Three-to-one internal resonances in coupled harmonic oscillators with cubic nonlinearity

L. Di Gregorio, W. Lacarbonara

Abstract

We investigate a general system of two coupled harmonic oscillators with cubic nonlinearity, a model relevant to various structural engineering applications. As a concrete example, we consider the case of two oscillators obtained from the reduction of the wave propagation equations representing a cellular hosting structure with 1-dof resonators in each cell. Without damping, the system is Hamiltonian, with the origin as an elliptic equilibrium characterized by two distinct linear frequencies. To understand the dynamics, it is crucial to derive explicit analytic formulae for the nonlinear frequencies as functions of the physical parameters involved. In the small amplitude regime (perturbative case), we provide the first-order nonlinear correction to the linear frequencies. While this analytic expression was already derived for non-resonant cases, it is novel in the context of resonant or nearly resonant scenarios. Specifically, we focus on the 3:1 resonance, the only resonance involved in the first-order correction. Utilizing the Hamiltonian structure, we employ Perturbation Theory methods to transform the system into Birkhoff Normal Form up to order four. This involves converting the system into action-angle variables (symplectically rescaled polar coordinates), where the truncated Hamiltonian at order four depends on the actions and, due to the resonance, on one “slow” angle. By constructing suitable nonlinear and not close-to-the-identity coordinate transformations, we identify new sets of symplectic action-angle variables. In these variables, the resulting system is integrable up to higher-order terms, meaning it does not depend on the angles, and the frequencies are obtained from the derivatives of the energy with respect to the actions. This construction is highly dependent on the physical parameters, necessitating a detailed case analysis of the phase portrait, revealing up to six topologically distinct behaviors. In each configuration, we describe the nonlinear normal modes (elliptic/hyperbolic periodic orbits, invariant tori) and their stable and unstable manifolds of the truncated Hamiltonian. As an application, we examine wave propagation in metamaterial honeycombs with periodically distributed nonlinear resonators, evaluating the nonlinear effects on the bandgap particularly in the presence of resonances.

Acknowledgments Project ECS 0000024 Rome Technopole, CUP B83C22002820006, National Recovery and Resilience Plan (NRRP) Mission 4 Component 2 Investment 1.5, funded by the European Union - NextGenerationEU.

Funder Project funded under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.5 - Call for tender No. 3277 of 30 December 2021 of the Italian Ministry of University and Research funded by the European Union - NextGenerationEU.

1 Introduction

Let us briefly recall the model introduced in [SW23jsv]. Figure 1 shows schematic view of the orthotropic plate model with the periodically distributed spider-web resonators. Each multi-frequency resonator should be meant as the multi-mass-spring system resulting from the multi-dof modal reduction of the infinite-dimensional resonator (i.e., the spider webs with a central mass, here represented in the figure, for the sake of graphical clarity, by a single mass-spring system instead of a set of mass-spring systems). The modal reduction is performed via the Galerkin projection method employing a number of mode shapes of the distributed-parameter resonators. Each resonator is represented by equivalent modal masses and modal springs.

Refer to caption — Figure 1: Schematic view of the orthotropic plate model with the periodically distributed spider-web resonators, see [SW23jsv] as reference.

The adopted plate theory (see [W]) with the elastic constants of the equivalent, homogenized orthotropic material describes the motion of the honeycomb with the attached resonators. By the Floquet-Bloch Theorem, which states that the solutions of the corresponding linear periodic resonators-plate system are quasi-periodic in space with the fundamental periodicity provided by the lattice period, the plate equation of motion can be projected onto the unit cell domain (i.e., the periodically repeated lattice unit). Then one obtains a system of $2N$ coupled second order ODEs, $N$ being the number of retained resonators modes. For the metamaterial lattice with an array of equally spaced single-dof resonators, i.e., N = 1, equations reduce to the following system of second order ODEs

\left(\begin{array}[]{cc}\tilde{M}_{H}(\tilde{k}_{1},\tilde{k}_{2})&\tilde{M}\\ \tilde{M}&\tilde{M}\end{array}\right)\left(\begin{array}[]{c}\ddot{\tilde{w}}_{0}\\ \ddot{\tilde{z}}_{0}\end{array}\right)+\left(\begin{array}[]{cc}\tilde{K}_{H}(\tilde{k}_{1},\tilde{k}_{2})&0\\ 0&\tilde{K}\end{array}\right)\left(\begin{array}[]{c}{\tilde{w}}_{0}\\ {\tilde{z}}_{0}\end{array}\right)=-\left(\begin{array}[]{c}0\\ \tilde{N}^{(3)}{\tilde{z}}_{0}^{3}\end{array}\right)\,,

(1)

where ${\tilde{w}}_{0}$ and ${\tilde{z}}_{0}$ denote the nondimensional plate deflection and resonator relative motion at the origin of the fixed frame;

\tilde{M}_{H}(\tilde{k}_{1},\tilde{k}_{2}):=\frac{4\sqrt{3}\sin\left(\frac{\tilde{k}_{1}}{2}\right)\sin\left(\frac{1}{4}\left(\tilde{k}_{1}+\sqrt{3}\tilde{k}_{2}\right)\right)}{\tilde{k}_{1}\left(\tilde{k}_{1}+\sqrt{3}\tilde{k}_{2}\right)}

(2)

and

\begin{split}&\tilde{K}_{H}(\tilde{k}_{1},\tilde{k}_{2})=\tilde{K}_{H}(\tilde{k}_{1},\tilde{k}_{2};\tilde{D}_{12},\tilde{D}_{66},\tilde{D}_{22}):=\tilde{M}_{H}(\tilde{k}_{1},\tilde{k}_{2})\left[\tilde{k}_{1}^{4}+2\tilde{k}_{1}^{2}\tilde{k}_{2}^{2}(\tilde{D}_{12}+2\tilde{D}_{66})+\tilde{k}_{2}^{4}\tilde{D}_{22}\right]\end{split}

(3)

are the nondimensional modal mass and stiffness as functions of the nondimensional wave numbers $(\tilde{k}_{1},\tilde{k}_{2})$ , which stay within the irreducible Brillouin triangle $\triangle$ (see Figure 2):

moreover

\tilde{D}_{12}=0.0815599,\qquad\tilde{D}_{22}=12.48,\qquad\tilde{D}_{66}=0.0000247357\,,

are the nondimensional plate bending stiffness coefficients; finally $\tilde{N}^{(3)}$ is the nondimensional nonlinearity.

