J Nonlinear SciDOI 10.1007/s00332-008-9018-x

Weak Coupling Limit and Localized Oscillationsin Euclidean Invariant Hamiltonian Systems

Guillaume James · Pascal Noble

Received: 24 October 2006 / Accepted: 28 January 2008© Springer Science+Business Media, LLC 2008

Abstract We prove the existence of time-periodic and spatially localized oscilla-tions (discrete breathers) in a class of planar Euclidean-invariant Hamiltonian sys-tems consisting of a finite number of interacting particles. This result is obtained inan “anticontinuous” limit, where atomic masses split into two groups that have dif-ferent orders of magnitude (the mass ratio tending to infinity) and several degrees offreedom become weakly coupled. This kind of approach was introduced by MacKayand Aubry (Nonlinearity 7:1623–1643, 1994) (and further developed by Livi et al.in Nonlinearity 10:1421–1434, 1997) for one-dimensional Hamiltonian lattices. Weextend their method to planar Euclidean-invariant systems and prove the existence ofreversible discrete breathers in a general setting. In addition, we show the existenceof nonlinear normal modes near the anticontinuous limit.

Keywords Breathers · Euclidean invariance · Anticontinuous limit · Continuationof periodic orbits

Mathematics Subject Classification (2000) 70F10 · 70K42 · 70H09

Communicated by R.S. MacKay.

G. James (�)Département de Mathématiques, INSA de Toulouse, Institut de Mathématiques de Toulouse, UMR5219, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, Francee-mail: Guillaume.James@insa-toulouse.fr

P. NobleCNRS UMR 5208, Institut Camille Jordan, Université de Lyon, Université Lyon 1,43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex, Francee-mail: noble@math.univ-lyon1.fr

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1 Introduction

Discrete breathers are time-periodic, spatially localized oscillations that occur in anonlinear system of interacting particles. They are also called intrinsically localizedmodes (ILM) in distinction to Anderson modes triggered by disorder. In their pioneer-ing work, Sievers and Takeno (1988) argued for their existence in a one-dimensionalchain of nonlinearly coupled particles (the Fermi–Pasta–Ulam model with a harden-ing interaction potential), and suggested ILM existence should be a generic picturein anharmonic periodic crystals. Since then, several experiments have given strongevidence of the existence of breathers in various physical systems such as quasi-one-dimensional solids (Swanson et al. 1999), antiferromagnetic materials (Schwarz etal. 1999), molecular crystals (Edler and Hamm 2002) and uranium crystals (Man-ley et al. 2006). These recent experiments concern large ensembles of atoms, but theexistence of breathers or “local modes” in small molecules was studied during the1930s (see e.g. the references in A.C. Scott’s book 2003, Chap. 1). Figure 1 showsthe structure of such a local mode in benzene (C6H6), where the CH stretching os-cillation can remain mainly localized on a single bond. The properties of breathersare also analyzed in artificial systems such as Josephson junction arrays (Binder andUstinov 2002), micromechanical cantilever arrays (Sato et al. 2003) and coupled op-tical waveguides (Mandelik et al. 2003). For a review of some properties of discretebreathers and their experimental realizations see Flach and Willis (1998), Flach andGorbach (2007).

A mathematical proof of the existence of breathers was obtained by MacKay andAubry (1994) for Hamiltonian lattices having an uncoupled or “anticontinuous” limit(a better term might be anticontinuum) in which breathers trivially exist. The proofwas worked out for a one-dimensional chain of nonlinear oscillators linearly coupledto their nearest neighbors (Klein–Gordon lattices), but the results apply in a muchmore general setting. In Klein–Gordon lattices, an uncoupled limit corresponds tofixing the nearest-neighbors coupling constant to 0. In this limit (where the modelreduces to an array of uncoupled anharmonic oscillators), the simplest type of discretebreather consists of a single particle oscillating while the others are at rest. Undersome nondegeneracy conditions, it was proven that this solution could be continued tosmall coupling between the oscillators (this is a consequence of the implicit functiontheorem).

This approach was extended by Livi et al. (1997) to diatomic Fermi–Pasta–Ulamchains. In this model, two types of atoms with different masses alternate on a one-dimensional lattice and interact anharmonically with their nearest neighbors. In con-trast to the case of Klein–Gordon lattices, no breather solution exists when the an-harmonic interaction potential is set to 0 since the lattice simply reduces to a set of

Fig. 1 Illustration of a localmode of the CH stretchingoscillation in a benzenemolecule

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free particles. However, an uncoupled limit is obtained when the mass ratio betweenheavy and light atoms goes to infinity. In this limit, light masses do not interact andoscillate in a local potential corresponding to immobile heavy masses. Then a sin-gle breather can be constructed and continued down to large and finite mass ratio.One should note that the continuation process is slightly more technical in that case,because the implicit function theorem cannot be applied directly. Indeed, breathersexist (at fixed frequency ω) in one-parameter families due to the invariance of thesystem under translations. The result of MacKay and Aubry (1994) does not applydirectly because the phonon spectrum includes 0 (translational invariance), so nω be-longs to the phonon spectrum for n = 0, which violates the nonresonance conditionassumed in MacKay and Aubry (1994). It is worthwhile mentioning that the mathe-matical approach of Livi et al. is in accordance with experimental results concerninglocal modes in small molecules. Indeed it has been observed (Halonen et al. 1998)that the existence of local modes in tetraedric molecules is linked with their atomicmass ratio.

More mathematical difficulties arise for lattices of higher dimension invariant un-der translations. The existence of breathers in this type of system was addressed byAubry (1998) with some simplifying assumptions. Aubry’s model consists of a crys-tal described by a Hamiltonian system, where the crystal configuration is describedby two kinds of variables. On the one hand one has “optical” variables (e.g. the inter-nal coordinates of molecules arranged on a lattice) which evolve anharmonically andare unchanged by translations. On the other hand there are “acoustic” variables whichare uniformly shifted when the crystal is translated (they correspond, e.g., to the dis-placement from equilibrium of the center of mass of each molecule, and additionaldegrees of freedom describing atoms in their neighborhoods). An essential assump-tion in the analysis is that the part of the Hamiltonian involving the acoustic variablesis harmonic. This allows us to explicitly eliminate the acoustic variables with respectto optical ones, which results in a nonlinear equation for the optic variables wherethe degeneracy due to translations is eliminated. For infinite systems invariant undertranslations, the elimination of acoustic variables yields some complications even inthe harmonic case (see Aubry 1998 for more details). Breather solutions of the non-degenerate equation for optic variables are obtained with the usual anticontinuouslimit approach (MacKay and Aubry 1994).

For showing the existence of breathers in more general Euclidean-invariant Hamil-tonian systems, an interesting strategy was described by MacKay (2000b), whichgeneralizes to larger lattice dimensions the former approach of Livi et al. (1997) fordiatomic chains. The models include the invariance under rotations and do not as-sume the harmonicity of acoustic variables. In the case of diatomic systems, the firststep consists of fixing the mean position of heavy atoms and “eliminating” the othervariables (positions of light atoms and deviation of heavy atoms from mean position)with respect to the mean positions. This elimination is based on the anticontinuouslimit introduced in Livi et al. (1997), where a breather obtained for infinite mass ratiois continued down to large and finite mass ratio. The solution obtained in this waysmoothly depends on the mean position of heavy atoms, which are treated as para-meters. The second step consists of solving the “reduced” static problem for thesemean positions, a problem which comes out as a result of step 1. This static problem

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inherits the Euclidean invariance of the original Hamiltonian system. A local methodfor solving static problems of this type was developed by MacKay in a companionpaper (2000a). More precisely, a method is developed to study the persistence ofa basic equilibrium of a network with a Euclidean-invariant energy function whensmall defects are introduced. Two configurations related by translations or rotationsare considered as equivalent and a suitable norm is introduced on this quotient space(MacKay 2000a). The static problem is solved around a given unperturbed solutionusing the implicit function theorem, and in some cases the choice of the norm al-lows us to obtain persistence uniformly in the system size. This approach should beapplicable to the context of MacKay (2000b), where the unperturbed solution of thereduced static problem would correspond to heavy mass positions for a given equi-librium of the diatomic lattice. The perturbed situation would correspond to a largemass ratio and a small amplitude breather solution close to the equilibrium.

In this paper we use the same kind of approach to prove the existence of breathersin a large class of planar Euclidean-invariant systems. More precisely, we considera Hamiltonian system consisting of N + n masses γMi (i = 1, . . . ,N , N ≥ 2), mi

(i = 1, . . . , n) interacting in an anharmonic potential V , which is assumed sufficientlysmooth (at least C4 on (R2)N+n) and Euclidean-invariant. The parameter γ is pos-itive and we shall consider the anticontinuous limit γ → +∞ where N masses aremuch heavier than the others. We denote by Pi,pi the momenta conjugate to positionvectors Qi,qi in a fixed reference frame. The Hamiltonian is defined by

H =N∑

i=1

1

2γMi

P 2i +

n∑

i=1

1

2mi

p2i + V (Q1, . . . ,QN,q1, . . . , qn), (1)

where Qi(t), qi(t),Pi(t),pi(t) ∈ R2. We assume that the system has one or several

families of equilibria (translations and rotations of any equilibrium generate a con-tinuous family of equilibria).

In this context we study the existence of nonlinear normal modes and discretebreathers, this second class of solutions being our main concern. We denote by non-linear normal modes the nonlinear continuations of linear modes of the linearizationaround equilibrium solutions near the anticontinuous limit. Discrete breathers cor-respond here to continuations of time-periodic solutions of the anticontinuous limitwith fewer oscillating particles than the total number n of light particles.

