Digital Object Identifier (DOI) 10.1007/s00205-006-0423-8Arch. Rational Mech. Anal. 182 (2006) 77–123

Asymptotic Resonance, Interaction of Modesand Subharmonic Bifurcation

Giuseppe Molteni, Enrico Serra, Massimo Tarallo

& Susanna Terracini

Communicated by C.A Stuart

Abstract

We study the existence of small amplitude oscillations near elliptic equilibria ofautonomous systems, which mix different normal modes. The reference problem isthe Fermi-Pasta-Ulam β-model: a chain of nonlinear oscillators with nearest-neigh-borhood interaction. We develop a new bifurcation approach that locates secondarybifurcations from the unimodal primary branches. Two sufficient conditions forbifurcation are given: one involves only the arithmetic properties of the eigen-values of the linearized system (asymptotic resonance), while the other takes intoaccount the nonlinear character of the interaction between normal modes (nonlinearcoupling). Both conditions are checked for the Fermi-Pasta-Ulam problem.

1. Introduction

The principal purpose of this work is to provide a theoretical explanation of cer-tain bifurcation phenomena that appear in some classes of Lagrangian systems. Asa consequence, our main results will propose a new method for the detection of sec-ondary bifurcations that relies on very simple and explicitly computable conditions.

Our interest in these topics was originally motivated by the paper [2], wherethe authors look for (odd) periodic solutions in the N -particle Fermi-Pasta-Ulamβ-model [7]. The dynamics are governed by the equation

x + Ax +W ′(x) = 0, (1)

where

Ax · x =N∑

i=0

(xi+1 − xi)2,

W(x) = 1

4

N∑

i=0

(xi+1 − xi)4,

78 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

0.195 0.196 0.197 0.198 0.199 0.2 0.201 0.2020

1

2

3

4

5

6

frequency

norm

A

B

C

D

E

F

H

I

J

K

G

Fig. 1. The bifurcation diagram of [2].

and x = (x1, . . . , xN) ∈ RN denotes the displacement of the particles from equi-librium. In [2] the spatially periodic case is considered, but essentially the sameresults hold for the simpler fixed ends problem, where x0 = xN+1 = 0, which weuse as a model throughout this paper.

After the time scaling

x(t) = u(ωt), with ω = 2π

T,

the search for T -periodic solutions to (1) is formulated by the boundary valueproblem

{ω2u+ Au+W ′(u) = 0u 2π − periodic,

(Pω)

the frequency ω now being a free (bifurcation) parameter.By means of computer-assisted arguments and numerical methods, Arioli,

Koch & Terracini produce the bifurcation diagram of odd periodic solutionsreported in Figure 1. See, also, [10] for similar results in the context of waterwaves.

Since the N -positive eigenvalues µ2j of A are explicitly known ([19], for in-

stance) and satisfy the fundamental propertyµi

µj/∈ Q for all i �= j, (Q)

([5], or Section 8), it is a standard fact from bifurcation theory [6, 1] that everyfrequency ωjk = µj/k, j = 1, . . . , N , k ∈ N+, is a bifurcation point for thesystem from the trivial line (ω, 0).

The local primary branches that depart from these points are small deformationsof the normal modes of oscillation of the linearization at zero, and carry solutions

Interaction of Modes and Subharmonic Bifurcation 79

with bounded minimal periods. Such is the case, in Fig. 1, for the branches labelledbyF ,H , J andK , which in [2] are then numerically continued until other branchesbifurcate from them.

The analysis of Fig. 1, and of the results in [2], poses a number of problemsfrom a theoretical point of view. First Arioli, Koch & Terracini explain that theapproach they follow to construct the diagram in the figure is driven by a conjec-ture: the appearance of secondary bifurcation branches is related to anomalies inthe distribution of the frequencies ωjk on the real line. In the plot, for example,the bifurcation frequencies of the primary branches F , H , J and K appear as atight cluster among all other frequencies, and those of H and J as an even tightersubcluster.

A further analysis of the harmonic energy along the secondary branches showsthat these, in contrast to the primary ones, carry solutions that “mix modes”; in thiscontext, another role of the frequencies involved in the cluster is that they appearto permit only certain modes to be mixed along a given secondary branch. Themixing, however, turns out to be quite asymmetric and the analysis of [2] leads tobelieve that prevalently a primary branch in a cluster interacts and mixes modeswith other branches located close to it and at its left.

These considerations are the result of mainly numerical computations and callfor a theoretical understanding. This need is the aim of the present paper, for aclass of systems including the Fermi-Pasta-Ulam model. The main questions weaddress are the mechanism of formation of secondary bifurcations, the reasonsfor the asymmetry in the interactions, and the discussion of the notions we willintroduce to prove the main results.

Of course, we will test the outcome on the model but, we believe, the instrumentwe construct is flexible enough to be applied to more general problems.Our results are formulated for problem (1) where now theN ×N matrix A and thepotential W : RN → R satisfy the assumptions

(A) A is symmetric, positive definite and has only simple eigenvalues whose squareroots are pairwise independent over Q;

(W) W is an homogeneous polynomial of degree four and W > 0 except at zero.

We denote by

µ1 < · · · < µN and e1, . . . , eN ,

respectively, the square roots of the eigenvalues of A and their corresponding nor-malized eigenvectors. Moreover, we call characteristic values the numbers (fre-quencies)

ωjk = µj

k, j = 1, . . . , N, k ∈ N+.

The same arguments as for the Fermi-Pasta-Ulam problem show that from every(ωjk, 0) there departs a primary branch �jk of even solutions to (Pω).

In all the paper we will deal with even functions, so that all the results we willpresent refer only to even periodic solutions. Clearly, exactly the same results holdfor odd solutions too; however, we preferred to treat the even case in view of apossible removal of the evenness of W in a future work. It should be pointed out

80 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

that, while restricting to some given class of symmetric functions is of course veryconvenient form a technical point of view, in principle it may give rise to weakerresults: our feeling is that this is not the case here, but the proof certainly needsfurther investigations.

Our first concern is to establish sufficient conditions for the existence of bifur-cation points on the branches �jk , or, in other words, of secondary bifurcations.We enter here a word of caution. In contrast to other possible approaches, we donot construct a global extension of �jk , and, even worse, we will be forced towork as close as possible to the base point (ωjk, 0). Close to this point, unique-ness arguments prevent the existence of secondary bifurcations, so that a delicateconstruction must be carried out to obtain a sort of “local but not too local” result.

We restrict our attention to a pair of (close) characteristic values

ωih < ωjk, (2)

with i �= j , and to the associated primary branches �ih and �jk . Our main resultsgive a sufficient condition for a secondary bifurcation to appear on the right branch�jk , or on the left branch �ih, respectively, because of their mutual interaction.We stress that there is no reason to expect symmetric results: the fact that all theprimary branches turn to the right makes it evident that the choice between left andright is not immaterial.

The next theorems are our main results. We denote by ωR and ωL the nearestcharacteristic values respectively to the right and to the left of ω.

Theorem 1 (Bifurcation from the right branch). Under the assumptions (A) and(W), suppose moreover that

6µ2i W(ej )− µ2

jW′′(ej ) · e2

i < 0 (3)

for some i �= j . Then, there exists σ > 0 such that, if for some h, k ∈ N+, thereresults

ωih < ωjk andωjk − ωih

ωRjk − ωjk< σ, (4)

then the primary branch �jk supports a secondary bifurcation in the interval(ωjk, ω

Rjk).

Theorem 2 (Bifurcation from the left branch). Under the assumptions (A) and (W),suppose moreover that

6µ2jW(ei)− µ2

i W′′(ei) · e2

j > 0 (5)

for some for some i �= j . Then, there exists σ > 0 such that, if for some h, k ∈ N+,there results

ωih < ωjk andωjk − ωih

ωih − ωLih< σ, (6)

then the primary branch �ih supports a secondary bifurcation in the interval(ωjk, 2ωih − ωLih).

Interaction of Modes and Subharmonic Bifurcation 81

As we will see, in the second theorem the range for ω could have been equiva-lently replaced by (ωih, 2ωih−ωLih), making the statement as symmetric as possibleto the first one. Anyway, in both cases nothing interesting may happen too close tothe feet of the involved branches. The proofs will clearly demonstrate that second-ary bifurcations appear on�jk and�ih at a distance from the respective feetωjk andωih which is much bigger than their mutual distance ωjk − ωih, but much smallerthan their “directional” distance to the remaining part of the set of frequencies,namely

ωRjk − ωjk and ωih − ωLih

respectively.The presence of secondary bifurcations will be detected via a result by Kiel-

höfer, [11], implying that along an analytic branch of solutions, every point ofdiscontinuity for the Morse index of the solutions (seen as critical points of theusual action functional) is either a bifurcation or a turning point; the last possibilitywill be ruled out by trivial local arguments.

In the statement of the theorems we do not provide any estimate for the quan-tity σ . Thus, the practical way to check the assumption is to locate cases in whichthe fractions in (4), (6) are not bounded away from zero along some divergentsequences. This concept plays a fundamental role in our research and motivates thefollowing definition.

Definition 1. We say that µi is right asymptotically resonant with µj , if there existdiverging sequences hn, kn ∈ N such that

ωihn < ωjkn ∀n and limn→∞

ωjkn − ωihn

ωRjkn − ωjkn= 0. (7)

Left asymptotical resonance is defined similarly to restate (6) (see Section 5).

The notions of right and left asymptotic resonance make precise the informalidea of clustering appearing in [2], for an interaction involving a pair of primarybranches. Essentially, they are closeness assumptions on the two frequencies ωjknand ωihn , whose relative distance has to be much smaller than their distance fromthe remaining characteristic values (on the right or left, respectively). The conceptdepends only on the arithmetical properties of the numbers µj , and is thereforerelated to the linear part of the problem.

The other ingredients of Theorems 1 and 2 are the nonlinear coupling conditions(3) and (5). These conditions fully account for the asymmetry in the interactionsalready mentioned. Indeed defining

Wij = 6µ2i W(ej )− µ2

jW′′(ej ) · e2

i , i, j = 1, . . . , N,

we will see that the term Wij accounts for the effect of �ih on the Morse indexalong �jk , while the reverse effect involves the symmetric termWji . Precisely, thetheorems state that in the presence of the correct asymptotic resonance a secondarybifurcation appears on the right branch �jk if Wij < 0, or on the left branch �ihif Wji > 0. The proofs should also demonstrate that we cannot reasonably expectsecondary bifurcations (in an appropriate range) when the signs are reversed in the

82 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

previous assumptions; however, some further work is needed to make this remarkeffective.

In summary, it is the simultaneous occurrence of the appropriate nonlinear cou-pling with the appropriate asymptotic resonance that produces a secondary bifur-cation; we believe that the understanding of this phenomenon is the core of ourwork.

As a corollary of the previous theorems, we obtain the following Birkhoff-Lewistype result, which also accounts for the subharmonic character of the secondarybifurcations.

Corollary 1. Under the assumptions (A) and (W), suppose moreover that

Wij = 6µ2jW(ei)− µ2

i W′′(ei) · e2

j �= 0 (8)

for some for some i �= j . If, according to Wij < 0 or Wij > 0, either µi is rightasymptotically resonant withµj , orµj is left asymptotically resonant withµi , thenthere exists a sequence of periodic solutions to (1), whose C2 norms tend to zeroand whose minimal periods tend to infinity.

In the Fermi-Pasta-Ulam problem our results will show that this Birkhoff-Lewistype theorem holds for every dimensionN . The same conclusion could be obtainedas a by-product of a result in [16], by showing that the fourth-order Birkhoff nor-mal form is nonresonant and nondegenerate (in the sense of Kolmogorov). Wepoint out that this approach may fail in the more general framework identified byassumptions (A) and (W) for two reasons. First, there might be fourth-order res-onances among the µj s, preventing the normal form to be nonresonant; this factseems to be unrelated to the notion of asymptotic resonance. Moreover, even inthe absence of resonances, the Kolmogorov nondegeneracy condition need not befulfilled; although this condition holds generically, it is very difficult to check, forone should know the explicit expression of the normal form, which rarely happens.This has to be compared with the simple and computable form of the nonlinearcoupling conditions (3) and (5).