Actually we consider the more general system of ODEs

\mathtt{M}\left(\begin{array}[]{c}\ddot{v}\\ \ddot{y}\end{array}\right)+\mathtt{K}\left(\begin{array}[]{c}v\\ y\end{array}\right)=-\left(\begin{array}[]{c}M_{3}v^{3}\\ N_{3}y^{3}\end{array}\right)\,,

(4)

where $v(t),y(t)$ are unknown scalar functions, $M_{3},N_{3}$ are real coefficients, $\mathtt{M}$ is a symmetric positive definite $2\times 2$ real matrix and $\mathtt{K}$ is a diagonal positive definite $2\times 2$ real matrix.

Note that (1) is a particular case of (4) taking $v=\tilde{w}_{0}$ , $z=\tilde{z}_{0}$ , $M_{3}=0,$ $N_{3}=\tilde{N}^{(3)}$ and

\mathtt{M}=\left(\begin{array}[]{cc}\tilde{M}_{H}(\tilde{k}_{1},\tilde{k}_{2})&\tilde{M}\\ \tilde{M}&\tilde{M}\end{array}\right)\,,\qquad\mathtt{K}=\left(\begin{array}[]{cc}\tilde{K}_{H}(\tilde{k}_{1},\tilde{k}_{2})&0\\ 0&\tilde{K}\end{array}\right)\,,

(5)

with $\tilde{M}_{H}(\tilde{k}_{1},\tilde{k}_{2})$ and $\tilde{K}_{H}(\tilde{k}_{1},\tilde{k}_{2})$ defined in (2) and (3), respectively.

The existing literature on Hamiltonian and dissipative systems covers various topics, including bifurcations, invariant manifolds, and homoclinic and heteroclinic orbits. In [Fontich23], the authors study a one-parameter family of 2-DOF Hamiltonian systems with an equilibrium point undergoing a Hamiltonian-Hopf bifurcation. They focus on invariant manifolds and the behavior of the splitting of 2D invariant manifolds in the presence of homoclinic orbits. Similarly, [Celletti13] presents a KAM theory for conformally symplectic dissipative systems, demonstrating that solutions with a fixed n-dimensional (Diophantine) frequency can be found by an a-posteriori approach adjusting the parameters.

In [Llave06], the authors develop numerical algorithms to compute invariant manifolds in quasi-periodically forced systems, focusing on invariant tori and their asymptotic invariant manifolds (whiskers). These algorithms utilize Newton’s method and power-matching expansions of parameterizations. [Cabre05] describes a method to establish the existence and regularity of invariant manifolds, simplifying the proof of the stable manifold theorem near hyperbolic points by using the implicit function theorem in Banach spaces.

[H16] proposes a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach defines nonlinear normal modes (NNMs) and spectral submanifolds, emphasizing the importance of damping for accurate conclusions about them, and the reduced-order models they produce. Lastly, [HW95], [HW96] and [HW93] develop methods to detect orbits asymptotic to slow manifolds in perturbed Hamiltonian systems, revealing complex chaotic behaviors and the creation of homoclinic orbits in resonant Hamiltonian systems through geometric singular perturbation theory and Melnikov-type methods.

1.1 Main results

We are interested here in small amplitude solutions of (4). In the first approximation the system is linear with linear frequencies $\omega_{-}$ and $\omega_{+}$ and the nonlinearity is a third order perturbation. If the linear frequencies are non vanishing, distinct and satisfy the non resonance condition $3\omega_{-}\neq\omega_{+}$ the system can be integrated, for instance, using the multiple scales method, up to a smaller fifth order nonlinear remainder, see [SW23jsv]. In particular, [SW23jsv] provides explicit expressions for the nonlinear frequencies of the truncated system (obtained by disregarding the fifth order perturbation) as functions of the initial amplitudes. Moreover the effects on the bandgap were explored.
In [DL], we analytically estimated the applicability threshold of the perturbative argument, specifically the maximal admissible amplitude for which the above formula is valid. It was found that this applicability threshold decays to zero in the presence of resonances, more precisely when the ratio between the optical and acoustic frequencies is close to 3; indeed the 3:1 resonance is the only involved resonance in the first order correction.
The methodology used is based on techniques from Hamiltonian Perturbation Theory. Since the system is conservative, we study it as a Hamiltonian system. The origin is an elliptic equilibrium and we put the system in (complete) Birkhoff Normal Form up to order 4 (3 in the equations of motion). The Birkhoff Normal Form is a powerful tool in Hamiltonian Perturbation Theory that, through a suitable symplectic, close-to-the-identity nonlinear change of coordinates, simplifies the Hamiltonian. More precisely, after introducing action-angle variables¹¹1Essentially rescaled polar coordinates., in the non resonant case, the truncated system at order four is integrated, meaning its Hamiltonian depends only on the actions, which are constant of motion, and not on the angles. As a consequence the phase space of the truncated Hamiltonian is completely foliated by nonlinear normal modes (NNMs), which are two dimensional invariant tori filled with periodic/quasi-periodic orbits depending on whether the frequency ratio is rational/irrational. Moreover such tori are (constant) graphs over the angles. Finally the nonlinear frequencies of the truncated Hamiltonian are easily evaluated as the derivatives of the Hamiltonian, i.e. the energy, with respect to the two actions. This procedure, being perturbative in nature, only works in a ball of small radius $\varepsilon$ around the origin. More precisely in [DL] we proved that there exists a constant $c_{1}$ , which was explicitly estimated as function of the physical parameters, such that the smallness condition reads

\varepsilon\leq c_{1}\sqrt{|\sigma|}\,,\qquad\mbox{where}\qquad\sigma:=\omega_{+}-3\omega_{-}\,.

