Chaotic-Like Transfers of Energy in Hamiltonian PDEs

We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^2$$\end{document}T2 and we prove the existence of different types of solutions which exchange energy between Fourier modes in certain time scales. This exchange can be considered “chaotic-like” since either the choice of activated modes or the time spent in each transfer can be chosen randomly. The key point of the construction of those orbits is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form truncation of those equations.


Introduction
A fundamental question in nonlinear Hamiltonian Partial Differential Equations (PDEs) on compact manifolds is to understand how solutions can exchange energy among Fourier modes as time evolves. A way to capture such behaviors is to analyze the invariant objects of the equation (or a "good approximation of it"), such as periodic orbits or invariant tori, and to understand how they structure the global dynamics through their stable and unstable manifolds and their possible intersections. This "dynamical systems" approach works very well, for instance, for PDEs on the torus T n . Such equations can be seen as infinite dimensional systems of ODEs for the Fourier coefficients and classical perturbative arguments can be adapted to the infinite dimensional context for the analysis of stability and instability phenomena. This approach has been classically applied to the analysis of stable motions, that is KAM Theory (the literature is huge, we refer to [4] for an overview on the subject and to the reference therein). However, its application to exchange of energy phenomena is much more recent.
In the last decade there has been a lot of activity in building exchange of energy behaviors in different Hamiltonian PDEs almost exclusively for the nonlinear Schrödinger equation. They can be classified into two groups. The first one are the so-called beating solutions [19][20][21]28,29,42]. Those are orbits that are essentially supported on a finite numbers of modes and whose energy oscillates between those modes in a certain time range.
The other group are those addressing the problem of Sobolev norm explosion. That is, constructing orbits whose energy is transferred to increasingly higher modes as time evolves [6,8,9,16,17,[22][23][24][25][26][27]31,32,35,38,39]. Those are solutions whose dynamics is essentially supported in a large number of modes and it is related to weak turbulence. J. Bourgain considered this problem one of the key questions in Hamiltonian PDEs for the XXI century [7].
Most of these results rely on analyzing certain truncations of the Hamiltonian PDEs (its first order Birkhoff normal form truncation) and building invariant objects for such models. Note that these first order Birkhoff normal forms are typically non-integrable Hamiltonian systems (at least in dimension greater or equal than 2) with very complicated dynamics. Nevertheless, restricted to suitably chosen invariant subspaces those models are integrable (they have "enough" first integrals in involution), and therefore one can have a very precise knowledge of their orbits in such invariant subspaces. Most of the results cited above strongly rely on this integrability on subspaces to construct unstable motions and exchange of energy solutions. This is somewhat surprising from the point of view of (finite dimensional) dynamical systems where usually unstable motions and drifting orbits must rely on non-integrability and transverse homoclinic orbits.
Can one take advantage of the non-integrability and chaoticity of a normal form truncation to construct new types of beating solutions? Can one exploit this chaoticity/nonintegrability to build new type of dynamics in Hamiltonian PDEs? This is the goal of this paper. We consider three different PDEs, a nonlinear Wave equation, a nonlinear Beam equation and the Hartree equation (see (1.1), (1.2) and (1.10) below) and we are able to show the non-integrability and chaoticity (symbolic dynamics) of its Birkhoff normal form. This allows us to obtain different types of exchange of energy behaviors for the actual PDEs in some time scales. In particular, • Solutions which exchange energy in a chaotic-like way between a given set of modes. By chaotic-like we refer to orbits which oscillate between being supported in two different sets of modes and the "oscillation times" can be chosen "randomly", see Theorem 1.3 below for the precise statement. • Chaotic-like transfer of energy phenomenon: those orbits are essentially supported in a finite number of modes and the support is changing as follows. At each transition two modes get deactivated (their modulus becomes essentially constant) and we can choose randomly which new two modes are activated (their modulus starts oscillating) among certain set. See Theorem 1.4 below for the precise statement.
These results provide different types of beating solutions which are significantly different from the previous results [19,20,29]. The beating solutions in these papers exchange energy periodically in time and they rely on integrability and existence of action-angle variables. On the contrary, in the present paper the oscillations can be "randomly" chosen: in the first one with respect to the time and in the second one with respect to the choice of activated modes. Our second result leads to transfer of energy. However, the transfer does not involve arbitrarily high modes and therefore does not lead to explosion of Sobolev norms. The methods in [8] for the construction of solutions exhibiting growth of the norms seem to fit very well for the NLS model [22][23][24][25][26][27]. Nevertheless, it is not clear how to apply it to other PDEs. We think that the present work could represent a first step to strengthen the strategy in [8] so that is applicable to other PDEs by incorporating tools and mechanisms inspired by the theory of Arnold diffusion. In Sect. 1.2 we relate our results to the approach developed in [8].
The key point to obtain the results in this paper is to consider certain first order truncations of the PDEs which can be treated as nearly integrable Hamiltonian systems. Then, one can apply classical methods in dynamical systems such as Melnikov Theory, shadowing arguments (Lambda lemma), hyperbolic invariant sets and symbolic dynamics. (1.1) and the cubic nonlinear Beam equation

Main results. Consider the completely resonant cubic nonlinear Wave equation on the 2-dimensional torus
We prove the existence of special beating solutions for such PDEs, namely solutions that exhibit transfer of energy between Fourier modes. Such solutions u(t, x) are mainly Fourier supported on a finite set of 4-tuples of resonant modes with N ≥ 2, in the sense that where R(t, x) is small in some Sobolev norm. The transfers of energy between modes in are chaotic-like, in the following sense. Either (a) one can prescribe a finite sequence of times t 1 , . . . , t n and find a solution that exists for long but finite time exhibiting transfers of energy among the modes in at the prescribed times t 1 , . . . , t n or (b) one can prescribe a sequence of resonant tuples {n (r n ) j } n=1,...,k j=1,...,4 ⊆ and find a solution and a sequence of times t 1 , . . . , t k such that at time zero many modes are "switched off" (modulus of the modes almost constant) and at times t n the modes (n (r n ) 1 , n (r n ) 2 , n (r n ) 3 , n (r n ) 4 ) are "switched on", in the sense that they start to exchange between them.
Those phenomena are consequence of the presence of (partially) hyperbolic, finite dimensional manifolds which are approximately invariant for the Eqs. (1.1), (1.2) and possess stable and unstable invariant manifolds that intersect transversally within some energy level.
We look for beating solutions in the subspace of functions Fourier supported on Z 2 odd := ( j (1) , j (2) ) ∈ Z 2 : j (1) odd , j (2) even , (1.4) which is invariant under the flow of the Eqs. (1.1), (1.2) (see [40]). We restrict the equations to this subspace so that the origin becomes an elliptic fixed point and the variational equation areü j + λ 2 j u j = 0 j ∈ Z 2 odd (1.5) where λ j = | j| [for the Wave Eq. (1.1)] and λ j = | j| 2 [for the Beam Eq. (1. 2)]. Note that, if one does not restrict to Z 2 odd , one has to deal with the harmonic 0 which is not elliptic.
Restricted to Z 2 odd , the variational equations (1.5) are superposition of decoupled harmonic oscillators, hence all solutions are periodic/quasi-periodic/almost-periodic in time and there is no transfer of energy between the linear modes when time evolves. This implies that the existence of beating solutions (if any) depend on the presence of the nonlinearities. To catch the nonlinear effects in a neighborhood of an elliptic equilibrium we perform a Birkhoff normal form analysis. Namely we construct changes of coordinates 1 that transform the Hamiltonian of the Eqs. (1.1), (1.2) into a Hamiltonian of the form K = K (2) + K (4) + R, (1.6) where K (i) are homogenous terms of degree i and R is a function that can be considered as a small perturbation. Then, one can consider the truncated system N := K (2) + K (4) , (1.7) called normal form [see (3.7) below for the explicit formulas], as a model which describes the effective dynamics of Eqs. (1.1), (1.2) for a certain range of times. The normal form Hamiltonian N possesses many finite-dimensional, symplectic, invariant subspaces of the form V := {u j = 0 ∀ j / ∈ }, where ⊂ Z 2 odd is a finite set. We shall prove the following. Theorem 1.1. Let N ≥ 2. There exist sets 2 ⊂ Z 2 odd of cardinality 4N such that V is invariant by the dynamics of N and the following holds.
(i) Let N = 2. Then, the flow t associated to N in V has the following property.
There exists a section transverse to the flow t such that the induced Poincaré map P : U =Ů ⊂ → has an invariant set X ⊂ U which is homeomorphic to × T 5 where = N Z is the set of sequences of natural numbers. Moreover, the dynamics of P : X → X is topologically conjugated to the following dynamics where σ is the usual shift (σ ω) k = ω k+1 and f : → R 5 is a continuous function. Namely P has a Smale horseshoe of infinite symbols as a factor. (ii) There exist N partially hyperbolic 2(N + 1)-dimensional tori T 1 , . . . , T N invariant for the restriction of the normal form Hamiltonian N at the subspace V which have the following property. Take arbitrarily small neighborhoods V i of T i and any sequence { p i } i≥1 ⊂ N N . Then, there exists an orbit u(t) and a sequence of times 1 It is well known that the existence of such changes of coordinates cannot be always guaranteed because of the presence of small divisor problems and / or derivatives in the nonlinear terms. At this stage, one can consider the normal form truncation as a formal "good first order" of the full equation. To show that is truly a good first order in the regions of the phase space that we consider, we adopt the strategy of performing a weak version of the Birkhoff normal form which does not remove all the non-resonant terms but a finite number of them. 2 Actually there exist "many sets" with such properties. See Remark 1.2 below.
Those tuples form a parallelogram inscribed on an ellipse with foci at F 1 = 0 and F 2 = n 1 + n 2 and semi-major axis a = (|n 1 | + |n 2 |)/2. Let us explain in which sense there are many sets ⊂ Z 2 for which Theorem 1.1 (and also Theorems 1.3 and 1.4 below) are satisfied. Theorem 1.1 relies on proving the transverse intersection of certain invariant manifolds. This transversality is proven by perturbative methods and, therefore, we need N | to be close to integrable. For the Wave (1.1) and Beam (1.2) equations this relies on choosing appropriate sets . The precise statement goes as follows. Fix ε > 0 (which will measure the closeness to integrability). Then, for any R 1, one can choose the resonant tuples in the set generically in the annulus Generically means that one has to exclude the zero set of a finite number of algebraic varieties (and the number of those is independent of ε and R).
The items (a) and (b) above are consequences respectively of items (i) and (ii) in Theorem 1.1. Let us make some remark on the type of dynamics for the normal form Hamiltonian N .
• Item (i) of Theorem 1.1 gives the existence of invariant sets for the Birkhoff normal form truncation which possess chaotic dynamics. Such chaotic dynamics is obtained through the classical Smale horseshoe dynamics for a suitable Poincaré map. This invariant set is constructed in the neighborhood of homoclinic points to an invariant tori orbit (which becomes a periodic orbit for a suitable symplectic reduction). The (infinite) symbols codify the closeness to the invariant manifolds of the periodic orbit, and therefore the larger the symbol is the longer the return time to the section is. In particular, one can construct orbits which take longer and longer time to return for higher iterates.
Even if the theorem, as stated, gives the existence of one invariant set, one actually can construct a Smale horseshoe at each energy level. • Item (ii) of Theorem 1.1 gives orbits which visit (possibly infinitely many times) a given set of invariant tori in any prescribed order. The construction of such orbits follows the classical strategy of Arnold Diffusion [1]. That is, is a consequence of the existence of a chain of invariant tori (again periodic orbits in a suitable symplectic reduction) connected by transverse heteroclinic connections (see Fig. 1) plus a classical shadowing argument (Lambda lemma, see for instance [14]). This is radically different from the approach in [8,24]. In these papers, the authors consider the normal form associated to the nonlinear cubic Schrödinger equation.
This normal form has "extra integrability", due to the symmetries of the model, and the considered heteroclinic orbits are not transverse. Therefore, the associated shadowing arguments are more delicate. We refer to [10] for a thorough analysis of non-transverse shadowing arguments. In particular, the authors of this paper show that the number of dimensions needed for the shadowing depend on the number of tori the orbits have to visit (what they called the dropping the dimension mechanism). As for item (i) one can obtain the explained behavior at each energy level. Indeed, the invariant tori come in families parameterized by the energy level and therefore one can obtain this shadowing behavior at each energy level as well.
Note that the knowledge of the orbits obtained in 1 such that for all T ≥ T 0 there exists M 0 > 0 such that for all M ≥ M 0 there exists δ 0 = δ 0 (M, ε, T) > 0 such that ∀δ ∈ (0, δ 0 ) the following holds.
Choose any k ≥ 1 and any sequence {m j } k j=1 such that m j ≥ M 0 and 3 k j=1 m j ≤ M − k. Then, there exists a solution u(t, x) of (1.1), (1.2) for t ∈ [0, δ −2 MT] of the form where κ = 1 for the Wave equation (1.1) and κ = 2 for the Beam equation (1.2), and and has the following behavior.

