ∞ ≤ C(ε, δ, E)γ ε , because ε ∈ 0, 1 2 .
To extend the second bound in (48), we write for all τ ∈ τ0 ,

1 4 T 0 , | Ż(τ )| = 2(H(τ ) -V (γ -1/2 Z(τ )) -Z 2 (τ )) ≤ 2H(τ ) = √ 2E + O(γ ε ), (50) | Ẋ(τ )| = 2(E -V (γ -1/2 X(τ )) -X 2 (τ )) ≤ √ 2E, ( 51
)
where we have used bounds (39) and (46). By bounds ( 40) and ( 49), hence we have

X H 1 (τ0,T0/4) + Z H 1 (τ0,T0/4) ≤ C(ε, δ, E)γ 1/4 . ( 52
)
As a conclusion, estimate (28) follows from estimates (48) and (52). Estimate (29) follows from ( 48), (50), and (51) for γ small enough.

Remark 5. A T 0 -periodic solution of the perturbed oscillator equation ( 24) may not exist if the constraint ( 27) is not met. For instance, if V ≡ 0 (this corresponds to the case a 0 = 0) and F (τ ) = cos( √ 2τ ), the perturbed linear oscillator (24) has no T 0 -periodic solutions because of the resonance of the linear oscillator and the T 0 -periodic force for T 0 = √ 2π. Since ∆ 0 = 0, expansion (35) yields the classical constraint

π 2 √ 2 0 F (τ ) cos( √ 2τ )dτ = 0
for the existence of a √ 2π-periodic solution under symmetric periodic forcing. The constraint is not satisfied if

F (τ ) = cos( √ 2τ ).
For the arguments in the proof of Theorem 1 based on the Schauder fixed-point theorem, we will need a continuity of the nonlinear map F → Z from L 2 e to H 2 e . Lemma 4. Under Assumptions of Lemma 3, for all γ ∈ (0, γ 0 ) and all F ∈ B δ , the solution Z = G γ,ǫ (F ) ∈ H 2 e is continuous with respect to F ∈ L 2 e . Proof. From Lemma 3, we know that for given F 1 ∈ B δ and F 2 ∈ B δ , there exist roots E * = E 1 and E * = E 2 of equation ( 43) and if

F 1 → F 2 in L 2 e , then E 1 → E 2 .
In what follows, we denote by W 1,2 , Z (1),(2) 0 , and τ 1,2 the variables W 0 , Z 0 , and τ 0 introduced in Lemma 3, in the case when E * = E 1,2 . Expansion (35) implies that

∃C > 0 : |τ 1 -τ 2 | ≤ Cγ 1/2 |E 1 -E 2 | + γ ε F 1 -F 2 L 2 (0,T0) .
Let τ0 = min(τ 1 , τ 2 ). From the solution (32) of the linear oscillator equation (31) and the expansion (34), we obtain

∃C > 0 : Z 1 -Z 2 H 2 (0,τ0) ≤ C |E 1 -E 2 | + γ ε+1/2 F 1 -F 2 L 2 (0,T0) . ( 53
)
Consequently, if

F 1 → F 2 in L 2 e , then E 1 → E 2 and Z 1 → Z 2 in H 2 (0, τ0
). There remains to complete (53) by a continuity result on the time interval [τ 0 , τ * ], on which we rewrite the differential equation (36) in variables

z 1,2 (t) = γ -1/2 Z 1,2 (τ ), f 1,2 (t) = F 1,2 (τ ), t = γ -1/2 (τ -τ0 ), ( 54
) or explicitly, z1,2 (t) + 2γz 1,2 (t) + V ′ (z 1,2 (t)) = γ ε+1 f 1,2 (t), t ∈ [0, t0 ], ( 55
)
where t0 = γ -1/2 (τ * -τ0 ) = O(1) thanks to (30) and τ * = T 0 /4. For definiteness, let us assume that τ 1 > τ 2 so that τ0 = τ 2 . The initialvalue problems for differential equations (55) are started with (z 1 (0), ż1 (0)) and (z 2 (0), ż2 (0)), where z 2 (0) = a 0 , ż2 (0) = Ż2 (τ 2 ) = W 2 , and z 1 (0) = γ -1/2 Z 1 (τ 2 ) > a 0 , ż1 (0) = Ż1 (τ 2 ) are determined by (32). Since V ′ is globally Lipschitzian by the assumptions (P1) and (P4), Gronwall's inequality implies the existence of C > 0 such that

z 1 -z 2 C 1 (0, t0) ≤C |z 1 (0) -z 2 (0)| + | ż1 (0) -ż2 (0)| + γ ǫ+1 t0 0 |f 1 (t) -f 2 (t)|dt ≤C |Z 1 (τ 2 ) -Z 2 (τ 2 )| γ 1/2 + | Ż1 (τ 2 ) -Ż2 (τ 2 )| + γ ǫ+1 t1/2 0 f 1 -f 2 L 2 (0, t0) .
When

F 1 → F 2 in L 2
e , the first equation of (33) yields Z

(1)

0 → Z (2) 0 (since W 1 → W 2 and τ 1 → τ 2 ), which implies Z 1 (τ 2 ) → Z 2 (τ 2 ) and Ż1 (τ 2 ) → Ż2 (τ 2 ) thanks to expres- sion (32). Consequently we obtain z 1 -z 2 C 1 (0, t0) → 0 and Z 1 -Z 2 C 1 (τ0,τ * ) → 0.
Combining this result with (53), we see that if

F 1 → F 2 in L 2 e then Z 1 → Z 2 in H 1 e . Now observing that ∆ = d 2 dτ 2 : H 2 e → L 2
e is invertible and considering equation ( 24) that defines G γ,ǫ (F ) implicitly, Z 1,2 = G γ,ǫ (F 1,2 ) satisfy the equality

Z 1,2 = ∆ -1 [ γ ε+1/2 F 1,2 -2Z 1,2 -γ -1/2 V ′ (γ -1/2 Z 1,2 ) ].
Hence continuity in H 1 e norm implies continuity in H 2 e norm.

4. Small-amplitude oscillations on other sites. In our construction, the largeamplitude breather bifurcating from infinity as γ → 0 is localized at a single site n = 0, and close to the periodic solution of Lemma 1. The other sites for n = 0 display the oscillatory motion guided by the oscillation at the central site n = 0 and powered by a small amplitude in γ. By symmetry of the discrete Laplacian, we may assume that x n = x -n , n ≥ 1. Since the amplitudes of oscillations for n = 0 are small, we shall consider the linearized discrete Klein-Gordon equation (1) at the zero solution to study possible resonances with the oscillatory motion at n = 0.

To this end, let us introduce the function spaces X := L 2 per ((0, T 0 ); l 2 (N)), D := H 2 per ((0, T 0 ); l 2 (N)), and denote ω 0 = 2π T0 . Let us use assumption (P2) and consider the inhomogeneous linear problem

γU ′′ 1 (τ ) + κ 2 U 1 (τ ) = γ(U 2 (τ ) -2U 1 (τ )) + F 1 (τ ), γU ′′ n (τ ) + κ 2 U n (τ ) = γ(U n+1 (τ ) + U n-1 (τ ) -2U n (τ )) + F n (τ ), n ≥ 2, (56)
where τ = γ 1/2 t denotes the rescaled time. We denote the sequence {F n (τ )} n∈N ∈ X by F and look for solution U = {U n (τ )} n∈N ∈ D of the inhomogeneous linear equations (56). We shall prove the following.