(6)

In contrast, the main aim of the present paper is to investigate what happens in the complementary regime, namely when the linear frequencies are in, or almost in, 3:1 resonance, specifically when

c_{1}\sqrt{|\sigma|}<\varepsilon

(7)

and $\varepsilon$ is small enough. In this case, only a resonant BNF is available. This means that, after introducing action-angle variables and a linear symplectic change of coordinates, the truncated Hamiltonian at order four, $\hat{\mathbb{H}}_{\rm res}$ (see (38)), depends on the actions and on one “slow” angle (as its associated frequency is small). The phase portrait becomes more complicated and interesting; its topology strongly depends on the values of the physical parameters. The phase space is still foliated by two dimensional NNMs (invariant tori) but many of them are no longer graphs over the angles as in the nonresonant case, exhibiting different topologies. Moreover, one dimensional NNMs appear such as: elliptic periodic orbits or even hyperbolic ones with their two dimensional (coinciding) stable and unstable manifolds. As the parameters vary, six possible topologically different phase portraits appear. An example is given in Figure 3.

Let us denote by ${J_{2}}$ the action conjugated to the other angle, the “fast” one, which does not appear in $\hat{\mathbb{H}}_{\rm res}$ . Then ${J_{2}}$ is a constant of motion for $\hat{\mathbb{H}}_{\rm res}$ . For every fixed value of ${J_{2}}$ , $\hat{\mathbb{H}}_{\rm res}$ evaluated at ${J_{2}}=const$ in the reduced bidimensional phase space containing only the slow angle and its conjugated action is a 1-degree-of-freedom Hamiltonian system. In this reduced system, the above two dimensional NNMs (invariant tori) correspond to one dimensional NNMs (periodic orbits), one dimensional NNMs (elliptic/hyperbolic periodic orbits) correspond to zero dimensional NNMs (elliptic/hyperbolic fixed points) and, finally, two dimensional (coinciding) stable and unstable manifolds correspond to one dimensional (coinciding) stable and unstable separatrices, respectively. Some examples are shown in Figures 3, 4 and 5.

Up to the singular²²2We call it singular since it is formed by all the points whose energy is singular, namely corresponds to some critical value of the Hamiltonian. set formed by the union of zero dimensional NNMs (equilibria) and one dimensional separatrices, the phase space of the reduced Hamiltonian is separated into two or four³³3According to the different values of the parameters. In Figure 3 a case with four regions is shown. open connected components having different topologies. Since the reduced system has one degree of freedom, on such connected components one can introduce suitable new action-angle coordinates, integrating the system. Recollecting, in these new variables, $\hat{\mathbb{H}}_{\rm res}$ depends only on the new actions and the nonlinear frequencies are simply obtained as the derivatives of the Hamiltonian with respect to the actions.

However, we note that, at this stage, the nonlinear frequencies take the form of elliptic integrals, which are not simple to explicitly evaluate since both the integrating functions and the domains strongly depend on parameters. Nevertheless, we calculate them by using suitable Moebius transformations.

Finally, having the explicit formulas available, we study the nonlinear bandgap in the resonant regime. We found that, while the nonlinearity far from resonances can significantly change the bandgap, in the resonant case, the effect of resonances results in a less pronounced variation in the bandgap.

Here we study in details the truncated Hamiltonian giving a very precise description of its phase space and explicitly integrating the system. The case of the complete Hamiltonian is different since the system is genuinely two dimensional and, therefore, not integrable⁴⁴4Since the fast angle appears at higher order terms and, therefore, its conjugated action ${J_{2}}$ is not more a constant of motion.. However, using methods of KAM Theory one can prove the persistence of hyperbolic periodic orbits with their (local) stable and unstable manifolds as well as of the majority of invariant tori. Indeed, our analysis can bee seen as a necessary preparatory step toward applying KAM techniques in the resonant zones (see Remark 9).

Finally, we stress that our analysis is not limited to the case of the honeycomb metamaterials but applies directly to a wide range of problems modeled by two harmonic oscillators coupled with cubic nonlinearity as in equation (4).

1.2 Summary of the paper

Section 2: the resonant Birkhoff Normal Form

We reinterpret the problem as a Hamiltonian system (see (11)). In Subsection 2.1, we put the system, close to the origin, in resonant BNF. Then, we examine the Hamiltonian truncated at fourth order, which is equivalent to third order in the equations of motion, as it captures the essential characteristics of the overall motion. Upon introducing action-angle variables it becomes evident that the truncated, or “effective”, Hamiltonian, after a suitable linear change of variables (see (36)), also depends on one angle, known as the “slow” angle (see (38)), as its associated frequency is small or even zero on the exact resonance.

After a suitable rescaling, the effective Hamiltonian, depending on the slow angle $\psi\in[0,2\pi)$ and on the non-dimensional action $x\in(0,1)$ , takes the form $F(\psi,x)=\frac{1}{2}a_{2}x^{2}+a_{1}x+b(x)\cos\psi$ , where $b(x)=\sqrt{(1-x)^{3}x}$ and $a_{1},a_{2}$ depend on the physical parameters and on the other action (which is a constant of motion); see Subsection 2.2.