The above results are contained in Sections 2–6. The remaining part of thepaper is devoted to the analysis of conditions used in the main theorems and tothe application of this analysis to the Fermi-Pasta-Ulam problem (for recent resultson this problem see [16, 17] and references therein). Here we use some numbertheoretical arguments which may by unfamiliar to the analyst.

In particular, a complete study of the notion of asymptotic resonance shows thatit is equivalent to a set of Diophantine equations depending on the coefficients ofthe ternary relations (null linear combinations with integer coefficients involvingexactly three µj s) that may be present.

As a conclusion, µi is proved to be asymptotically resonant with any other µj ,whenever the ternary relations involving them, if any exist, are of a special type(see Section 7).

The abstract Diophantine equations are then fully solved for the Fermi-Pasta-Ulam model, obtaining the complete list of asymptotic resonances for every valueof N . As a particular case we obtain that µi is asymptotically resonant with anyother µj as soon as N + 1 is not a multiple of 3.

Interaction of Modes and Subharmonic Bifurcation 83

Finally, we analyze the validity of the nonlinear coupling conditions (3) and (5)for the Fermi-Pasta-Ulam model. We obtain, for every N , that

Wij is

{> 0 if i + j = N + 1

< 0 otherwise.

The relative preponderance of negative terms explains, we believe, numerical resultssuch as those of [2].

The format of the paper is as follows. In Section 2 we establish the abstractsetting and we construct the primary bifurcation branches with their asymptoticexpansions and properties. In Section 3 we begin the study of the Morse indexalong the primary branches. Section 4 introduces the main asymptotic argumentsand establishes that a variation of the Morse index takes place along a right branchunder certain conditions. In Section 5 the same is done for the left branches. Theproofs of the main results are given in Section 6. The concept of asymptotic reso-nance is studied for the abstract case in Section 7, where necessary and sufficientconditions for its validity are established. The Diophantine equations for asymptoticresonance are completely solved in Section 8 for the Fermi-Pasta-Ulam problemand finally, in Section 9, for the same problem, the nonlinear coupling conditionsare established.

Notation. We denote by �a� and {a} the integer and fractional parts, respectively,of a real number a. Ifm, n ∈ N, the symbol (m, n) stands for the greatest commondivisor of m and n, while m | n means that m divides n. Finally xn yn meansthat axn � yn � bxn definitely holds for suitable positive constants a, b.

2. Primary bifurcations and Lyapunov orbits

Solutions of (Pω) are critical points of the analytic action functional

J (ω, u) = Jω(u) = ω2

2

∫ 2π

0|u|2 dt − 1

2

∫ 2π

0Au · u dt −

∫ 2π

0W(u) dt (9)

on the Hilbert space H 12π = {u ∈ H 1

loc((0, 2π); RN) | u(t + 2π) = u(t) ∀t }.In view of the development of our arguments and, in particular, in order to

gain a uniqueness property for the (primary) bifurcation branches, we restrict thefunctional Jω to the space

H = {u ∈ H 12π | u is even }.

The space H is a natural constraint for the functional Jω, in the sense that criticalpoints of Jω constrained toH are free critical points of Jω considered as a functionalon the whole space H 1

2π , and hence solutions of problem (Pω). These statementsderive from the fact that problem (Pω) is autonomous and are easy to check.

The principal scope of this section is to locate and describe all the bifurcationbranches from the trivial critical line (ω, 0), ω ∈ R+.

84 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

It is well known that J ′ω is a Fredholm map of index zero. Denoting by

Lωv = −ω2v − Av

the derivative of Jω at u = 0, we have that up to the Riesz isomorphism, Lω is asymmetric operator and we can write H = kerLω ⊕ Im Lω. A standard analysisshows that Lω is not invertible precisely for

ω = ωjk = µj

k, j = 1, . . . , N, k ∈ N+.

In this case, moreover, condition (Q) implies that kerLωjk is one-dimensional andis spanned by

ϕjk(t) = 1√π

cos(kt)ej .

The functions ϕj1 are usually called normal modes of oscillation, while the ϕjkswith k > 1 are the associated higher harmonics; the choice we made of the coeffi-cients makes all these functions normalized in L2.

Notice that, for instance by a result of Stuart [18], all the elements of the set

� =N⋃

j=1

�j , where �j = {µjk

| k ∈ N+},

are bifurcation points without any restriction on the spectrum of the matrixA. How-ever, we cannot generally assume that there is a branch originating at each pointof�. The most direct way to obtain a branch is the application of the Crandall-Ra-binowitz bifurcation theorem (see [6, 1]). The required transversality condition isimmediate to check, so that we obtain the following classical statement:

Proposition 1 (Primary bifurcations). Assume that (A) and (W) hold. Then, foreveryωjk ∈ �, the set of nontrivial critical points of Jω around (ωjk, 0) is a uniqueanalytic curve ξ �→ (ωξ , uξ ) whose expansion near ξ = 0 satisfies

ωξ = ωjk + 3W(ej )

2πµjkξ2 +O(ξ3) in R

uξ = ξϕjk +O(ξ2) in H.(10)

Notice that since W > 0 outside zero, every ωjk is a supercritical bifurcationpoint. The previous result is just a particular case of theorems that can be found,for instance, in [1, 12]; the analyticity of the branch is explicitly considered in [3].These branches (especially when k = 1) are sometimes called Lyapunov orbitsbecause they can also be constructed by the Lyapunov Center Theorem; also in thiscase though, property (Q) is essential.

From this point forth, we are interested in bifurcations from the primary branches,or, in other words, in secondary bifurcations. However, these are clearly impossibleto find unless the primary branches are extended outside the uniqueness regions

Interaction of Modes and Subharmonic Bifurcation 85

defined by Proposition 1. As we will see shortly, the way to carry out these exten-sions is first to focus on the principal primary branches, namely the N branchescorresponding to k = 1, and then to act on them with the map

{ω �→ ω/k

u(t) �→ u(kt),(11)

which sends solutions of (Pω) into solutions of (Pω/k). This map transforms thebranch originating at (µj , 0) into a branch originating at (ωjk, 0), which mustcoincide locally with that branch already found in Proposition 1. The scaled branchhowever may differ from the other one for ω far from ωjk , and this is where sec-ondary bifurcations may take place.

In order to carry out our proposed program of work, it is convenient to repara-metrize the primary branches using the global variable ω instead of the local one ξ .This reparametrization may be done locally, splitting each branch into two branchesaccording to the sign of ξ . We do it first for the principal primary branches, pointingout at the same time some useful properties.

Looking at (10) for k = 1, we see that we can choose small positive numbersr, R such that in

Br(µj )× BR(0) ⊂ R ×H, (12)

the curve of nontrivial critical points is subject to the uniqueness property statedin Proposition 1 and is free of turning points, apart from the trivial one (µj , 0).Reparametrizing the curve using ω instead of ξ , we obtain two branches:

ω ∈[µj , µj + r) �→ ± uω ∈ BR(0),according to the sign of ξ . The symmetry of the two branches comes from theoddness of W ′ and it is not relevant for our approach.

The function uω depends analytically onω (except atω = µj ). Moreover, sincechoosing ξ > 0 in (10) and inverting we obtain

ξ =(

2πµj3W(ej )

(ω − µj )

)1/2

+O(ω − µj ),

the expansion of uω as ω → µ+j is given by

uω = (cj (ω − µj ))1/2ϕj1 +O(ω − µj ) (13)

in H , where we have set

cj = 2πµj3W(ej )

.

For future reference we also notice from (13) that inBr(µj )×BR(0) every functionlying on �j has minimal period 2π . We are now ready to introduce the objects wewill deal with from this point forth, namely convenient versions of the primarybranches, restricted to a neighborhood where all the properties stated above holdtrue.

86 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

Definition 2. For every j = 1, . . . , N , we denote by �j the graph of the function

ω ∈[µj , µj + r) �→ uω ∈ BR(0)

and, for every integer k, by �jk the image of �j trough the map (11).

Of course each�jk is a branch of solutions emanating from (ωjk, 0); its domainis estimated simply, as a function of k, by

ωjk � ω < ωjk + r

k.

The branch�jk shares the properties of�j , including the analyticity and the absenceof turning points; these properties will all be used in Section 6.

The scaling procedure shows that

�jk ⊂ Br/k(ωjk)× BkR(0),

and moreover makes the dependence on k in the expansion of �jk clear. This isreported in the following lemma.

Lemma 1. For every j = 1, . . . , N and every k ∈ N+, the branch �jk has theexpansion

uω = (cj k(ω − ωjk))1/2ϕjk +O(k(ω − ωjk)) (14)

as k(ω − ωjk) → 0+, where O(k(ω − ωjk)) is in the L∞ norm.

Proof. Expansion (13) says that on �j

u(t) = (cj (µ− µj ))1/2 1√

πej cos t + R(t;µ− µj ),

where ||R(· ; µ − µj )|| � C|µ − µj | as µ → µ+j , for a suitable constant C

independent of j . Evaluating this expansion at kt and setting ω = µ/k we seethat the principal part is the same as in (14), while the remainder takes the formR(kt, k(ω − ωjk)). Since

supt

|R(kt, k(ω − ωjk))| = supt

|R(t, k(ω − ωjk))| � C||R(·, k(ω − ωjk))||,

the statement follows. ��The reason for the substitution of H by L∞ in the evaluation of the remainderterm is that in this way we gain a property of uniformity: the remainder does notgrow as k → ∞. All the estimates of the next sections will be based on L∞ typeexpansions.

Interaction of Modes and Subharmonic Bifurcation 87

3. Morse index estimates

In this section we begin the computation of the Morse index of the solutionslying on a given branch �jk , namely we will look for subspaces of H where thequadratic form

J ′′ω(uω) · w2 =

∫ 2π

0Lωw · w dt −

∫ 2π

0W ′′(uω) · w2 dt

is negative definite; here, as in the previous section, uω describes �jk as ω varies.We set

Ejk = span {ϕih | ωih > ωjk} ⊂ H and νjk = dimEjk < ∞and we identify the subspace of constant functions with RN . We also recall that ωL

(resp. ωR) is the largest (resp. smallest) element of � at the left (resp. right) of ω.Denoting the Morse index of uω bym(uω), the main result of this section is the

following:

Theorem 3. Under assumptions (A), and (W), for every j = 1, . . . , N and everyk ∈ N+, the equality

m(uω) = νjk +N + 1.

eventually holds as ω → ω+jk .

Proof. For a given ωjk we take ω ∈ (ωjk, ωRjk) and we first prove that J ′′ω(uω) is

negative definite on Ruω ⊕Ejk ⊕ RN provided ω is close to ωjk . This amounts toshowing that for every α ∈ R, every ϕ ∈ Ejk , and every c ∈ RN (not all zero)

J ′′ω(uω) · (αuω + ϕ + c)2 < 0.

We will do this by expanding every term in

J ′′ω(uω) · (αuω + ϕ + c)2 = α2J ′′

ω(uω) · u2ω + J ′′

ω(uω) · ϕ2 + J ′′ω(uω) · c2

+2αJ ′′ω(uω) · (uω, ϕ)

+2αJ ′′ω(uω) · (uω, c)+ 2J ′′

ω(uω) · (ϕ, c), (15)

as k → ∞ and ω → ωjk+ in a sense that we now make precise.

In this section we could work with a fixed k and estimate the terms asω → ω+jk;

however, in the next section we will need to take a subsequence kn → ∞ and, foreach n, an ωn such that kn(ωn − ωjkn) → 0+. Therefore, we prefer to make thedependence on k explicit by expressing the estimates from the beginning as asymp-totic relations as k(ω − ωjk) → 0+. This action, even if not needed in the presentsection, will spare some computations later. Of course k(ω−ωjk) → 0+ is satisfiedwhen k is fixed and ω → ω+

jk .Thus, every term in (15) will be now estimated asymptotically as k(ω−ωjk) →

0+. We have split the computations into a series of lemmas; assumptions (A), and(W) have been taken for granted in each lemma.