Lemma 5. Assume the non-resonance condition (9) to be satisfied for given values of κ, γ and ω 0 , that is,

κ 2 + 2γ(1 -cos q) = m 2 γω 2 0 , ∀ m ∈ Z, ∀q ∈ [-π, π]. (57)
Then, for all F ∈ X, equations (56) admit a unique solution U ∈ D. Moreover, the solution satisfies

U X ≤ C(γ, ω 0 , κ) F X , (58)
where

C(γ, ω 0 , κ) = inf m∈Z, q∈[-π,π] κ 2 -m 2 γω 2 0 + 2γ(1 -cos q) -1 < +∞. If in addition F ∈ H 1 per ((0, T 0 ); l 2 (N)) then U H 1 per ((0,T0);l 2 (N)) ≤ C(γ, ω 0 , κ) F H 1 per ((0,T0);l 2 (N)) . (59)
Proof. To solve (56), we expand F and U using Fourier series in the time variable t and band-limited Fourier transforms in the discrete spatial coordinate n. Defining

fm (q) = 1 πT 0 n∈N T0 0 F n (τ ) sin(nq) e -imω0τ dτ , ûm (q) = 1 πT 0 n∈N T0 0 U n (τ ) sin(nq) e -imω0τ dτ ,
we have thus

F n (τ ) = m∈Z π -π fm (q) sin(nq)e imω0τ dq, U n (τ ) = m∈Z π -π
ûm (q) sin(nq)e imω0τ dq.

Note that the Fourier transform defines an isometric isomorphism between the Hilbert spaces X and X and between D and D, where X := ℓ 2 (Z; L 2 per (0, 2π)), D := ℓ 2 2 (Z; L 2 per (0, 2π)), and the latter denotes the usual Hilbert space consisting of sequences {û m } m∈Z in L 2 per (0, 2π) for which {m 2 ûm } ∈ ℓ 2 (Z; L 2 per (0, 2π)). In the Fourier space, differential equations (56) reduce to the set of algebraic equations

[ κ 2 -m 2 γω 2 0 + 2γ(1 -cos q) ] ûm (q) = fm (q), (60)
where f ∈ X and we look for û ∈ D. Thanks to condition (57), for all m ∈ Z, equation (60) has a unique solution ûm = Ĥm fm ∈ L 2 per (0, 2π), where Ĥm (q) = [ κ 2 -m 2 γω 2 0 + 2γ(1 -cos q) ] -1 is 2π-periodic and analytic over R. To check that û ∈ D, we use the estimate

m∈Z m 4 ûm 2 L 2 per ≤ m∈Z m 4 Ĥm 2 L ∞ fm 2 L 2 per ≤ sup m∈Z (m 2 Ĥm L ∞ ) 2 f 2 X , (61)
where

Ĥm L ∞ = 1 inf q∈[-π,π] |κ 2 -m 2 γω 2 0 + 2γ(1 -cos(q))| . For |m| ≥ ω -1 0 (4 + γ -1 κ 2 ) 1/2 one has Ĥm L ∞ = 1 m 2 γω 2 0 -κ 2 -4γ
and m∈Z m 4 ûm 2 L 2 per < ∞ according to (61). Consequently, there exists a unique solution U ∈ D of the inhomogeneous system (56) in the form

U n (τ ) = m∈Z π -π fm (q) sin(nq)e imω0τ κ 2 -m 2 γω 2 0 + 2γ(1 -cos q) dq. (62)
Moreover, we have

U 2 X = m∈Z ûm 2 L 2 per ≤ m∈Z Ĥm 2 L ∞ fm 2 L 2 per ≤ sup m∈Z Ĥm L ∞ 2 F 2 X . ( 63
)
This proves estimate (58). Estimate (59) follows by differentiating (56) with respect to τ and using (58).

Now we come back to the case in which ω 0 depends in fact on γ and on the fixed energy E of the excitation at site n = 0. More precisely, we recall that

ω 0 (E, γ) = 2π T (E, γ) γ -1/2 = √ 2 + O(γ 1/2 ) as γ → 0. ( 64
)
Non-resonance conditions (57) are satisfied if γ belongs to the disjoint set

C E = ∪ m≥m0 (Γ m , γ m )
, where Γ m and γ m are roots of equations [START_REF] Mackay | Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators[END_REF] for m large enough.

For each γ ∈ C E , a unique solution of the inhomogeneous system (56) exists in the form (62). However, the norm U X diverges as γ approaches the boundary of C E , according to estimate (58). Following the approach introduced in [7], we quantify this divergence when γ tends towards 0 but remains in a well-chosen subset CE,ν of C E far enough from resonances. for m ≥ m 0 , and satisfy as m → +∞

Γm = κ 2 2m 2 1 + κλ(E) πm - 2 -ν m + O(m -2 ) , γm = κ 2 2m 2 1 + κλ(E) πm - ν m + O(m -2 ) .
There exist γ 0 (ν) > 0 and C 0 (ν) > 0 such that for any γ ∈ CE,ν ∩ (0, γ 0 (ν)) and any F ∈ X, the solution U ∈ D of the inhomogeneous equation (56) satisfies

U X ≤ C 0 (ν)γ -1/2 F X . (66)
Moreover, if in addition F ∈ H 1 per ((0, T 0 ); l 2 (N))

U H 1 per ((0,T0);l 2 (N)) ≤ C 0 (ν)γ -1/2 F H 1 per ((0,T0);l 2 (N)) . (67)
Proof. Equations (65) can be solved for m large enough (say m ≥ m 0 (E)) and small Γm , γm by combining expansion (64) and the implicit function arguments. To deduce estimates (66) and (67) from Lemma 5, we need a lower bound for

M (n, q) = κ 2 -n 2 γω 2 0 + 2γ(1 -cos(q)), n ∈ Z, q ∈ [-π, π]. (68)
Let us assume γ ∈ ( Γm , γm ) with m large enough. In what follows we will show that M (m, q) > 0 and M (m + 1, q) < 0. For n ≤ m -1 it follows that M (n, q) > M (m, q) > 0, and for n ≥ m+2 we have also M (n, q) < M (m+1, q) < 0, hence the infimum of |M (n, q)| will be reached for n = m or n = m + 1.