Section 3: the six possible phase portraits

The behavior of the system depends on the number and on the nature of the critical points of $F$ , which, in turn, depends on the values of $a_{1}$ and $a_{2}$ . The gradient of $F$ can vanish only on the lines $\{\psi=0\}$ , when $a_{2}x+a_{1}+b^{\prime}(x)=0$ , or $\{\psi=\pi\}$ , when $a_{2}x+a_{1}-b^{\prime}(x)=0$ . At this point studying the solutions of these equations, as $a_{1}$ and $a_{2}$ vary, is crucial (see Figure 9). This identifies six zones in the plane $(a_{1},a_{2})$ , as detailed in Proposition 1, Lemma 5 and Figure 15. Correspondingly we have six possible configurations. When reached, the maximum of $F$ is attained on the line $\{\psi=0\}$ , conversely the minimum is attained on the line $\{\psi=\pi\}$ . E.g. let us briefly describe the scenario $a_{1}+a_{2}<0$ . By studying $x\to F(0,x)$ we have three possible cases: no critical points, a maximum and a minimum with negative energy, a maximum and a minimum with positive energy. On the other hand $x\to F(\pi,x)$ has a minimum. Note that the maximum of $F(0,x)$ corresponds to a maximum for $F(\psi,x)$ , the minimum of $F(0,x)$ corresponds to a saddle for $F(\psi,x)$ and the minimum of $F(\pi,x)$ corresponds to a minimum of $F(\psi,x)$ . Analogously, the complementary case $a_{1}+a_{2}>0$ gives rise to three additional configurations.

Section 4: construction of the integrating action variable

Since the action conjugated to the “fast” angle is a constant of motion, the truncated system has two independent conserved quantities (the other one is the energy) and, therefore, is integrable (by the Arnold-Liouville Theorem), in the sense that one can find a new set of symplectic action-angle variables in which the new Hamiltonian depends only on the actions. Although the theoretical construction of the integrating action is classic, finding an explicit analytical expression as a function of all the physical parameters involved is rather complicated.

For every value of the energy $E$ , the new integrating action ${\rm I}_{1}$ is given by the area enclosed by the level curve $F(\psi,x)=E$ divided by $2\pi$ , see Section 4. Such level curves are closed and can either wrap around the cylinder $[0,2\pi)\times(0,1)$ or remain confined to its surface without wrapping around it; see Figures 3 and 6.

Since $F$ is even in $\psi$ we can restrict to consider $(\psi,x)\in[0,\pi]\times(0,1)$ . In this set the level curves are graphs over $x$ and the area enclosed by them can be computed by an integral over $x$ , whose endpoints are the $x$ -coordinate of their intersections with the lines $\{\psi=0\}$ and $\{\psi=\pi\}$ . It turns out that these correspond to the roots $0<x_{j}(E)<1$ , with $j=1,2,3,4$ , of the quartic polynomial $\mathbf{P}(x;E)=\big{(}\frac{1}{2}a_{2}x^{2}+a_{1}x-E\big{)}^{2}-(b(x))^{2}$ , see (55) and Figure 25. As the energy $E$ varies, it is necessary to distinguish whether $\mathbf{P}$ has $4,2$ or $0$ real roots⁵⁵5Note that, excluding the degenerate case of multiple roots, the number of real roots is even. and whether a root corresponds to an intersection with $\{\psi=0\}$ or $\{\psi=\pi\}$ . Explicit formulae for the roots are given in Subsection 3.5, see Figure 14.

Once we have defined the integrating action ${\rm I}_{1}$ as a function of $E$ (and of the “dumb” action, let us say, ${\rm I}_{2}$ ), the resulting integrated Hamiltonian will be its inverse $E=E({\rm I}_{1},{\rm I}_{2})$ . The nonlinear frequencies are given by the derivatives of the energy with respect to the actions, see (100), expressed through integrals, see Proposition 2. Such integrals are evaluated by suitable Moebius transformations in terms of elliptic functions, see Subsections 4.3 and 4.4.

Section 5: evaluation of the nonlinear bandgap for the honeycomb metamaterial

Finally, having the explicit formulas for the nonlinear frequencies available, we discuss the nonlinear bandgap for the honeycomb metamaterial, especially in the resonant regime. We found that, while nonlinear effects far from resonances can significantly alter the bandgap, in the resonant case the nonlinear frequencies, especially the acoustic one, closely align with the linear frequencies, resulting in a less pronounced variation in the bandgap.

2 The Hamiltonian structure and resonant BNF

In this section, after introducing optical and acoustic modes, we identify the system in (4) as Hamiltonian, see (11) below, and we evaluate the coefficients of the Hamiltonian, see (22). Set

\mathbf{\Lambda}:=\left(\begin{array}[]{cc}\omega_{-}^{2}&0\\ 0&\omega_{+}^{2}\end{array}\right)\,,

where $\omega_{-}^{2}<\omega_{+}^{2}$ are the positive eigenvalues of $\mathtt{M}^{-1}\mathtt{K}$ and $0<\omega_{-}<\omega_{+}$ . Since $\mathtt{M}$ is symmetric and $\mathtt{K}$ is diagonal, there exists a $2\times 2$ matrix $\mathbf{\Phi}$ such that

\mathbf{\Phi}^{T}\mathtt{M}\mathbf{\Phi}=\mathbf{I}\,,\qquad\mathbf{\Phi}^{T}\mathtt{K}\mathbf{\Phi}=\mathbf{\Lambda}\,,\qquad\mathbf{\Phi}=\left(\begin{array}[]{cc}\phi_{1}^{-}&\phi_{1}^{+}\\ \phi_{2}^{-}&\phi_{2}^{+}\end{array}\right)\,,

(8)

where $\mathbf{I}$ is the identity matrix. Consider the change of variables

\left(\begin{array}[]{c}v\\ y\end{array}\right)=\mathbf{\Phi}\mathbf{q}\,,\qquad\mathbf{q}:=\left(\begin{array}[]{c}q_{1}\\ q_{2}\end{array}\right)\,.

(9)

By Lemma 8 the system in (4) is transformed into

\ddot{\mathbf{q}}+\mathbf{\Lambda}\mathbf{q}=\mathbf{c}(\mathbf{q})\,,\qquad\mathbf{c}(\mathbf{q})=\left(\begin{array}[]{c}c_{1}\\ c_{2}\end{array}\right):=-\mathbf{\Phi}^{T}\left(\begin{array}[]{c}M_{3}(\phi_{1}^{-}q_{1}+\phi_{1}^{+}q_{2})^{3}\\ N_{3}(\phi_{2}^{-}q_{1}+\phi_{2}^{+}q_{2})^{3}\\ \end{array}\right)\,.