• First resonant tuple (Periodic transfer of energy):
There exists a T-periodic function Q(t), independent of δ and satisfying min [0,T] Q(t) < ε and max [0,T] |Q(t)| > 1 − ε, such that • Second resonant tuple (Chaotic-like transfer of energy): There exists a sequence of times {t j } k j=0 satisfying t 0 = 0 and Moreover, there exists another sequence {t j } j=1...k satisfying t j <t j < t j+1 such that, Note that the first order {δa n i } i=1...8 are the trajectories obtained in Theorem 1.1-(i) which belong to the horseshoe. This phenomenon is genuinely nonlinear since for the linear equation the actions |a n i (t)| 2 = constant.
The first resonant tuple has a periodic beating behavior similar to [20]. On the contrary, the behavior of the second resonant tuple is radically different. The modulus of the modes a n i , i = 5, 6, 7, 8 "oscillate" from being O(ε) to being O(ε)-close to 1 (see Fig. 2). However, the sequence of times {t j } in which all the modes in the tuple have the same modulus, that is (and the modulus of a n 5 and a n 7 is increasing) can be chosen randomly as any (large enough) integer multiple of T.
Finally, let us explain the role of the constant T in the theorem. To build the horseshoe in Theorem 1.1, we apply a symplectic reduction to N | [see (1.7)] which leads to a 2 degree of freedom Hamiltonian. For this Hamiltonian we construct a periodic orbit with transverse invariant homoclinic orbits. The time T is the period of this periodic orbit and can be taken arbitrarily big. Now we state the second main result of this paper, which gives solutions of Eqs. 1.1 and 1.2 which (approximately) behave as those obtained in Item (ii) of Theorem 1.1.

Fig. 2.
An example of the evolution of the energy |a n 5 (t)| 2 as time evolves. The energy is a multi-bump like function. It assumes the value 1/2 at the "random" times t = t j and also at t =t j . The randomness in the t j 's prescribes the separation of the bumps. The larger is the increment t j+1 − t j , the more separated are the corresponding bumps. This shows that one can obtain very complicated energy transfer behaviors for the second resonant tuple Then for a large choice of a set such that for any δ ∈ (0, δ 0 ) and any sequence ω = (ω 1 , . . . , ω k ), ω i ∈ {1, . . . , N }, there exists a solution u(t, x) of the (1.1), (1.2) of the form where κ = 1 for the Wave Eq. (1.1) and κ = 2 for the Beam Eq.
x) H s (T 2 ) s δ 3/2 for all s ≥ 0, and the first order {a n } n∈ , has the following behavior: There exist some α p , β p satisfying such that {a n } n∈ satisfies: • In the beating-time intervals I p , there exists t p > 0 such that where Q(t) is the periodic function given by Theorem 1.3.
• In the transition-time intervals J p, p+1 , The solutions obtained in this theorem are approximations of those obtained in Item (ii) of Theorem 1.1 and possess two different regimes. The orbits of Theorem 1.1 are obtained by shadowing a sequence of invariant tori (periodic orbits for a suitable symplectic reduction) connected by transverse heteroclinic orbits. Then, what we call beating-time intervals are the time intervals where the orbit is in a small neighborhood of each of the periodic orbits. In this regime, (the moduli of) some modes oscillate periodically, whereas the others are at rest. The transition-time intervals correspond to time intervals in which the orbit is "traveling" along a heteroclinic orbit and is "far" from all periodic orbits. In this regime, all modes undergo a drastic change to drift along the heteroclinic connection (see Fig. 3).
Hartree equation. Similar results hold true also for the Hartree equation and assuming the following hypothesis. Once fixed the set ⊂ Z 2 , the Fourier coefficients V j of the potential with j = n 1 − n 2 for some n 1 , n 2 ∈ satisfy V j = 1 + εγ j with ε 1. (1.12) Assume that the coefficients γ j satisfy a non-degeneracy condition which is of codimension 1 and take ε small enough. Then, the Hartree equation has solutions of the form where the first order {a n } and the remainder R satisfy the statements given either in  • Beating partially hyperbolic quasiperiodic tori: The Smale horseshoe obtained in Theorem 1.1 possesses a dense set of periodic orbits. Even if the horseshoe may not persist for the Eqs. 1.1, 1.2, 1.10, KAM Theory should give the persistence of these periodic orbits. In [28], the authors prove the existence of beating KAM Tori. The tori in [28] are elliptic whereas those coming from the horseshoe would be partially elliptic and partially hyperbolic. • Non-integrability of N | in (1.7): Theorem 1.1 (and therefore Theorems 1.3 and 1.4) relies on the fact that N | is not integrable and admits invariant tori with transverse homoclinic orbits. On the other hand, the Birkhoff normal form truncation associated to the cubic Nonlinear Schrödinger equation is such that N | is integrable. Therefore, the invariant manifolds of the invariant tori coincide and one cannot construct the orbits given in Theorems 1.3 and 1.4 for this equation (at least not with the tools used in the present paper). • Unbounded nonlinearities and higher dimensions: The PDEs analyzed in this paper are semilinear PDEs, namely they have bounded nonlinearities, on the two dimensional torus. However we expect that our results could be extended (with opportune modifications) to models with unbounded nonlinearities and to higher dimensional tori. Usually the unboundedness of the nonlinearity creates problems with the convergence of the normal form transformations. The same issue can arise if one increases the dimension of the spatial domain due to the possible loss of derivatives coming from the small divisors. The reduction to the resonant model N | is obtained by means of a weak version of the Birkhoff normal form procedure, that is not affected by the aforementioned problems of convergence. This method is described in Sect. 3 and it is well established in the KAM theory for quasi-linear PDEs (see for instance [2,12]). Unfortunately, such transfer of energy does not lead to growth of Sobolev norms [7,8,24]. We would like to devote this section to relate our results to that of [8]. In [8], the authors obtain orbits undergoing growth of Sobolev norms for the defocusing nonlinear Schrödinger equation on T 2 . One of the key points of their proof is to construct, for the Birkhoff normal form truncation, a chain of invariant tori (periodic orbits in certain symplectic reduction, named toy model) which are connected by non-transverse heteroclinic orbits. To obtain such connections, they strongly rely on the following fact. Even if this toy model is not integrable, it is integrable once restricted to certain invariant subspace (what can be called two generations model following [8]). Then the orbits undergoing growth of Sobolev norms are well approximated by orbits which shadow (follow closely) this chain of periodic orbits.
If one wants to use their ideas to obtain similar behavior in other equations such as the Wave (1.1), Beam (1.2) and Hartree (1.10) equations, one has to face several challenges.
First of all, in these equations, the two generations model is not integrable (for the Hartree equation it is not for a generic potential). 4 This is not surprising. Indeed, typically (at least in finite dimensional Hamiltonian systems) unstable motion (Smale horseshoes, Arnold diffusion) is related to non-integrability. Still, even if non-integrability should "help " to achieve growth of Sobolev norms it makes the analysis considerably more difficult. The present paper is a first attempt to understand this regime (for the two generations model).
The models we consider are carefully chosen so that they are close to integrable and therefore can be analyzed through perturbative methods. Unfortunately, for the Wave and Beam equation, to be close to integrable we have to choose the modes in with very similar modulus and therefore it seems difficult to use the analysis done in this paper to construct orbits undergoing growth of Sobolev norms. For the Hartree equation, one should expect that the ideas developed in this paper could lead to growth of Sobolev norms for a generic potential satisfying (1.11), (1.12).
A second fundamental difference between NLS and the PDEs considered in this paper is about the chain of tori connected by heteroclinic connections considered in [8]. Such structure is not structurally stable in the following sense: to have such heteroclinic connections one certainly needs that the connected invariant tori belong to the same level of energy (and to the samel level of other first integrals that the finite dimensional reduction possesses). This does not happen to be the case in other equations besides NLS. Indeed, for the Hartree equation (1.10) with a generic potential V the tori considered in [8] belong to different level of energy and the same happens for the Wave and Beam equations for a generic choice of resonant tuples. Therefore, to achieve growth of Sobolev norms for those equation one certainly needs to consider other invariant objects. The tori considered in Theorem 1.1 are radically different from those in [8]. These tori come in families of higher dimension which are transverse to the first integrals. Moreover, they are indeed connected by heteroclinic orbits. These connections are transverse and, therefore, they are robust. We believe that such objects could play a role if one wants to implement [8] to other PDEs.