Let us start with the case n = m. It follows from (64) that

∂ γ (γω 2 0 (E, γ)) = 2 + O( √ γ) > 0
for small γ > 0. This property implies that the minimum of M (m, q) occurs at γ = γm and q = 0. Using the definition of γm by (65), we obtain

M (m, q) ≥ κ 2 -m 2 γm ω 2 0 (E, γm ) = νκ 2 m + ν > 0.
Next, for n = m + 1 one has for m large enough

∂ γ (4γ -(m + 1) 2 γω 2 0 (E, γ)) = 4 -2(m + 1) 2 + O(m 2 γ 1/2 ) = -2m 2 + O(m) < 0.
Therefore, the maximum of M (m + 1, q) occurs at γ = Γm and q = π, where

M (m + 1, q) ≤ κ 2 -(m + 1) 2
Γm ω 2 0 (E, Γm ) + 4 Γm . Using the definition of Γm by (65), we obtain

κ 2 -(m + 1) 2 Γm ω 2 0 (E, Γm ) + 4 Γm = - ν(m + 1)κ 2 ω 2 0 (E, Γm ) ω 2 0 (E, Γm )((m + 1) 2 -ν(m + 1)) -4
, and consequently

M (m + 1, q) ≤ - ν(m + 1)κ 2 ω 2 0 (E, Γm ) ω 2 0 (E, Γm )((m + 1) 2 -ν(m + 1)) -4 < - νκ 2 m + 1 -ν < 0.
As a result of the above analysis, we have

inf n∈Z,q∈[-π,π] | M (n, q) | ≥ νκ 2 m + 1 .
Since γ ≤ γm ≤ κ 2 m 2 for m large enough, we get finally for all γ ∈ CE,ν small enough inf

n∈Z,q∈[-π,π] | M (n, q) | ≥ νκ 2 γ 1/2 . (69)
Estimates (66) and (67) follow directly from Lemma 5 and estimate (69).

Remark 6. If the set CE,ν = ∪ m≥m0 ( Γm , γm ) ⊂ C E is defined by

κ 2 + 4 Γm (m + 1) 2 -ν(m + 1) q = Γm ω 0 (E, Γm ), κ √ m 2 + νm q = γm ω 0 (E, γm ),
for some q ∈ (0, 2) and ν > 0, then Γm and γ m satisfy as

m → +∞ Γm = κ 2 2m 2 1 + κλ(E) πm - 2 m + ν m 2-q + O(m -2 ) , γm = κ 2 2m 2 1 + κλ(E) πm - ν m 2-q + O(m -2 ) .
From the requirement that Γm < γm for large m ≥ m 0 , we can see that either q ∈ (0, 1) and ν > 0 or q = 1 and ν ∈ (0, 1). On the other hand, we have from the proof of Lemma 6 that inf

n∈Z,q∈[-π,π] | M (n, q) | ≥ C(ν)γ (2-q)/2 . ( 70
)
For q ∈ (0, 1), the interval ( Γm , γm ) converges faster to the interval (Γ m , γ m ) for the price of losing too much power of γ in the bound (67) by inverting of (70). Therefore, the estimate of Lemma 6 is sharp in this sense.

5. Proof of Theorem 1. Let us represent

x 0 (t) = 1 γ 1/2 X 0 (τ ), x -n (t) = x n (t) = X n (τ ), n ≥ 1,
where {X n } n∈Z is a new set of unknowns in time τ = γ 1/2 t. From the discrete Klein-Gordon equation (1), we obtain

Ẍ0 + 2X 0 + γ -1/2 V ′ (γ -1/2 X 0 ) = 2γ 1/2 X 1 , (71)
γ Ẍ1 + κ 2 X 1 + N (X 1 ) = γ(X 2 -2X 1 ) + γ 1/2 X 0 , (72)
γ Ẍn + κ 2 X n + N (X n ) = γ(X n+1 -2X n + X n-1 ), n ≥ 2, ( 73
)
where N (X) := V ′ (X) -κ 2 X. Let B δ ⊂ H 1 e be a ball of small radius δ > 0 centered at 0 ∈ H 1 e . By assumption (P2), N (X) :

B δ → H 1 e is a C 5 map. Moreover, expansion V ′ (x) = κ 2 x + O(x 5 ) near x = 0 implies the existence of C > 0 such that for all δ > 0 small enough we have ∀X ∈ B δ , N (X) H 1 e ≤ C X 5 H 1 e , (74)
∀X 1 , X 2 ∈ B δ , N (X 1 ) -N (X 2 ) H 1 e ≤ C δ 4 X 1 -X 2 H 1 e . (75)
From system (72) and (73), we can see that oscillations near the zero solution would involve inverting the linearized operator in the inhomogeneous system (56). By estimate (67), we are going to loose γ 1/2 , which is the size of the inhomogeneous term γ 1/2 X 0 . This would prevent us from using the contraction mapping theorem in the neighborhood of the zero solution. To overcome this obstacle, we introduce the near-identity transformation

X 1 = Y 1 + γ 1/2 κ -2 X 0 , X n = Y n , n ≥ 2 and rewrite system (71)-(73) in the equivalent form Ẍ0 + 2X 0 + γ -1/2 V ′ (γ -1/2 X 0 ) = 2γκ -2 X 0 + 2γ 1/2 Y 1 , (76)
γ Ÿ1 + κ 2 Y 1 -γ(Y 2 -2Y 1 ) + N (Y 1 + γ 1/2 κ -2 X 0 ) = -γ 3/2 κ -2 Ẍ0 + 2X 0 ,(77) γ Ÿ2 + κ 2 Y 2 -γ(Y 3 -2Y 2 + Y 1 ) + N (Y 2 ) = γ 3/2 κ -2 X 0 , (78)
γ Ÿn + κ 2 Y n -γ(Y n+1 -2Y n + Y n-1 ) + N (Y n ) = 0, n ≥ 3. (79)
Extracting Ẍ0 + 2X 0 from (76), we can rewrite (77) in the equivalent form

γ Ÿ1 + κ 2 Y 1 -γ(Y 2 -2Y 1 ) + N (Y 1 + γ 1/2 κ -2 X 0 ) = -2γ 2 κ -2 Y 1 -2γ 5/2 κ -4 X 0 + γκ -2 V ′ (γ -1/2 X 0 ). ( 80
)
We shall solve the above system in two steps, using the contraction mapping theorem to solve (78)-(80) at fixed X 0 , and then Schauder's fixed point theorem to solve (76). In the latter case, we shall consider equation (76) similar to equation (24) with X 0 ∈ H 1 e being close to the solution X ∈ H 1 e of Lemma 1 rescaled by [START_REF] Peyrard | Nonlinear analysis of the dynamics of DNA breathing[END_REF]. The source term depends on Y 1 ∈ B δ ⊂ H 1 e and X 0 , and it will be proved that δ = O(γ ε ) is small as γ → 0.