(10)

In particular

\begin{array}[]{ll}c_{1}=-\phi_{1}^{-}M_{3}(\phi_{1}^{-}q_{1}+\phi_{1}^{+}q_{2})^{3}-\phi_{2}^{-}N_{3}(\phi_{2}^{-}q_{1}+\phi_{2}^{+}q_{2})^{3}\\ c_{2}=-\phi_{1}^{+}M_{3}(\phi_{1}^{-}q_{1}+\phi_{1}^{+}q_{2})^{3}-\phi_{2}^{+}N_{3}(\phi_{2}^{-}q_{1}+\phi_{2}^{+}q_{2})^{3}\,.\end{array}

Introducing the momenta $\dot{\mathbf{q}}=\mathbf{p}=\left(\begin{array}[]{c}p_{1}\\ p_{2}\end{array}\right)$ , the system in (10) is Hamiltonian with Hamiltonian

H(\mathbf{p},\mathbf{q})=\frac{1}{2}(p_{1}^{2}+p_{2}^{2})+\frac{1}{2}\omega_{-}^{2}q_{1}^{2}+\frac{1}{2}\omega_{+}^{2}q_{2}^{2}+f(q)\,,

(11)

where

f(\mathbf{q}):=\frac{1}{4}M_{3}(\phi_{1}^{-}q_{1}+\phi_{1}^{+}q_{2})^{4}+\frac{1}{4}N_{3}(\phi_{2}^{-}q_{1}+\phi_{2}^{+}q_{2})^{4}\,.

(12)

Indeed it is immediate to see that the Hamilton’s equations $\dot{\mathbf{p}}=-\partial_{\mathbf{q}}H$ , $\dot{\mathbf{q}}=\partial_{\mathbf{p}}H=\mathbf{p}$ are equivalent to the system in (10). Since $f(\mathbf{q})$ is a homogeneous polynomial of degree $4$ we write

f(\mathbf{q})=\sum_{i+j=4}f_{i,j}q_{1}^{i}q_{2}^{j}\,,\qquad\mbox{with}\quad f_{i,j}:=\frac{6}{i!j!}\Big{(}(\phi_{1}^{-})^{i}(\phi_{1}^{+})^{j}M_{3}+(\phi_{2}^{-})^{i}(\phi_{2}^{+})^{j}N_{3}\Big{)}\,.

(13)

Introducing coordinates $\mathbf{Q}=(Q_{1},Q_{2})$ , $\mathbf{P}=(P_{1},P_{2})$ through

p_{1}=\sqrt{\omega_{-}}P_{1}\quad p_{2}=\sqrt{\omega_{+}}P_{2}\quad q_{1}=\frac{1}{\sqrt{\omega_{-}}}Q_{1}\quad q_{2}=\frac{1}{\sqrt{\omega_{+}}}Q_{2}

(14)

we have that the Hamiltonian in the new variables reads

\mathtt{H}(\mathbf{P},\mathbf{Q}):=\omega_{-}\frac{P_{1}^{2}+Q_{1}^{2}}{2}+\omega_{+}\frac{P_{2}^{2}+Q_{2}^{2}}{2}+f\left(\frac{Q_{1}}{\sqrt{\omega_{-}}},\frac{Q_{2}}{\sqrt{\omega_{+}}}\right)\,.

(15)

In complex coordinates, $\mathrm{i}=\sqrt{-1}\in\mathbb{C}$ , $\mathbf{z}=(z_{1},z_{2})\in\mathbb{C}^{2}$

z_{j}=\frac{Q_{j}+\mathrm{i}P_{j}}{\sqrt{2}}\quad\bar{z}_{j}=\frac{Q_{j}-\mathrm{i}P_{j}}{\sqrt{2}}\qquad j=1,2

(16)

the Hamiltonian reads

\mathtt{H}(\mathbf{z},\bar{\mathbf{z}})=\mathtt{N}(\mathbf{z},\bar{\mathbf{z}})+{\mathtt{G}}(\mathbf{z},\bar{\mathbf{z}})

(17)

where

\mathtt{N}(\mathbf{z},\bar{\mathbf{z}}):=\omega_{-}z_{1}\bar{z}_{1}+\omega_{+}z_{2}\bar{z}_{2}\,,\qquad{\mathtt{G}}(\mathbf{z},\bar{\mathbf{z}}):=f\left(\frac{z_{1}+\bar{z}_{1}}{\sqrt{2\omega_{-}}},\frac{z_{2}+\bar{z}_{2}}{\sqrt{2\omega_{+}}}\right)\,.

(18)

Note that in complex coordinates the Hamilton’s equations of motion are

\dot{z}_{j}=-\mathrm{i}\partial_{\bar{z}_{j}}\mathtt{H}\,,\ \ \dot{\bar{z}}_{j}=\mathrm{i}\partial_{z_{j}}\mathtt{H}\,.

(19)

In the following we use the multi-index notation

P(\mathbf{z},\bar{\mathbf{z}})=\sum_{({\bm{\alpha}},{\bm{\beta}})\in\mathbb{N}^{2}\times\mathbb{N}^{2}}P_{{\bm{\alpha}},{\bm{\beta}}}\mathbf{z}^{\bm{\alpha}}\bar{\mathbf{z}}^{\bm{\beta}}

(20)

for suitable coefficients $P_{{\bm{\alpha}},{\bm{\beta}}}\in\mathbb{C}$ with $\mathbf{z}^{\bm{\alpha}}=z_{1}^{\alpha_{1}}z_{2}^{\alpha_{2}}$ (analogously for $\bar{\mathbf{z}}^{\bm{\beta}}$ ). In these notation, recalling (13) and (18), we rewrite ${\mathtt{G}}$ as⁶⁶6Where, for integer vectors ${\bm{\alpha}}=(\alpha_{1},\alpha_{2}),{\bm{\beta}}=(\beta_{1},\beta_{2})$ we set $|{\bm{\alpha}}+{\bm{\beta}}|:=\alpha_{1}+\alpha_{2}+\beta_{1}+\beta_{2}$ .