Heuristics and Description of the Paper
The general argument we use in the proofs of Theorems 1.3 and 1.4 follows some of the ideas in the literature [8,[22][23][24]27]. The steps are the following. First, a weak Birkhoff normal form procedure simplifies the infinite dimensional Hamiltonian defined by the PDE, removing some non-resonant terms. Second, the normal form is truncated. The truncated normal form admits finite dimensional invariant subspaces. Third, a choice of these subspaces is made, defining a finite dimensional approximation of the PDE, that we call resonant model. Some particular finite dimensional orbit of the finite dimensional model is found. Fourth and final, a true solution of the original PDE, close to the finite dimensional one for long enough time, is found. We will use this scheme, with particular choices in each step, particularly when considering the finite dimensional model.
In [8,27], in the third step, the particular orbit found in the resonant model is obtained relying on the fact that the resonant model is integrable. More precisely, some invariant manifolds of different hyperbolic objects, coincide. Our approach is essentially different, because our resonant model is non-integrable in the sense that the invariant manifolds of several invariant objects-fixed points or periodic orbits-intersect transversally. We take advantage of the non-integrable dynamics of the finite dimensional model to obtain solutions of the truncated normal form with prescribed behavior; indeed, non-integrable dynamics is richer than the integrable one. More details are given below.
As a matter of fact, the proofs of Theorems 1.3 and 1.4 share all these common ingredients and only differ in the finite dimensional phenomena arisen by non-integrability.
Let us give more details concerning our implementation of the strategy. We refer to [18] for further details.
Step 1: Each of the PDEs under consideration has a Hamiltonian structure. Let us denote by H the Hamiltonian. Given a complete (see Definition 3.3) finite subset ⊂ Z 2 of resonant modes, to be chosen later, a weak normal form scheme is applied to the Hamiltonian. This weak normal form only "removes" a finite number of monomials of degree 4 of the Hamiltonian. Hence it is well defined (the normal form transformation is defined by the flow of a system of ODEs). The monomials to be killed are related to the set . Although the normal form procedure is not complete and many non-resonant terms of degree 4 are left untouched, for suitable the truncated normal form admits a finite dimensional invariant subspace supported on . This is done in Sect. 3.
Once the Hamiltonian is written in the normal form coordinates, we consider the truncated normal form, disregarding the terms of degree 6 or more. We call this truncated normal form the resonant model.
Step 2: This step is the core of the paper and can be divided as follows.
• Construction of the Set (Sect. 4): The set is chosen in such a way that its associated subspace of modes (see V in (3.11)) is invariant by the flow of the resonant model, but of course satisfies other requirements. Its precise definition depends on the PDE model we consider, but all three instances (Wave, Beam and Hartree equations) of the set share some common features. They have exactly 4N elements which, using the terminology introduced in [8], encompass two generations (see [29] for the analysis of transfer of energy for a different two generations model given by the quintic NLS). The elements of the set are organized in groups of four, pairwise disjoint, each of them forming a parallelogram. The choice of the modes is such that each individual parallelogram is invariant. It also happens that the dynamics of a single parallelogram is integrable, that is, if the rest of the modes are at 0, the dynamics of the four modes in a parallelogram is integrable. At this point is where our choice of the modes differs from other examples in the literature.
• The dynamics of the finite dimensional model (Sects. 5, 6, 7): First, we choose our modes in such a way that the dynamics of the resonant model is close to integrable, where closeness to integrability is measured through some parameter ε. The nearly integrability is obtained choosing properly the modes in . The unperturbed system (where ε = 0) possesses certain invariant objects, namely hyperbolic fixed points and hyperbolic periodic orbits, whose invariant manifolds form heteroclinic or homoclinic separatrices. Our second (generic) condition on the modes is sufficient to ensure that these heteroclinic or homoclinic manifolds split for small ε = 0, giving rise to horseshoes and instability phenomena from which we deduce the existence of certain types of orbits. The splitting of these manifolds is measured by means of a suitable set of Melnikov integrals [36]. There is a wide literature on the application of Melnikov Theory to construct homoclinic solutions in PDEs (see, for instance [34,41,43]). However, note that in the present paper we apply Melnikov Theory to a finite dimensional model, not to the full PDE.
• The infinite symbols Smale horseshoe (Sect. 6): The orbits in Theorem 1.3 give rise from a horseshoe of infinite symbols that can be constructed close to a hyperbolic periodic orbit whose invariant manifolds intersect transversally. The construction of this horseshoe follows the ideas in [37]. The horseshoe can be described as follows. Let = {1, 2, 3, . . . } be a denumerable set of symbols and the space of bi-infinite sequences, with the product topology. Notice that, unlike what happens when is a finite set, is not compact. The shift σ : → is the homeomorphism on defined by (σ (s)) i = s i−1 . Following the construction of Moser in [37], given a hyperbolic periodic orbit whose invariant manifolds intersect transversally, it is possible to find a set of coordinates-one of the coordinates is time, in T-, a suitable section S that defines a return map φ and a set Q in this section with φ(Q) = Q, such that there exists a homeomorphism τ : The set Q is in fact the intersection of forward and backward images by φ of a set of disjoint closed bands {V j , j ∈ N}, where the index j denotes precisely the time between to consecutives passes through S and hence measures the distance to one of the invariant manifolds of the set V j . In this way, V j tends to the invariant manifold when j tends to infinity. The set Q is not compact because the return map is not defined in the invariant manifolds. The Moser [37] construction implies that the bigger s i is [see (2.1)], the longer it takes the orbit to come back to . Since the sequence s can be taken randomly, the return times as well.
• Shadowing of a sequence of periodic orbits (Sect. 7): the orbits in Theorem 1. 4 travel along a chain of periodic orbits connected by transverse heteroclinic orbits, following the diffusion mechanism described originally by Arnold [1]. This mechanism consists of a sequence -finite or infinite -of partially hyperbolic periodic orbits 5 , Here, since the system we are considering is autonomous, transversally means transversality in the energy level, which implies that the intersection of the manifolds is, locally, a single heteroclinic orbit. If a nondegeneracy condition is met, this transversality is sufficient to have a Lambda Lemma that implies that [15]), which in turn implies that for any i, One can then choose arbitrary small neighborhoods of the tori T i and orbits that visit these neighborhoods according to an increasing sequence of times. It is worth to remark that the orbits found in the resonant model do exist for any positive time. In the case of the horseshoe with infinite symbols, one obtains orbits that arrive closer and closer to the periodic orbit, in randomly chosen times. In the case of the diffusion orbits, one obtain solutions that wander along the chain of periodic orbits for any positive time, and can be chosen to arrive closer and closer to each periodic orbit.
Step 3: The last step of the proof consists in finding a true solution of each PDE shadowing for long enough time the chosen solution of the resonant model. This is accomplished by a standard Gronwall and bootstrap argument. This relies on the Approximation argument given in Sect. 3 with the analysis of the dynamics of the Birkhoff normal form truncation of Sects. 6, 7. In this final step is crucial that the Eqs. (1.1), (1.2) have been restricted to Z 2 odd (see (1.4)) since this implies that the u = 0 is an elliptic critical point.