Let us now describe our functional setting in more detail. Given µ ∈ (0, 1 2 ) and γ > 0 small enough, we define

Dµ,γ = X0 ∈ H 1 e ∩ C 1 e : X0 C 1 ≤ 3 √ E, X0(τ ) ≥ a0γ 1/2 , 0 ≤ τ ≤ T0 4 -γ 1/2-µ . ( 81
)
When γ is small enough, Corollary 1 and Lemma 2 imply that X ∈ D µ,γ . Moreover, D µ,γ defines a closed, bounded and convex subset of C 1 per (0, T 0 ). Repeating the same arguments as in the proof of Corollary 2, we obtain

∃C > 0 : ∀X 0 ∈ D µ,γ : V ′ (γ -1/2 X 0 ) H 1 e ≤ Cγ -(1+2µ)/4 . (82)
For sufficiently small γ in the set Cω0,ν for fixed ν ∈ (0, 1), we can rewrite system (78), (79), and (80) in the form

Y + L -1 N(Y, X 0 ) = L -1 F(Y 1 , X 0 ),
where L -1 is the Green operator of Lemma 6 solving the linear inhomogeneous problem (56), N(Y, X 0 ) : H 1 e ((0, T 0 ); l 2 (N)) × D µ,γ → H 1 e ((0, T 0 ); l 2 (N)) is the nonlinear operator at the left side of (78)-(80), and

F(Y 1 , X 0 ) : B δ × D µ,γ → H 1
e ((0, T 0 ); l 2 (N)) is the right side of (78)-(80). To use the estimates of Lemma 6, we assume γ small enough in Cω0,ν . Using (67) and (82), we obtain that

∃M > 0 : ∀Y 1 ∈ B 1 , ∀X 0 ∈ D µ,γ : L -1 F(Y 1 , X 0 ) H 1 e ((0,T0);l 2 (N)) ≤ M 2 γ ǫ , (83)
where ǫ = 1 4 -µ 2 . Now let us denote by B δ the ball of radius δ = M γ ǫ centered at 0 in H 1 e ((0, T 0 ); l 2 (N)). Using (67) and (74), we obtain ∃C > 0 : ∀Y ∈ B δ , ∀X 0 ∈ D µ,γ : L -1 N(Y, X 0 ) H 1 e ((0,T0);l 2 (N)) ≤ Cγ 5ǫ-1/2 . (84) Let us further assume µ ∈ 0, 1 4 , which implies ǫ ∈ 1 8 , 1 4 . From (83)-(84) and the triangle inequality, it follows that the map L -1 (N-F)(•, X 0 ) maps B δ into itself for γ small enough in Cω0,ν and for all X 0 ∈ D µ,γ . Moreover, thanks to bound (75) and Lemma 6, for all sufficiently small γ in Cω0,ν and for all X 0 ∈ D µ,γ , the map

L -1 (N -F)(•, X 0 ) is a contraction in B δ , with Lipschitz constant O(γ 4ǫ- 1
2 ). By the contraction mapping theorem (and using the fact that N and F are in addition locally Lipschitzian with respect to X 0 ∈ H 1 e ), there exists a unique continuous map D µ,γ ∋ X 0 → Y ∈ H

We can now substitute Y 1 from solutions of system (78)-(80) to equation (76). Applying Lemma 3, we rewrite equation (76) in the form

X 0 = F γ,µ (X 0 ), ( 87
)
where F γ,µ : D µ,γ → C 1 e is defined by F γ,µ (X 0 ) = G γ,ǫ (γ 1/2-ǫ 2κ -2 X 0 + 2Y 1 (X 0 )γ -ǫ ), G γ,ǫ is the nonlinear Green operator of Lemma 3 solving equation (24), and Y 1 (X 0 ) is defined from the map (85). By Lemma 4 and the continuity of the map (85), the map F γ,µ is continuous. Moreover, thanks to the estimates of Lemma 3 and the fact that µ > 0, F γ,µ maps D µ,γ into itself when γ is small enough. Observing that ∆ = d 2 dτ 2 : H 3 e → H 1 e is invertible and considering equation ( 24) that defines G γ,ǫ (F ) implicitly, we have the equality G γ,ǫ (F ) = ∆ -1 [ γ ε+1/2 F -2G γ,ǫ (F ) -γ -1/2 V ′ (γ -1/2 G γ,ǫ (F )) ].

(88)

Since the embedding of H 3 e into C 1 e is compact, it follows that G γ,ǫ :

B δ ⊂ H 1 e → C 1
e is compact, hence F γ,µ : D µ,γ → C 1 e is compact. Consequently, by the Schauder fixed-point theorem, there exists a solution X 0 ∈ D µ,γ of equation (87) for sufficiently small γ > 0. Moreover, Lemma 3 ensures the existence of θ > 0 such that

X 0 (τ ) ≥ a 0 √ γ for 0 ≤ τ ≤ T 0 4 -θγ 1/2 .
Repeating the same estimates as above with µ = 0, we obtain ∃M > 0 : ∀X 0 ∈ D 0,γ : Y H 1 e ((0,T0);l 2 (N)) ≤ M γ 1/4 . (89)

Fixing now ǫ = 1 4 in Lemma 3, estimate (28) yields finally ∃C > 0 :

X 0 -X H 1 e ≤ Cγ 1/4 . ( 90
)
Combining all transformations above with bounds (89) and (90) as well as using embedding of H 1 e into L ∞ e and of l 2 (N) into l ∞ (N), we obtain bound (13). Remark 7. If assumption (P2) is relaxed with the expansion V ′ (x) = κ 2 x + O(x 3 ) near x = 0, then the map L -1 N(Y, X 0 ) is a contraction operator with respect to Y in a ball of radius δ = γ ε for any ε > 1 4 . In this case, the inhomogeneous term L -1 F(Y 1 , X 0 ) with the bound (83) is critical with ε = 1 4 and prevent us to close the arguments of the contraction mapping theorem.

Remark 8. The Lipschitz continuity of the map (85) can also be established but the Lipschitz constant may have a bad behavior as γ → 0 because of the factor γ -1/2 in the last term V (γ -1/2 X 0 ) of equation (80). This obstacle prevents us from the use of the contraction mapping theorem for equation (87).

6. Exponential decay on Z. Our last result is to show that the large-amplitude discrete breather constructed in Theorem 1 decays exponentially in n on Z. The arguments repeat those of reference [START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF], to which we shall refer for some standard steps of the proof. where M (k, q) is defined in (68). Using estimate (69) we get consequently

A -1 γ L(H 1 e ) = O(γ -1/2 ) ( 92
)
when γ → 0 in CE,ν . Now we rewrite system (79) as

Y n = A -1 γ [ γ(Y n+1 -2Y n + Y n-1 ) -N (Y n ) ], n ≥ 3. ( 93

Figure 1 .

 1 Figure 1. Morse potential (dotted line) and modified double-well potential (solid line).

Figure 2 .

 2 Figure 2. The phase plane (x, ẋ) (top) and the frequencyamplitude diagram (ω, a) (bottom) for the potential (2).

ωFigure 3 .

 3 Figure 3. Resonance tongues on the plane (γ, ω) (delimited by dotted and dashed lines), and breather frequency curve ω = √ γω 0 (solid line) for κ = 1. No resonances occur for γ ∈ ∪ m≥m0 (Γ m , γ m ), where Γ m and γ m correspond to the intersections of ω = √ γω 0 with ω = √ κ 2 +4γ m(dotted lines) and ω = κ m (dashed lines) respectively.

Figure 4 .

 4 Figure 4. Non-resonance coefficient Q versus energy E for the potential (2).

Lemma 6 .

 6 Fix E > V L and ν ∈ (0, 1). Let γ ∈ CE,ν = ∪ m≥m0 ( Γm , γm ) ⊂ C E , where m 0 ≥ 1 is large enough and Γm , γm are found from the equationsκ 2 + 4 Γm (m + 1) 2 -ν(m + 1)= Γm ω 0 (E, Γm ), κ √ m 2 + νm = γm ω 0 (E, γm ), (65)

  1 e ((0, T 0 ); l 2 (N))(85) such that {Y n } n≥1 solves (78)-(80) and satisfies the bound∃M > 0 : ∀X 0 ∈ D µ,γ : Y H 1 e ((0,T0);l 2 (N)) ≤ M γ ǫ .