{\mathtt{G}}(\mathbf{z},\bar{\mathbf{z}})=\sum_{i+j=4}\frac{f_{i,j}}{4(\sqrt{\omega_{-}})^{i}(\sqrt{\omega_{+}})^{j}}(z_{1}+\bar{z}_{1})^{i}(z_{2}+\bar{z}_{2})^{j}=\sum_{|{\bm{\alpha}}+{\bm{\beta}}|=4}{\mathtt{G}}_{{\bm{\alpha}},{\bm{\beta}}}\mathbf{z}^{\bm{\alpha}}\bar{\mathbf{z}}^{\bm{\beta}}

(21)

where

{\mathtt{G}}_{{\bm{\alpha}},{\bm{\beta}}}:=\frac{f_{\alpha_{1}+\beta_{1},\alpha_{2}+\beta_{2}}}{4(\sqrt{\omega_{-}})^{\alpha_{1}+\beta_{1}}(\sqrt{\omega_{+}})^{\alpha_{2}+\beta_{2}}}\frac{(\alpha_{1}+\beta_{1})!}{\alpha_{1}!\beta_{1}!}\frac{(\alpha_{2}+\beta_{2})!}{\alpha_{2}!\beta_{2}!}\,.

(22)

Note that ${\mathtt{G}}_{{\bm{\alpha}},{\bm{\beta}}}={\mathtt{G}}_{{\bm{\beta}},{\bm{\alpha}}}\in\mathbb{R}$ .

2.1 Resonant BNF

The aim of the BNF is to construct a symplectic change of variables that “simplifies” the Hamiltonian $\mathtt{H}$ in (17). First note that a Hamiltonian $H$ depending only on $|z_{1}|^{2}$ and $|z_{2}|^{2}$ writes $H=\sum_{{\bm{\alpha}}}H_{{\bm{\alpha}},{\bm{\alpha}}}|\mathbf{z}|^{2{\bm{\alpha}}}$ and is integrable; in particular $|z_{1}|^{2}$ and $|z_{2}|^{2}$ are constants of motion. In light of the above considerations we guess if it is possible to find, in a sufficiently small neighborhood of the origin

\|\mathbf{z}\|\leq\epsilon\,,

(23)

a close-to-the-identity symplectic transformation that “integrates” $\mathtt{H}$ up to terms of degree 6 in $(\mathbf{z},\bar{\mathbf{z}})$ , which are smaller. This amounts to transform $\mathtt{H}$ into ${\mathtt{N}}+\bar{\mathtt{H}}_{4}+O(\|\mathbf{z}\|^{6})$ , with

\bar{\mathtt{H}}_{4}:=\sum_{|{\bm{\alpha}}|=2}{\mathtt{G}}_{{\bm{\alpha}},{\bm{\alpha}}}|\mathbf{z}|^{2{\bm{\alpha}}}={\mathtt{G}}_{(2,0),(2,0)}|z_{1}|^{4}+{\mathtt{G}}_{(1,1),(1,1)}|z_{1}|^{2}|z_{2}|^{2}+{\mathtt{G}}_{(0,2),(0,2)}|z_{2}|^{4}\,,

(24)

where, recalling (22),

$\displaystyle{\mathtt{G}}_{(2,0),(2,0)}$	$\displaystyle=$	$\displaystyle\frac{3f_{4,0}}{2\omega_{-}^{2}}=\frac{3}{8\omega_{-}^{2}}\Big{(}(\phi_{1}^{-})^{4}M_{3}+(\phi_{2}^{-})^{4}N_{3}\Big{)}\,,$
$\displaystyle{\mathtt{G}}_{(1,1),(1,1)}$	$\displaystyle=$	$\displaystyle\frac{f_{2,2}}{\omega_{-}\omega_{+}}=\frac{3}{2\omega_{-}\omega_{+}}\Big{(}(\phi_{1}^{-})^{2}(\phi_{1}^{+})^{2}M_{3}+(\phi_{2}^{-})^{2}(\phi_{2}^{+})^{2}N_{3}\Big{)}\,,$
$\displaystyle{\mathtt{G}}_{(0,2),(0,2)}$	$\displaystyle=$	$\displaystyle\frac{3f_{0,4}}{2\omega_{+}^{2}}=\frac{3}{8\omega_{+}^{2}}\Big{(}(\phi_{1}^{+})^{4}M_{3}+(\phi_{2}^{+})^{4}N_{3}\Big{)}\,.$	(25)

As well known, this is possible if the nonresonance condition $\omega_{+}k_{1}+\omega_{-}k_{2}\neq 0$ is satisfied for every couple of integers $k_{1},k_{2}$ with $|k_{1}|+|k_{2}|=4$ and $\epsilon$ is small enough. It is simple to show (see, e.g. Proposition 1 in [DL]) that

\min_{|k_{1}|+|k_{2}|=4}|\omega_{+}k_{1}+\omega_{-}k_{2}|\geq\min\{\omega_{-},\omega_{+}-\omega_{-},|3\omega_{-}-\omega_{+}|\}\,.