88 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

For clarity of notation in the proofs we set

s = ω − ωjk,

so that the expansion of �jk in the L∞ norm, given by (14), reads

uω = (cj ks)1/2ϕjk +O(ks) (16)

as ks → 0+. ��The first lemma specifies the asymptotic behavior of some quantities that appear

repeatedly in the proofs of this, and of the next, section. This lemma is the mainpoint where the homogeneity properties of W come into play.

Lemma 2. Let uω be as in (14), and let v,w ∈ H . Then

W(uω) = π−2(cj k(ω − ωjk))2 cos4(kt)W(ej )

+O((k(ω − ωjk))52 ), (17)

W ′(uω)·v = π− 32 (cj k(ω − ωjk))

32 cos3(kt)W ′(ej )·v

+O((k(ω − ωjk))2)|v|, (18)

W ′′(uω)·(v,w) = O(k(ω − ωjk))|v||w|, (19)

where the remainders are in the L∞ norm.

Proof. By homogeneity ofW , it is easily seen using (16) and the definition of ϕjkthat

W(uω) = 1

4!W′′′′(0) · u4

ω

= 1

4!W′′′′(0) · ((cj ks)1/2ϕjk +O(ks))4

= 1

4! (cj ks)2 1

π2 cos4(kt)W ′′′′(0) · e4j +O((ks)5/2)

= 1

π2 (cj ks)2 cos4(kt)W(ej )+O((ks)5/2),

which is (17). Using the very same argument we see that

w′(uω) · v = 1

6W ′′′′(0) · (u3

ω, v)

= 1

6W ′′′′(0) · ((cj ks)1/2ϕjk +O(ks))3, v)

= 1

π3/2 (cj ks)3/2 cos3(kt)W ′(ej ) · v + |v|O((ks)2),

which yields (18). The proof of (19) is even simpler, thus we choose to omit it fromthis paper. ��

Interaction of Modes and Subharmonic Bifurcation 89

In the following lemma we group the estimates of the simplest terms in (15).

Lemma 3. As k(ω − ωjk) → 0+ we have

J ′′ω(uω) · u2

ω � −2µjcj (k(ω − ωjk))2, (20)

J ′′ω(uω) · c2 � −πµ1|c|2, (21)

J ′′ω(uω) · (uω, ϕ) = O((k(ω − ωjk))

3/2)||ϕ||2, (22)

J ′′ω(uω) · (uω, c) = O((k(ω − ωjk))

2)|c|, (23)

J ′′ω(uω) · (ϕ, c) = O(k(ω − ωjk))||ϕ||2 |c|. (24)

Proof. We start by estimating

J ′′(uω) · u2ω =

∫ 2π

0Lωuω · uω dt −

∫ 2π

0W ′′(uω) · u2

ω dt.

Since uω is a solution, and by homogeneity, Lωuω ·uω = W ′(uω) ·uω = 4W(uω);still by homogeneity, W ′′(uω) · u2

ω = 12W(uω). Therefore, by (17) we obtain

J ′′ω(uω) · u2

ω = −8∫ 2π

0W(uω) dt

= −8

((cj ks)

2 1

π2

3π

4W(ej )

)+O((ks)5/2)

= −4µjcj (ks)2 +O((ks)5/2),

where we have also used the definition of cj . Hence (20) holds for ks small enough.

Passing to (21), with the elementary properties of A and (19) with v = w = c,we see immediately that

J ′′ω(uω) · c2 = −2πAc · c −

∫ 2π

0W ′′(uω) · c2 dt � −2πµ1|c|2 + |c|2O(ks),

and we obtain the statement for ks small enough.

We now turn to (22) and (23). Let v be any element ofH ; sinceLωuω = W ′(uω)and W ′′(uω) · (uω, v) = 3W ′(uω) · v, we see that

J ′′ω(uω) · (uω, v) =

∫ 2π

0W ′(uω) · v dt −

∫ 2π

0W ′′(uω) · (uω, v) dt

= −2∫ 2π

0W ′(uω) · v dt.

Applying (18) with v = ϕ ∈ Ejk yields

∫ 2π

0W ′(uω) · ϕ dt = O((ks)3/2)||ϕ||2,

90 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

which proves (22). If v = c ∈ RN , thenW ′(ej )·c is a constant, so that the vanishingof the integral of cos3(kt) cancels the first term of (18), giving

∫ 2π

0W ′(uω) · c dt = O((ks)2)|c|,

which also proves (23).

Finally, for (24), we notice that Lωϕ ∈ Ejk , namely it is a linear combinationof cosines; therefore,

∫ 2π

0Lωϕ · c dt = 0,

so that

J ′′ω(uω) · (ϕ, c) =

∫ 2π

0Lωϕ · c dt −

∫ 2π

0W ′′(uω) · (ϕ, c) dt

= −∫ 2π

0W ′′(uω) · (ϕ, c) dt.

Applying (19) with v = ϕ and w = c, this last term is O(ks)||ϕ||2 |c|, and theproof is complete. ��We conclude the first set of estimates with the analysis of J ′′

ω(uω) · ϕ2.The behavior of this term, which will play a central role in our argument,

depends on the properties of the function defined by

δ(ω) = min{h2(ω2ih − ω2) | ωih > ω }, 0 < ω < µN. (25)

This function will be estimated from below in the next section. For now we list itssimplest properties.

Lemma 4. The function δ is strictly positive in (0, µN); moreover,

limω→ω+

jk

δ(ω) = δ(ωjk) and limω→0+ δ(ω) = 0.

Proof. Positivity is obvious. To check the first limit, notice that

h2(ω2ih − ω2

jk) > h2(ω2ih − ω2) = h2(ω2

ih − ω2jk)+O(ω − ωjk).

Since the equality {ωih | ωih > ω} = {ωih | ωih > ωjk} holds for ω → ω+jk ,

minimization on this set yields

δ(ωjk) � δ(ω) � δ(ωjk)+O(ω − ωjk),

which establishes the first limit.For any given ω, let now m ∈ N be such that µ1/(m+ 1) � ω < µ1/m. Then

δ(ω) � m2(ω21m − ω2) = m2(ω1m − ω)(ω1m + ω) � 2µ2

1

m+ 1,

and of course m → ∞ if ω → 0. ��

Interaction of Modes and Subharmonic Bifurcation 91

We can now conclude the estimates.

Lemma 5. As k(ω − ωjk) → 0+,

J ′′ω(uω) · ϕ2 � −δ(ω)||ϕ||22 +O(k(ω − ωjk))||ϕ||22.

Proof. By definition of Lω and ϕih we have

Lωϕih = Lωihϕih − (ω2 − ω2ih)ϕih

= h2(ω2 − ω2ih)ϕih,

since ϕih ∈ kerLωih .Therefore, if we write as above ϕ = ∑

i,h aihϕih, with the sum extended to all(i, h) such that ωih > ωjk , we see that

∫ 2π

0Lωϕ · ϕ dt =

∑

i,h

h2(ω2 − ω2ih)a

2ih � −δ(ω)

∑

i,h

a2ih

= −δ(ω)||ϕ||22.Then

J ′′ω(uω) · ϕ2 =

∫ 2π

0Lωϕ · ϕ dt −

∫ 2π

0W ′′(uω) · ϕ2 dt

� −δ(ω)||ϕ||22 +O(ks)||ϕ||22,since the second integral can be treated in the same manner as analogous terms inthe previous lemmas. ��Conclusion of the proof of Theorem 3. We are now ready to replace all the termsin (15) with their expansions obtained in the preceding lemmas. We find, as ks →0+,

J ′′ω(uω) · (αuω + ϕ + c)2 � −2µjcj (ks)2α2 − (δ(ω)+O(ks))||ϕ||22

−πµ1|c|2 +O((ks)3/2)|α|||ϕ||2+O((ks)2)|α||c| +O(ks)||ϕ||2|c|

(26)

We think of the right-hand side as a bilinear form in (|α|, ||ϕ||2, |c|) representedby the matrix

M3 =

−2µjcj (ks)2 O((ks)3/2) O((ks)2)

O((ks)3/2) −δ(ω)+O(ks) O(ks)

O((ks)2) O(ks) −πµ1

.

In order to show that M3 is negative definite we check that (−1)� detM� > 0 for� = 1, 2, 3, where the M�s are the principal square submatrices. Now − detM1 =2µjcj (ks)2 > 0, while

detM2 = 2µjcj (ks)2[ δ(ω)+O(ks)] ,

− detM3 = 2πµ1µjcj (ks)2[ δ(ω)+O(ks)] .

92 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

For each fixed k ∈ N+, when s → 0+, namely when ω → ω+jk , we have

δ(ω) → δ(ωjk) > 0 by Lemma 4 and, of course, ks → 0; therefore, detM2and − detM3 are both positive as ω → ωjk .

We have thus proved that for every j = 1, . . . , N and every k ∈ N+, the qua-dratic form J ′′

ω(uω) is negative definite on Ruω ⊕ Ejk ⊕ RN provided that ω isclose enough to ωjk (from the right); in other words, for all such ωs,

m(uω) � νjk +N + 1. (27)

The proof will be complete when we establish the reversed inequality.

To this aim it suffices to find a subspace of H of codimension νjk +N + 1 onwhich J ′′

ω(uω) is positive definite. We claim that one such subspace is

Fjk = cl span {ϕih | ωih < ωjk}(closure taken in H ). Notice that since H = Fjk ⊕ Rϕjk ⊕ Ejk ⊕ RN anddim(Rϕjk ⊕Ejk ⊕ RN) = νjk +N + 1, the space Fjk has the right codimension.We simply have to show that J ′′

ω(uω) is positive definite on Fjk .

Now if ϕ ∈ Fjk , then ϕ = ∑i,h aihϕih, where the series is extended to all i, h

such that ωih < ωjk . Then with the same computations as in Lemma 5:∫ 2π

0Lωϕ · ϕ dt =

∑

i,h

h2(ω2 − ω2ih)a

2ih � ρ(ω)

∑

i,h

a2ih = ρ(ω)||ϕ||22,

where ρ(ω) = inf{h2(ω2 − ω2ih) | ωih < ωjk}. Notice that for every ω � ωjk we

have ρ(ω) � ω2jk − (ωLjk)

2, a number that does not depend on ω. Moreover, with(by now) standard computations,

∫ 2π

0W ′′(uω) · ϕ2 dt = O(ks)||ϕ||22.

Therefore,

J ′′ω(uω) · ϕ2 � ((ω2

jk − (ωLjk)2)+O(ks))||ϕ||22,

by standard arguments; this is enough to prove that J ′′ω(uω) is also positive definite

on Fjk for the H topology, for ω close to ωjk . ��

4. Asymptotic resonance and Morse index jumps

The analysis of the Morse index was carried out in the previous section ona single primary branch, independently of its position among the other branches.From now on we consider pairs of primary branches �ih and �jk , originating atclose characteristic values

ωih < ωjk,

Interaction of Modes and Subharmonic Bifurcation 93

with the aim of describing their mutual interaction. More precisely, in this sectionwe will study under which circumstances the presence of �ih affects the Morseindex along �jk; the analysis of the symmetric case will be dealt with in the nextsection.

The main result will show that forω far enough fromωjk the function ϕih shiftsfrom the positive to the negative eigenspace of J ′′

ω(uω). Since this type of shift can-not occur for linear problems (W ≡ 0), it must be an effect of the nonlinearity thatignites the change of the index. In our case it takes the very simple and computableform

6µ2i W(ej )− µ2

jW′′(ej ) · e2

i < 0,

to which we refer as a nonlinear coupling of the frequency µi with µj . We willstudy this inequality in detail for the Fermi-Pasta-Ulam model in Section 9.

The key point in this section is the choice of the right ω to compute the indexof J ′′

ω(uω); note, for example, that in order to use the expansion (14) we must workwithω very close toωjk , while the property we need can only hold forω far enoughfrom ωjk . The fulfillment of these competing requirements forces us to consider,instead of a fixed pair of characteristic frequencies, sequences of pairs

ωihn < ωjkn with hn, kn → ∞.

We now make a preliminary analysis of these sequences, in order to establish theright tuning for ω = ωn. Recall that ωR denotes the smallest element of � at theright of ω.