Lemma 7 .

 7 Let x(t) ∈ l 2 (Z, L ∞ per (0, T )) be the solution in Theorem 1. There exists a constant D 0 > 0 such thatsup t∈[0,T ] |x n (t)| ≤ (D 0 γ) (2n-1)/4 , n ∈ N,(91)for all sufficiently small γ ∈ CE,ν .Proof. The operatorA γ = γ d 2 dτ 2 + κ 2 : H 3 e → H 1e is unbounded, closed and selfadjoint. Its spectrum consists of simple eigenvalues at {κ 2 -γk 2 ω 2 0 } k∈Z , henceA -1 γ L(H 1 e ) = 1 inf k∈Z |κ 2 -γk 2 ω 2 0 | = 1 inf k∈Z |M (k, 0)| ,

)

  By estimate (89) we have Y n H 1 e = O(γ 1/4 ) uniformly in n ∈ N, which in conjunction with (74) yields N (Y n ) H 1 e ≤ C γ Y n H 1 e .Using this estimate and the bound (92) in (93), one finds M > 0 such that for all γ ∈ CE,ν small enough and for all n ≥ 3Y n H 1 e ≤ M γ 1/2 ( Y n+1 H 1 e + Y n-1 H 1 e ). (94)A simple application of the discrete maximum principle yields (see[START_REF] James | Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing[END_REF], Lemma 3.3)Y n H 1 e ≤ (2M γ 1/2 ) n-2 Y 2 H 1 e , n ≥ 3. (95)Using equation (78), estimates (92) and (95) with n = 3, the fact that Y 1 H 1 e = O(γ 1/4 ) and X 0 H 1 e = O(1) (direct consequence of (90) and Corollary 1), we getY 2 H 1 e = O(γ 3/4 ). (96)Then one completes the proof by putting estimates (95) and (96) together and using the continuous embedding of H 1 e in L ∞ e .


Najat Bara
email: bara@emines.um6p.ma

Frédéric Gautier

Vincent Giard

Frédéric Fontane

Economic evaluation in decision models: a critical review and methodological propositions

Keywords: Industrial management, mathematical model, economic model

Decision models of industrial management articles are often based on an economic criterion to find the proposed solution. They use economic parameters that are generally imported from the firm cost accounting system. When cost information is not adapted to the decision, the obtained solution of the model may be invalid. In this article, we deal with a critical literature review to report the methodological problems encountered in industrial management articles vis-à-vis the used costs. Finally we suggest methodological propositions to be kept in mind by authors when they are using costs in decision models.

Introduction

We are interested in the management of a supply chain (SC) or a sub-system of a SC. It concerns strategic, tactical and operational decisions taken to design, and run production units. To improve their decision making process, managers call in decision models that attempt to reproduce, through a simplified physical model, the functioning of the studied productive sub-system. In these physical models, decisions to take are represented by decision variables that determine, through a number of causal relationships, their impact on some physical characteristics of the studied system (flow, stock, resource use …). Decision variables may be quantitative (capacity of a new plant, reorder points…) or qualitative (opening or closing facilities, production orders assignment…). Across a modelling, we focus more particularly on certain physical quantities calculated by this model and linked to the decision variables, because they condition the performance of the studied system. In modelling, these physical quantities are generally called state variables and are used with costs to obtain an economic evaluation of the considered alternative decisions [START_REF] Giard | Methodological problems in defining costs used in industrial management decision models[END_REF].

The physical model is bounded to a studied physical subsystem exchanging flows of goods and/or services with its environment. In a decision model, evaluation of alternative decisions lays on a criterion in relation with the global performance of the studied subsystem. Such criterion may be physical (machines utilization rate, % of unsatisfied demand…) and, in this case, it's supposed that the variation of the company financial performance (cost or margin) is linked with the one of the considered physical criteria and is assumed to vary always in the same direction (or in the opposite one). The decision criterion may be economic and called on costs information provided by a cost model. In that case, the relevance of the solution provided by the decision model depends on the compatibility of the implicit assumptions of the physical system functioning, used by the cost system with the one used in the decision model. A significant discrepancy invalids the relevance of the proposed solution. Most often, authors focus on the physical model development techniques and resolution methods, without paying attention to the consistency of the implicit functioning assumptions of the used cost model with the one of the physical model used by the decision model. It is assumed that the modelling teams can work independently of those of management accounting. The costs regularly provided by a firm management accounting system, cannot be suitable to any type of use.

This paper present a critical literature review to see to what extent authors check the adequacy of the used cost model with their developed decision model. In each article we dissect the economic function to highlight the considered costs, the way they were defined vis-à-vis the considered time and space horizons and whether the authors studied under which conditions, the proposed solution remains valid, especially when the given solution impacts the system design and the use of its productive capacity.

The rest of this paper is organized as follows, we begin with the theoretical framework, next we present the adopted methodology of the review. Subsequently we report the encountered methodological issues. And finally propositions of contributions and the conclusion are presented.

Theoretical framework

We focus here on a description and a literature review of the main theoretical concepts, that are necessary for the discussions dealt in the present paper. Firstly, we deal with highlighting the decision model characteristics, thereafter, we present some cost accounting approaches and briefly some economic concepts and costing theories.

Decision model underlying characteristics

Decision models are developed under the assumption that we can explain and capture the behavior of a physical system and/or a decision making process dealt by managers. Causal relationships exist between the considered variables of the model. These causal relationships enable the model to predict the future state of the modeled physical system [START_REF] Bertrand | A robust optimization model for cellular manufacturing system into supply chain management[END_REF].

A decision model has two key characteristics, they are space and time granularities [START_REF] Giard | Methodological problems in defining costs used in industrial management decision models[END_REF]. The space granularity concerns the covered perimeter of the studied productive system and the degree of details considered which in turn, determines the granularity of information provided for the user. The time granularity, defines whether the model deals with an operational, tactical or strategic issue. Space and time granularities are not independent; in general, the granularity is fine in operational decisions analysis and more aggregated in other cases. Granularity conditions bill of materials, production routings, product ranges and the definition of resources used in the physical model. It has an essential impact on the relevance of costs to be mobilized.

Concerning decision models development approaches, we can distinguish two main approaches, mathematical (operation research (OR)) and simulation modelling. The first one refers to i) optimisation that gives the best solution for a particular issue, it's a very visible and influential topic in the field of operations management and, ii) heuristics which are intelligent rules leading to "good" solution and are used when optimization is impossible or too expensive. The second approach refers to all simulation techniques (discrete event simulation, agent-based simulations, system dynamics).