Our approach splits up into different steps. In Sect. 2 we set up the equationsof motion corresponding to the Hamiltonian system (1) in a suitable rotating coor-dinate system, the total angular momentum being fixed and treated as a parameter.This process eliminates the rotational degeneracy of solutions and yields a lower-dimensional system of differential equations.

Next, in Sect. 3 we perform a convenient scaling to describe the limiting caseγ = +∞, where heavy masses are assigned to some static positions and light onesoscillate in the resulting potential. As for one-dimensional lattices (Livi et al. 1997),the evolution problem for light masses is coupled to a static problem for heavy masspositions. We consider solutions of this coupled system with time-periodic oscilla-tions of the light masses.

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This well-posed limiting problem allows for the numerical continuation of time-periodic oscillations from infinite to finite mass ratio. We give an example of suchcomputations for a triatomic system in a companion paper (James et al. 2007). Thiscontinuation procedure is a useful technique for computing breather solutions in dis-crete systems, as shown in Marin and Aubry (1996) for one-dimensional lattices. Inour case the numerical continuation to γ < +∞ yields in general relative periodicorbits of (1), i.e. solutions which appear time-periodic in a reference frame rotating atconstant velocity (in a fixed reference frame, these oscillations are not time-periodic,but only quasi-periodic due to rotational motion).

In addition, in Sect. 4 we perform an analytical study of the limiting problem withγ = +∞, and in Sect. 5 we study the continuation of its solutions to large and finitemass ratio. The analysis of Sect. 5 is restricted to time-periodic orbits of (1) in a fixedreference frame and time-reversible oscillations. In this paper this expression refersto solutions even in time, although it can be given a more general meaning whenadditional reflectional symmetries are present. For γ = +∞, we show in Sects. 4.2and 4.3 the existence of small amplitude reversible nonlinear normal modes (Theo-rem 1) and breathers (Theorem 2) under general conditions on V . These results arelocal, and based on the Lyapunov center theorem and the implicit function theoremto solve the coupled dynamical and static problems. Next, under certain nondegener-acy conditions, we show that reversible periodic orbits of (1) for γ = +∞ persist forlarge and finite values of γ (Theorem 3, Sect. 5). This result is based on the implicitfunction theorem but the limiting solution is not required to have a small amplitude.For the Hamiltonian system (1) with large values of γ , this result is used (in conjunc-tion with Theorems 1 and 2) to prove the existence of small amplitude, time-periodic,and reversible nonlinear normal modes and breathers (Theorems 4 and 5, Sect. 5).

In this paper we do not treat the more delicate continuation of nonreversible os-cillations and relative periodic orbits. We solve this problem in James et al. (2007)for a general class of triatomic systems with two identical heavy atoms and a lightone. Under some nondegeneracy conditions, it is shown that a given family of rel-ative periodic orbits existing for γ = +∞ (parameterized by phase, rotational de-gree of freedom and period) persists for large values of γ and nearby angular ve-locities (this result is valid for small angular velocities). Our proof is based on amethod initially introduced by Sepulchre and MacKay (1997) and further developedby Muñoz-Almaraz et al. (2003) for the continuation of “normally degenerate” peri-odic orbits in Hamiltonian systems. Our result provides in particular a global branchof relative periodic orbits for the triatomic system with large mass ratio. Note thatthe mathematical theory of relative equilibria and relative periodic orbits is verywell developed (see e.g. Roberts and Sousa Dias 1997; Montaldi and Roberts 1999;Ortega 2003 and the references therein), but these results do not cover the infinitemass ratio limit of reference (James et al. 2007). It is also worthwhile stressing thatsome local techniques used in James et al. (2007) could be applied to an arbitrarynumber of particles.

Another extension of the present work would be to consider the three-dimensionalcase. The scaling and the analytical tools employed here would remain mainly un-changed, but suitable internal coordinates should be introduced to keep the resultingequations of motion in a convenient form. There exists a vast literature on this topic,

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in particular the basic works of Eckart (1934, 1935) (see also Louck and Galbraith1976), Littlejohn et al. 1995, 1997, 1998, Chapuisat and Nauts 1991 and Kupper-mann 1996 (see Yanao et al. 2007 for additional references and a concise review onthis topic). These aspects provide directions for future work.

2 Equations of Motion and Reduction Via Symmetries

In this section we set up the equations of motion corresponding to the Hamil-tonian system (1) in a suitable rotating coordinate system, the total angular momen-tum being fixed and treated as a parameter. We note Q = (Q1, . . . ,QN)t ∈ R

2N ,q = (q1, . . . , qn)

t ∈ R2n and note the components of P ∈ R

2N,p ∈ R2n in a similar

way. The equations of motion read

dQ

dt= ∇P H,

dP

dt= −∇QH,

dq

dt= ∇pH,

dp

dt= −∇qH,

or equivalently

γMd2Q

dt2= F, m

d2q

dt2= f, (2)

where F = −∇QV , f = −∇qV , and M,m denote the (diagonal) mass matrices M =diag(M1,M1, . . . ,MN,MN), m = diag(m1,m1, . . . ,mn,mn).

Due to the Euclidean invariance of (2), the angular momentum

J =N∑

i=1

γMiQi ∧ dQi

dt+

n∑

i=1

miqi ∧ dqi

dt

and the linear momentum∑N

i=1 γMidQi

dt+∑n

i=1 midqi

dtare conserved quantities (we

note Q1 ∧ Q2 = det(Q1,Q2)).We shall consider time-periodic solutions of (2) modulo isometries, which leads

us to eliminating translational and rotational degrees of freedom. On the one hand,we fix the center of mass to 0

N∑

i=1

γMiQi +n∑

i=1

miqi = 0, (3)

which is equivalent to working in a reference frame in translation at constant velocity.On the other hand, we work in a rotating reference frame fixed to a specific degree offreedom, where the horizontal axis is directed towards the mass M1. More precisely,we set Q1 = �(cos θ, sin θ)t and use the new coordinates Xi (i = 1, . . . ,N ), xi (i =1, . . . , n) defined by

Qi = RθXi, qi = Rθxi,

Rθ =(

cos θ − sin θ

sin θ cos θ

).

(4)

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One has in particular X1 = (�,0)t . We shall note in the following that X⊥i = Rπ/2Xi

and X = (X1, . . . ,XN)t , x = (x1, . . . , xn)t . The equations of motion become

γM

(d2X

dt2+ d2θ

dt2X⊥

)= F + Fcen + Fcor, (5)

m

(d2x

dt2+ d2θ

dt2x⊥

)= f + fcen + fcor, (6)

where Fcen = ( dθdt

)2γMX, fcen = ( dθdt

)2mx correspond to the centrifugal force and

Fcor = −2 dθdt

γM dXdt

⊥, fcor = −2 dθ

dtmdx

dt

⊥denote the Coriolis force. In addition, the

angular momentum can be split into

J = Jvib + Idθ

dt, (7)

where Jvib = ∑Ni=1 γMiXi ∧ dXi

dt+ ∑n

i=1 mixi ∧ dxi

dtand I is the moment of inertia

I =N∑

i=1

γMiX2i +

n∑

i=1

mix2i . (8)

Moreover, condition (3) (which fixes the center of mass) becomes

N∑

i=1

γMiXi +n∑

i=1

mixi = 0. (9)

In the following we derive a differential system for the reduced variable (�,Z,x),where �∈R is the component of the horizontal vector X1 and Z=(X2, . . . ,XN−1)

t ∈R

2N−4. This system depends on the additional parameter J . Note that the variableZ is absent from the subsequent equations in the particular case N = 2. Equation (9)defines XN as a function

XN(�,Z,x, γ ) = − 1

MN

N−1∑

i=1

MiXi − 1

γMN

n∑

i=1

mixi, (10)

and (7) defines dθdt

as a function

dθ

dt= Ω

(�,Z,x,

d�

dt,dZ

dt,dx

dt, γ, J/γ

). (11)

System (5)–(6) becomes

γM1d2�

dt2= G, (12)

γ M

(d2Z

dt2+ dΩ

dtZ⊥

)= F , (13)

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m

(d2x

dt2+ dΩ

dtx⊥

)= f , (14)

where

Fi = Fi

((�,0)t ,Z,XN,x

) + Ω2γMiXi − 2ΩγMi

dXi

dt

⊥, i = 2, . . . ,N − 1,

fi = fi

((�,0)t ,Z,XN,x

) + Ω2mixi − 2Ωmi

dxi

dt

⊥, i = 1, . . . , n,

G = F(1)1

((�,0)t ,Z,XN,x

) + Ω2γM1�,

F(1)1 denotes the first component of F1 and M the reduced mass matrix M =

diag(M2,M2, . . . ,MN−1,MN−1). Consequently one obtains a (2N + 2n − 3)-dimensional system of second-order differential equations, with the additional para-meter J and the mass parameter γ . The system has the invariance (t, J ) → (−t,−J ).

One should note that a time-periodic solution of (12)–(14) corresponds to a so-lution of (2) being time-periodic in a frame rotating at some constant velocity. Suchsolutions are termed relative periodic orbits. The angular velocity is equal to the meanvalue of dθ

dtgiven by (11) (this follows from (4)). In the particular case J = 0 and if

Jvib does not vanish, one can see from (7) that the angular velocity does not vanishin general.