First of all notice that if kn is a diverging sequence of integers, then

ωRjkn − ωjkn � µj

kn − 1− µj

kn= O

(1

k2n

). (28)

This means not only that ωRjkn lies in the domain of �jkn , (see Definition 2), butalso guarantees the stronger property

kn

(ωRjkn − ωjkn

)→ 0+. (29)

In other words, the expansion

uωn = (cj kn(ωn − ωjkn))1/2ϕjkn +O(kn(ωn − ωjkn)) (30)

makes sense at every ωn satisfying ωjkn < ωn < ωRjkn .We next introduce a key concept in our work:

Definition 3. We say that µi is right asymptotically resonant with µj if there existdiverging sequences hn, kn ∈ N such that

ωihn < ωjkn ∀n and limn→∞

ωjkn − ωihn

ωRjkn − ωjkn= 0. (31)

94 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

Notice that the conditions in (31) are not symmetric with respect to µi and µj .The existence of frequencies that satisfy (31) is determined only by the arith-

metical properties of the set of eigenvalues of the matrix A, and is therefore aproperty of the linear part of the problem. In Section 7 we characterize the asymp-totic resonance for a general set of frequencies, and in Section 8 we particularizethe result to the Fermi-Pasta-Ulam problem. For the moment we simply assumethat a frequency µi is right asymptotically resonant with µj , and we deduce someconsequences. The first of which concerns the relations between hn and kn hiddenin (31).

Lemma 6. Assume condition (31) holds. Then i �= j . Moreover, for all n largeenough, ωjkn is the first element of �j at the right of ωihn ; symmetrically, ωihn isthe first element of �i at the left of ωjkn ; and finally

kn

hn→ µj

µi. (32)

Proof. As already computed, the denominator in (31) is estimated from above bythe quantity µj/kn(kn − 1). On the other hand, if i = j , or (along a subsequence)i �= j but ωjkn is not the first element of �j at the right of ωihn , then

ωjkn − ωihn � µj

kn− µj

kn + 1= µj

kn(kn + 1).

In both casesωjkn − ωihn

ωRjkn − ωjkn� kn(kn − 1)

kn(kn + 1)→ 1,

violating (31). Therefore i �= j andµj

kn + 1� µi

hn<µj

kn,

from which the limit (32) follows. It remains to prove that ωihn is the first elementof �i at the left of ωjkn . Indeed, observe that if (along a subsequence) this is false,then

ωjkn − ωihn � µi

hn − 1− µi

hn= µi

hn(hn − 1).

Since we already know that hn has the same order as kn, this leads to the samecontradiction as before. ��Remark 1. A further consequence of Definition 3, via (28), is the fact that

ωjkn − ωihn = o(ωRjkn − ωjkn

)= o

(1

k2n

).

Thus, although we cannot have an exact resonance involving µi and µj because of(Q), the preceding relation shows that along the integer sequences kn and hn wehave

hn µj − kn µi = o(1),

namely an “asymptotic resonance”. This justifies the term adopted in the abovedefinition.

Interaction of Modes and Subharmonic Bifurcation 95

The introduction of the notion of right asymptotic resonance allows us to com-plete the required balance between “how close” and “how far” ωn should be fromωjkn . Indeed we choose a frequency ωn such that

ωjkn < ωn < ωRjkn, (33)

ωjkn − ωihn = o(ωn − ωjkn), (34)

ωn − ωjkn = o(ωRjkn − ωjkn), (35)

simultaneously hold as n → ∞.Asymptotic resonance guarantees that such choice for ωn is possible, as for

instance the position

ωn = ωjkn + (ωjkn − ωihn) logωRjkn − ωjkn

ωjkn − ωihn

makes evident.The sense of the requirements (34) and (35) is most easily visualized if one

thinks of the three adjacent intervals determined by

ωihn < ωjkn < ωn < ωRjkn .

The choice of ωn makes the central interval (asymptotically) much larger than theleft one and much smaller than the right one.

We will evaluate the Morse index precisely at the points ωn; in view of futurecomputations, we establish the following results.

Lemma 7. Assume that (31) holds and chooseωn according to (33) and (34). Then,as n → ∞,

kn(ωn − ωjkn)

hn(ωn − ωihn)→ µj

µi.

Proof. Using Lemma 6 we compute

kn(ωn − ωjkn)

hn(ωn − ωihn)= kn

hn

ωn − ωjkn

(ωn − ωjkn)+ (ωjkn − ωihn)

=(µj

µi+ o(1)

)1

1 + o(1). ��

The second property provides an estimate from below of the rate of vanishingof the function δ, defined by (25), when computed at ωn.

Lemma 8. If ωn is chosen according to (33) and (35) then, as n → ∞,

δ(ωn)

kn(ωn − ωjkn)→ +∞.

96 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

Proof. For every α = 1, . . . , N , denote the first element of�α at the right of ωjknby µα/pαn; by (33) it is also the first element of �α at the right of ωn. It is easy tosee that pαn kn as n → ∞.

Notice that for fixed α and n, the function p �→ µ2α − p2ω2

n is decreasing.Therefore,

δ(ωn) = min{µ2α − p2

αnω2n | α = 1, . . . , N}.

Now for every α = 1, . . . , N ,

µ2α − p2

αnω2n = p2

αn(µα/pαn − ωn)(µα/pαn + ωn)

� 2ωjknp2αn(ω

Rjkn

− ωn) kn(ωRjkn

− ωn).

Dividing by kn(ωn − ωjkn) and using (35) we obtain the required estimate. ��We are now ready to state the main result of this section. In its formulation

notice the two assumptions involving the linear, and nonlinear, phenomena dis-cussed earlier.

Theorem 4. Assume that (A) and (W) hold, and suppose, moreover, that

(i) µi is right asymptotically resonant with µj ,(ii) 6µ2

i W(ej )− µ2jW

′′(ej ) · e2i < 0 .

Then, for every choice of hn and kn according to (31), and every choice of ωnaccording to (33), (34) and (35), we have

m(uωn) � νjkn +N + 2

for every n large enough.

Proof. We follow closely the proof of Theorem 3, in the sense that with a seriesof estimates we will show that for n large, the quadratic form J ′′

ωn(uωn) is negative

definite on the subspace ofH given by Ruωn ⊕ Rϕihn ⊕Ejkn ⊕ RN . This is easilyseen to have dimension νjkn +N + 2.

The relation to check is thus

J ′′ωn(uωn) · (αuωn + βϕihn + ϕ + c)2 < 0

for n large, and for every α, β ∈ R, every ϕ ∈ Ejkn , and every c ∈ RN (not allzero).

Writing

J ′′ωn(uωn) · (αuωn + βϕihn + ϕ + c)2

= J ′′ωn(uωn) · (αuωn + ϕ + c)2 + β2J ′′

ωn(uωn) · ϕ2

ihn(36)

+2βJ ′′ωn(uωn) · (ϕihn, ϕ)+ 2βJ ′′

ωn(uωn) · (ϕihn, c)

+2αβJ ′′ωn(uωn) · (uωn, ϕihn),

we see that the first term in the right-hand side has already been estimated in (26),so that we can concentrate on the remaining four terms.

Interaction of Modes and Subharmonic Bifurcation 97

As in the proof of Theorem 3 we divide the computations into a series of lemmas,where the conditions (A) and (W) are similarly taken for granted.

In the proofs of these lemmas, as we did in the previous section to simplifynotation, we will set

εn = ωjkn − ωihn,

�n = ωRjkn − ωjkn,

sn = ωn − ωjkn,

σn = ωn − ωihn .

With these conventions, right asymptotic resonance takes the form εn = o(�n),while the requirements on ωn are expressed by

0 < sn < �n, εn = o(sn), sn = o(�n) .

Moreover, the expansions of �jkn reads

uω = (cj knsn)1/2ϕjkn +O(knsn) as knsn → 0+,

and, owing to Lemma 7,

knsn

hnσn→ µj

µi.

This last relation allows us to substitute hnσn (to any power) with O(knsn) (to thesame power) in the computations below.

Finally, we notice that

Lωnϕihn = h2n(ω

2n − ω2

ihn) ϕihn = hnσn(2µi + hnσn) ϕihn, (37)

which will be used repeatedly.We begin with the estimate of the mixed terms, which are more simple. ��

Lemma 9. As k(ω − ωjk) → 0+ we have

J ′′ωn(uωn) · (ϕihn, ϕ) = O(kn(ωn − ωjkn))||ϕ||2, (38)

J ′′ωn(uωn) · (ϕihn, c) = O(kn(ωn − ωjkn))|c|, (39)

J ′′ωn(uωn) · (uωn, ϕihn) = O((kn(ωn − ωjkn))

2). (40)

Proof. Concerning (38) and (39), notice that for every w ∈ H we have

∫ 2π

0Lωnϕihn · w dt = hnσn(2µi + hnσn)

∫ 2π

0ϕihn · w dt = O(knsn)||w||2,

while using (19) we obtain

∫ 2π

0W ′′(uωn) · (ϕihn, w) dt = O(knsn)||w||2.

98 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

To prove the last statement, since uωn is a solution, and by homogeneity,

J ′′ωn(uωn) · (uωn, ϕihn) = −2

∫ 2π

0W ′(uωn) · ϕihn dt.

Now, using (18), we obtain

∫ 2π

0W ′(uωn) · ϕihn dt

= 1

π2 (cj knsn)32 (W ′(ej ) · ei)

∫ 2π

0cos3(knt) cos(hnt) dt +O((knsn)

2)

Since kn/hn → µj/µi /∈ Q, we know that for all n large, hn �= kn and hn �= 3kn.Therefore,

∫ 2π

0cos3(knt) cos(hnt) dt = 1

4

∫ 2π

0(cos(3knt)+ 3 cos(knt)) cos(hnt) dt = 0

for all large n, and (40) follows. ��We conclude this set of estimates with the analysis of J ′′

ωn(uωn) · ϕ2

ihn. As we

anticipated, this term plays a central role since it is where the nonlinear couplingcoefficient

Wij = 6µ2i W(ej )− µ2

jW′′(ej ) · e2

i (41)

comes into play.

Lemma 10. As n → ∞,

J ′′ωn(uωn) · ϕ2

ihn= Wij

3µjW(ej )(kn(ωn − ωjkn))+ o(kn(ωn − ωjkn)).

Proof. By (37) we have

∫ 2π

0Lωnϕihn · ϕihn dt = hnσn(2µi + hnσn),

while, with the usual arguments,

∫ 2π

0W ′′(uωn) · ϕ2

ihndt

= 1

2

∫ 2π

0W ′′′′(0) · (u2

ωn, ϕ2ihn) dt

= 1

π2 (cj knsn)W′′(ej ) · e2

i

∫ 2π

0cos2(knt) cos2(hnt) dt +O((knsn)

3/2).

Interaction of Modes and Subharmonic Bifurcation 99

Since the integral equals π/2 unless hn = kn, which we can exclude for n large by(32) and (Q), we obtain

J ′′ωn(uωn) · ϕ2

ihn= hnσn(2µi + hnσn)− 1

2π(cj knsn)W

′′(ej ) · e2i

+O((knsn)

3/2).

Recalling that hnσn = knsnµi/µj + o(1) and substituting, we find

J ′′ωn(uωn) · ϕ2

ihn

=(

2µ2i

µj− 1

2πcjW

′′(ej ) · e2i

)knsn + o(knsn)

= 1

3µjW(ej )

(6µ2

i W(ej )− µ2jW

′′(ej ) · e2i

)knsn + o(knsn),

where we have used the definition of cj for the last equality. This completes theproof and the series of estimates we need. ��

For further use, we notice that since the coefficient of the principal part of theterm J ′′

ωn(uωn) · ϕ2

ihnis negative by assumption, for knsn small we can write

J ′′ωn(uωn) · ϕ2

ihn� Nij (knsn),

where

Nij = Wij

6µjW(ej )< 0.

Conclusion of the proof of Theorem 4. We are ready to replace every term in (36)with its asymptotic expansion. For the last four terms we use the preceding lemmas,while for J ′′

ωn(uωn) · (αuωn +ϕ+ c)2 we use the results of Section 3, given in (26),

and particularized here to k = kn, ω = ωn, so that also s = sn.