Cost accounting

A cost system refers to means by which organizations calculate costs of products and services to make sales profitable. Moreover, cost system concerns the modeling of costs of organizations resources and operations in order to support internal decision making process. Cost information should be useful to help managers in managing the performance and the profitability of products, customers, productions processes… (Lawson, 2018). Here we distinguish the traditional cost accounting and the Activity Based Costing (ABC) approaches:

The traditional cost accounting system in a company aims to allocate, at the end of each month, costs to products and services. It's oriented toward a retrospective calculation of historical costs. It's based on a succession of spills of analysis centers expenses to other cost centers by means of volume key distribution. ABC is a cost accounting approach developed from the eighties. It's based on the principle that activities consume resources, and products and services consume activities. ABC focuses on the activities that take place within the company, the cost of carrying out each activity and the factors generating these activities. It goes beyond the traditional system by allocating resources, based not only on volume units, but also on the diversity and degrees of complexities of the products (Cooper and Kaplan, 1998). Hence, the ABC approach distinguishes different cost pools (activity drivers): unit-level activities are performed each time a unit is produced, batchlevel activities are performed whenever a batch is processed, product-level activities (process engineering, design, etc.) are costs which come into play whenever a particular product (order) is manufactured and facility sustaining activities concern costs related for example to rent, utilities, maintenance, and facility management.

Furthermore, an activity cost may evolve in the long run. Value chain analysis is the way to analyse activities in order to achieve competitive advantages by performing these activities better than competitors. Each activity has its structure and behavior, which determine its cost. Porter calls these determinants "cost drivers". He distinguishes ten possible causes of an activity cost evolution: i) economies or diseconomies of scale (costs sensitive to volume), ii) learning phenomenon (unit costs depend on time), iii) capacity cost utilization (impacts the fixed unit costs), iv) the linkage between activities that makes an activity cost dependent of other activities costs, v) interrelationships (resources share between different strategic units), vi) the degree of vertical integration (impacts transaction costs), vii) the timing (first-mover reward), viii) discretionary policies that refer to products or services design, ix) localisation and x) change in institutional and regulatory factor that may affect unit costs. [START_REF] Hergert | Accounting data for value chain analysis[END_REF] [START_REF] Bouquin | Comptabilité de gestion[END_REF].

The absence of the concepts of activity drivers and cost drivers in the traditional cost system is one the reasons that make it inappropriate for the study and analysis of operational and strategic decisions respectively. The traditional system is characterised by a lack of cause and effect traceability [START_REF] Stratton | Activity-based costing: is it still relevant?[END_REF], that makes it inadequate to understand how activities are performed and makes it weak in cost traceability.

Cost concepts

There is a multitude of cost concept whose use relevance depends on the studied context:

 Opportunity cost is a concept that dates back to the nineteenth century. It is proposed by Friedrich von Wieser, (1851-1926) who was interested in determining the production factors value [START_REF] Burch | Opportunity and incremental cost : attempt to define in systems terms : a comment[END_REF]. He defined the opportunity cost as the net income generated by this factor of production in its best use. [START_REF] Andreani | Le coût d'opportunité[END_REF] linked it to the existence of a conflict between opportunities that could lead to a shortfall. Opportunity costs are the revenues to be gained from possible actions, but lost because other actions have been taken to achieve a particular goal, [START_REF] Andreani | Le coût d'opportunité[END_REF], [START_REF] Vera-Munoz | The effects of accounting knowledge and context on the omission of opportunity costs in resource allocation decisions[END_REF].

 Marginal cost is the cost incurred for the production of an additional unit. The marginal cost differs according to whether it is decided to momentarily increase the volume of production by one unit, or if it is decided to increase it durably by one unit [START_REF] Boiteux | La tarification au coût marginal et les demandes aléatoires[END_REF]. In the first case, the additional unit is satisfied by the current capacity, but the second case may require an increase in capacity.

 Variable costs: the traditional system accounting defines variable costs as those who change with production volume and fixed costs those who do not change with volume.

Cooper and Kaplan propose to use short-term variable cost, they vary with production volume, long-term variable costs which do not vary with production volume but do vary with other activity measures (handling, setup…) and fixed costs that do not vary; in a given period; with any activity driver.

 Capacity costs are defined by the set of resources (facilities, staff...) in which the company is committed to achieve a given level of performance. This latter is linked to the estimated maximum demand, the intended degree of flexibility to respond to unforeseen demand and the desire to ensure products diversity. These capacity costs, also called fixed costs, are generated form strategic decisions defining the company's strategy and vision [START_REF] Bouquin | Comptabilité de gestion[END_REF]. "Investments in resource capacity are made based on the expected demand because instantaneous adjustment of capacity is impossible and/or extremely costly" (Balakrishnan et al, 2007). The difference between the maximum capacity and the actual activity level is called unused capacity.

Different economic criterions in decision models

Several economic criterions may be used to assess decisions, their relevance depends on the studied decision nature. Here we retain the three criterions of costs, profit and discounted cash flows (DCF) through Net Present Value (NPV). Investment decisions refer to long-run decisions which focus on studying different alternatives of investment (outflows) in one or several periods to generate returns (inflows) in one or several future periods. They differ from operational decisions in the way that their study requires the consideration of the value of money evolution over time [START_REF] Shillinglaw | The concept of attributable cost[END_REF] [START_REF] Lucey | Management accounting, 5 th Ed, Continuum[END_REF]. Hence, the NPV is the appropriate measure to study investment decisions, it calculates the present values of expected inflows and outflows. Operational and tactical decisions study is based on the anticipated current cash flow only. The profit is the sum of revenues minus costs. When the revues are constant (sales and prices are independent of decision variables), profit maximization can replaced by costs minimization. Accounting profits are not suitable for investment decisions, they are period oriented (quarter, year…) [START_REF] Lucey | Management accounting, 5 th Ed, Continuum[END_REF].

Costs and spatial perimeter

Issues dealt by decision models may belong to any SC activity: production, inventory management, transportation, pricing, design, relationship management… We note then that a decision perimeter may either belong to the internal firm perimeter (production scheduling, investment, transportation…) or extends the firm boundaries (interrelationships management). The former case settles for the firm system accounting (intra costing) whereas the latter one necessitates an inter-organizational costing (inter costing). In fact a cost system is contingent to the studied physical system characteristics.

Cost accounting systems of two cooperating companies are seldom similar. The difference between the two approaches creates inaccurate costs and ineffective practices [START_REF] Kulmala | The role of cost management in network relationships[END_REF]. Thus, creating relevant approaches for partnership relations is one of cost accounting challenges. Furthermore cooperation between a supplier and customer is based on trust and their readiness to open their book cost accounting. But this willingness to share information (reliability) is not enough, there should be dedicated approaches to generate relevant cost information [START_REF] Kulmala | The role of cost management in network relationships[END_REF].

Costs at service of SC professionals

Previous research works has pointed out that often SC managers are not satisfied by the cost information provided by their firm accounting system. [START_REF] Gurowka | Selecting the right costing tool for your business needs[END_REF] presented a road map to help managers in choosing the appropriate costing approach. When cost information is not correct and accurate, the accounting system has no value for the company, worst it can damage the organization decision making.

According to a survey done on 2003 by the Institute of Management Accountants (IMA)/Ernst and Young, only 23% of the interviewed managers are satisfied by their accounting information, 92% of them state that this information is distorted. [START_REF] Gurowka | Selecting the right costing tool for your business needs[END_REF] say that if cost information still irrelevant, decisions are irrelevant and over time, departments managers may stop relying on financial criterion and make decisions based solely on non-financial data (more accurate).