3 A Large Mass Ratio Asymptotics

In this section we consider time-periodic solutions (with fixed period T ) of system(12)–(14) in the limit γ → +∞. We obtain a suitable rescaling of (12)–(14) such thatthe problem is well posed for γ = +∞, in the sense that certain nontrivial solutionsexist (Sect. 4) and can be smoothly continued by the implicit function theorem tolarge (but finite) values of γ (Sect. 5). A suitable rescaling was initially introducedin Livi et al. (1997) in a one-dimensional case (for diatomic Fermi–Pasta–Ulam lat-tices). The idea is to split heavy-mass motions into a O(1) static part and a smalloscillatory part (having zero mean) as γ → +∞. More precisely, we define the smallparameter ε = γ −1/2 and set Z = X + εY , � = � + εr , with

∫ T

0Y(t) dt = 0,

∫ T

0r(t) dt = 0. (15)

We shall note X = (X2, . . . , XN−1)t ∈ R

2N−4 and Y = (Y2, . . . , YN−1)t ∈ R

2N−4,with, by definition, Xi = Xi + εYi for i = 2, . . . ,N − 1. Using (10) one finds in addi-tion XN = XN + εYN , where the functions XN(�, X) and YN(r,Y, x, ε) are definedby

MNXN = −M1(�,0)t −N−1∑

i=2

MiXi (16)

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and MNYN = −M1(r,0)t − ∑N−1i=2 MiYi − ε

∑ni=1 mixi. In addition we rescale the

angular velocity Ω defined by (7)–(11) by setting ω = γ 1/2Ω . One has

ω = I−1

(εJ −

N∑

i=2

MiXi ∧ dYi

dt− ε

N∑

i=2

MiYi ∧ dYi

dt− ε

n∑

i=1

mixi ∧ dxi

dt

), (17)

where the rescaled moment of inertia I = I /γ takes the form

I = I + 2ε

(M1�r +

N∑

i=2

MiXiYi

)+ ε2

(M1r

2 +N∑

i=2

MiY2i +

n∑

i=1

mix2i

),

(18)

I = M1�2 +

N∑

i=2

MiX2i .

System (12)–(14) now reads in rescaled form

M1d2r

dt2= εG(�, X, r, Y, x, ε, εJ ), (19)

M

(d2Y

dt2+ dω

dtX⊥ + ε

dω

dtY⊥

)= εF (�, X, r, Y, x, ε, εJ ), (20)

m

(d2x

dt2+ ε

dω

dtx⊥

)= f (�, X, r, Y, x, ε, εJ ), (21)

with εG = εF(1)1 ((�,0)t + ε(r,0)t , X + εY, XN + εYN,x) + εω2M1(� + εr)

(F (1)1 denotes the first component of F1), the ith component of εF being (i =

2, . . . ,N − 1)

εFi = εFi

((�,0)t + ε(r,0)t , X + εY, XN + εYN,x

)

+ εω2Mi(Xi + εYi) − 2εωMi

dYi

dt

⊥,

and the ith component of f having the form (i = 1, . . . , n)

fi = fi

((�,0)t + ε(r,0)t , X + εY, XN + εYN,x

) + ε2ω2mixi − 2εωmi

dxi

dt

⊥.

We observe that the force field at the right side of (19)–(20) is O(ε). Here this doesnot mean that r, Y are slow variables since we look for time-periodic solutions withfixed period T , but we shall later obtain (r, Y ) = O(ε) via the implicit function theo-rem.

Note that time-periodic solutions of (15)–(21) satisfy simple compatibility rela-tions obtained by integrating (19), (20), (21) over one period T . More precisely, oneobtains for (19) and (20) (the integrals of the two first terms at the left side of (20)

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cancel and one integrates the third term by parts)

G(�, X, r, Y, x, ε, εJ ) = 0, (22)

F (�, X, r, Y, x, ε, εJ ) = 0, (23)

where the scalar function G and F = (F2, . . . , FN−1)t are given by

T G =∫ T

0F

(1)1

((�,0)t + ε(r,0)t , X + εY, XN + εYN,x

)dt + M1�

∫ T

0ω2 dt

+ εM1

∫ T

0ω2r dt,

T Fi =∫ T

0Fi

((�,0)t + ε(r,0)t , X + εY, XN + εYN,x

)dt + MiXi

∫ T

0ω2 dt

+ εMi

∫ T

0ω2Yi dt − Mi

∫ T

0ω

dYi

dt

⊥dt.

These equations will be used for obtaining the static part (�, X) of heavy-mass os-cillations. As one searches for time-periodic solutions for ε → 0 (Sect. 4) it is neces-sary to consider the extended system (15)–(23), because (22)–(23) cannot be deducedfrom (19)–(20) for ε = 0.

At this stage one can distinguish two different weak coupling limits. On the onehand, the case of bounded angular momenta is obtained by fixing the value of J

and letting ε → 0. On the other hand, the case of large angular momenta followsby setting J = j/ε in (15)–(23), keeping j fixed and letting ε → 0. In that case j

denotes a continuous parameter which has the meaning of a renormalized angularmomentum. Note that centrifugal effects will persist for ε = 0 in the second weakcoupling limit.

4 Limit Case of Infinite Mass Ratio

In this section we consider the limit case ε = 0 in (15)–(23) and obtain the existenceof different types of periodic solutions: nonlinear normal modes close to an equilib-rium (Sect. 4.2) and breather solutions (Sect. 4.3). For this purpose, the equations ofmotion are prepared in a convenient form in Sect. 4.1.

4.1 Heavy Mass Motion

In this section we show that, considering time-periodic oscillations for ε = 0, heavymass motion is purely static if a certain geometric nondegeneracy condition is satis-fied.

We begin by recalling the notations X = (X2, . . . , XN−1)t ∈ R

2N−4 and Y =(Y2, . . . , YN−1)

t ∈ R2N−4. System (19)–(21) yields for ε = 0

d2r

dt2= 0, (24)

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d2Y

dt2+ dω

dtX⊥ = 0, (25)

md2x

dt2= f ∗(�, X, x), (26)

where the ith component of f ∗ has the form f ∗i (�, X, x) = fi((�,0)t , X, XN , x) for

i = 1, . . . , n and XN is defined by (16). In addition the frequency ω in (25) now takesthe form ω = [M1�

2 + ∑Ni=2 MiX

2i ]−1(j − ∑N

i=2 MiXi ∧ dYi

dt), where the function

YN(r,Y ) is defined by

MNYN = −M1(r,0)t −N−1∑

i=2

MiYi, (27)

and we set j = 0 in the limit with bounded angular momenta and j = 0 in the limitwith large angular momenta.

Now we turn to conditions (22) and (23). Setting ε = 0 yields

∫ T

0Fi

((�,0)t , X, XN , x

)dt + MiXi

∫ T

0ω2 dt − Mi

∫ T

0ω

dYi

dt

⊥dt = 0 (28)

for i = 2, . . . ,N − 1 and

∫ T

0F

(1)1

((�,0)t , X, XN , x

)dt + M1�

∫ T

0ω2 dt = 0. (29)

Let us solve (24) and (25). Since r, Y are T -periodic and have 0 time-average, weobtain r = 0 and

Y − X⊥[M1�

2 +N∑

i=2

MiX2i

]−1 N∑

i=2

MiXi ∧ Yi = 0 (30)

after integrating twice. For solving (30) we note that all solutions are by definitioncolinear to X⊥. Setting Y = λX⊥ in (27) and using (16) yields MNYN = MNλX⊥

N +M1λ(0, �)t since r = 0. It follows

N∑

i=2

MiXi ∧ Yi = λ

(N∑

i=2

MiX2i + M1XN · (�,0)t

)(31)

(the symbol · denotes the canonical scalar product). By substitution of (31) into (30)and after some elementary algebra, one obtains Y = 0 is the unique solution of (30)if the following nondegeneracy condition is satisfied

(XN − (�,0)t

) · (�,0)t = 0. (32)

This condition can be rewritten

� = 0, XN · (1,0)t = �. (33)

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In that case, one has YN = 0 and ω = [M1�2 +∑N

i=2 MiX2i ]−1j . Substitution in (28)

and (29) yields the simplified system (36)–(37) given below. In conclusion, the abovecomputations yield the following result.

Lemma 1 System (15)–(23) with ε = 0, supplemented by the nondegeneracy condi-tion

� = 0, XN · (1,0)t = �, (34)

yields necessarily r = 0, Y = 0 and can be rewritten

md2x

dt2= f ∗(�, X, x), (35)

1

T

∫ T

0Fi

((�,0)t , X, XN , x

)dt + MiXi I

−2j2 = 0, i = 2, . . . ,N − 1, (36)

1

T

∫ T

0F

(1)1

((�,0)t , X, XN , x

)dt + M1�I−2j2 = 0, (37)

where x : R → R2n is T -periodic, X = (X2, . . . ,XN−1)

t ∈ R2N−4, � ∈ R, I =

M1�2 + ∑N

i=2 MiX2i and

XN(�, X) = − M1

MN

(�,0)t −N−1∑

i=2

Mi

MN

Xi.

One has f ∗i (�, X, x) = −∇qi

V ((�,0)t , X, XN , x) for i = 1, . . . , n. One fixes j = 0in the limit with bounded angular momenta and j = 0 in the limit with large angularmomenta.

4.2 Nonlinear Normal Modes for Infinite Mass Ratio

In this section we solve the system (34)–(37) locally, in the neighborhood of a sta-tic solution corresponding to an equilibrium of (2). We first solve (35) locally withrespect to x, considering (�, X) as parameters. Families of small amplitude time-periodic solutions x(�, X) are obtained by the Lyapunov center theorem (see, e.g.,Buzzi and Lamb 2005b; Kielhöfer 2004 for a recent account). As a second step, sta-tionary mass displacements (�, X) are obtained from (36) and (37) using the implicitfunction theorem. The solutions obtained in this way correspond to relative nonlinearnormal modes of the system, in the limit of infinite mass ratio.