We obtain, as knsn → 0+,

J ′′ωn(uωn) · (αuωn + βϕihn + ϕ + c)2

� −2µjcj (knsn)2α2− (δ(ω)+O(knsn))||ϕ||22

−πµ1|c|2 +O((knsn)3/2)|α|||ϕ||2

+O((knsn)2)|α||c| +O(knsn)||ϕ||2|c|+β2Nij (knsn)+O(knsn)|β|||ϕ||2+O((knsn)2)|α||β| +O(knsn)|β||c|,

100 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

which we consider as a bilinear form in (|α|, ||ϕ||2, |c|, |β|), represented by thematrix

M4 =

−2µjcj (knsn)2 O((knsn)3/2) O((knsn)

2) O((knsn)2)

O((knsn)3/2) −δ(ω)+O(knsn) O(knsn) O(knsn)

O((knsn)2) O(knsn) −πµ1 O(knsn)

O((knsn)2) O(knsn) O(knsn) Nij (knsn)

.

We now show that M4 is negative definite. Once again we prove that the principalsquare submatrices M� satisfy (−1)� detM� > 0, this time for � = 1, . . . , 4. Wetake advantage of the computations carried out in the previous section, which arevalid for any j, k and ω.

Obviously, − detM1 = 2µjcj (knsn)2 > 0. By the results of the previoussection we have

detM2 = 2µjcj (knsn)2 [δ(ωn)+O(knsn)]

= 2µjcj (knsn)3[δ(ωn)

knsn+O(1)

],

− detM3 = 2πµ1µjcj (knsn)2[ δ(ωn)+O(knsn)]

= 2πµ1µjcj (knsn)3[δ(ωn)

knsn+O(1)

],

which are both positive, for all large n, by Lemma 8.

The evaluation of detM4 is rather boring, though elementary. We expand detM4along the last column denoting by Mικ the matrix obtained from M4 erasing theι-th row and the κ-th column. With straightforward computations we obtain

detM14 = O(knsn)5/2 +O(knsn)

2(δ(ωn)+O(knsn));since δ(ωn) → 0 as n → ∞, we immediately recognize that

detM14 = O(knsn)2.

Next, without even using properties of δ, we obtain

detM24 = O(knsn)3,

and, with the same argument as for M14,

detM34 = O(knsn)3.

Finally, we can read detM44 from the results of the previous section, to find

detM44 = −2πµ1µjcj (knsn)2[ δ(ωn)+O(knsn)] .

Interaction of Modes and Subharmonic Bifurcation 101

We obtain therefore the estimate of detM4:

detM4 = −2πµ1µjcjNij (knsn)3[ δ(ωn)+O(knsn)] +O(knsn)4.

Since this can be written

detM4 = −2πµ1µjcjNij (knsn)4[δ(ωn)

knsn+O(1)

],

using Lemma 8 and recalling that by assumption Nij < 0, we see that detM4 ispositive for all n large enough.

These sign relations prove that the quadratic form J ′′ωn(uωn) is negative definite

on Ruωn ⊕ Rϕihn ⊕ Ejkn ⊕ RN , provided that ωn = ωjkn + sn ∈ (ωjkn, ωRjkn) ischosen according to the procedure described. We have therefore obtained that

m(uωn) � νjkn +N + 2,

and the proof is complete. ��

5. Bifurcation from the left branch

The aim of this section is to carry out an analysis of the Morse index on the leftbranches (those departing from ωih) similar to the one developed in the previoussection for the right branches. Here, instead of repeating computations which areanalogous to those of the previous section, we simply point out the differences, themain one being that we now expect the Morse index to decrease along the branches.

From this point forth, the assumptions (A) and (W) are taken for granted every-where.

Similarly to Definition 3 we now have:

Definition 4. We say that µi is left asymptotically resonant with µj if there existdiverging sequences hn, kn ∈ N such that

ωihn < ωjkn ∀n and limn→∞

ωjkn − ωihn

ωihn − ωLihn

= 0. (42)

It is easy to see that the conclusions of Lemma 6 again hold true with the very sameproof.

Denoting by vω the parametrization of �ihn we notice that for every fixed n,the arguments of Section 3, and in particular Theorem 3, apply to show that

m(vω) = νihn +N + 1

as ω → ω+ihn

, where we recall that νihn = dim span {ϕβq | ωβq > ωihn}.We next show that, provided the correct nonlinear coupling condition is satis-

fied, condition (42) allows us to choose ωn such that

J ′′ωn(vωn) > 0 on Fihn ⊕ Rϕjkn, (43)

102 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

where Fihn = cl span{ϕβq | ωβq < ωihn}, thereby proving that the Morse indexof vω decreases to νihn + N when ω passes from ωihn to ωn. As in the previoussection, the correct choice of ωn is obtained by balancing competing requirements,which now lead to the conditions

ωihn < ωn < ωihn + (ωihn − ωLihn), (44)

ωjkn − ωihn = o(ωn − ωihn), (45)

ωn − ωihn = o(ωihn − ωLihn). (46)

which are satisfied, because of (42), for example, by setting

ωn = ωihn + (ωjkn − ωihn) logωihn − ωLihn

ωjkn − ωihn.

We also notice that (44) implies hn(ωn−ωihn) → 0+, which guarantees the validityof the expansion

vωn = (cihn(ωn − ωihn))1/2ϕihn +O(hn(ωn − ωihn)).

Moreover, in order to prove that J ′′ωn(vωn) · ϕjkn > 0 we need ωn > ωjkn . This last

inequality follows from (45), which yields

ωn − ωjkn = (ωn − ωihn)+ (ωihn − ωjkn) ∼ ωn − ωihn;therefore, ωn − ωjkn is a positive quantity for n large. Another consequence of thechoice of ωn is that the conclusion of Lemma 7 again holds true.

The last remark concerns the relative position of ωn and ωRjkn : depending on

how close ωRjkn is to ωjkn , it may possibly be that ωn > ωRjkn . In such a case thesecondary bifurcation on �ihn may appear after the branch has gone beyond a num-ber of other branches. This is in contrast with the picture for the right branch �jkn ,where the secondary bifurcation appears before the birth of other branches.

The analogue of the function δ used in the previous section is now given by

ρn(ω) = inf{p2(ω2 − ω2αp) | ωαp < ωihn}.

The key estimate is provided by the following lemma.

Lemma 11. If ωn is chosen according to (44) and (46), then, as n → ∞,

ρn(ωn)

hn(ωn − ωihn)→ +∞ .

Proof. For every α = 1, . . . , N , denote the first element of �α at the left of ωihnby µα/qαn. It is easy to check that qαn hn as n → ∞.

Notice that for fixed α and n, the function q �→ q2ω2n − µ2

α is increasing.Therefore,

ρn(ωn) = min{q2αn ω

2n − µ2

α | α = 1, . . . , N}.

Interaction of Modes and Subharmonic Bifurcation 103

Now for every α = 1, . . . , N ,

q2αn ω

2n − µ2

α = q2αn(ωn − µα/qαn)(ωn + µα/qαn)

> 2µαqαn(ωihn − ωLihn) hn(ωihn − ωLihn).

Dividing by hn(ωn − ωihn) and using (46) we obtain the required estimate. ��We are finally ready to state the main result of the section.

Theorem 5. Assume that (A) and (W) hold, and suppose, moreover, that

(i) µi is left asymptotically resonant with µj ,(ii) 6µ2

jW(ei)− µ2i W

′′(ei) · e2j > 0 .

Then, for every choice of hn and kn according to (42), and every choice of ωnaccording to (44), (45) and (46), we have

m(vωn) � νihn +N

for every n large enough.

Proof. We only sketch the proof, since it depends on the very same arguments usedin the previous section. Denoting by ψ the generic element of Fihn , we can checkthat

J ′′ωn(vωn) · ψ2 �

[ρn(ωn)+O(hn(ωn − ωihn))

] ||ψ ||22, J ′′ωn(vωn) ·

(ψ, ϕihn) = O(hn(ωn − ωihn)).

Moreover, using (45), we obtain

J ′′ωn(vωn) · ϕ2

jkn= Wji

3µiW(ei)(hn(ωn − ωihn))+ o(hn(ωn − ωihn)),

where

Wji = 6µ2jW(ei)− µ2

i W′′(ei) · e2

j > 0.

The positivity of J ′′ωn(vωn) on Fihn ⊕ Rϕjkn then follows from Lemma 11. ��

6. Proof of the main results

In this section we deduce the results stated in the Introduction from Theorems 3and 4, already proved. To this aim we will use the structure of the �j ’s describedin Section 2, to which we refer for notation and properties.

104 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

Proof of Theorems 1 and 2. For definiteness, we prove Theorem 1; the same argu-ment can be used for Theorem 2.

To say that the theorem is false means that there exist i, j such that µi is rightasymptotically resonant with µj , but there are no bifurcation points on �jkn forω ∈ (ωjkn, ωRjkn). Here kn is the sequence provided by Definition 3. A well knownresult of Kielhöfer ([11], see also [4]) implies that if at some ω0 there results

limω→ω−

0

m(uω) �= limω→ω+

0

m(uω),

then (ω0, uω0) is either a turning point or a bifurcation point. In our case the curve�jkn is free of turning points in (ωjkn, ω

Rjkn) by construction. Moreover since the

dependence on ω is analytic, the Morse index along �jkn is locally constant whereit is defined. For all ω close to ωjkn we havem(uω) = νjkn +N+1, by Theorem 3;however, by Theorem 4 we know that at some ωn ∈ (ωjkn, ω

Rjkn) there results

m(uω) � νjkn + N + 2. Therefore, there must be a point in (ωjkn, ωn) where theleft and right limits of m(uω) are different. This point is thus a bifurcation point,contradicting the assumption. ��

We now turn to the proof of the Birkhoff-Lewis type result, namely Corollary 1;this is where the subharmonic nature of the secondary bifurcations becomes mani-fested. In what follows we denote the minimal period of a continuous nonconstantperiodic function f by T (f ).

Proof of Corollary 1. We show the proof for right asymptotic resonance only. Thenotation and the properties listed in Section 2 will be used repeatedly, and we set

�jk = {±uω | uω ∈ �jk}.Let i, j be such thatµi is right asymptotically resonant withµj , and let kn be the cor-responding divergent sequence as defined in Definition 3. By Theorem 1 we knowthat for every n (up to subsequences) there is a bifurcation point (ωn, un) ∈ �jkn ,with ωn satisfying

kn(ωn − ωjkn) → 0. (47)

This means that for every n we can choose

(θn, vn) ∈ Br/kn(ωjkn)× BknR(0)

such that

(i) (θn,±vn) /∈ �jkn ;(ii) vn is a solution of problem (Pθn);

(iii) ||vn − un|| + kn|θn − ωn| → 0.

Moreover, since un has minimal period 2π/kn by construction (it lies on �jkn ), itis easy to see that (θn, vn) can be taken so close to (ωjkn, un) that

T (vn) = pnT (un) = pn2π

kn(48)

for some pn ∈ N+. Notice that since vn is 2π periodic, pn must divide kn.

Interaction of Modes and Subharmonic Bifurcation 105

We prove that the sequence xn of solutions of (1) defined by xn(t) = vn(θnt)

satisfies the conclusions of the corollary.First, we notice that

||xn||∞ = ||vn||∞ � ||un||∞ + ||vn − un||∞ → 0,

via the usual expansion (30), since kn(ωn − ωjkn) → 0; therefore, xn → 0 in C2.We now show that T (xn) → ∞. If this is false, then up to subsequences

T (xn) = 1

θnT (vn) = 2πpn

θnkn

is bounded. By (47) we see that knωn → µj , so that by our choice of θn we deducethat knθn → µj ; hence pn must be bounded. Still passing to subsequences, ifnecessary we can assume that pn = p ∈ N+ for all n.

Consider, then, the functions

wn(t) = vn

(p

knt

).

These are 2π periodic functions by (48) and are solutions to problems (Pθnkn/p),with θnkn

p→ µj

p. Plainly, ||wn|| → 0. Thus, we see that

(θnkn

p,wn

)→

(µj

p, 0

)in R ×H.