The results of a recent survey [START_REF] Lawson | Working together to Enhance Supply Chain Management with Better Costing Practices[END_REF] shows that supply chain professionals often view their company's managerial costing system as being only slightly useful for generating questions and for making managerial decisions. According to [START_REF] Lawson | Working together to Enhance Supply Chain Management with Better Costing Practices[END_REF], three main causes explain this result: i) many organizations rely on externally oriented financial accounting systems to produce the information supporting internal business decision making; ii) many organizations use outdating cost models; these latter are not appropriate for the current organizations challenges and environment; iii) the third cause is the finance and accounting professionals resistance to change. This is amplified by little pressure from managers; who use costs; to improve data accuracy and relevance [START_REF] Kilcarr | Heads up: New supply chain costing methods are being called for[END_REF].

In this paper, we complete this previous works, by a critical literature review of articles dealing with decisions models to highlight methodological problems related to the used costing models. It's to our knowledge, the first article that fully addresses this literature review.

Methodology

The adopted methodology leads to the selection and the evaluation of selected sources. We followed the same methodology adopted by [START_REF] Maestrini | Supply chain performance measurement systems: A systematic review and research agenda[END_REF]. We first i) identify articles ii) we select articles, and finally iii) we evaluate and analyze them.

Articles identification

For the sources search we relied on the Business Source Complete base (BSC). Since we are interested in the study of the coherence between the developed decision model and the used economic model, we used the key words "industrial management", "mathematical model", "simulation model" and "economic model". To broaden the search range of articles, we used two key words combinations, in the first one we used "industrial management", "mathematical model" and "economic model", and in the second combination we replaced "mathematical model" by "simulation model".

The use of two key words combinations gave a total of 1225 articles. We relied on SJR site, we investigate the journal scope and H Index. We considered only IJPE, IJPR, POM, OMEGA and PPC whose H Index varies between 61 and 141. We also limited the period of publication, we focused on the period between 2013 and 2018. We therefore get a sample of 202 articles.

Articles selection

We set boundaries to select the relevant articles to be considered in the present review. We select articles in which a decision model is developed to help users in making decisions related to a sub-system of the considered SC, hence there should be explicit variables that determine characteristics (functioning or design) of this system. The selection criterions are discussed and defined by authors of the present paper, and the task is done by two authors based on the abstract, the conclusion and eventually the introduction. Consequently, we selected a sample of 113 articles.

Articles evaluation and analysis

To evaluate the selected articles we started with an analysis grid. This grid attributes were defined based on previous literature review works [START_REF] Maestrini | Supply chain performance measurement systems: A systematic review and research agenda[END_REF], a SC taxonomy developed by [START_REF] Capar | A taxonomy for supply chain management literature[END_REF] and additional attributes deduced from the theoretical framework above.

We begin with articles journal of publication. In the second stage we present the modelling approach on which the decision model is based [START_REF] Bertrand | A robust optimization model for cellular manufacturing system into supply chain management[END_REF]. Thereafter, we highlight the considered time granularity in each article [START_REF] Anthony | Planning and Control Systems: a framework for analysis[END_REF], [START_REF] Ansoff | Strategic issue management[END_REF] and the considered perimeter that implicitly determines whether it's an inter or intra costing. We additionally mention the considered objective function physical or economic and whether it's costs, profit or NPV and the mobilized cost concepts. A grid summary is presented in table 1 bellow.

Findings

In this section we will present the mains results obtained from the reviewed articles analysis. We begin by reporting the articles journals of publication. In the second stage, we report the decision criteria. And finally, we focus on articles in which an economic model is used, in order to highlight the main encountered methodological issues.

Publication journal

The final number of the reviewed articles is 113. The largest percentage 49% of articles belong to IJPE, succeeded by 40% that bong to IJPR , and succeeded by 7%, 3% and 2%, that belong to OMEGA, PPC and POM respectively.

Decision criteria 13 out of 113 articles deal with problematics in which the developed decision model gives the solution based on the minimization or the maximization of a physical criteria. Most of these articles (11 out of 13) deal with operational and tactical levels issues 12% of articles consider both an economic and physical criteria. In the rest (76 %), the choice of the proposed solution is based solely on an economic criterion. Hereafter, we will exclude articles which are limited to the study of a physical criteria, we retain 100 article.

Encountered methodological issues

In this stage we based on the grid analysis to report the widespread methodological problems encountered in each article. This figure presents for each decision level the SC activities dealt with. Bellow, we discuss issues related to cost content explanation (activity drivers), costs traceability, inter or intra costing nature, appropriateness of used economic criteria, capacity costs, cost drivers in the long run and finally the eventual impact of the obtained solution.

In the quasi totality of the reviewed articles, the authors gave no explanation about the used economic model parameters. These latter, are considered as given parameters provided by an external source which is generally the firm accounting system. They do not discuss the relevance of the used cost content according to the spatiotemporal perimeter. This suppose the assumption that modelling and accounting teams can work independently, as if costs could be absolute. No article uses activity drivers to define the relevant content of the used cost parameters. In some articles the authors give details about some used costs, for instance in [90] the unit transportation cost "includes costs of fuel, salaries, wages, operating supplies, insurance, and depreciation", [62] explain that the unit inventory cost is due to "inventory obsolescence costs and capital costs related to slow-moving parts", and that the unit manufacturing cost excludes all capacity costs (e.g. machines depreciation). Generally a firm system accounting has its standards, it regularly calculates costs at the end of a predefined period (month). When a decision model is developed, it may necessitate the introduction of some specific costs that the firm system accounting is regularly not interested in. So here we identify a problem of traceability. In [69] and [105], the salvage value is difficult to be determined a priori, elsewhere there should be a sufficient data to can to estimate it. [37], [66]…models use penalty cost of unsatisfied demand, [103] developed a model for a retailer replenishment under supplier prices discount, and he uses a penalty cost induced whenever the retailer change his order quantity. [12], [33] mobilize the opportunity costs which are future costs that need to be known a priori. In [98] costs of non-conformity and rework cost are considered. Furthermore, some papers deal with batch or order level costs, if the firm system accounting is not based on ABC approach, these costs are not provided. Here then, we mention that there are several costs that are needed by the developed decision models in a company but for which there is no traceability in the system accounting.

The objective in [69] is the comparison between different relationship scenarios between a manufacturer, a distributor centre (DC) and a retailer ( i) not using contract, ii) using buyback contract or iii) revenue sharing contract). [105] developed a model to coordinate between a supplier and retailer, it determines e-concession as a decision variable (a concession that both retailer and supplier need to make to maximize the whole system profit). Their model mobilizes the following economic parameters the supplier production cost, the retailer wholesale price, retailer salvage value and the retailer shortage penalty. [START_REF] Boiteux | La tarification au coût marginal et les demandes aléatoires[END_REF] focus on the comparison between centralized and decentralized strategies between a retailer and its supplier. These examples of articles and others that compare different relationship strategies between different SC players imply that the applicability of the presented models is conditioned by the involvement and willingness of the considered firms to collaborate and share their costs information with other firms, which is not evident in practice. Through this inter costing necessity we signal the problem of the model perimeter which presents a new methodological issue.