As a simple illustration, consider a Hamiltonian model for the (classical) dynamicsof a planar molecule A2 − B − C, where the masses of atoms A,B,C are denotedmA,mB,mC , respectively, with mB/mA � 1, mC/mA � 1, and mB,mC having thesame order of magnitude. In order to stick to our previous notations, we define m1 =mA, γ0 = mB/mA, M1 = mA and M2 = mAmC/mB. We denote by Q1,Q2 andP1,P2, respectively, the positions and momenta of the two heavy atoms B,C in afixed reference frame, and use the similar notation q1, q2,p1,p2 for the two light

J Nonlinear Sci

Fig. 2 Equilibriumconfiguration (left) andsymmetric bending mode (right)of a trigonal planar molecule.This normal mode is representedin the limit of infinite mass ratiobetween light and heavy atoms.Dotted lines denote chemicalbonds

atoms A. The Hamiltonian reads

H =2∑

i=1

(1

2γMi

P 2i + 1

2m1p2

i

)+ V (Q1,Q2, q1, q2), (38)

where γ = γ0, the interaction potential V is Euclidean-invariant and admits someequilibrium configuration depicted in Fig. 2 (left).

Since γ0 � 1, the limiting problem with γ = +∞ described by system (34)–(37) is expected to approximate certain solutions of the original model with γ = γ0.For example, Fig. 2 (right) represents a typical nonlinear normal mode solution forγ = +∞, provided by the results of the present section (see Theorem 1). As wewill show in Sect. 5, under some nondegeneracy conditions this solution persists forvalues of γ in the Hamiltonian (38) taken in some unbounded interval γ > γc. Ifγ0 > γc, following the normal mode family with respect to γ down to γ0 will providea nonlinear normal mode solution of the original system. From a numerical point ofview, this allows us to compute nonlinear normal modes by path-following, startingfrom the simpler (lower-dimensional) system (34)–(37).

Now we turn to the general case of Hamiltonian (2) and detail the theory for solv-ing the limiting problem (34)–(37). We shall work in the neighborhood of an equi-librium (Q, q) of (2). More precisely, there exists a suitable translation and rotationof (Q, q) having the form (Q∗, q∗) = ((�∗,0)t ,X∗,X∗

N,x∗), such that the centerof mass corresponding to heavy masses is fixed to 0, i.e., X∗

N = XN(�∗,X∗). Thisyields a particular (stationary) solution Γ ∗ = (�∗,X∗, x∗) of system (35)–(37) forj = 0. For j ≈ 0 and under certain nonresonance conditions, we shall prove the exis-tence of a stationary solution and time-periodic solutions close to Γ ∗.

In what follows we assume the following.

Hypothesis 1 The equilibrium (Q∗, q∗) satisfies the geometric nondegeneracy con-dition

�∗ = 0, X∗N · (1,0)t = �∗. (39)

J Nonlinear Sci

Condition (39) corresponds to the nondegeneracy condition (33), and can be alwaysfulfilled (except in the degenerate case Q∗ = 0) by relabeling heavy masses if neces-sary.

Since translations and rotations of (Q∗, q∗) generate a continuous family ofequilibria, the second derivative D2V (Q∗, q∗) is at least triply degenerate. Indeed,let us consider two vectors E1,E2 ∈ R

2(n+N) defined by Ei = (ei, . . . , ei)t , with

e1 = (1,0)t , e2 = (0,1)t . The relation V ((Q,q) + tEi) = V (Q,q) holds for anyt ∈ R and (Q,q) ∈ R

2(n+N), and implies (differentiate with respect to t at t = 0)DV (Q,q)Ei = 0, and thus

D2V (Q,q)Ei = 0 (40)

for all (Q,q) ∈ R2(n+N) (D2V denotes the Hessian matrix). Now we consider the

degeneracy due to rotations and define Rθq = (Rθq1, . . . ,Rθqn)t (RθQ is defined

in a similar way). The relation V (RθQ,Rθq) = V (Q,q) holds for any θ ∈ R and(Q,q) ∈ R

2(n+N), and yields DV (Q,q)(Q⊥, q⊥)t = 0 for all (Q,q) ∈ R2(n+N) (dif-

ferentiate with respect to θ at θ = 0). Now differentiating this relation at (Q,q) =(Q∗, q∗) and using the fact that (Q∗, q∗) is a critical point of V , one obtains

D2V (Q∗, q∗)(Q∗⊥, q∗⊥

)t = 0. (41)

Equations (40) and (41) show that D2V (Q∗, q∗) is at least triply degenerate. In whatfollows one makes the following assumption.

Hypothesis 2 The kernel of D2V (Q∗, q∗) is three-dimensional.

Now we solve (35) locally with respect to x, considering (�, X) ≈ (�∗,X∗) asparameters. In the sequel we shall note U = (�, X)t , U∗ = (�∗,X∗)t . One makes thefollowing assumption on the potential (depending on static heavy masses) in whichlight masses oscillate.

Hypothesis 3 The partial second derivative D2qV (Q∗, q∗) is invertible and possesses

p positive real eigenvalues (p ≥ 1).

Note that D2qV (Q∗, q∗) is in general invertible because fixing U in (35) breaks the

Euclidean invariance.For U = U∗, (35) linearized at the equilibrium x = x∗ reads

d2x

dt2= −L∗x, (42)

where L∗ = m−1D2qV (Q∗, q∗). Since L∗ has the same number of positive and neg-

ative eigenvalues as D2qV (Q∗, q∗) (m is positive definite), L∗ has in particular p

positive eigenvalues ω∗1

2, . . . ,ω∗p

2 (counted with their multiplicity) with associatedeigenvectors ζ ∗

1 , . . . , ζ ∗p .

Equation (42) admits solutions in the form of normal modes x(t) = ae±iω∗k t ζ ∗

k . Aswe shall see in the following, these normal modes have nonlinear analogues for sys-tem (35) under a nonresonance condition. More precisely, we consider one particular

J Nonlinear Sci

mode frequency ω∗1 and make the following nonresonance assumption, which will

allow us to apply the Lyapunov center theorem.

Hypothesis 4 The mode frequency ω∗1 satisfies the nonresonance condition

kω∗1 = ω∗

2, . . . ,ω∗p, for all k ∈ Z.

In particular, we have assumed that the eigenvalue ω∗1

2 of L∗ is simple.Now we rewrite (35) in a suitable form for applying the Lyapunov center theorem.

Since D2qV (Q∗, q∗) is invertible, for U ≈ U∗ the equilibrium x∗ of (35) can be con-

tinued into a (unique) family of equilibria x(�, X) by the implicit function theorem,with x(�∗,X∗) = x∗. One has in addition for all U = (�, X)t

Dx(U∗)U = −[D2

qV (Q∗, q∗)]−1∇qDQV (Q∗, q∗)

((�,0)t , X, XN (U)

)(43)

(one uses the fact that XN is a linear function). Moreover, let us define Q(U) =((�,0)t , X, XN (U))t for U ≈ U∗. The linearized map L = m−1D2

qV (Q(U), x(U ))

has a simple eigenvalue ω12 close to ω∗

12, with an associated eigenvector ζ1 close

to ζ ∗1 , where ω1, ζ1 smoothly depend on (�, X) ≈ (�∗,X∗). Setting x = x + y, (35)

becomes

md2y

dt2= g∗(�, X, y), (44)

where g∗ takes the form g∗(�, X, y) = −∇yV (�, X, y) with V (�, X, y) = V ((�,0)t ,

X, XN , x + y). By construction we have g∗(�, X,0) = 0, i.e. y = 0 is an equilibriumof (44) for all values of U ≈ U∗.

Now applying the Lyapunov center theorem to (44), y = 0 is contained in a two-dimensional invariant manifold consisting of a two-parameter family of periodic so-lutions whose frequency tends to ω1 as they approach the equilibrium (and there areno other periodic solutions with frequency close to ω1 near y = 0). These solutionssmoothly depend on (�, X) for (�, X) ≈ (�∗,X∗). Returning to the original variablex, the corresponding solutions of (35) take the form

x(t;α, �, X) = x(�, X) + α cos (Ωt + φ)ζ1 + O(α2), (45)

where α ≈ 0 is a real parameter measuring the amplitude, Ω(α, �, X) denotes thefrequency of x (Ω is even in α) and Ω = ω1 + hα2 + o(α2). Note that x is an evenfunction of t if we fix φ = 0 (this originates from the invariance t → −t of (35)).