By the uniqueness properties in Proposition 1, it must be(θnkn

p,wn

)∈ �jp

for every n large enough. This means, scaling (recall that p divides kn), that(θn, vn) ∈ �jkn , which contradicts the initial choice. Therefore, pn → ∞, andso does T (xn). ��

7. The notion of asymptotic resonance

This section is devoted to the study of the abstract notions of right and leftasymptotic resonance, with the aim of deriving necessary and sufficient conditionsfor their validity. These conditions will be obtained with the aid of some number-theoretical arguments. In the next section we will use the conditions to determinecompletely the set of asymptotically resonant frequencies in the Fermi-Pasta-Ulamproblem.

For the time being we consider a set of N � 2 positive frequencies

µ1 < · · · < µN,

under the only assumption (Q), namely that they are pairwise independent over therationals.

We consider also a stronger notion, that combines those introduced in Defini-tion 3 and 4. In its statement, a ∧ b = min(a, b).

106 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

Definition 5. We say thatµi is asymptotically resonant withµj if there exist diverg-ing sequences hn, kn ∈ N such that

ωihn < ωjkn ∀n and limn→∞

ωjkn − ωihn(ωRjkn − ωjkn

)∧(ωihn − ωLihn

) = 0. (49)

Roughly speaking, the definition asserts that the characteristic values in the interval[ωihn, ωjkn ] are (asymptotically) isolated from the remaining part of �; this is thecase in most of the situations we will describe later on.

The first lemma rewrites the various notions of asymptotic resonance in a moresuitable form for the computations.

Lemma 12. Let N � 3 and assume that (Q) holds. Let two indices i �= j begiven. Then µi is right asymptotically resonant with µj if, and only if, there existsa sequence of positive integers hn such that for all α �= i, j ,

limn→∞

{hnµj

µi

}

{knµα

µj

} = 0, (50)

where

kn =⌊hnµj

µi

⌋. (51)

Left asymptotic resonance is equivalent to require the existence of a sequence hnsuch that, for every α �= i, j ,

limn→∞

{hnµj

µi

}

1 −{hnµα

µi

} = 0, (52)

while for asymptotic resonance the two above limit conditions must hold with thesame sequence hn.

Notice that the validity of (50) or (52) yields in particular

limn→∞

{hnµj

µi

}= 0, (53)

which, contrary to (50) and (52), makes sense also for N = 2. It will be clear fromthe proof that when N = 2 the latter condition is equivalent to the fact that µi isasymptotically resonant with µj , in any of the given definitions. Since µj/µi isirrational, this condition can always be satisfied along suitable sequences of inte-gers. Summing up, when N = 2 and the two frequencies have an irrational ratio,then each is asymptotically resonant with the other.

Interaction of Modes and Subharmonic Bifurcation 107

Proof. We give the details only for right asymptotic resonance, the approach beingthe same in other cases.

For a given µi/h, let µj/kh be the first element of �j at its right; because of(Q), it must be

kh =⌊hµj

µi

⌋.

Likewise, defining

pαh ={⌊khµαµj

⌋if α �= j,

kh − 1 if α = j,

we see that µα/pαh is the smallest element of �α at the right of µj/kh, so that

(µj

kh

)R= minα=1,...,N

µα

pαh.

If we now introduce the distances

εh = µj

kh− µi

hand �αh = µα

pαh− µj

kh,

then Lemma 6 states that µi is asymptotically resonant with µj exactly when

εh

�αh→ 0 ∀α = 1, . . . , N (54)

along some suitable diverging sequence h = hn. This suggests that the proof of thestatement depends on the asymptotic behavior of εh and �αh as h → +∞, whichwe now determine. For clarity we set

µα = µα/µi and µα = µα/µj .

First notice that since kh ∼ hµj , we have

εh = µi

h kh

(hµj − kh

) ∼ µ2i

µj

1

h2

{hµj

}and �jh = µj

kh (kh − 1)∼ µ2

i

µj

1

h2 ,

which yield

εh

�jh∼ {

hµj}. (55)

By (54) this shows that (53) holds.

When α �= j , similar arguments assure that pαh ∼ khµα ∼ hµα , so that

�αh = µj

kh pαh(kh µα − pαh) ∼ µ2

i

µα

1

h2{khµα} .

108 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

Hence,

εh

�αh∼ µα

{hµj

}

{khµα} ∀α �= j,

and, again by (54), we see that (50) is satisfied. This proves the only if part of thestatement. Concerning the if part, assume that (50) holds, and notice that togetherwith the previous formula it yields (54) for α �= i, j . Moreover, in particular{hµj

} → 0 which, owing to (55), establishes the validity of (54) also for α = j .Looking now at α = i, observe that

khµi = ⌊hµj

⌋µi = h− µi

{hµj

},

eventually yields {khµi} = 1 − µi{hµj

}. This implies that �ih ∼ µi/h

2, andthen

εh

�ih∼ µi

{hµj

} → 0,

proving the validity of (54) also for α = i. ��Some comments about condition (50) are in order. Looking more closely at the

denominator in (50), notice that (51) yields

knµα

µj= hn

µα

µi− µα

µj

{hnµj

µi

}=

{hnµα

µi

}+

⌊hnµα

µi

⌋− µα

µj

{hnµj

µi

},

which in turn implies the congruence{knµα

µj

}=

{hnµα

µi

}− µα

µj

{hnµj

µi

}mod (1) (56)

for every α = 1, . . . , N . In order to compute the fraction in (50), we have to deducefrom (56) some exact equality on the fractional parts. This is done in the next lemmafor a particular case, and it leads to a useful, sufficient condition for (50).

Lemma 13. Assume that (Q) holds and that for some α �= i, j

limn→∞

{hnµj

µi

}

{hnµα

µi

} = 0 (57)

along a diverging integer sequence hn. Then condition (50) holds for the same α.

Proof. Because of (57), the corresponding right-hand side in (56) can be rewrittenas

{hnµα} − µα{hnµj

} = {hnµα} (1 − εn),

where 0 < εn → 0 (the bar and the tilde have the same meaning as in the previouslemma). Hence, for large n, this quantity lies in (0, 1), so that the equality in (56)holds without the mod (1) limitation. This yields {knµα} ∼ {hnµα}, proving thatcondition (50) is satisfied for the considered α. ��

Interaction of Modes and Subharmonic Bifurcation 109

The previous arguments suggest that the classical Kronecker Theorem on Dio-phantine approximation is the main tool when looking for asymptotic resonances.Hereafter we present a slightly refined version of this theorem, in a form suited forour purposes.

Theorem 6 (Kronecker). Assume that the real numbers 1, α1, . . . , αM are lin-early independent over Z, and that P is a given natural number. Then, for everyc1, . . . , cM ∈[ 0, 1] and every p0, p1, . . . , pM ∈ N there exists an integer se-quence hn such that

limn→∞

{hnαj

} = cj ∀j,hn = p0 mod (P ) ∀n,⌊hnαj

⌋ = pj mod (P ) ∀n, ∀j.The conclusion about the fractional parts is the standard one (Theorems 442–

443 of [8]). Notice that by a diagonal process we may also prescribe that in eachlimit the convergence holds from above (or from below). We will use this furtherfreedom in the proof of the next proposition. By similar arguments, for every j , wemay also decide at which rate we want cj to be approximated by

{hnαj

}; this will

be used in the proof of Proposition 3.The new part in the statement is the prescription on the integer parts and it will

be used in Proposition 4. As it will be clear after the proof, the above theorem is infact equivalent to the classical one.

Proof. We will prove the (new part of the) statement for 0 � cj < 1 only, sinceby a standard diagonal process we can also recover the case cj = 1.

Assume now that hn = p0 + P�n for suitable integers �n, and then use theclassical Kronecker Theorem to choose �n such that

limn→∞

{�nαj

} ={cj + pj − p0αj

P

}+∀j.

Notice that for every j ,

hnαj = (p0 + P�n)αj = cj + P

({�nαj

} −{cj + pj − p0αj

P

})+ pj mod (P ),

so that the choice of �n yields

{hnαj

} = cj + P({�nαj

} −{cj+pj−p0αj

P

})→ cj ,⌊

hnαj⌋ = pj mod (P ),

which prove the claim. ��Returning to asymptotic resonances, in the next proposition we deal with the

particular case in which the frequencies are independent over the integers as awhole, not only pairwise as required by (Q). Although this covers a generic choiceof frequencies, we will see in the next section that this is not always the case forthe Fermi-Pasta-Ulam model.

110 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

Proposition 2. Assume that µ1, . . . , µN are linearly independent over Z. Thenany µi is asymptotically resonant with any other µj with j �= i.

Proof. Using the same notation as in the proof of Lemma 12, we see that thenumbers µ1, . . . , µN are also independent over Z. Since by construction µi = 1,the Kronecker Theorem allows us to choose an integer sequence hn such that, forinstance,

{hnµj

} → 0 and {hnµα} → 1/2 ∀α �= i, j

as n → +∞, the second condition to be considered only whenN � 3. To concludewe apply Lemma 12, via Lemma 13. ��

When the frequencies are dependent over Z, we cannot expect that the sameresult is true. Consider, for instance, a case where (Q) holds, preventing dependenceon pairs, but

µα = pµi + µj (58)

for a given choice of indexes i, j, α and some (nonzero) p ∈ Z. The equality{hnµα

µi

}=

{hnµj

µi

},

holds for every choice of the integer hn. Looking now at kn defined by (51), noticethat the right-hand side of (56) takes the form

{hnµα

µi

}− µα

µj

{hnµj

µi

}=

(1 − µα

µj

){hnµj

µi

}= −p µi

µj

{hnµj

µi

}.

Choose the integer sequence hn such that (53) is satisfied. If p < 0, we obtain{knµα

µj

}= −p µi

µj

{hnµj

µi

},

which prevents condition (50) being satisfied, so that µi cannot be right asymptot-ically resonant with µj .

The next proposition states that the presence of ternary relations over Z such as(58), namely relations involving exactly three frequencies are the only obstructionto asymptotic resonance.

Proposition 3. Assume that (Q) holds, and let i �= j be fixed. Suppose that no ter-nary relation over Z involvesµi andµj at the same time. Thenµi is asymptoticallyresonant with µj .

Proof. Denote the dimension of the Q-vector space generated by the Q-relationsamong the frequencies µ1, . . . , µN by d . Then d of them, say µα where α ∈ A,may be expressed as a function of the remaining N − d, say µβ where β ∈ B.Namely,

µα =∑

β∈Bcαβ µβ ∀α ∈ A, (59)

Interaction of Modes and Subharmonic Bifurcation 111

for some suitable cαβ ∈ Q. As a consequence of (59), the frequencies µβ ’s mustbe linearly independent over Q. The idea is now to apply the Kronecker Theoremto obtain some control on the µαs.

First, we notice that because of (Q), it is not restrictive to assume that i, j ∈ B.Indeed, assume for instance that i ∈ A and j ∈ B; in the relation correspondingto α = i in (59), because of µi/µj �∈ Q, it must be cil �= 0 for some l ∈ B \ {j}.Hence with a trivial manipulation, we can take µl away from all the right-handsides, the final effect being to swap the position of l and i. Similar arguments applystarting from i, j ∈ A.

Recalling that there are no ternary relations involving µi and µj , we also seethat the set B cannot reduce to {i, j}; more precisely, for every α ∈ A the set

Bα = {β ∈ B \ {i, j} | cαβ �= 0}must be nonempty. We denote its smallest element by βα .

Rewriting (59) using integer coefficients, namely as

aαµα =∑

β∈Bbαβ µβ ∀α ∈ A, (60)

with aα, bαβ ∈ Z, we can assume without loss of generality that for every α ∈ A,aα > 0 and the integer numbers in the set {aα} ∪ {bαβ | β ∈ B} have no commondivisors.

As a consequence of (60), if hn is any integer sequence, then

aα {hn µα} =∑

β∈B\{i}bαβ

{hn µβ

} +∑

β∈Bbαβ

⌊hn µβ

⌋mod (aα)

for every α ∈ A; here the bar denotes, as usual, the division by µi .