While, by hypothesis, $\omega_{-},\omega_{+}-\omega_{-}>0$ , $\sigma=\omega_{+}-3\omega_{-}$ (introduced in (6)) could be zero or small. It turns out that there exists a constant $C_{1}$ (see [DL] for a proof and the evaluation of $C_{1}$ ) such that, if

\epsilon\leq C_{1}\sqrt{|\sigma|}\,,

(26)

then it is possible to construct a symplectic transformation putting $\mathtt{H}$ in (complete) BNF up to order 4, namely ${\mathtt{N}}+\bar{\mathtt{H}}_{4}+O(\|\mathbf{z}\|^{6})$ . Otherwise, if $|\sigma|$ is too small with respect to⁷⁷7In particular we can assume that $|\sigma|\leq\min\{\omega_{-},\omega_{+}-\omega_{-}\}$ . $\epsilon$ , namely if $\epsilon>C_{1}\sqrt{|\sigma|}$ , but $\epsilon$ still satisfies a suitable (weaker⁸⁸8With $C_{2}>C_{1}\sqrt{|\sigma|}$ .) smallness condition $\epsilon\leq C_{2}$ , only a resonant BNF is available. This means that, in the case

C_{1}\sqrt{|\sigma|}\leq\epsilon\leq C_{2}\,,

(27)

through a symplectic transformation, the Hamiltonian takes the form ${\mathtt{N}}+\bar{\mathtt{H}}_{4,{\rm res}}+O(\|\mathbf{z}\|^{6})$ , where

\bar{\mathtt{H}}_{4,{\rm res}}:=\bar{\mathtt{H}}_{4}+\mathtt{G}_{(0,1),(3,0)}z_{2}\bar{z}_{1}^{3}+\mathtt{G}_{(3,0),(0,1)}z_{1}^{3}\bar{z}_{2}\stackrel{{\scriptstyle\eqref{Gab}}}{{=}}\bar{\mathtt{H}}_{4}+\frac{f_{3,1}}{4(\sqrt{\omega_{-}})^{3}\sqrt{\omega_{+}}}(z_{2}\bar{z}_{1}^{3}+z_{1}^{3}\bar{z}_{2})\,.

(28)

Remark 1.

The construction of the above symplectic transformation in the resonant case was given in [DL], where the remainder $O(\|\mathbf{z}\|^{6})$ was explicitly estimated. This means that we found a concrete constant $c_{*}$ depending on the parameters such that $O(\|\mathbf{z}\|^{6})\leq c_{*}\epsilon^{6}$ .

Remark 2.

$\epsilon$ introduced in (23) is simply related to $\varepsilon$ introduced in (6) by the change of variables (9), (14), (16). This means that there exist two constants $\underline{c}<\bar{c}$ such that $\underline{c}\leq\epsilon/\varepsilon\leq\bar{c}$ . Then (26) and (27) justify (6) and (7), respectively.

We now introduce action-angle variables⁹⁹9 $\mathbb{T}:=\mathbb{R}/{2\pi\mathbb{Z}}$ , $\mathbb{T}^{2}:=\mathbb{R}^{2}/{2\pi\mathbb{Z}^{2}}$ . $(\mathbf{I},{\bm{\varphi}})=(I_{1},I_{2},\varphi_{1},\varphi_{2})\in\mathbb{R}^{2}\times\mathbb{T}^{2}$ through the transformation

z_{j}=\sqrt{I_{j}}e^{-\mathrm{i}\varphi_{j}}\,,\qquad I_{j}>0\,,\qquad j=1,2.

(29)

Remark 3.

Note that the above map is singular at $\mathrm{z}_{1}$ or $\mathrm{z}_{2}=0$ and is defined for $I_{1},I_{2}>0$ .

In the symplectic variables in (29) the truncated Hamiltonians ${\mathtt{N}}+{\mathtt{H}}_{4}$ and ${\mathtt{N}}+\bar{\mathtt{H}}_{4,{\rm res}}$ take the final forms

$\displaystyle\hat{\mathcal{H}}_{\rm res}(\mathbf{I},{\bm{\varphi}})$	$\displaystyle:=$	$\displaystyle\hat{\mathcal{H}}(\mathbf{I},{\bm{\varphi}})+\frac{f_{3,1}}{2(\sqrt{\omega_{-}})^{3}\sqrt{\omega_{+}}}\sqrt{I_{1}^{3}I_{2}}\cos(\varphi_{2}-3\varphi_{1})\,,\,,$	(30)
$\displaystyle\hat{\mathcal{H}}(\mathbf{I},{\bm{\varphi}})$	$\displaystyle:=$	$\displaystyle\omega_{-}I_{1}+\omega_{+}I_{2}+\mathcal{H}_{4}(\mathbf{I})$	(31)
$\displaystyle\mathcal{H}_{4}(\mathbf{I})$	$\displaystyle:=$	$\displaystyle{\mathtt{G}}_{(2,0),(2,0)}I_{1}^{2}+{\mathtt{G}}_{(1,1),(1,1)}I_{1}I_{2}+{\mathtt{G}}_{(0,2),(0,2)}I_{2}^{2}\,.$	(32)

The frequencies of the integrable nonresonant truncated Hamiltonian $\hat{\mathcal{H}}$ in (31) are the derivatives of the energy with respect to the actions, namely, by (32),

	$\displaystyle\omega_{-}^{\rm nl}$	$\displaystyle:=$	$\displaystyle\partial_{I_{1}}\hat{\mathcal{H}}=\omega_{-}+2{\mathtt{G}}_{(2,0),(2,0)}I_{1}+{\mathtt{G}}_{(1,1),(1,1)}I_{2}\,,$
	$\displaystyle\omega_{+}^{\rm nl}$	$\displaystyle:=$	$\displaystyle\partial_{I_{2}}\hat{\mathcal{H}}=\omega_{+}+2{\mathtt{G}}_{(0,2),(0,2)}I_{2}+{\mathtt{G}}_{(1,1),(1,1)}I_{1}\,,$