The hardness coefficient h(�, X) = 12

∂2Ω

∂α2 (0, �, X) depends on m and the derivatives

Dpy V (�, X,0) up to order 4. Using a perturbative expansion of x(.;α, �, X) up to

order 3 in α, one obtains after lengthy but classical computations (see e.g. Kielhöfer2004, Theorem I.10.3, p. 51)

h = 1

16ω1

(m−1η1

) · χ1, (46)

where η1 ∈ Ker (Lt − ω21I ), η1 · ζ1 = 1,

J Nonlinear Sci

χ1 = g∗2

(ζ1,

[2g∗

1−1 + (

g∗1 + 4mω2

1

)−1]g∗

2(ζ1, ζ1)) − g∗

3(ζ1, ζ1, ζ1)

and g∗i = Di

yg∗(�, X,0). Note that (46) is the multidimensional extension of the clas-

sical formula

Ω =√

V ′′(0)

m+ hα2 + o

(α2), h = (V ′′(0))−3/2

16√

m

[V ′′(0)V (4)(0) − 5

3

(V (3)(0)

)2]

for the frequency Ω(α) of a one-dimensional oscillator in a potential V .The next step is to solve (36)–(37) with respect to (�, X), where x is the function

of (�, X) defined by (45). Setting T = 2π/Ω and x(t) = x(Ωt) in (36)–(37) yieldsthe equation

G(j,α, �, X) = 0, (47)

where G : R2 × R

2N−3 → R2N−3 is given by G = (G

(1)1 ,G2, . . . ,GN−1), G

(1)1 ∈ R,

Gi ∈ R2,

Gi = 1

2π

∫ 2π

0Fi

((�,0)t , X, XN , x(t;α, �, X)

)dt + MiXi I

−2j2, (48)

G(1)1 = 1

2π

∫ 2π

0F

(1)1

((�,0)t , X, XN , x(t;α, �, X)

)dt + M1�I−2j2 (49)

and I is given by (18) (note that G does not depend on the phase φ of x).For α = 0 and j = 0, (47) has the particular solution (�, X)t = U∗ = (�∗,X∗)t

corresponding to the stationary solution Γ ∗ = (�∗,X∗, x∗) of system (35)–(37). Tosolve (47) in the vicinity of U∗ we shall use the implicit function theorem. For thatpurpose one has to show that DUG(0,0,U∗) is invertible (the derivative is taken withrespect to U = (�, X)t ). This follows from Hypotheses 1 and 2 and the followingresult.

Lemma 2 Let U = (�, X)t such that

DUG(0,0,U∗)U = 0 (50)

and Q = ((�,0)t , X, XN(U))t , q = Dx(U∗)U . Then if Hypothesis 1 is satisfied onehas D2V (Q∗, q∗)(Q, q)t = 0.

The proof of Lemma 2 follows from lengthy but straightforward algebra. One hasto use (50) with the relations (41) and

N∑

i=1

∇QiDV (Q∗, q∗) +

n∑

i=1

∇qiDV (Q∗, q∗) = 0 (51)

(reformulation of (40)) which originate from the invariance of (2) under rotations andtranslations. The second condition in (39) is used to solve a linear system appearingin the computation of D2V (Q∗, q∗)(Q, q)t .

J Nonlinear Sci

Now the invertibility of DUG(0,0,U∗) follows immediately. Due to the form ofthe vector Q defined in Lemma 2 (fixed direction of its first component Q1 andcenter of mass set to 0), (Q, q)t does not belong to the (three-dimensional) ker-nel of D2V (Q∗, q∗) unless (Q, q)t = 0. Consequently (50) implies U = 0 and thusDUG(0,0,U∗) is invertible.

For α ≈ 0 and j ≈ 0, (47) has consequently a unique solution (�, X)t = U (α, j)

close to U∗, with U (α, j) = U∗ +O(α2 + j2) (U is even in α, since changing α into−α in (45) is equivalent to shifting φ by π ). Moreover, condition (34) is satisfied bycontinuity for (α, j) ≈ 0 due to condition (39).

For j ≈ 0 the above analysis yields different solutions of system (34)–(37), closeto the stationary solution Γ ∗ = (�∗,X∗, x∗) existing for j = 0. First, system (34)–(37) possesses a (locally unique) stationary solution Γ (j) close to Γ ∗ and definedby Γ (j) = (U(0, j), x[U (0, j)]). One has in addition Γ (j) = Γ ∗ + O(j2). Second,there exists a two-parameter family of time-periodic solutions Γ (α, j) close to Γ (j)

(the two parameters are the amplitude α and phase shift). These solutions are definedby Γ (t;α, j) = (U (α, j), x[t;α, U(α, j)]). For (α, j) ≈ 0, they can be expanded as

Γ (t;α, j) = Γ ∗ + (0, α cos (Ωt + φ)ζ ∗

1

) + O(α2 + j2), (52)

with Ω = ω∗1 + O(α2 + j2). Generically these solutions can be also parameterized

by their period (instead of their amplitude α) close to 2π/ω∗1 (and locally above or

below).As a conclusion we have proved the following result.

Theorem 1 Assume Hypotheses 1–4 are satisfied. For each j ≈ 0, system (34)–(37)possesses the following solutions, close to the stationary solution Γ ∗ = (�∗,X∗, x∗)existing for j = 0:

(i) A locally unique stationary solution Γ (j) = Γ ∗ + O(j2), corresponding to arelative equilibrium of (1) in the limit of infinite mass ratio;

(ii) A two-parameter family of time-periodic solutions Γ (α, j) close to Γ (j) (the twoparameters are the amplitude α and phase shift) with frequency Ω[α, U(α, j)] =ω∗

1 +O(α2 + j2). These solutions have the form (52) and are even in t for φ = 0.They correspond to relative nonlinear normal modes of (1) associated with Γ (j)

in the limit of infinite mass ratio.

It should be stressed that our analysis could be extended to other contexts wherethe nonresonance condition (Hypothesis 4) is not satisfied, a common situation be-ing the existence of a double semisimple eigenvalue ω∗

1 in a symmetric Hamiltoniansystem. In that case, the structure of the set of periodic solutions is more compli-cated (Montaldi et al. 1988, 1990; Buzzi and Lamb 2005a) and a theorem of the sametype as theorem 1 could be derived, providing multiple branches of relative equilibria(each one parameterized by j ) and relative nonlinear normal modes.

4.3 Breathers for Infinite Mass Ratio

In this section we consider a case when some degrees of freedom of x are uncoupledto the remaining ones at infinite mass ratio. More precisely, we split the light atoms

J Nonlinear Sci

positions x into two component groups x = (ξ0, ξ1)t , with ξi ∈ R

2ni and n0 +n1 = n.The interaction potential V is assumed of the form

V (X,x) = V0(X, ξ0) + V1(X, ξ1). (53)

We use the term breather for a time-periodic oscillation involving principally a fewdegrees of freedom (say ξ0) while the others (say ξ1) oscillate with a much smalleramplitude. We shall obtain such solutions at infinite mass ratio by adapting the localanalysis of the previous section (in the infinite mass ratio limit we simply assumeX and ξ1 at equilibrium, while only ξ0 oscillates). In Sect. 5.3 we shall treat thepersistence of such solutions at finite mass ratio and for a weak coupling betweenξ0, ξ1.

As a simple illustration, consider a Hamiltonian model for the (classical) dynam-ics of a planar molecule CAB − BAC, where the masses of atoms A,B,C are de-noted mA,mB,mC , respectively, with mB/mA � 1, mC/mA � 1, and mB,mC hav-ing the same order of magnitude. As in the previous section we define m1 = mA,γ0 = mB/mA, M1 = M3 = mA and M2 = M4 = mAmC/mB. We denote by Q =(Q1, . . . ,Q4)

t and P = (P1, . . . P4)t , respectively, the positions and momenta of the

four heavy atoms B,C,B,C in a fixed reference frame, and use the similar notationq1, q2,p1,p2 for the two light atoms A. The Hamiltonian reads

H =4∑

i=1

1

2γMi

P 2i +

2∑

i=1

1

2m1p2

i + V0(Q,q1) + V1(Q,q2), (54)

where γ = γ0 and the interaction potentials Vi are Euclidean-invariant. Consequently,the motions of the two atoms A are not directly coupled in this model, and interactonly indirectly through the heavy atoms. The system admits some equilibrium con-figuration depicted in Fig. 3 (left). In the notation described above we have simplyξ0 = q1 and ξ1 = q2, but for more complex molecules ξ0, ξ1 will involve groups ofatoms.

Since γ0 � 1, the limiting problem with γ = +∞ described by system (34)–(37)is expected to approximate certain solutions of the original model with γ = γ0. Foran infinite mass ratio, heavy atoms are static and the motions of the two atoms A

become uncoupled. This allows for the existence of localized oscillations as shownin Fig. 3 (right). Here one light atom A oscillates in the potential generated by four

Fig. 3 Equilibriumconfiguration (left) and localizedstretching mode (right) of aplanar molecule. The localizedmode is represented in the limitof infinite mass ratio betweenlight and heavy atoms. Dottedlines denote chemical bonds

J Nonlinear Sci

heavy atoms, and in turn produces a change in their static positions. The secondatom A remains at an equilibrium configuration determined by static atoms. In thissection we analytically obtain such solutions in a general setting (see Theorem 2).As we will show in Sect. 5, under some nondegeneracy conditions such localizedoscillations persist for values of γ in the Hamiltonian (38) taken in some unboundedinterval γ > γc. In models where γ0 > γc, following the breather family with respectto γ down to γ0 will provide a breather solution of the original system. Persistenceresults are also available when a sufficiently small interaction energy between the twoatoms A is added in the Hamiltonian.

Now we return to the general case of Hamiltonian (2) and describe how to ob-tain breather solutions to the limiting problem (34)–(37) with the interaction poten-tial (53). We use the same notations as in Sect. 4.2 and work in the neighborhoodof the equilibrium (Q∗, q∗) of (2) considered earlier. As in Sect. 4.2 we make Hy-potheses 1 and 2. We shall note q∗ = (ξ∗

0 , ξ∗1 )t light mass coordinates at the trivial

equilibrium. As previously we also note U = (�, X)t , U∗ = (�∗,X∗)t .Now we solve (35) locally with respect to x, considering (�, X) ≈ (�∗,X∗) as

parameters. Equation (35) can be split into

m0d2ξ0

dt2= f ∗

0 (�, X, ξ0), (55)

m1d2ξ1

dt2= f ∗

1 (�, X, ξ1), (56)

where f ∗i (�, X, ξi) = −∇ξi

Vi((�,0)t , X, XN , ξi) and m0, m1 are the diagonal massmatrices corresponding to each component ξ0, ξ1.