Now choose the sequence hn such that

{hn µj

} ∼ 1

nj+N(61)

and

{hn µβ

} ∼ 1

nβ∀β ∈ B \ {i, j} (62)

as n → +∞. This is possible due to the Kronecker Theorem, since the µβ ’s are Qindependent and µi = 1.

Notice that condition (52) is satisfied trivially inB \{i, j}, while condition (57)(and hence (50)) is satisfied because the fractional part in (61) vanishes faster thanall the ones in (62).

To conclude the proof we have to show that the same occurs for every µα withα ∈ A. To this aim notice that by definition of βα , we have bαβα �= 0 and

∑

β∈B\{i}bαβ

{hn µβ

} ∼ bαβα

nβα.

112 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

Moreover, up to subsequences, there is an integermα with 0 � mα < aα , such that

∑

β∈Bbαβ

⌊hn µβ

⌋ = mα mod (aα)

for every α ∈ A and every n. If mα > 0 then

{hn µα} = 1

aα

(mα + bαβα

nβα(1 + o(1))

),

so that {hn µα} → mα/aα > 0. The previous relation also takes place whenmα = 0 and bαβα > 0, proving that in this case

{hn µα} = bαβα

aα nβα(1 + o(1)).

Finally, if mα = 0 but bαβα < 0, we have

{hn µα} = 1 + bαβα

aα nβα(1 + o(1)) → 1.

In all these cases, conditions (57) and (52) follow immediately. ��Next we discuss the effect of ternary relations. As an example we notice that

although relation (58) is poor for p < 0 (it prevents right asymptotic resonance),a change in the coefficient of µα may remove the obstruction to the asymptoticresonance of µi with µj for all p. For example, it can be proved that the ternaryrelation 2µα = pµi + µj , with a nonzero p ∈ Z, is always harmless, for everynotion of asymptotic resonance.

To settle the problem in general we begin by noticing that because of (Q), givenany three different frequencies, up to multiplication by integers, there is at mostone ternary relation involving them.

Assume, then, that

aαµα = bαµi + cαµj

aα > 0, (aα, bα, cα) = 1α ∈ � (63)

is the complete list of the ternary relations with integer coefficients involving µiand µj at the same time, normalized to exclude repetitions. Notice that all thefrequencies µα in the left-hand side must be distinct.

The next proposition is the main result of this section; it provides a simplecriterion for asymptotic resonance.

Proposition 4. Assume that (Q) holds, and that (63) lists all the normalized ter-nary relations involving µi and µj . Then µi is right asymptotically resonant withµj if, and only if, there exist integer numbers p, q such that

bαp + cαq �= 0 mod (aα) (64)

Interaction of Modes and Subharmonic Bifurcation 113

for every α ∈ � for which

bα < 0 and cα > 0. (65)

For left asymptotic resonance the same condition is to be considered for the α ∈ �for which

cα < 0, (66)

while for asymptotic resonance the α’s to be considered are those which satisfyeither (65) or (66).

In case no relation in (63) satisfies (65) or (66), the corresponding conclusionsare automatically true.

Proof. We prove only the statement concerning right asymptotic resonance, sincebecause of the form of the denominator in (50), it is the most delicate. The othercases follow similarly. To prove the if part, we modify the proof of the previousproposition, maintaining the same notation. Consider then the list of relations (60)and the associated partition A, B of the set {1, . . . , N}. Recall that this is a veryspecial base for the relations, since it is adapted to the choice i, j ∈ B.

We claim that, as a consequence, all the relations listed in (63) must appear inthe list (60). Now fix a ternary relation in (63), corresponding to some α0 ∈ �.Since i, j ∈ B and the µβs with β ∈ B are independent over the integers, we musthave α0 ∈ A. Now if the relation in (60) corresponding to α0 does not coincidewith the one fixed in (63), then a suitable linear combination of the two yields anontrivial relation among the µβs, which is impossible.

Now choose an integer sequence hn according to (61) and (62).We have already proved that this choice rules out the relations in (60) corre-

sponding to the α’s lying B \ {i, j} or in A \ �. It remains to look at �, which wepartition into the three subsets

�1 = {α ∈ � | cα < 0},�2 = {α ∈ � | cα > 0, bα < 0},�3 = {α ∈ � | cα > 0, bα > 0}.

First we show that �1 is ruled out by (53), which holds owing to choice of hnsatisfying (61). Indeed, by using now standard computations, for every α ∈ � wehave

aα {hn µα} = cα{hn µj

} + bαhn + cα⌊hn µj

⌋mod (aα)

= cα{hn µj

} +mnα mod (aα)

for a unique integer satisfying 0 < mnα � aα . As usual, the bar denotes the divisionby µi . Owing to (61), when α ∈ �1 this yields (for large n) the exact equality

{hn µα} = 1

aα

(mnα + cα

{hn µj

}).

114 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

As a consequence,

lim infn→+∞ {hn µα} � 1

aα> 0,

which proves condition (57) and then the claim.

We now use condition (64) to rule out the relations corresponding to �2 ∪ �3.DefineP = ∏

α∈�2∪�3aα and assume that, in addition to (61) and (62), the sequence

hn also satisfies

hn = p mod (P ) and⌊hn µj

⌋ = q mod (P ), (67)

where the integers p and q are exactly those considered in (64). This is alwayspossible because of Theorem 6.

With standard manipulations, we see that

aα {hn µα} = cα{hn µj

} + bαp + cαq mod (aα)

holds for every α ∈ �2 ∪ �3. Here, we used the fact that any equality modulo Pis also true modulo aα for every α ∈ �2 ∪ �3. Denote by mα the unique integersatisfying

0 � mα < aαmα = bαp + cαq mod (aα).

Since cα > 0 and (61) holds, the exact equality

{hn µα} = 1

aα

(mα + cα

{hn µj

})

must be true for large values of n.Now, if mα > 0 for some α, then we obtain {hn µα} → mα/aα > 0, which

rules out the corresponding relations by means of (57).

Because of (64) and (65), this is certainly the case at least for every α ∈ �2.Thus it remains to consider only the α ∈ �3 for which mα = 0, in which case

{hn µα} = cα

aα

{hn µj

}. (68)

Condition (57) is not satisfied, and we have to look at the most general condition(50). The right-hand side of (56) is

{hnµα} − µα{hnµj

} = cαµj − aαµα

aαµj

{hnµj

}mod (1),

= −µi bαaα

{hnµj

}mod (1),

where the tilde denotes the division by µj .

Interaction of Modes and Subharmonic Bifurcation 115

Since bα > 0 due to α ∈ �3, from (56) we obtain that the equality

{knµα} = 1 − µibα

aα

{hnµj

},

which must hold for large n. Hence {knµα} → 1, so that condition (50) is satisfied.We turn to the only if part of the statement. Let hn be an integer sequence satisfying(53). Up to subsequences we may certainly assume that

hn = p∗ mod (P ) and⌊hn µj

⌋ = q∗ mod (P )

hold for some suitable integers p∗ and q∗, where, as above, P = ∏α∈�2∪�3

aα .Assume now that (64) is false, and choose α ∈ �2 such that bαp∗ + cαq∗ = 0mod(aα). Since

aα {hn µα} = cα{hn µj

} + bαhn + ⌊hn µj

⌋mod (aα)

and cα > 0, we deduce that the equality (68) holds also in this case. Repeating thesame arguments as above, with bα < 0, now leads to the equality

{knµα} = −µi bαaα

{hnµj

},

which prevents condition (50) being satisfied. Hence µi cannot be asymptoticallyresonant with µj , and the proof is complete. ��

8. Asymptotic resonance in the Fermi-Pasta-Ulam problem

In this section we specialize the general results on asymptotic resonance to thecase of the Fermi-Pasta-Ulam model withN particles and fixed ends considered inthe introduction. In this case:

Ax · x =N∑

j=0

(xj+1 − xj )2,

(where x0 = xN+1 = 0) and it is well known (see for instance [19]) that theeigenvalues µ2

j of A and the corresponding eigenvectors ej satisfy

µj = 2 sin

(jπ

2(N + 1)

)and (ej )k = sin

(kjπ

N + 1

)(69)

for j, k = 1, . . . , N . In this section we are interested in theµj ’s only, and in partic-ular in the properties of the Q-vector space of their Q-relations, which we denoteby MN .

The next lemma characterizes the very favorable situation whereµj ’s are ratio-nally independent as a whole (namely MN is trivial); though we are really inter-ested in ternary relations only, we report the situation for sake of completeness. Therational independence is due to Hemmer [9] and it is well known in connection tothe Fermi-Pasta-Ulam problem. Since the proof is unpublished, for the convenienceof the reader we present here a slightly simplified version.

116 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

Lemma 14. MN is trivial if and only ifN+1 is a power of 2 or a prime; otherwiseits dimension is N − ϕ(2(N+1)), where ϕ is the Euler function.

We recall that the Euler function is defined on the natural numbers by

ϕ(n) = # {1 � a � n | (a, n) = 1}.Notice that ϕ(1) = 1 and ϕ(n) < n for all n > 1. If n is prime, ϕ(n) = n − 1;otherwise the inequality ϕ(n) < n − 1 holds. The practical computation of ϕdepends on the well-known properties

{ϕ(ab) = ϕ(a) ϕ(b) if (a, b) = 1ϕ(ph+1) = ph(p − 1) if p is prime, h � 0,

(70)

where a, b and h are all naturals (see [8], Chapter V).The other object we will use in the proof is �n(x), the cyclotomic polynomial

of order n. We recall that, by definition, it is the element of minimal degree in Q[ x]having e2πi/n as a root, normalized by taking the leading coefficient, equal to one.It is well known (see Theorem 4.10 in [15]) that it can be represented as

�n(x) =∏

(a,n)=1

(x − e2πia/n), (71)

so that its degree is exactly ϕ(n).

Proof. Let

N∑

j=1

ajµj = 0

be a nontrivial element of MN , and let m be the smallest index such that aj = 0when j > m. If we set ζ = exp(2πi/4(N+1)), then µj is (the double of) theimaginary part of ζ j . Hence, defining

G(x) =m∑

j=1

aj (xm+j − xm−j ) ,

we can rewrite our relation in the more convenient polynomial formG(ζ) = 0. Ofcourse, G ∈ Q[ x] and, since am �= 0 by definition of m, its degree is ∂G = 2m.Moreover, the manifest symmetry of G yields the formula

x∂GG(1/x) = −G(x).Since G(ζ) = 0, the polynomial G(x) must be divisible by �4(N+1)(x). We canthen write G(x) = f (x)�4(N+1)(x) for some f ∈ Q[ x]. Using the properties ofG, it is not difficult to check that

∂f = 2m− 2 ϕ(2(N+1)), (72)

x∂f f (1/x) = −f (x). (73)

Interaction of Modes and Subharmonic Bifurcation 117

Formula (72) depends on the fact that ϕ(4(N+1)) = 2ϕ(2(N+1)), which followsfrom (70) after decomposing N+1 into its odd and even parts. Concerning (73),we use the representation (71) to prove that �4(N+1)(x) has the same symmetry asG but for the sign.

The key point is now to realize that necessarily ∂f � 2; indeed, f cannot beconstant because of (73), and it has even degree due to (72). We conclude thatN � m > ϕ(2(N+1)). In case

N � ϕ(2(N+1)), (74)

we obtain a contradiction, and therefore MN must be trivial.

On the other hand, if N > ϕ(2(N+1)), then for every r = 1, . . . , N −ϕ(2(N+1)) we consider the polynomial f ∈ Q[ x] defined by

f (x) = x2r − 1,

which has even degree and satisfies (73). Then, by simply reversing the previ-ous construction, we obtain a nontrivial element in MN . Moreover, since everypolynomial f (x) of even degree and satisfying (73) uniquely decomposes as

f (x) =∂f/2∑

j=1

cj x∂f/2−j (x2j − 1)

for suitable cj ∈ Q, the relations obtained in this way are clearly a base for MN .This proves the claim concerning the dimension of MN .

To conclude the proof, we show that (74) holds if, and only if,N + 1 is a primeor a power of 2. To this aim, begin by writing

N + 1 = 2hD,

with h and D integers and D odd; of course, if h = 0 then necessarily D > 1.Using (70), condition (74) can be rewritten as

D − 1

2h� ϕ(D). (75)

If h > 0 this condition holds if, and only if, ϕ(D) � D, which is equivalent toD = 1; this is the case where N + 1 is a power of 2.