Weak Birkhoff
Hamiltonian structure of equations (1.1), (1.2) In the following we use the parameter κ ∈ {1, 2} to treat both cases at the same time. More precisely, κ = 1 if we refer to the Wave equation (1.1) or κ = 2 when we consider the Beam equation (1.2). By setting v :=u, we can express these equations as the following system of two first order equations We recall the subset Z 2 is well defined on U odd and it transforms the system (3.2) into the following one The vector field in (3.5) is Hamiltonian with respect to the 2-form := id ∧ d and Hamiltonian By considering the Fourier expansion = j∈Z 2 a j e i j·x , we can consider (3.6) We observe that the Hamiltonians (3.1) and (3.6) have the form 6 We remark that [using (1.12) for the Hartree equation] the coefficients C σ 1 σ 2 σ 3 σ 4 j 1 j 2 j 3 j 4 are such that (3.10)

Weak Birkhoff normal form.
In this section we apply a Birkhoff normal form argument to the Hamiltonian (3.7). We consider the symplectic form = i j∈Z 2 * da j ∧da j . We denote by ad H (2) the adjoint action of the Hamiltonian H (2) .
. . a σ n j n is a homogenous, momentum preserving Hamiltonian of degree n we have then We denote by Ker(H (2) ) the projection on the kernel of ad H (2) .
Since there are no regularity issues in what follows we decide to work on the phase space of analytic sequences. We fix ρ > 0 and define We denote by B ρ (δ) the open ball of radius δ > 0 centered at the origin of W ρ . We use the notation A B to denote A ≤ C B where C > 0 is a constant possibly depending on the fixed ρ.
Let be a finite subset of Z 2 * . We consider the following splitting Given a homogenous n-degree, momentum preserving Hamiltonian F = . . a σ n j n , we denote by F (n,k) the projection of F onto the monomials a σ 1 j 1 . . . a σ n j n with exactly k indices j i / ∈ . Thus F (n,k) is the part of the Hamiltonian F which is Fourier supported on S n,k .
We denote by We refer to F (n,≤k) (and F (n,≥k) ) the part of the Hamiltonian H which is Fourier supported on S n,≤k (and S n,≥k ).

Remark 3.2.
Since we assume that is finite, the preservation of momentum implies that the Hamiltonians F (n,≤1) have compact Fourier support.

Definition 3.3.
We say that a subset ⊂ Z 2 is complete if the following holds: given a * be finite and complete and consider the Hamiltonian H in (3.7). Then, where R satisfies Proof. Let us consider the 4-degree homogenous Hamiltonian The function F solves the homological equation (3.14) By Remark 3.2 the vector field generated by F has just a finite number of non zero components. Hence t F is the flow of a ODE with a smooth vector field. We call = ( t F ) | t=1 the time-one flow map of F. By Remark 3.2 the denominators in (3.13) have a uniform lower bound, hence by (3.10) the coefficients defined in (3.13) are uniformly bounded. Then by Young's inequality it is easy to see that X F (a) ρ a 3 ρ for all a ∈ W ρ . This implies that for δ > 0 small enough So we have proved item (ii). After the change of coordinates the Hamiltonian (3.7) transforms into (3.14) = H (2) Then, the completeness of implies that Ker(H (2) ) H (4,≤1) = Ker(H (2) ) H (4,0) .
Moreover, we can take R := {H (4) , We observe that {H (4) , F} is a homogenous Hamiltonian of degree 6 . Regarding the integral term, we have that t F is smooth and {{H, F}, F} is the sum of two homogenous Hamiltonians of degree at least 6, hence it is an analytic function on B ρ (δ) that can be Taylor expanded at a = 0. The first term of the Taylor expansion of the vector field is a polynomial of degree 5 and the remainder is smaller in a sufficiently small neighborhood of the origin. Again, by the uniform boundness of the coefficients of H and F, one can obtain the estimate in item (i) by using Young's inequality.
Let us consider the time-dependent change of coordinates This change of coordinates leaves resonant monomials unchanged, that is Ker(H (2) ) F• = Ker(H (2) ) F. Then, we have that Moreover, the functions Q and R satisfy Now, if one considers a complete set (see Definition 3.3), the associated subspace V [see (3.11)] is left invariant by X H Res . Moreover, on V , X H Res = X Ker(H (2) ) H (4,0) . This Hamiltonian is scaling invariant in the sense that if r (t) is a trajectory of this vector field also is. Taking δ 1, in certain time scales, this trajectory r δ (t) stays close to the trajectory of the Hamiltonian (3.16) with the same initial condition. Then there exists δ 2 = δ 2 (T 0 ) ≤ δ 1 (where δ 1 is given in Proposition 3.4) such that the following holds: for all 0 < δ ≤ δ 2 the rescaled solution r δ of the Hamiltonian H Res given by (3.18) and the solution u(t) of the Hamiltonian system H Res + R with initial condition u(0) = r δ (0) = δr (0) satisfy By the estimate in item (i) of Proposition 3.4, the fact that H Res is a homogenous Hamiltonian of degree 4 and (3.17) we have the following estimates (3.20) Now we use a bootstrap argument to conclude the proof. We assume temporarily that We already know that this is true for t = 0 since ξ(0) = 0. Then by Minkowsky inequality, the fact that r δ ρ δ and (3.20) we have that Thus integrating (3.21) and by using Gronwall lemma we get and since 0 < t ≤ δ −2 T 0 we have Since T 0 is independent from δ, we can choose δ small enough such that ξ(t) ρ δ 5/2 for t ∈ [0, δ −2 T 0 ]. Since this bound is stronger than the bootstrap assumption we can drop such hypothesis and the proof is concluded.

Lambda set.
We introduce a suitable finite and complete (see Definition 3.3) resonant set of modes ⊂ Z 2 , whose construction is based on the ideas of [8]. This set is constructed such that the associated subspace is invariant under the flow associated to the Hamiltonian H Res in (3.16). Later on we study the dynamics of the Hamiltonian H Res restricted to initial data supported on V . First we introduce the set of resonant tuples for the nonlinear Beam equation (those of the Hartree equation are a subset of it), and a subset of it, which is going to be used to build the set , Analogously, one can define the resonant tuples for the Wave equation and the corresponding associated subset We consider sets whose modes form resonant tuples in A bh / A w . One could choose the resonant tuples in the bigger sets A bh / A w . However, the particular form of the tuples in A bh / A w simplifies the analysis of the dynamics in V . Indeed, the Hamiltonian H Res restricted to V will have first integrals which correspond to the mass associated to each resonant tuple (see Sect. 4.3).
Let N ≥ 2 be an integer, and let A be either A bh or A w (analogously be A either A bh or A w ). We define a set ⊂ Z 2 which consists of two disjoint generations, = 1 ∪ 2 , | 1 | = | 2 | = 2N . Define a nuclear family to be a set (n 1 , n 2 , n 3 , n 4 ) ∈ A whose elements are ordered, such that n 1 and n 3 (known as the parents) belong to the first generation 1 , and n 2 and n 4 (known as the children) belong to the second generation 2 . Note that if (n 1 , n 2 , n 3 , n 4 ) is a nuclear family, then so are (n 1 , n 4 , n 3 , n 2 ), (n 3 , n 2 , n 1 , n 4 ) and (n 3 , n 4 , n 1 , n 2 ). These families are called trivial permutations of the family (n 1 , n 2 , n 3 , n 4 ). The first conditions to impose on the set were already imposed in the paper [8].
1 (Closure) If n 1 , n 2 , n 3 ∈ and there exists n ∈ Z 2 such that (n 1 , n 2 , n 3 , n) ∈ A (or any permutation of it), then n ∈ . In other words, if three members of a nuclear family are in , so is the fourth one. This is a rephrasing of the completeness condition (see Definition 3.3). 2 (Existence and uniqueness of spouse and children) For any n 1 ∈ 1 , there exists a unique nuclear family (n 1 , n 2 , n 3 , n 4 ) (up to trivial permutations) such that n 1 is a parent of this family. In particular, each n 1 ∈ 1 has a unique spouse n 3 ∈ 1 and has two unique children n 2 , n 4 ∈ 2 (up to permutation). 3 (Existence and uniqueness of sibling and parents) For any n 2 ∈ 2 , there exists a unique nuclear family (n 1 , n 2 , n 3 , n 4 ) (up to trivial permutations) such that n 2 is a child of this family. In particular each n 2 ∈ 2 has a unique sibling n 4 ∈ 2 and two unique parents n 1 , n 3 ∈ 1 (up to permutation). 4 (Faithfulness) Apart from the nuclear families, does not contain any other set (n 1 , n 2 , n 3 , n 4 ) ∈ A.
Note that the resonant tuples in belong to A. However, condition 4 requires that no other resonant tuples in the (larger set) A are possible in .
In the next two propositions we construct a set for the three considered PDEs. In some of the cases we need further conditions. Recall that the Eqs. (1.1), (1.2) have been restricted to Z 2 odd (see (1.4)) so that u = 0 is an elliptic critical point. Therefore, for these equations the set is constructed in Z 2 odd .
Proposition 4.1. Let N ≥ 2 and take A = A bh . Then there exists a set ⊂ Z 2 , with = 1 ∪ 2 and | j | = 2N , which satisfies properties 1 -4 and the following additional property: any n k , n k , n h , n h ∈ such that n k = n h and n k = n h satisfy Then there exists a set ⊂ Z 2 odd , with = 1 ∪ 2 and | j | = 2N , which satisfies conditions 1 -4 and the following additional condition. Take any n, n ∈ , then |n| = |n |. (4.6) Moreover, if one takes 0 < ε 1, there exists R = R(ε) 1 so that can be chosen to satisfy also ||n| − R| < Rε, for all n ∈ . (4.7) Let us make some comments on the extra conditions imposed on in these propositions. Condition (4.6) below is required to apply Melnikov Theory in Sect. 5. Condition (4.7) is used to obtain Hamiltonian systems on V [see (4.1)] which are close to integrable for the Beam and Wave equations. For the Beam and Wave equation we also require that the first component of the modes in is odd. This is fundamental in the approximation argument (Proposition 3.5) to avoid interactions with the mode n = 0 which is not elliptic.
We defer the proof of the above propositions to the Appendix 7.3.
Proof. The particular form of Hamiltonian (H Res ) | V is a direct consequence of the Properties 1 -4 satisfied by the set and the definition of H Res in (3.16). The definition of the coefficients C j 1 j 2 j 3 j 4 is given in (3.8), (3.9) and their estimates are consequence of (1.11), (1.12) and (4.7).
We use the symmetries of the Hamiltonian (4.8) to remove some of the monomials by a gauge transformation. Indeed, since the mass M := n∈ |a n | 2 is a conserved quantity for (H Res ) | V , we can consider the change of coordinates and time reparametrization α n = a n e iGt and for some g ∈ R to be chosen. The new system is Hamiltonian with respect to (4.10) Choosing the constant g in (4.9) as where A = (A i, j ) ∈ R 4N ×4N is a symmetric matrix given by and (C h ) h=1,...,N satisfies The equations of motion read as where k, i, l, j ∈ {4(h − 1) + 1, 4(h − 1) + 2, 4(h − 1) + 3, 4h} give the four modes forming a resonant tuple, l + k is even (namely, following [8], n k and n l belong to the same generation) and h ∈ {1, . . . , N }.