We replace Hypothesis 3 by the following one.

Hypothesis 5 The partial second derivatives D2ξ0

V0(Q∗, ξ∗

0 ) and D2ξ1

V1(Q∗, ξ∗

1 ) are

invertible. Moreover, D2ξ0

V0(Q∗, ξ∗

0 ) possesses p positive real eigenvalues (p ≥ 1).

Since D2ξ1

V1(Q∗, ξ∗

1 ) is invertible, for U ≈ U∗ the equilibrium ξ∗1 of (56) can

be continued into a (unique) family of equilibria ξ1(�, X) by the implicit functiontheorem, with ξ1(�

∗,X∗) = ξ∗1 . One has in addition for all U = (�, X)t

Dξ1(U∗)U = −[

D2ξ1

V1(Q∗, ξ∗

1 )]−1∇ξ1DQV1(Q

∗, ξ∗1 )

((�,0)t , X, XN(U)

). (57)

In the following we fix ξ1 = ξ1(�, X) in order to construct a breather solution.Now we turn to (55). For U = U∗, (55) linearized at the equilibrium ξ0 = ξ∗

0 reads

d2ξ0

dt2= −L∗ξ0, (58)

where L∗ = m−10 D2

ξ0V0(Q

∗, ξ∗0 ). Since L∗ has the same number of positive and

negative eigenvalues as D2ξ0

V0(Q∗, ξ∗

0 ), L∗ has in particular p positive eigenval-

ues ω∗1

2, . . . ,ω∗p

2 (counted with their multiplicity) with associated eigenvectors

J Nonlinear Sci

ζ ∗1 , . . . , ζ ∗

p . We consider one particular mode frequency ω∗1 and make the non-

resonance assumption 4, which allows us to apply the Lyapunov center theo-rem.

Since D2ξ0

V0(Q∗, ξ∗

0 ) is invertible, for U ≈ U∗ the equilibrium ξ∗0 of (55) can

be continued into a unique family of equilibria ξ0(�, X) by the implicit functiontheorem, with ξ0(�

∗,X∗) = ξ∗0 . For all U = (�, X)t we have

Dξ0(U∗)U = −[

D2ξ0

V0(Q∗, ξ∗

0 )]−1∇ξ0DQV0(Q

∗, ξ∗0 )

((�,0)t , X, XN(U)

). (59)

Moreover, considering Q(U) = ((�,0)t , X, XN (U))t for U ≈ U∗, the linearized mapL = m−1

0 D2ξ0

V0(Q(U ), ξ0(U)) has a simple eigenvalue ω21 close to ω∗2

1 , with an

associated eigenvector ζ1 close to ζ ∗1 , where ω1, ζ1 smoothly depend on (�, X) ≈

(�∗,X∗).By the Lyapunov center theorem, for U ≈ U∗ the equilibrium ξ0 is contained in

a two-dimensional invariant manifold consisting of a two-parameter family of peri-odic solutions whose frequency tends to ω1 as they approach the equilibrium. Thesesolutions smoothly depend on (�, X) for (�, X) ≈ (�∗,X∗) and take the form

ξ0(t;α, �, X) = ξ0(�, X) + α cos (Ωbt + φ)ζ1 + O(α2), (60)

where α ≈ 0 is a real parameter measuring the amplitude, Ωb(α, �, X) denotes thefrequency of ξ0 and Ωb = ω1 + hbα

2 + o(α2). Note that ξ0 is an even function oft if we fix φ = 0. As in the computation of Sect. 4.2, the coefficient hb(�, X) =12

∂2Ωb

∂α2 (0, �, X) depends on m0 and the derivatives of V0 up to order 4. More preciselywe have

hb = 1

16ω1

(m−1

0 η1) · χ1, (61)

where η1 ∈ Ker (Lt − ω21I ), η1 · ζ1 = 1,

χ1 = g∗2

(ζ1,

[2g∗

1−1 + (

g∗1 + 4m0ω

21

)−1]g∗

2(ζ1, ζ1)) − g∗

3(ζ1, ζ1, ζ1)

and g∗i = Di

ξ0f ∗

0 (�, X, ξ0(�, X)).

The next step is to solve (36)–(37) with respect to (�, X), where x(t) =(ξ0(t;α, �, X), ξ1(�, X))t . Setting T = 2π/Ωb and ξ0(t) = ξ0(Ωbt) in (36)–(37)yields the equation

G(j,α, �, X) = 0, (62)

where G : R2 × R

2N−3 → R2N−3 is given by G = (G

(1)1 , G2, . . . , GN−1), G

(1)1 ∈ R,

Gi ∈ R2,

Gi = 1

2π

∫ 2π

0Fi

((�,0)t , X, XN , ξ0(t;α, �, X), ξ1(�, X)

)dt

+ MiXi I−2j2, (63)

J Nonlinear Sci

G(1)1 = 1

2π

∫ 2π

0F

(1)1

((�,0)t , X, XN , ξ0(t;α, �, X), ξ1(�, X)

)dt

+ M1�I−2j2 (64)

and I is given by (18).For α = 0 and j = 0, (62) has the particular solution (�, X)t = U∗ = (�∗,X∗)t

corresponding to the stationary solution Γ ∗ = (�∗,X∗, x∗) of system (35)–(37). Tosolve (62) in the vicinity of U∗ we again use the implicit function theorem. For thatpurpose one has to check that DUG(0,0,U∗) is invertible. This result follows fromHypothesis 1, 2 as in Sect. 4.2 (the result of Sect. 4.2 directly applies since G and G

are equal for α = 0). For α ≈ 0 and j ≈ 0, (62) has consequently a unique solution(�, X)t = Ub(α, j) close to U∗, with Ub(α, j) = U∗ + O(α2 + j2). Moreover, wehave Ub(0, j) = U(0, j) (U (α, j) is the unique local solution of G(j,α, U) = 0, seeSect. 4.2).

For j ≈ 0, the above analysis yields a two-parameter family of time-periodic so-lutions of (34)–(37), close to the stationary solution Γ (j) = Γ ∗ + O(j2) determinedin Sect. 4.2. The two parameters are the oscillation amplitude α and phase shift.We shall denote these solutions by (�, X, x) = Γb(α, j), and we have Γb(t;α, j) =(Ub(α, j), ξ0[t;α, Ub(α, j)], ξ1(Ub(α, j))). For (α, j) ≈ 0, they can be expanded as

Γb(t;α, j) = Γ ∗ + (0, α cos (Ωbt + φ)ζ ∗

1 ,0) + O

(α2 + j2), (65)

with Ωb = ω∗1 + O(α2 + j2). Generically these solutions can be parameterized by

their period (instead of their amplitude α) close to 2π/ω∗1 (and locally above or be-

low). They correspond to small amplitude breathers since oscillations are restrictedto component ξ0 while the other components remain at equilibrium. As a conclusion,we have proved the following result.

Theorem 2 Assume Hypotheses 1, 2, 4 and 5 are satisfied. Then for each j ≈ 0,system (34)–(37) possesses a two-parameter family of time-periodic solutions(�, X, x) = Γb(α, j) (the two parameters are the amplitude α and phase shift), closeto the stationary solution Γ (j) = (�∗,X∗, x∗) + O(j2). These solutions have theform (65), are even in t if we fix φ = 0, and have a frequency Ωb[α, Ub(α, j)] =ω∗

1 + O(α2 + j2). They correspond to breather solutions (time-periodic in a framerotating at constant velocity) for the Hamiltonian system (1) in the limit of infinitemass ratio.

5 Persistence of Reversible Periodic Solutions at Finite Mass Ratio

The purpose of this section is to prove analytically the persistence for ε ≈ 0 of certainsolutions found in Sect. 4. The analysis is based on the application of the implicitfunction theorem as in MacKay and Aubry (1994), Livi et al. (1997), Aubry (1998).

In the following, the angular momentum J is fixed to 0 and we restrict our at-tention to T -periodic solutions of (19)–(23) even in time. Consequently the masspositions (X,x) in the rotating coordinate system are even in time. Since J = 0, the

J Nonlinear Sci

angular velocity θ defined by (7) is odd and T -periodic, hence θ is T -periodic andeven in time. It follows that mass positions (Q,q) in the fixed reference frame arealso T -periodic and even in time. As a conclusion, our analysis applies to T -periodicsolutions of (2) for which the velocity of all atoms vanishes at t = 0 (this impliesJ = 0 and the evenness of (Q,q)).

In what follows we also allow the potential V (Q,q,η) to depend on an additionalparameter η, which can be slightly varied around a reference value (for simplicity weassume V to be C4 on (R2)N+n × R, and assume η ≈ 0 without loss of generality).Indeed it is useful to obtain the regularity of solutions with respect to additional para-meters. This is especially useful for breather solutions, since many model-supportingdiscrete breathers involve a small coupling parameter (we shall consider this situationin Sect. 5.3).