In several of the reviewed articles, the authors deal with some top strategic issues.

[37] developed a decision model for a network design, it determines facilities (production plants, warehouses and DC…) to be opened, the supplier selection. [66] models defines production and inventory capacities in a correctional institution… [92] presents a model to compare different strategies to meet demand satisfaction, one of which consists on capacity investment. Even if all these issues imply investment with long term impact, the authors used costs rather than the NPV. They mix in one economic function costs related investment decisions and recurrent costs related to operating decisions. Here we identify the problem of discounting that should be considered when we deal with investment decisions whose impact is recorder along several periods. In the strategic level costs aren't the relevant to solely be considered, but they still the criteria adopted by the majority of authors. Only 3 of all reviewed articles [41], [34] and [104] consider NPV as the choice criteria.

In this paragraph we are interested in reporting the methodological issues related to capacity costs. We separate strategic form operational and tactical levels:  Strategic level: in 85% of the studied articles consider full costs. However, since the activity levels are not necessarily the capacity level, they are decision variables of the model, the use of full costs leads to sank costs that penalizes the considered cost objects (products, orders… [58] the model determines the production orders scheduling in parallel machines. In all these examples, the authors consider full costs, which are not relevant to all kind of uses as the case for operational decisions.

Only 6 out of 16 articles consider variable or marginal costs rather than full costs to evaluate operational decisions. This full costs lead to biased economic evaluation. Concerning cost drivers, none of the reviewed article considers cost drivers implying the eventual evolution of unit costs on the long run.

A new methodological problem is related to the impact of the obtained solution on the productive system. [93], [97], [37], [78], [26]… present developed models that define SC network designs. [1] developed a model for cellular manufacturing design, in each plant the model determines installed capacity in terms of machines number of workers… This kind of models, start from an initial physical system from which we deduce a cost model, and then the model use these information to obtain an optimal solution. The decision variables in these cases may affect the physical system, which calls into question the used cost parameters and then the obtained optimal solution. None of studied articles, studies the impact of the obtained solution on the used cost parameters.

Contributions

The presented results reveal that indeed authors neglect the costs relevance which leads to the reported methodological issues that affect the relevance of the proposed solution.

In this section, we suggest some methodological guides that industrial management researchers need to keep in mind when they define costs in their decisions models  Need of reprocessing the firm accounting system information: One of the main revealed findings, is that authors consider costs as given parameters without any explanation. A cost calculation makes sense if it is only adapted to the decision-making system that will use it. As a result, a decision model cannot take the output of full cost calculation without analysis and reprocessing. This means that decision models need to be provided by an ad hoc economic evaluation adapted to their spatiotemporal perimeter. If not, these decision models do not really improve the decision making, since the solution search relies on this economic evaluation. Therefore, the costs introduced in the decision models cannot be considered as valid physical parameters regardless of the physical system that they seek to improve. They are contingent to physical and functioning characteristics.  Costs specification and appropriate cost approach: To ensure a relevant economic evaluation, the one need to determine charges that are impacted by the studied decision and exclude from the evaluation all charges that do not steam form this decision. The understanding of costs behaviour; through the use of relevant activity drivers; and the use of appropriate cost concepts help to establish incremental cash flow report that measures the real impact of each scenario on the company income statement. Our findings also explore the use of specific costs (nonconformity costs, product or batch level cost…) or specific contexts (inter costing) which generally are not included in traditional system accounting. The companies need to change or complete their cost accounting by new costing approaches. If not, an inappropriate costing system will produce distorted information leading to a poor decision making [START_REF] Gurowka | Selecting the right costing tool for your business needs[END_REF].  Collaboration between cost accounting and production management: As a result of the two previous points, cost accounting, modelling and SC managers cannot be independent. They need to collaborate in order to determine and calculate the adequate costs to each studied context. This can help to overcome the resistance of accounting professionals to integrate new costing methodologies.  Impact of the obtained solution: Concerning decisions related to production system design, one starts from a basic scenario from which he deduces cost parameters. He feeds a decision model by these cost information to obtain a new design solution.

Since, costs are contingent to the organization physical characteristics, the solution of the decision model can lead to inconsistencies between the initial costs used and hypothesis, and the physical system new characteristics. This may make the obtained solution unworkable.

Conclusion

This article is grounded on the question: to what extent authors of industrial management articles study the adequacy between their decision models and the used cost system.

To answer this question we led a critical literature review whose finding reveal several methodological issues encountered when defining costs in decision models. i) In almost all articles authors give no explanation about the used economic parameters content, they assume costs as given parameters provided by an external source; ii) the reviewed decisions models use some costs that are not necessary traceable in the firm system accounting; iii) some articles suggest models that involve different companies, the applicability of this kind of models depends on the willingness of these companies to share their cost information; iv) the majority of search articles which study investment decisions having multi periods impact still base their decision criterion on costs rather than NPV; v) the majority of articles use full costs which generate inconsistencies related to capacity costs.

This work is intended to attract industrial management specialists to pay more attention about the relevance of the costs they use in decision models. The full costs of traditional system accounting cannot be used without reprocessing. To provide relevant and accurate costs, accounting professionals need to be open to integrate new costing methodologies. Modelling and cost accounting teams should be interconnected and work together to provide ad hoc economic evaluation for each decision model and finally we emphasize that decision models are a priori decision making oriented they need to integrate relevant costs concepts and to use activity drivers to can compare the existing alternative decisions and measure the impact on the company income statement.

  ).[37] ,[79],[78]… determine the production level in each plant and the inventory level in each SC node (e.g. DC, warehouse).[START_REF] Boiteux | La tarification au coût marginal et les demandes aléatoires[END_REF], [109]…present models that determine economic order quantity (EOQ). Full costs include capacity costs that are not generated by the chosen activity level but rather by the firm strategy. These sunk costs are due to unused capacity. One article [93] considered unused capacity costs concept.  Operational and tactical level: capacity costs are not controlled in the operational and tactical levels. Decisions related for example to week production planning, or orders scheduling during a month don't affect this category of costs.[106] model determines products scheduling in production machine.[71], [67] developed a model that define the orders distribution planning, they consider as decision variables, each product start and end dates, number of storage days… In

Table 1 :

 1 Grid summary

	Decision		Operational level			Tactical level					Startegic level
	level		16			38					46
	Modeling	Math	Simulation	Combined	Math	Simulation	Combined	Math			Simulation		Combined
	approach	12	2		2	31			7	37			5			4
	Economic	Costs	Profit	NPV	Costs	Profit	NPV	Costs			Profit		NPV
	criteria	16				26	12			27			17			3
	Cost concepts	Variable costs 6	Full costs 12	Marginal costs	Opportunity costs	Variable costs 7	Full costs 28	Marginal costs 1	Opportunity costs 4	4 Variable costs		Full costs 39	Patial cost 1	Direct cost 1	Marginal costs 2	Opportunity costs 2
	Methodological	Explanation Traceability Inter costing Capacity costs Explanation Traceability Inter costing Capacity costs Explanation Traceability Inter costing Capacity costs discounting	Cost drivers
	issue	16	14	4	12	38	27	14	24	45	24	36	17	17		11