Invariant subspaces and first integrals
One can easily check that all those subspaces are invariant under the flow associated to Eq. (4.14). Let us study the corresponding dynamics. For V 1 (and analogously for V 2 ) one obtains the equation Therefore, on V 1 , |α n r | 2 are constants of motion and the phase space is foliated by invariant tori It can be checked (see Sect. 4.3 below) that these invariant tori are hyperbolic and thus have stable and unstable invariant manifolds. The dynamics on V i 1 ,...,i k is given as well by Eq. (4.14) just considering the interactions between the modes in the rectangles R i 1 , . . . R i k .
Hamiltonian H Res in (4.12) has the first integrals ) and the symplectic form | V becomes the standard one dθ ∧ d I = 4N k=1 dθ k ∧ d I k . The Hamiltonian system (4.19) has 4N degrees of freedom. We perform a symplectic reduction that leads to an N degrees of freedom system. In particular first we consider the restriction of (4.12) to and then we further reduce it to the manifold We adopt the following notation: we denote by 0 n the null matrix of dimension n × n and by I n the identity matrix of dimension n × n. We consider the symplectic linear change of variable : The new Hamiltonian does not depend on the angles {φ i } i∈{4k+1,4k+2:k=0,...,N −1} and it reads as ndalign* The second symplectic reduction is obtained by considering the symplectic linear change of variable : After the reparametrization of time t → −4 t, the restriction of the transformed Hamiltonian H • to the subspace is given (up to constants) by where the coefficients a j , b j and d j can be written in terms of the entries of the matrix A in (4.12) in the following way (4.26) From the particular form of this Hamiltonian, it is clear that {x j = y j = 0} is invariant under the associated flow. In particular the point is a saddle (for small ε) with N dimensional stable and unstable manifolds. One can analogously blow down {K j = 1} by considering the coordinates and one also obtains that, for ε small enough, P + = {x j = 0, y j = 0, j = 1 . . . N } is a saddle with N dimensional stable and unstable manifolds. This saddle is the "blow down" of {K 1 = . . . = K N = 1}.

Dynamics of the Resonant Model
The reduced Hamiltonian (4.22) for N = 2 is of the form (5.1) Note that the only term which couples the two unperturbed Hamiltonians H (1) 0 , H (2) 0 is d 12 K 1 K 2 . The Hamiltonian H is reversible with respect to the involution (see Fig. 4). For 0 < K 1 < 1, the dynamics of the Hamiltonian H (1) 0 can be also analyzed easily. Consider the "half" of the phase space (−2π/3, 2π/3) × (0, 1) ⊂ T × (0, 1) limited by the heteroclinic orbits (the other "half" is symmetric). It has an elliptic points at (ψ 1 , K 1 ) = (0, 1/2) and the rest is foliated by periodic orbits When h → 0, the periodic orbits "tend" to the sequence of heteroclinics and K 1 = 0, 1 and therefore their period T h → +∞. Hence, the dynamics of the 2-dof Hamiltonian H 0 in (5.1) has the following features. The invariant tori connected by the following heteroclinic manifolds (5.7) In particular γ + connects the points e − . The trajectories in the heteroclinic manifolds are just given by γ ± (τ 1 + t, τ 2 + t), t ∈ R.

Transversal heteroclinic orbits to saddles
The first step to prove Theorem 5.2 is to prove the existence of heteroclinic intersections between the saddles e (0) ± and e (1) ± . This step is certainly not necessary to obtain homoclinic intersections. Nevertheless, it will make considerably easier the computation of the Melnikov function associated to the homoclinic intersections. To obtain the mentioned heteroclinic intersections, one certainly needs that the saddles belong to the same energy level, that is, H(e Proof of Proposition 5.3. Thanks to the symmetry (5.2) of the system (5.1), one of the intersections implies the other one. We just deal with the first one.
0 , H 1 } • (γ + (τ 1 + t, τ 2 + t)) dt (5.13) is the so-called Melnikov function (see [36]). Since the Hamiltonian system (5.1) is autonomous, the Melnikov function M + depends just on the one-dimensional variable τ 1 − τ 2 . That is, there exists a function . By Lemma 5.4, we will deduce Theorem 5.3 by proving that there exists a non-degenerate zero of the function M (0) + in (5.13). It is convenient to introduce the Melnikov potential L + : R 2 → R, since it is usually easier to compute. It is defined, up to constants, as a primitive of the Melnikov function, namely We have Recall that we are assuming (5.10), which implies H 1 (e (0) + ) = 0. Therefore, the integrand decays exponentially to zero as t → ±∞.
The Melnikov potential satisfies L + (τ ) = L (0) Hence we shall look for non-degenerate critical points of L where τ 0 = τ 1 − τ 2 and the constantη ∈ R is given bỹ , (5.15) which gives (5.14). Therefore, we have that If (a 1 + b 1 )(a 2 + b 2 ) > 0 (see (5.11)) the reduced Melnikov potential L (0) + has at least one critical point. Moreover, By (5.11) this function has constant sign since Therefore L (0) + is either convex or concave (depending on the sign of a 1 + b 1 + a 2 + b 2 ) and its critical points are non-degenerate.

Transversal homoclinic orbits to saddles: Proof of Theorem 5.2
We use the computation of the heteroclinic Melnikov potential in Lemma 5.5 to prove the existence of homoclinic transversal intersections given by Theorem 5.2.
Since the Hamiltonian (5.1) with ε = 0 does not have connections between e (0) ± , we cannot apply directly Melnikov Theory to obtain such connections for ε > 0. Instead, we exploit the usual technique of considering a modified unperturbed Hamiltonian and using two parameters ε and δ.
We consider the Hamiltonian 16) and If one takes δ = ε, this Hamiltonian coincides with (5.1). Nevertheless, for now we consider δ and ε independent parameters. Later one we will take δ = ε. If δ = 0, then the dynamics of H 0 is the same described in Sect. 5.1. If δ = 0, the tori defined in (5.5) are H 0 -invariant; moreover, they belong to different energy levels, since The equilibrium points contained in T 0 are the saddles e  − (see Fig. 6). Such orbit corresponds to a homoclinic to the saddle P − in (4.27) (expressed in the "blow down" coordinates (4.25)).
Proof. Using that H (1) 0 is zero when restricted to T 0 we get K 1 = 1 + 2 cos(ψ 1 ) 1 + 2 cos(ψ 1 ) + δ . (5.19) When the angle ψ 1 ∈ [− * , * ] the numerator in (5.19) is positive. Hence K 1 ∈ (0, 1) if δ > 0. Plugging (5.19) in the equation for ψ 1 we havė ψ 1 = −(1 + 2 cos(ψ 1 )), which leads to By using (5.19) and the trigonometric identity cos(2 arctan(x)) = (1 − x 2 )/(1 + x 2 ) we have . (5.21) Reasoning in the same way for (ψ 2 , K 2 ) we get that the homoclinic orbit to T 0 is given by (5.18). By reasoning as in the proof of Theorem 5.3 we have that the distance between the manifolds in a suitable section is given by (5.22) where the Melnikov function is given by It can be easily checked that the O C 1 (K) (ε 2 ) are uniform for δ small enough. The associated Melnikov potential is As before we consider the reduced Melnikov potential We want to deduce that L (0) 0 has non-degenerate critical points by using the information on the Melnikov potentials (5.14) of the heteroclinic case.
The proof of this proposition is deferred to Sect. 5.2.3. To complete the proof of Theorem 5.2 it is enough use (5.22) and Proposition 5.8 and take δ = ε. Indeed, the transverse homoclinic points are ε-close to the non-degenerate critical point for L (0) 0 . By Proposition 5.8, L (0) 0 has a non-degenerate critical point ε ν 0 -close to τ 0 = 0. Proposition 5.8 Thanks to the exponential convergence of the homoclinic orbit to the equilibrium points e (0) ± we have that (recall (5.18))
Since the homoclinic orbit (5.18) is "close" to the concatenation of γ + and γ − in (5.7), we show that there exists ν 0 > 0 such that the integral in (5.25) satisfies The estimate for the error is proved in the following lemma. To state it, we define (5.26) Lemma 5.9. Let K be a compact subset of R 2 . There exists δ 0 > 0 small, such that ∀δ ∈ (0, δ 0 ) and τ ∈ K there exists a positive constant ν 0 ∈ (0, 1) such that the following holds This lemma implies Proposition 5.8. We devote the rest of this section to prove Lemma 5.9.
Proof of Lemma 5.9. We write the function O F in (5.26) as Note that the shifts by the vector ( p, p) do not alter the value of the integral. These shifts are useful to bound the integrand. To obtain such estimates, we need the following lemmas.
Proof. The lemma follows by straightforward estimates and the hyperbolicity of the equilibria.
We split R = 5 k=1 I k We have By the symmetry of the problem it is sufficient to provide bounds for |T i | with i = 1, 2, 3. The idea is to use the exponentially fast convergence of the orbits γ 0 , γ ± to the saddles (see Lemma 5.11) to get bounds on the integrals over the unbounded intervals and to exploit the closeness of such orbits on the compact intervals using Lemma 5.10.
• Bound for T 2 (see (5.33)): We note that I 2 is of the form (5.29). By using the compactness of the orbits and the mean value theorem as in the previous step, we can apply Lemma 5.10 and obtain where ν ∈ (0, 1) is given by Lemma 5.10. • Bound for T 3 (see (5.34)): We use that F(1, 1) = 0. Let us denote m := −(b− p) = c − p > 0 (see (5.28) for the definition of p). By translating the variable t we obtain
± ) = 0. Now we can repeat the same strategy to get the bounds for the associated T i . The only difference is that when we compare the orbits γ 0 , γ ± on compact intervals we need to use also (5.31). Then we obtain O G C 0 (K) δ ν |ln δ| (recall (5.26)).
Regarding the second derivatives in τ 1 we have We observe that on a compact set |{G, H 0 }| δ. Then we can consider the function E := {G, H (1) 0 } and repeat the same arguments above to prove that O E C 0 (K) has a bound like (5.27). We conclude by noting that