Let us now define the periodic solutions of (19)–(23) as zeros of the map F :X × R

2 → Y, (U, ε, η) �→F(U, ε, η) with U = (�,X, r,Y, x) and

F(U, ε, η) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

G,

F,

M1d2r

dt2 − ε(G − G),

M(

d2Y

dt2 + dωdt

X⊥ + ε dω

dtY⊥) − ε(F − F),

m(

d2x

dt2 + ε dωdt

x⊥) − f ,

where we set J = 0 in the definition of ω (see (17)). The Hilbert spaces X and Y

are X = {U ∈ R2N−3 × (H 2

0 (T))2N−3 × (H 2(T))2n, U(t) = U(−t)}, Y = {U ∈R

2N−3 × (L20(T))2N−3 × (L2(T))2n, U(t) = U(−t)}, endowed with their usual

norms. Here T = R/T Z and H 2(T) is a classical Sobolev space of T -periodic func-tions. The subspaces L2

0(T) and H 20 (T) correspond to functions in L2(T) and H 2(T),

respectively, having zero mean. Now the T -periodic solutions of (19)–(23) (even in t ,with J = 0) are the zeros of F .

Let us suppose that for ε = 0 and η = 0, the system (19)–(23) possesses aT -periodic solution U0 = (�0, X0,0,0, x0), i.e. a zero of F(.,0,0). Such solutionswere obtained in Sect. 4 in the form of small amplitude nonlinear normal modes (The-orem 1) and breathers (Theorem 2) (see also James et al. 2007 for global existenceresults in the case of a triatomic system).

In Sect. 5.1 we give appropriate nondegeneracy conditions allowing to continuethe solution U0 to ε ≈ 0 and η ≈ 0 (we use the implicit function theorem). Sections5.2 and 5.3 focus, respectively, on nonlinear normal modes and breather solutions.Using the abstract persistence theorem obtained in Sect. 5.1, we prove persistence ofthe solutions of Theorems 1 and 2 for ε ≈ 0 and η ≈ 0.

5.1 A General Persistence Theorem

In order to apply the implicit function theorem, we show that DUF(U0,0,0)

is invertible. Let us choose g1 ∈ R, g2 ∈ R2N−4, h ∈ (L2(T))2n, p ∈ L2

0(T),

J Nonlinear Sci

q ∈ (L20(T))2N−4. Then we have to find a unique (δ�, δX, δr, δY, δx) ∈ X such that

DUF(U0,0,0)(δ�, δX, δr, δY, δx) = (g1, g2,p, q,h). (66)

Let us now solve (66). First δr satisfies

M1d2

dt2δr = p. (67)

Equation (67) has a unique T -periodic solution δr satisfying∫ T

0 δr dt = 0 since∫ T

0 p dt = 0 (this is immediate using Fourier series).Now the equation for δY is given by

Md2

dt2

(δY − 1

I0

(N∑

i=2

MiX0i ∧ δYi

)X⊥

0

)= q, (68)

with I0 = M1�20 + ∑N

i=2 MiX20i , MNδYN = −M1(δr,0)T − ∑N−1

i=2 MiδYi and

MNX0N = −M1(�0,0)t −N−1∑

i=2

MiX0i . (69)

Let us denote by Q the unique solution of d2Q

dt2 = q lying in (H 20 (T))2N−4 (recall q is

T -periodic with zero mean). System (68) is equivalent to an algebraic linear systemAδY = B obtained by integrating twice (68) with respect to time, with

AδY = M

(δY + 1

I0

(N−1∑

i=2

Mi(X0N − X0i ) ∧ δYi

)X⊥

0

),

B = Q − M

(M1

I0X0N ∧ (δr,0)t

)X⊥

0 .

(70)

Under the assumption that the solution U0 verifies the nondegeneracy condi-tion (34), A is invertible (see the resolution of the homogeneous problem in Sect. 4.1)and thus δY is unique.

In addition, (66) yields the following equation satisfied by δx:

L(δx) = ∂�f ∗(�0, X0, x0(t))δ� + DXf ∗(�0, X0, x0(t)

)δX + h, (71)

where L(δx) = m d2

dt2 δx − Dxf∗(�0, X0, x0(t))δx. Recall that f ∗ is the function de-

fined by f ∗i (�, X, x) = fi((�,0)T , X, XN , x) with fi = −∇qi

V (we omit the fixedparameter η = 0 in notations). Due to the time translation invariance of (19)–(23), x0

lies in the kernel of the extension of L from (H 2(T))2n into (L2(T))2n. We make thefollowing assumption:

J Nonlinear Sci

Hypothesis 6 The kernel of L : (H 2(T))2n → (L2(T))2n is spanned by x0.

We deduce from Hypothesis 6 that KerL = {0} within the subspace of even func-tions of (H 2(T))2n to which δx belongs. Since the operator L is Fredholm with in-dex 0, we deduce that L restricted to even functions is invertible. Denote L−1 itsinverse. We then have

δx(t) = L−1(∂�f ∗(�0, X0, x0(t)))

δ�

+L−1(DXf ∗(�0, X0, x0(t)))

δX +L−1(h)(t). (72)

We shall note more simply

δx(t) = A0(t)δ� + B0(t)δX +L−1(h)(t). (73)

Now (66) yields the following system satisfied by (δ�, δX)

A1δ� + B1δX = g1 − 1

T

∫ T

0DxF

(1)1

((�0,0)t , X0, X0N,x0(t)

)L−1(h)(t) dt,

Aiδ� + BiδX = g2i − 1

T

∫ T

0DxFi

((�0,0)t , X0, X0N,x0(t)

)L−1(h)(t) dt,

(74)

where A1 ∈ R, B1 ∈ M1,2N−4(R) and Ai ∈ R2, Bi ∈ M2,2N−4(R) for i = 2, . . . ,

N − 1. We have more explicitly

A1 = 1

T

∫ T

0∂�F

(1)1

((�0,0)t , X0, X0N,x0(t)

)

+ DxF(1)1

((�0,0)t , X0, X0N,x0(t)

)A0(t) dt,

Ai = 1

T

∫ T

0∂�Fi

((�0,0)t , X0, X0N,x0(t)

)

+ DxFi

((�0,0)t , X0, X0N,x0(t)

)A0(t) dt,

B1 = 1

T

∫ T

0DXF

(1)1

((�0,0)t , X0, X0N,x0(t)

)

+ DxF(1)1

((�0,0)t , X0, X0N, x0(t)

)B0(t) dt,

Bi = 1

T

∫ T

0DXFi

((�0,0)t , X0, X0N, x0(t)

)

+ DxFi

((�0,0)t , X0, X0N,x0(t)

)B0(t) dt.

J Nonlinear Sci

Now we make the following assumption:

Hypothesis 7 The matrix B ∈ M2N−3(R) defined by

B =

⎛

⎜⎜⎜⎝

A1 B1A2 B2...

...

AN−1 BN−1

⎞

⎟⎟⎟⎠

is invertible.

Then there exists a unique (δ�, δX) solving (74). As a consequence δx is com-pletely determined with (72).

At this step we have proved the invertibility of DUF(U0,0,0). Then the implicitfunction theorem allows us to resolve locally F(U, ε, η) = 0 into U = U(ε,η), withU(0,0) = U0. Consequently the solution U0 = (�0, X0,0,0, x0) of (19)–(23) exist-ing for ε = 0 and η = 0 can be locally continued to a unique solution for ε ≈ 0 andη ≈ 0. This in turn yields a solution of (12)–(14) for γ = ε−2. As a conclusion wehave proved the following theorem.

Theorem 3 Consider the Hamiltonian system (1), where one might allow the po-tential V to depend smoothly on an additional small parameter η. Suppose that theinfinite mass ratio problem (34)–(37) with j = 0 and η = 0 has an even, T -periodicsolution Γ0 = (�0, X0, x0). Under Hypotheses 6 and 7, for γ large enough and η ≈ 0,there exists a T -periodic solution Γγ,η = (�(t),Z(t), x(t)) of (12)–(14) with J = 0,satisfying limγ→+∞,η→0 Γγ,η = Γ0 in (H 2(T))2N−3 × (H 2(T))2n, and even in time.This solution is the unique local even continuation of Γ0 to finite values of γ andη ≈ 0. It corresponds (through the change of coordinates (4)) to a T -periodic andeven solution (Q,q) of (2).

5.2 Persistence of Nonlinear Normal Modes

According to Lemma 1, system (19)–(23) with ε = 0 reduces to system (34)–(37)for periodic light mass displacements and static heavy mass displacements. Conse-quently Theorem 1 provides a local family of solutions of (19)–(23) for ε = 0, andthese solutions can be fixed even in t by fixing the phase. Now let us apply Theorem3 to continue these solutions to ε ≈ 0 with J = 0 in (19)–(23).

Let us choose (�0, X0, x0) = Γ (t;α,0), where Γ (.;α,0) is an even solution of(34)–(37) (with amplitude α and j = 0) provided by Theorem 1, and having the form(52). In order to apply Theorem 3, we shall prove that Hypotheses 6 and 7 are satisfiedfor α ≈ 0, provided ∂Ω

∂α[α, U(α,0)] = 0 in order to satisfy Hypothesis 6. In particular,

the above condition is satisfied for all sufficiently small α = 0 if h(�∗,X∗) = 0 (see(46) for the definition of the hardness coefficient h).

We start by checking Hypothesis 7. For α ≈ 0, (�0, X0, x0) is close to the sta-tionary point (�∗,X∗, x∗) and we have B(α = 0, j = 0) = DUG(0,0,U∗), which isinvertible as a consequence of Lemma 2. Consequently, the matrix B(α, j = 0) isalso invertible for α ≈ 0 and Hypothesis 7 is satisfied.