On the other hand, if h = 0, then (75) can be rewritten as ϕ(D) � D−1 which,since D > 1, is equivalent to saying that D = N + 1 is prime. ��

Although Hemmer’s result covers completely some particular cases of impor-tance, and even provides an algorithm to construct a base for MN , its use is notreally advisable to handle the general case, mainly because it can be very difficult(especially for large N ) to isolate ternary relations from a base for MN .

118 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

An efficient alternative is provided by a result of Conway and Jones (see also [13,14]). Indeed, in [5], the authors set up an algorithmic way to solve the classical prob-lem of the vanishing sums of roots of unity, and make the explicit computations forlow-order sums. As a by-product of Theorem 7 in [5], it turns out that

{sin(π/3 + θ)− sin(π/3 − θ)− sin(θ) = 0, 0 < θ < π/6,sin(3π/10)− sin(π/10)− sin(π/6) = 0

(76)

is the complete list (up to multiplication by rationals) of the Q relations involvingup to three sines, in which the all the angles lie in (0, π/2) and are, moreover,rational with π .

The frequencies µ1, . . . , µN considered in (69) fit into this framework, forevery dimension N . As a first relevant consequence, there are no relations involv-ing two frequencies only, so that condition (Q) holds for every N .

Moreover, the result of Conway and Jones permits the complete classificationof the ternary relations among frequencies, for everyN . Straightforward computa-tions show that the µj ’s may satisfy the first relation in (76) only if 3 dividesN+1,and the second one when also 5 does. This yields the following proposition.

Proposition 5. There are no ternary relations among the frequencies in (69), unless3 | (N+1). In this case moreover, if 5 � (N+1) then all the ternary relations aredescribed (up to multiplication by rationals) by

µ 23 (N+1)+γ − µ 2

3 (N+1)−γ − µγ = 0, (77)

where the integer γ ranges over 1, . . . , (N+1)/3 − 1. Finally, if 5 |(N+1) also,then the special relation

µ 35 (N+1) − µ 1

5 (N+1) − µ 13 (N+1) = 0 (78)

must be added to the previous list.

We are finally ready to describe all the asymptotic resonances among the fre-quencies µj ’s, merging the above characterization together with the results provedin the last section.

For every N ∈ N+ and i ∈ {1, . . . , N} we can define the sets

�R(N, i) =

{i, 2

3 (N+1)+ i}

if 1 � i <(N+1)

3 , i �= (N+1)5

{i, 2

3 (N+1)+ i, 35 (N+1)

}if i = (N+1)

5

{i, 3

5 (N+1)}

if i = (N+1)3

{i, 4

3 (N+1)− i}

if (N+1)3 < i � 2(N+1)

3

{i} if 2(N+1)3 < i � N

Interaction of Modes and Subharmonic Bifurcation 119

and

�L(N, i) =

{i} if 1 � i � 2(N+1)3 , i �= 3(N+1)

5

{i, 1

5 (N+1), 13 (N+1)

}if i = 3(N+1)

5

{i, 4

3 (N+1)− i, i − 23 (N+1)

}if 2(N+1)

3 < i � N.

Theorem 7. Letµ1, . . . , µN be the frequencies of theN particle Fermi-Pasta-Ulamproblem with fixed ends. If N + 1 is not a multiple of 3, then µi is asymptoticallyresonant with µj for every j �= i. If N + 1 is a multiple of 3, then

(a) µi is right asymptotically resonant with µj if and only if j �∈ �R(N, i).(b) µi is left asymptotically resonant with µj if and only if j �∈ �L(N, i).(c) µi is asymptotically resonant with µj if and only if j /∈ �R(N, i) ∪ �L(N, i).Notice that when 3 | (N + 1), some of the forbidden values for j are integer onlywhen also 5 | (N + 1): to exclude them also when 5 � (N + 1) is just a trick todecrease the number of different cases to be considered.

Proof. When 3 does not divide N+1, the result follows from Propositions 3 and5. Let us then assume that 3 dividesN + 1 and apply Proposition 4 (in this case weonly prove the part concerning right asymptotic resonance).

Notice that in every ternary relation the coefficients are unitary. Hence, condi-tion (64) cannot be satisfied, and we have an obstruction to the asymptotic resonanceeach time there are ternary relations of the type (63) satisfying (65). Now, the list(63) depends, of course, on the choice of the indices i �= j . To prove the state-ment we must distinguish many different cases, each time deciding whether (65) isfulfilled or not.

First assume that

1 � i <1

3(N+1),

and notice that we can fit one of the ternary relations (77) only if γ = i. Moreover,if we set j = 2(N+1)/3+ i, then (65) is satisfied, preventing the asymptotic reso-nance ofµi withµj . On the contrary, (65) is not verified if we set j = 2(N+1)/3−i.This explains the first row in the definition of �R(N, i).

When also 5 divides N+1, then (78) comes into play, the only possible choicefor i being i = (N+1)/5. To explain the second row in �R(N, i), we use the verysame arguments as above to show that j = 3(N+1)/5 produces an obstruction,while j = (N+1)/3 does not. All the remaining cases can be worked out similarly,we thus leave these proofs for the reader to complete. ��

9. Nonlinear coupling in the Fermi-Pasta-Ulam problem

In this section we investigate the validity of the nonlinear coupling conditionused in Theorem 4 in the case of the Fermi-Pasta-Ulam β-model.

120 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

Without loss of generality we take β = 1, so that the nonlinear potential is

W(x) = 1

4

{x4

1 + (x2 − x1)4 + · · · + (xN − xN−1)

4 + x4N

}.

We also recall that in the case of the Fermi-Pasta-Ulam model, the matrixA admitsN eigenvalues µ2

j , with corresponding eigenvectors ej given by (69).Our aim is to describe for which N, i, j , the number

Wij = 6µ2i W(ej )− µ2

jW′′(ej ) · e2

i ,

is negative. In this case the nonlinear coupling condition between µj and µi issatisfied.

To begin with we notice that if we set, in agreement with (69), (ej )0 = 0 and(ej )N+1 = 0, then we can write more conveniently

W(ej ) = 1

4

N∑

k=0

((ej )k+1 − (ej )k

)4,

and

W ′′(ej ) · e2i = 3

N∑

k=0

((ej )k+1 − (ej )k

)2((ei)k+1 − (ei)k)

2 .

The computation of the numbers Wij requires only elementary algebraic manipu-lations. We begin with the following lemma, which will be used repeatedly; in itsstatement, δ denotes the Kronecker delta. We omit the proof, since it can be easilyobtained in many ways.

Lemma 15. Let q ∈ Z be such that −(N − 1) � q � 2(N + 1) and define

SN(q) =N∑

k=0

cos

((2k + 1)

qπ

N + 1

). (79)

Then SN(q) = (N + 1)(δq,0 − δq,N+1).

In particular, the cases we will use, with 1 � i, j � N , are

SN(j) = 0,

SN(2j) = −(N + 1)δ2j,N+1,

SN(j + i) = −(N + 1)δj+i,N+1,

SN(j − i) = (N + 1)δj−i,0 = (N + 1)δj,i .

We now turn to the computation of W(ej ) and W ′′(ej ) · e2i .

Lemma 16. We have

W(ej ) = N + 1

32µ4j (3 − δ2j,N+1). (80)

Interaction of Modes and Subharmonic Bifurcation 121

Proof. By an elementary trigonometric identity and the definition of µj , we canwrite

(ej )k+1 − (ej )k = sin

((k + 1)

jπ

N + 1

)− sin

(kjπ

N + 1

)

= µj cos

((k + 1

2

)jπ

N + 1

).

Therefore,

W(ej ) = 1

4µ4j

N∑

k=0

cos4((k + 1

2

)jπ

N + 1

).

Now, cos4 θ = 38 + 1

2 cos(2θ) + 18 cos(4θ), so that with the help of the previous

lemma,

W(ej ) = 1

4µ4j

N∑

k=0

(3

8+ 1

2cos

((2k + 1)

jπ

N + 1

)+ 1

8cos

((2k + 1)

2jπ

N + 1

))

= 1

4µ4j

(3

8(N + 1)

)+ 1

2SN(j)+ 1

8SN(2j)

)= N + 1

32µ4j (3 − δ2j,N+1).

��Lemma 17. We have

W ′′(ej ) · e2i = 3(N + 1)

8µ2i µ

2j (2 + δj,i − δj+i,N+1). (81)

Proof. By the same argument used at the beginning of the proof of the previouslemma, it is easy to see that

W ′′(ej ) · e2i = 3µ2

i µ2j

N∑

k=0

cos2((k + 1

2

)jπ

N + 1

)cos2

((k + 1

2

)iπ

N + 1

).

Applying the identity 2 cos2 θ = 1 + cos(2θ) we obtain, with the same argumentsas in Lemma 16

W ′′(ej ) · e2i = 3

4µ2i µ

2j

N∑

k=0

(1 + cos((2k+1) jπN+1 ))(1 + cos((2k+1) iπ

N+1 ))

= 3

4µ2i µ

2j

(N + 1 + SN(j)+ SN(i)

+N∑

k=0

cos((2k+1) jπN+1 ) cos((2k+1) iπ

N+1 )

)

= 3

4µ2i µ

2j

(N + 1 + 1

2

N∑

k=0

(cos((2k+1) (j+i)π

N+1 )+ cos((2k+1) (j−i)πN+1 )

))

= 3(N + 1)

8µ2i µ

2j (2 + δj,i − δj+i,N+1). ��

122 Giuseppe Molteni, Enrico Serra, Massimo Tarallo & Susanna Terracini

We are now ready to describe when the nonlinear coupling condition is satisfied.

Proposition 6. Let Wij = 6µ2i W(ej )− µ2

jW′′(ej ) · e2

i . Then

Wij < 0 if and only if

j + i �= N + 1 if N is even

j + i �= N + 1 or i = j = N+12 if N is odd.

(82)

Proof. By (80) and (81) we see immediately that

Wij = −3(N + 1)

16µ2i µ

4j (1 + δ2j,N+1 + 2δj,i − 2δj+i,N+1).

Then, if j + i �= N +1,Wij is negative, whatever the parity ofN . If j + i = N +1and i �= j , then also 2j �= N + 1, and Wij is positive. The only possibility left isj + i = N + 1 and i = j ; in this case of course 2j = N + 1, so that Wij is againnegative. This last case may take place only if N is odd. ��

In Theorem 4 the nonlinear coupling condition Wij < 0 is used in combina-tion with the requirement that µi be right asymptotically resonant with µj . Since,as we have pointed out in Section 4, in this case necessarily j �= i, the previousproposition yields the following simple result.

Corollary 2. Assume that µi is right asymptotically resonant with µj . Then

Wij < 0 if and only if j + i �= N + 1. (83)

The preponderance of negative terms explains, we believe, the numerical out-comes reported in [2].

As a final result, we report the specialization of Corollary 1 to the Fermi-Pasta-Ulam case.

Corollary 3. For every N � 2, the N particle Fermi-Pasta-Ulam problem withfixed ends admits a sequence of periodic solutions whose C2 norms tend to zeroand whose minimal periods tend to infinity.

Proof. We just treat N � 3. By Theorem 7, it is immediate to check that 2 /∈�R(N, 1) for everyN ; thereforeµ1 is right asymptotically resonant withµ2. Since1 + 2 = 3 < N + 1, the preceding corollary assures thatW12 < 0. The applicationof Corollary 1 completes the proof. ��

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Dipartimento di MatematicaUniversità di Milano

Via Saldini 5020133 Milano, Italy

e-mail: giuseppe.molteni@mat.unimi.ite-mail: enrico.serra@mat.unimi.it

e-mail: massimo.tarallo@mat.unimi.it

and

Dipartimento di Matematica e ApplicazioniUniversità di Milano Bicocca

Via Cozzi 5320125 Milano, Italy

e-mail: susanna.terracini@unimib.it

(Received April 1, 2005 / Accepted September 27, 2005)Published online April 3, 2006 – © Springer-Verlag (2006)