Transversal homoclinic orbits to saddles: N resonant tuples.
In this section we prove the generalization of Theorem 5.2 for the case of multiple resonant tuples. To break integrability we need to impose a non-degeneracy condition on the coefficients d i j in (4.23). To state it we introduce the matrix Remark 5.13. Note that condition (5.41) is satisfied for a generic choice of coefficients d i j . Indeed, the determinant of such matrix is a polynomial in the variables d i j . Then, it is enough to show that such polynomial is not identically zero. If one consider the matrix (5.40) is a multiple of the identity. This means that at some point the polynomial is not zero and therefore it is not-zero for almost every choice of d i j . In Sect. 7.2, we prove that condition (5.41) is satisfied for the resonant models associated to the Wave, Beam and Hartree equations that we consider.
Proof. We proceed as for the case N = 2 in Sect. 5 We proceed as in the proof of Theorem 5.2. For ε = 0 the dynamics is the same described in Sect. 5.2.2. In particular it is easy to see that, when ε = 0, one can consider the two saddle points (recall that * := 2π/3) We define the associated Melnikov potential We note that such function is the sum of terms of the form (5.24). Thanks to the autonomous nature of the system the potential, L 0,N depends just on τ 1 − τ N , . . . , τ N −1 − τ N . Thus one can consider the reduced Melnikov potential L (0) 0,N , which satisfies N (τ 1 , . . . , τ N ).
Classical Melnikov Theory ensures that non-degenerate critical points of this reduced Melnikov potential gives rise to transversal (within the energy level) intersections between W − ε (e (0) − ) and W + ε (e (0) + ). Denotingτ := (τ 1 − τ N , . . . , τ N −1 − τ N ), Proposition 5.8 implies that there exists a constant η ∈ R such that for some ν 0 > 0. Since x coth(( √ 3/2)x) is an even function, the origin (0, . . . , 0) ∈ R N −1 is a critical point of the first order of L (0) 0,N (that is, dropping the errors O C 2 (δ ν 0 )). The Hessian matrix of the first order of L (0) 0,N at the origin is where D is the matrix introduced in (5.40). Then, condition (5.41) implies det Hess = 0. The non-degeneracy of the Hessian implies that the reduced Melnikov potential L (0) 0,N has a non-degenerate critical point δ ν 0 -close toτ = 0. Then, taking δ = ε one can use classical Melnikov Theory to ensure the existence of the transverse intersection between invariant manifolds stated in Proposition 5.12.

Proof of Theorem 1.3
The goal of this section is to prove Theorem 1.3. The key point of the proof is to construct symbolic dynamics (an infinite symbols Smale horseshoe) for the resonant model (5.1) which has been derived from the equations (1.10), (1.1), (1.2). In Theorem 5.2 we have constructed transverse homoclinic orbits to saddles for (5.1). It is well known that the intersection of invariant manifolds of critical points in flows do not always lead to the existence of symbolic dynamics (see, for instance, [11]). Therefore, the first step of the proof is to obtain transverse homoclinic points to certain periodic orbits. This is done in Sect. 6.1. Then, following [37], in Sect. 6.2 we construct an invariant set of (a suitable Poincaré map of) the flow associated to the Hamiltonian (5.1) whose dynamics is conjugated to a shift of infinite symbols (see Sect. 2). Finally in Sect. 6.3 we complete the proofs of Theorem 1.3 by checking that the non-degeneracy conditions imposed on (5.1) are satisfied for the resonant models obtained from the PDEs (1.1), (1.10) and (1.2).

Transversality of invariant manifolds of periodic orbits.
The main result in this section is the following. (ii) The invariant manifolds W u (P +,0 h,ε ) and W s (P −,0 h,ε ) intersect transversally along orbits (within the energy level).
Note that in the coordinates introduced in (4.25), the periodic orbits P +,0 h,ε and P −,0 h,ε blow down to the same periodic orbit, which we denote by P 0 h,ε . In the coordinates (4.25), Proposition 6.1 can be restated as that the manifolds W u (P 0 h,ε ) and W s (P 0 h,ε ) intersect transversally within the energy level.
Proof of Proposition 6.1. To prove (i) is more convenient to use the cartesian coordinates {x j , y j } in (4.25) and therefore Hamiltonian H Res in (4.26) (with N = 2) to avoid the blow up of K j = 0. Then, the invariant subspace {K 2 = 0} corresponds to {x 2 = y 2 = 0}. The Hamiltonian on this invariant subspace is given by This Hamiltonian is integrable both for ε = 0 and ε > 0 and has the saddle (0, 0) at the energy level. Integrability and the particular form of H implies that the energy levels close to zero are given by periodic orbits. These periodic orbits are ε-close to those of the unperturbed problem (see (5.1)).
To prove (ii) we proceed as in Sect. 5 by doing approximations of several Melnikov functions and using an auxiliary parameter δ. We follow the notation of Sect. 5.2.2, In particular, we consider the Hamiltonians H 0 , H 1 in (5.16), which taking δ = ε also define the Hamiltonian H.

Lemma 6.3. Let K ⊂ R be a closed interval and let us define
There exists δ 0 > 0 small such that ∀δ ∈ (0, δ 0 ) there exist h 0 = h 0 (δ), positive and small, such that ∀h ∈ (0, h 0 ) there exists ν * ∈ (0, 1) such that the following holds Proof. We consider the splitting 0 , H 1 }( δ,h,0 (t, τ 0 + t)) dt, (6.4) where c is some positive constant. We observe that Then, by the exponential convergence of the flow to the hyperbolic saddles e ± 0 and the hyperbolic periodic orbits P ±,0 δ,h , the term on the r. h. s. of (6.3) is bounded by Cδ d where C, d > 0 are two constants independent of h. Let us call I := [−c|ln δ|, c|ln δ|]. By Remark 6.2 we have    We proceed as in [37]. Fix 0 < ε 1 and 0 < h 1. We build an invariant set with points arbitrarily close to the transverse homoclinic orbit to the T h -periodic orbit P 0 h,ε obtained in Proposition 6.1. We define the section within the energy level {H = h}, (6.5) for some small m > 0 (see Fig. 7). This section is transverse to the unperturbed flow (ε = 0) and therefore also transverse, for ε > 0 small enough, to the perturbed one. In particular, by Proposition 6.1, it contains points in W u (P 0 h,ε ) ∩ W s (P 0 h,ε ) (classical perturbative arguments ensure that the perturbed invariant manifolds are O(ε) to the unperturbed ones).
Denote by t H the flow associated to the Hamiltonian (5.1). For a point z ∈ S h , we define T (z) > 0 the first (forward) return time of the trajectory t H (z) to this section whenever it is defined. For those points whose forward trajectory never hits again S h we can take T (z) = +∞. Note that this happens in particular for the points in W s (P −,0 h,ε ) (note that by the perturbative results in Sect. 6.1 this intersection is not empty). Then Moreover, h −1 can be defined as follows. Fix z * ∈ Y and define ω * = h −1 (z * ). Associated to z one can define the sequence of hitting times Then, there exists C * ∈ N independent of z * such that where T h is the period of the periodic orbit P ±,0 h,ε .
This proposition gives symbolic dynamics for a Poincaré map associated to Hamiltonian (5.1). Note that it is constructed in a way that higher symbols in imply longer return times. In particular those can be unbounded. The proof of this proposition follows the same lines as the construction of symbolic dynamics done by Moser in Chapter 3 of [37]. Note that the natural C * in (6.6) is just to normalize and have as symbols N (since the horseshoe is build close the homoclinic orbit, the hitting times satisfy |t k − t k−1 | 1). We remark that condition (5.9) is necessary, indeed the term that breaks the integrability in the Hamiltonian (5.1) has the form d 12 K 1 K 2 (see for instance (5.16), (5.17)). Hence if the condition (5.9) does not hold then the Hamiltonian (5.1) is integrable.
We observe that the cardinality of I is bounded by 4N (4N −1)/2. Therefore, by condition (4.5), d 12 is a polynomial in the 4N (4N − 1)/2 variables γ k , k ∈ I . Such polynomial is not identically zero because if we set one of the γ k 's equal to one and all the others at zero then d 12 = 0.