J Nonlinear Sci

Now let us check Hypothesis 6. For that purpose, we consider the fundamentalsolution matrix M(t) ∈ R

4n (M(0) = I ) of the linearized equation

md2δx

dt2− Dxf

∗(�0, X0, x0(t))δx = 0, (75)

or equivalently dδxdt

= m−1δy,dδydt

= Dxf∗(�0, X0, x0(t))δx. The matrix M(T ) is

called the monodromy matrix of (75). Then δx ∈ KerL if and only if δx is solution of(75) and (δx, δy)(T ) = (δx, δy)(0), or equivalently (δx, δy)t (0) ∈ Ker(M(T ) − I ).First, we examine the case α = 0 where x0 = x∗ and T = 2π/ω∗

1 . Due to the non-resonance condition 4, for α = 0 the monodromy matrix has a double semi-simpleeigenvalue +1. Second, for sufficiently small α = 0, M(T ) possesses an eigenvalue+1 which is at least double and non-semisimple because ∂Ω

∂α[α, U(α,0)] = 0 (see

Sepulchre and MacKay 1997, p. 693). Using the above result for α = 0 in conjunc-tion with classical perturbation theory (Kato 1966), we deduce that the eigenvalue+1 is exactly double for α ≈ 0. Since the eigenvalue +1 is non-semisimple for smallα = 0, we thus have Ker(M(T ) − I ) = 〈(x0,mx0)

t (0)〉 and KerL = 〈x0〉. As a con-sequence Hypothesis 6 is satisfied for α ≈ 0 and α = 0.

Finally, Theorem 3 applies and one can continue the solutions provided by Theo-rem 1 (for α ≈ 0 and j = 0) to the case ε ≈ 0, or equivalently γ � 1. We sum up thisresult in the following theorem.

Theorem 4 Assume Hypotheses 1–4 are satisfied. Consider the family of nonlinearnormal modes Γ (t;α,0) = (�, X, x(t)) provided by Theorem 1, solution of (34)–(37)with j = 0 (we fix Γ (.;α,0) even in time and denote by T = 2π/Ω its period). As-sume ∂Ω

∂α[α, U(α,0)] = 0 for all sufficiently small α = 0 (this is the case in particular

if h(�∗,X∗) = 0). Then, for any fixed value of α ≈ 0 and γ large enough, there ex-ists a T -periodic solution Γγ = (�(t),Z(t), x(t)) of (12)–(14) with J = 0, satisfyinglimγ→+∞ Γγ = Γ (.;α,0) in (H 2(T))2N−3 × (H 2(T))2n. This solution is the uniquelocal even continuation of Γ (.;α,0) to finite values of γ . It corresponds (through thechange of coordinates (4)) to a T -periodic and even nonlinear normal mode (Q,q)

of (2).

5.3 Breather Persistence

In this section, we split the light atoms’ positions x into two component groups x =(ξ0, ξ1)

t with ξi ∈ R2ni and n0 + n1 = n. The interaction potential is assumed of the

form

V (X,x) = V0(X, ξ0) + V1(X, ξ1) + ηW(X,x), (76)

with η ≈ 0. For an infinite mass ratio (i.e. ε = 0) and in the zero coupling limitη = 0, Theorem 2 provides small amplitude breather solutions of system (19)–(23).For these solutions, ξ0 oscillates periodically (and can be fixed even in t) while X,ξ1are at rest. We shall prove that these solutions generically persist for ε ≈ 0 and η ≈ 0.

We begin by considering more general breather solutions (having not necessarily asmall amplitude) and study their persistence using Theorem 3. Let us suppose that forη = 0, the infinite mass ratio problem (34)–(37) with j = 0 has an even, T -periodic

J Nonlinear Sci

solution Γ0 = (�0, X0, x0), with x0(t) = (ξ00 (t), ξ0

1 )t , ξ01 being constant. Assume Hy-

potheses 6 and 7 are satisfied. By Theorem 3, for η ≈ 0 and γ large enough, thereexists a T -periodic solution Γγ,η = (�(t),Z(t), x(t)) of (12)–(14) with J = 0, sat-isfying limγ→+∞,η→0 Γγ,η = Γ0 in (H 2(T))2N−3 × (H 2(T))2n. The solution Γγ,η

is the unique local even continuation of Γ0 to finite values of γ and η ≈ 0. It corre-sponds to a T -periodic and even solution (Q,q) of (2).

Now we observe that Hypothesis 6 can be rewritten in a different form, due to thepartial decoupling of ξ0, ξ1 and the particular structure of Γ0 (ξ0

1 does not oscillate).In that case, the operator L reads (we use the notations introduced in (55)–(56))

L(δξ0, δξ1) =⎛

⎝m0

d2

dt2 δξ0 − Dξ0f∗0 (�0, X0, ξ

00 (t))δξ0

m1d2

dt2 δξ1 − Dξ1f∗1 (�0, X0, ξ

01 )δξ1

⎞

⎠ .

To simplify the notation we shall note L(δξ0, δξ1) = (L0δξ0,L1δξ1)t , with Li : {u ∈

(H 2(T))2ni , u(t) = u(−t)} → {u ∈ (L2(T))2ni , u(t) = u(−t)}. We also define Ω =2π/T , L1 = m−1

1 D2ξ1

V1((�0,0)t , X0, X0N, ξ01 ) (X0N is defined by (69)) and denote

by Sp(L1) the spectrum of L1. One can show the following result (the proof simplyfollows using Fourier series to expand the second component L1).

Lemma 3 Hypothesis 6 on the operator L is equivalent to{

KerL0 = 〈ξ00 〉,

n2Ω2 /∈ Sp(L1) for all n ∈ Z.(77)

Now let us apply Theorem 3 to the small amplitude breather solutions providedby Theorem 2, with V given by (76), η = 0 and j = 0. We choose (�0, X0, x0) =Γb(t;α,0), where Γb(.;α,0) is an even solution of (34)–(37) (with amplitude α andj = 0) provided by Theorem 2, and having the form (65). This yields a solutionof (19)–(23) for ε = 0 and η = 0, and we shall apply Theorem 3 to continue thissolution to ε ≈ 0 and η ≈ 0, with J = 0 in (19)–(23). In order to apply Theorem 3,one has to check Hypotheses 6 and 7 for α ≈ 0. As seen in Sect. 5.2 one has toassume ∂Ωb

∂α[α, Ub(α,0)] = 0 in order to satisfy Hypothesis 6. In particular, the above

condition is satisfied for all sufficiently small α = 0 if hb(�∗,X∗) = 0 (see (61) for

the definition of the hardness coefficient hb).The verification of Hypothesis 7 for α ≈ 0 follows exactly as in Sect. 5.2.

Moreover, checking Hypothesis 6 is equivalent to checking (77) according toLemma 3. Using the same arguments as in Sect. 5.2 (in particular the condition∂Ωb

∂α[α, Ub(α,0)] = 0) it follows that KerL0 = 〈ξ0

0 〉 for α sufficiently small and α = 0.There remains to check that n2Ω2 /∈ Sp(L1) for all n ∈ Z. This property holds truefor α small enough, provided we assume

n2ω∗1

2/∈ Sp(L1∗) for all n ∈ Z, (78)

where L1∗ = m−11 D2

ξ1V1(Q

∗, ξ∗1 ). Indeed one has Ω = ω∗

1 + O(α2), L1 is O(α2)-close to L1∗, and the continuity of the eigenvalues of L1 as α → 0 follows fromclassical perturbation theory (Kato 1966).

J Nonlinear Sci

As a conclusion, Theorem 3 applies and one can continue the solutions providedby Theorem 2 (for α ≈ 0, j = 0 and η = 0 in V ) to the case ε ≈ 0 (i.e. γ � 1) andη ≈ 0. We sum up this result in the following theorem.

Theorem 5 Consider the potential V defined by (76), depending on the small cou-pling parameter η. For η = 0, assume Hypotheses 1, 2, 4 and 5 are satisfied,and the nonresonance condition (78) holds true. Consider the family of breathersΓb(t;α,0) = (�, X, ξ0(t), ξ1) provided by Theorem 2, solution of (34)–(37) withj = 0 and η = 0 (we fix Γb(.;α,0) even in time and denote by T = 2π/Ωb its period).Assume ∂Ωb

∂α[α, Ub(α,0)] = 0 for all sufficiently small α = 0 (this is the case in par-

ticular if hb(�∗,X∗) = 0). Then, for any fixed α ≈ 0 , for γ large enough and η ≈ 0,

there exists a T -periodic solution Γγ,η = (�(t),Z(t), x(t)) of (12)–(14) with J = 0,satisfying limγ→+∞,η→0 Γγ,η = Γb(.;α,0) in (H 2(T))2N−3 × (H 2(T))2n. This so-lution is the unique local even continuation of Γb(.;α,0) to finite values of γ andη ≈ 0. It corresponds (through the change of coordinates (4)) to a T -periodic andeven breather solution (Q,q) of (2).

Acknowledgements This work was supported by the French Ministry of Research through the CNRSProgram ACI NIM (New Interfaces of Mathematics). The authors wish to thank Michel Peyrard for initi-ating this research. This work has started after the workshop Energy Localization: From Small PolyatomicMolecules to Large Biomolecules, held at CECAM (ENS Lyon, France) on September 2004, and organizedby S.C. Farantos, S. Flach and M. Peyrard. G.J. is grateful to Serge Aubry for his hospitality at the Labora-toire Léon Brillouin (CEA Saclay, France) where a part of this work has been carried out after the CECAMworkshop. Stimulating discussions with Serge Aubry and Robert MacKay are also acknowledged. The au-thors are grateful to the editors and one referee for their constructive remarks.

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