6.4.
End of the proof of Theorem 1.3. Lemmas 6.5, 6.6 imply that condition (5.9) holds and, therefore, Proposition 6.4 can be applied to the resonant models associated to the Wave (1.1), Beam (1.2) and Hartree (1.10) equations. This proposition gives certain orbits of these resonant models. These orbits will be the first order (up to changes of coordinates) of orbits of equations (1.1), (1.2) and (1.10). Fix 0 < ε 1 and 0 < h 1 and consider the periodic orbit P 0 h,ε given by Proposition 6.1, which has period T h . By Proposition 6.4 there exist a set Y ⊂ S h which is an invariant hyperbolic set (a Smale horseshoe) for the Poincaré map associated to the Hamiltonian H in (5.1). This set can be built arbitrarily close to homoclinic points of P 0 h,ε . Fix ω ∈ such that |ω k | ≥ M 0 T h , where M 0 satisfies M 0 log ε and T h is the period of the periodic orbit P 0 h,ε . Then, Proposition 6.4 ensures that there exists an orbit γ (t) of H with initial condition in Y , which satisfies the following. There exists a sequence of times {t k } k∈Z satisfying (6.6) such that γ (t k ) ∈ S h where S h is the section defined in (6.5). Note that, by (6.6), the times t k satisfy for some θ k ∈ (0, 1) and C * ∈ N.
By construction, there exists another sequence of times The Smale horseshoe, can be built arbitrarily close to the invariant manifolds of P 0 h,ε and therefore, one can ensure that there exist intervals • I k ⊂ (t k , t k+1 ) such that, for t ∈ I k , γ (t) belongs to a ε-neighborhood of P 0 h,ε ; • J k ⊂ (t k , t k ) such that for t ∈ J k the orbit γ (t) belongs to a O(ε)-neighborhood of K 2 = 1, since the homoclinic orbit obtained in Proposition 6.1 have points O(ε)close to K 2 = 1.
This behavior implies estimates (1.8) and (1.9) in Theorem 1.3, once we undo the symplectic reductions, the changes of coordinates and we add the error terms as it is explained below. By Proposition 6.1 the parameterization of the periodic orbit P 0 h,ε is ε-close to (5.8), hence we have that where Q(t) is the time parameterization of P 0 h,ε and thus is T h -periodic, and sup t∈[0,T ] |R 2 (t)| ≤ ε.
By the symplectic reduction performed in Sect. 4.3 there exists r (t) solution of H Res in (3.16) with Fourier support such that This can be seen using Remark 4.4, which gives also the behavior of the other actions. Since the solutions of H Res are invariant under the scaling (3.18), we can consider r δ (t) := δr (δ 2 t). Then, r δ (t) is also a solution of H Res for t ∈ [0, δ −2 T ]. Now it only remains to obtain an orbit for the Eqs. 1.1, 1.2 and 1.10 which is close (up to certain changes of coordinates) to r δ (t). First step is to apply Proposition 3.5. It ensures that there exists 0 < δ 2 1 such that for all δ ∈ (0, δ 2 ), there exists a solution w(t) of H • • = H Res + R such that w(t) = r δ (t) + R(t) with R(0) = 0, R(t) ρ δ 2 for t ∈ [0, δ −2 T ]. We note that, by Item (ii) of Proposition 3.4, the Birkhoff map is δ 3 -close to the identity. Finally the transformations (3.15) and (4.9) preserve the modulus of the Fourier coefficients. The last change of coordinates that one has to apply (for the Wave (1.1) and Beam (1.2) equations) is passing from complex coordinates (3.4) to the original ones. We remark that by (4.6) if n i ∈ then −n i / ∈ . Thus is ε-close to the periodic orbit P h (see (5.4)). When ε = 0, the invariant manifolds W u (P + 0,h,k ) and W s (P − 0,h,k ) coincide. Proposition 7.1. Take any i, j = 1, . . . , N , i = j. Assume that the condition (5.41) is satisfied (see (5.40), (4.23)). Then, there exists ε 0 > 0 such that for ε ∈ (0, ε 0 ) and h 0 > 0 such that for any h ∈ (0, h 0 ), the manifolds W u (P − ε,h,i ) and W s (P + ε,h, j ) intersect transversally within the energy level.
The transversality of the invariant manifolds allows to construct orbits which shadow them. Note that in the coordinates introduced in (4.25), the periodic orbits P + 0,h,k and P − ε,h,k blow down to the same periodic orbit, which we denote by P 0,h,k . In the coordinates (4.25), Proposition 7.1 can be restated as that the manifolds W u (P ε,h,i ) and W s (P ε,h, j ) intersect transversally along an orbit within the energy level. Definition 7.2. We will say that a family of hyperbolic periodic orbits {P } ∈N of a system of differential equations, is a transition chain if W u (P ) W s (P +1 ), for all ∈ N.
Note that Proposition 7.1 gives full transversality between the invariant manifolds on the energy level. Thus, recalling that H(P i ) = h, from now on, we restrict the flow to this energy level, which is a regular manifold. is invariant by the flow of H for any ε and δ (this is properly seen in the coordinates (4.25), since then π k corresponds to x = y = 0, = k, see (4.26)).
The dynamics on the π k plane is integrable and is given by the 1-d.o.f. Hamiltonian H (k) 0 + εH 1|π k . This Hamiltonian has two saddles ( * ±,ε, , 0) ε-close to (± * , 0) = (±2π/3, 0), at the zero energy level. For h > 0 small, the set {H (k) 0 + εH 1|π k = h} is a periodic orbit, whose period tends to infinity when h goes to 0. Let ( (h) k (τ k ), K (h) k (τ k )) be a time parametrization of this periodic orbit satisfying where ( k ) are components of the homoclinic manifold introduced in (7.2). Then, the Hamiltonian H possesses two hyperbolic periodic orbits P ± ε,h,k at the energy level h, whose time parametrization is given by When ε = 0, the invariant manifolds W u (P + 0,h,k ) and W s (P − 0,h,k ) coincide. This homoclinic manifold can be parameterized as are components of the homoclinic manifold introduced in (7.2). Now, fix i, j ∈ {1, . . . , N }. For small ε > 0, the periodic orbits P − ε,h,i , P + ε,h, j and their invariant manifolds, W s (P + ε,h,i ) and W u (P − ε,h, j ) persist slightly deformed. We show now that the perturbation allows them to intersect.
In order to analyze the possible intersection, we introduce a N -dimensional section in the following way. We define, taking into account (5.42), where 0 is integrable and H can be also written as where We consider the N -dimensional section 0 |γ 0 (τ ) , r = (r 1 , . . . , r N ) ∈ (−m, m) N (7.9) where γ 0 is the homoclinic manifold introduced in (7.2). Observe that γ 0 (τ ), which is Then we can express the right hand side of (7.15) in terms of the P r 's in the following way: Then, the determinant of the matrix D in (5.40) is of the form This determinant can be written as Indeed, it is enough to modify the matrix in two steps. First replace the last column by the sum of all columns. Then, the last column is the vector with all components equal to P N . Second, subtract the last row to the other rows. Then, it is very easy to obtain (7.17).
Recall that in the proof of Lemma 6.5 we have shown that the sets of Proposition 4.2 satisfy |n (r ) (6.10)). Moreover, in Proposition 4.2 it shown that they also satisfy (4.6). These three properties imply P k = 0 for all k = 1, . . . , N (7.18) Therefore, by (7.17), to prove det(D) = 0, it only remains to check that N k=1 P k = 0. (7.19) If the set obtained in Proposition 4.2 satisfies this property, the proof is complete. Now, we show that if the set obtained in these propositions satisfies N k=1 P k = 0, one can modify it slightly so that the new one satisfies (7.19). Assume thus that satisfies N k=1 P k = 0 and (7.18). Then, we modify the first resonant tuple (n (1)

).
Then, since P 1 = 0, P 1 is strictly decreasing in λ and therefore N k=1 P k = 0 can only happen for λ = 1. Thus, one can modify the first rectangle by taking λ ∈ Q arbitrarily close to 1 and then blowing up the N rectangles so that the new rectangles belong to Z 2 . It is clear that with this modification (for λ close enough to 1) the properties in Proposition 4.2 are still satisfied.
We note that, by Item (ii) of Proposition 3.4, the Birkhoff map is δ 3 -close to the identity. Finally the transformations (3.15) and (4.9) preserve the modulus of the Fourier coefficients. The last change of coordinates that one has to apply (for the Wave (1.1) and Beam (1.2) equations) is passing from complex coordinates (3.4) to the original ones. We remark that by (4.6) if n i ∈ then −n i / ∈ . Thus Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
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A the Set : Proof of Propositions 4.1 and 4.2
The proofs of Propositions 4.1 and 4.2 are modifications of the proof of the construction of the set ⊂ Z 2 in [8]. Note that the resonances of the cubic nonlinear Schrödinger equation considered in [8] are the tuples contained in A bh in (4.3). We summarize the ideas in that paper and explain the main modifications.
In [8], the set is first constructed in Q 2 and then scaled to Z 2 . The placement of the modes in Q 2 is done inductively: first one places the modes in 1 , then those in 2 , checking at each placement that conditions 1 -4 are fulfilled. To this end, one has to ensure that the imposed non-degeneracy conditions are open and dense in Q 2 and then for "most of the placements" are satisfied. More concretely, the placement goes as follows • First generation: In order to place the first generation we have to chose 2N points in Q 2 . We choose them inductively checking that they satisfy the non-degeneracy conditions. Condition 2 and 3 are satisfied if all the points are chosen different and 1 will be satisfied by construction. The condition 4 is equivalent to check that each new point does not make a right angle with two of the modes already placed. That is, consider any segment whose endpoints are two points already chosen. Then, this new point cannot belong either to a line orthogonal to this segment and containing one of the points nor to the circle having this segment as a diameter. • Second generation: The set 1 is divided into pairs of modes, which are the parents of the N nuclear families. For each of these pairs n 1 , n 3 ∈ 1 ⊂ Q 2 , we place a pair of points n 2 , n 4 ∈ 2 in such a way that they form a rectangle with the other pair. That is, we consider the circle having as a diameter the segment between n 1 and n 3 . Then, the new modes n 2 ,n 4 have to be endpoints of another diameter of this circle. To ensure that n 2 , n 4 ∈ Q 2 it is enough to chose an angle between the two diameters which has rational tangent. Note that those angles are dense. The choice is done checking that the non-degeneracy conditions are verified 1 -4 following the same arguments as for the first generation.
This placement is generic in the following sense 1. The first generation is placed generically in Q 2 , that is anywhere except in the zero set of one polynomial. 2. The placement angles θ for the second generation are any angle such that tan θ ∈ Q except a finite number of values.
We use this scheme developed in [8] to prove Proposition 4.1.
Proof of Proposition 4.1. It is a direct consequence of the scheme developed in [8].
Indeed, the only extra condition added with respect to [8] is (4.5), which is certainly satisfied by a generic placement in Q 2 . Indeed, in placing inductively the new points one has only to avoid a finite number of points.
Proof of Proposition 4.2 (Beam case: A = A bh ) The set from Proposition 4.2 has three differences with respect to the one in [8]: properties (4.6) and (4.7) and the fact that the condition 4 requires that the modes in do not satisfy any of the resonance conditions in A bh \ A bh in (4.2)-(4.3). One can easily check that a generic placement satisfies (4.6) and the 4 condition. Indeed, in placing the new modes one has to avoid circles centered at zero with radius equal to the norm of the already placed modes and the circles and hyperbolas defined by (4.2) when two modes are fixed.
To build a set having property (4.7), we also follow the ideas in [8]. We first construct a prototype embedding. That is a "bad" set 0 ∈ Q 2 which is the union of N rectangles but which however does not satisfy the non-degeneracy conditions. For instance, consider This embedding satisfies (4.7) but does not satisfy conditions 1 − 4 nor (4.6) (in particular is not injective). However, by genericity one can chose points in Q 2 which are ε/4-close to those of 0 which define a set satisfying that all points are different and also conditions 1 − 4 . Finally, one needs to apply a scaling and a translation to obtain a set ⊂ Z odd × Z. Indeed, consider R 1 such that R ⊂ Z 2 and Rε 1. Then, we define = 2R + (1, 0) Then, one can check that for n ∈ , which is of the form n = 2Rn + (1, 0) with n ∈ , taking R large enough, Proof of Proposition 4.2 (Wave case: A = A w ) To prove Proposition 4.2 one has to take into account that the resonance condition for the Wave equation (1.1) given in (4.4) is different from that of the cubic nonlinear Schrödinger, Hartree (1.10) and Beam (1.2) equations (see (4.2)). Now four resonant modes (n 1 , n 2 , n 3 , n 4 ) ∈ A w form a parallelogram whose vertices are on an ellipse with one focus at zero. Indeed, if one fixes the modes n 1 and n 3 , then n 2 , n 4 must belong to the ellipse defined by n ∈ Q 2 : |n| + |n − (n 1 + n 3 )| = |n 1 | + |n 3 | , (A.1) that is, the ellipse with foci at 0 and n 1 + n 3 and such that the sum of distances from any point of the ellipse to the two foci is given by |n 1 | + |n 3 |. Note that the case n 1 = −n 3 trivially corresponds to the circle with center 0 and radius |n 1 |. We need to consider N ellipses of this type with dense rational points to apply the genericity arguments as in the previous cases. The standard ellipse has dense rational points provided a, b ∈ Q. To obtain ellipses of the form (A.1) from (A.2) one needs to apply a translation (one could also apply a rotation, but there is no need for it). To ensure that the transformed ellipse has dense rational points it is enough to ensure that the foci of the standard ellipse (A.2) are rational. Assuming that a > b, the foci are given by F ± = (±c, 0) = (± √ a 2 − b 2 , 0). Therefore, to build ellipses E j with dense rational points , j = 1 . . . N , it is enough to consider N different rational Pythagorean triples {(a j , b j , c j )} N j=1 , that is a 2 j = b 2 j + c 2 j , a j , b j , c j ∈ Q, a j > b j . Then, one can apply a translation to place one of the foci at 0. Let us denote by F j the focus of the ellipse E j which is not at the origin.
Having fixed these ellipses, one can prove Proposition 4.2 following the scheme of [8] explained above. One first places each pair of the first generation in one of the ellipses. To place one pair E j it is enough to chose one rational point n j 1 ∈ E j . Then, the other mode is obtained through the equation n j 1 + n j 3 = F j (see (A.1)).
Since the ellipses have dense rational points, one can place the points such that the conditions 2 − 4 and (4.6) are satisfied as follows. Let us assume that we have placed all modes of the first generation for the ellipses E j , j = 1 . . . j * − 1 and we want to place the first generation modes in the ellipse E j * . We show that we only need to avoid a finite number of points.
1. For property (4.6), we need to avoid the intersection points of E j * with all the circles centered at the origin and radius equal to the norm of the already placed modes. 2. For properties 2 , 3 , we need to avoid the points at the intersection of E j * with the other ellipses E j , j = 1 . . . j * − 1, j * + 1 . . . N . 3. For property 4 , one needs to avoid placing a mode such that with two previous modes m, m and an extra mode may create a nuclear family. To this end we have to avoid the following points: • Case (i)-m, m are non adjacent vertices of the parallelogram: One has to avoid the intersection points between E j * and the ellipse defined by m, m , that is |n| + |n − (m + m )| = |m| + |m | Note that this ellipse is different from E j * since by Item 2 above, m, m ∈ E j * . • Case (ii)-m, m are adjacent vertices of the parallelogram: One has to avoid the intersection points between E j * and the hyperbolas defined by m, m , that is |n| − |n − (m + m )| = ±|m| ∓ |m |.
• One can deal analogously with the conditions which arise from avoiding the resonances conditions in A w given by n 1 + n 2 + n 3 − n 4 = 0, |n 1 | + |n 2 | + |n 3 | − |n 4 | = 0, which either define ellipses or hyperbolas. Note that the two new placed modes and one already placed mode cannot be part of a nuclear family since the already placed mode does not belong to the ellipse defined by the two new modes.
One can proceed analogously to place the second generation. Note that this construction implies Property 1 . To build a set satisfying also condition (4.7) it is enough to chose the rational Pythagorean triples {(a j , b j , c j )} N j=1 such that |a j − 1|, |b j − 1|, c j ε in such a way that the ellipses are ε-close to the unit circle. Note that this is possible since, in particular, rational Pythagorean triples are dense in the unit circle. This construction gives a set in Z 2 . Note that one cannot scale and translate to construct a set in Z 2 odd as in the proof of Proposition 4.2 for the Beam case. Indeed, the resonance condition (4.3) is not invariant by translation. Instead, we refine the construction of the set in Q 2 by choosing more carefully the modes. To this end, we recall that the rational modes on the unit circle are given by z = p 1 q , p 2 q = m 2 − n 2 m 2 + n 2 , 2mn m 2 + n 2 , m, n ∈ Z.
If one choses m odd and n even one obtains a point z ∈ Q 2 whose denominator is odd and their numerators are odd in the first component and even in the second component. Certainly such points are dense in the unit circle. After a blow up by q (or any odd multiple of it), one obtains a point in Z 2 odd . We show that one can construct a set ⊂ Q 2 as just done keeping track of the rational numbers to show that all of them can be chosen of the form Indeed, one can choose the ellipses E j with rational Pythagorean triples {(a j , b j , c j )} N j=1 , a j , b j , c j ∈ Q, such that a j , b j are of the form odd/odd and c j is even/odd. Then, the rational points on the ellipse E j are of the form z = c j + a j m 2 − n 2 m 2 + n 2 , b j 2mn m 2 + n 2 , m, n ∈ Z.
Choosing m odd and n even, one has a point z of the form (A.3). Since points of this form are dense in E j one can proceed the construction such that all points in ⊂ Q 2 are of the form (A.3). Finally, it only remains to multiply by the least common divisor of all points in to obtain a set in Z 2 odd and the same happens by the multiplication by any odd multiple of the least common divisor.