In particular, when $M_{3}=0$ , by (25) we have

	$\displaystyle\omega_{-}^{\rm nl}$	$\displaystyle=$	$\displaystyle\omega_{-}+N_{3}\left(\frac{3}{8\omega_{-}}(\phi_{2}^{-})^{4}a_{-}^{2}+\frac{3}{4\omega_{-}}(\phi_{2}^{-})^{2}(\phi_{2}^{+})^{2}a_{+}^{2}\right)\,,$
	$\displaystyle\omega_{+}^{\rm nl}$	$\displaystyle=$	$\displaystyle\omega_{+}+N_{3}\left(\frac{3}{8\omega_{+}}(\phi_{2}^{+})^{4}a_{+}^{2}+\frac{3}{4\omega_{+}}(\phi_{2}^{-})^{2}(\phi_{2}^{+})^{2}a_{-}^{2}\right)\,,$		(33)

where $a_{-},a_{+}>0$ are the initial amplitudes. Note that in the original variables $q_{1}$ and $q_{2}$ , one has

q_{1}(0)=a_{-}\,,\quad q_{2}(0)=a_{+}\,,\quad\dot{p}_{1}(0)=\dot{p}_{2}(0)=0\,,

(34)

that correspond, by (14), (16) and (29), in initial action-angle variables:

I_{1}(0)=\frac{1}{2}\omega_{-}a_{-}^{2}\,,\qquad I_{2}(0)=\frac{1}{2}\omega_{+}a_{+}^{2}\,,\qquad\varphi_{1}(0)=\varphi_{2}(0)=0\,.

(35)

Formula (2.1) was already known (see [SW23jsv] or [DL]), but it does not hold close to resonances. To obtain the analogous of formula (2.1) in the resonant case is much more complicated since one has to integrate the Hamiltonian $\hat{\mathcal{H}}_{\rm res}$ in (30). This is exactly what we are going to do in the following sections. The analogous of (2.1) in the resonant case are the formula (153)-(156) below.

Remark 4 (Reversibility).

Since the Hamiltonian $\hat{\mathcal{H}}_{\rm res}(\mathbf{I},{\bm{\varphi}})$ in (30) is even in ${\bm{\varphi}}$ the system is reversible, namely if $\big{(}\mathbf{I}(t),{\bm{\varphi}}(t)\big{)}$ is a solution the same holds true for $\big{(}\mathbf{I}(-t),-{\bm{\varphi}}(-t)\big{)}$ . In particular if ${\bm{\varphi}}(0)=0$ the solution is even in the actions and odd in the angles, namely $\mathbf{I}(t)=\mathbf{I}(-t)$ and ${\bm{\varphi}}(t)=-{\bm{\varphi}}(-t)$ .

2.2 The slow angle and the effective Hamiltonian

It is convenient to introduce the adimensional effective Hamiltonian $F$ depending solely on one angle $\psi_{1}$ , namely the “slow angle”. Let us consider the canonical transformation

\Phi_{*}:\mathbb{R}^{2}\times\mathbb{T}^{2}\to\mathbb{R}^{2}\times\mathbb{T}^{2}\qquad\Phi_{*}(J,\psi)=(I,{\bm{\varphi}}):=(\mathcal{M}^{T}J,\mathcal{M}^{-1}\psi)\qquad\mathcal{M}=\left(\begin{array}[]{cc}-3&1\\ 1&0\end{array}\right)

so that

\left\{\begin{array}[]{l}I_{1}=J_{2}-3J_{1}\\ I_{2}=J_{1}\,,\end{array}\right.\qquad\left\{\begin{array}[]{l}\varphi_{1}=\psi_{2}\\ \varphi_{2}=\psi_{1}+3\psi_{2}\,.\end{array}\right.

(36)

Note that $\mathcal{M}$ has integer entries and $\mathrm{det}\mathcal{M}=-1$ so that the inverse $\mathcal{M}^{-1}$ has also integer entries. This implies that $\psi=\mathcal{M}\varphi$ and its inverse $\varphi=\mathcal{M}^{-1}\psi$ are well defined on the torus $\mathbb{T}^{2}$ . Note also that, by (29), we have

J_{2}>3J_{1}\,,\qquad J_{1}>0\,.

(37)

Let us write $\hat{\mathcal{H}}_{{\rm res}}$ in (30) in the $(J,\psi)$ -variables

\hat{\mathbb{H}}_{\rm res}(J,\psi_{1}):=\hat{\mathcal{H}}_{{\rm res}}\big{(}\Phi_{*}(J,\psi_{1})\big{)}:=\omega_{-}J_{2}+\sigma J_{1}+\mathbb{H}_{4,{\rm res}}(J,\psi_{1})\,,

(38)

where

\mathbb{H}_{4,{\rm res}}(J,\psi_{1}):=\mathcal{H}_{4}(J_{2}-3J_{1},J_{1})+\frac{f_{3,1}}{2(\sqrt{\omega_{-}})^{3}\sqrt{\omega_{+}}}\sqrt{(J_{2}-3J_{1})^{3}J_{1}}\cos(\psi_{1})\,.

(39)

Note that $\hat{\mathbb{H}}_{\rm res}$ is reversible in the sense of Remark 4. Moreover it depends only on the “slow angle” $\psi_{1}$ , that evolves by a small frequency $\sigma+O(|J|)\sim 0$ (recall (7)), but does not depend on the “fast angle” $\psi_{2}$ , that, on the contrary, evolves by a frequency $\omega_{-}>0$ , which is definitively different from zero. So the partial derivative w.r.t. $\psi_{2}$ of $\hat{\mathbb{H}}_{\rm res}$ vanishes and, by the Hamilton’s equations, $\dot{J}_{2}=0$ , so that $J_{2}$ is a constant of motion, namely

J_{2}(t)=J_{2}(0)=:{J_{2}}\,.

Moreover the fast angle $\psi_{2}$ simply evolves as $\psi_{2}(t)=\omega_{-}t+\psi_{2}(0)$ . It remains to study the evolution of the $(J_{1},\psi_{1})$ variables.

Being $J_{2}$ a constant of motion the dynamic of the “resonant truncated Hamiltonian” $\hat{\mathbb{H}}_{\rm res}$ in (38) is simply generated by the one-degree-of-freedom “effective Hamiltonian”

\mathbb{H}_{{J_{2}}}(J_{1},\psi_{1}):=\sigma J_{1}+\mathbb{H}_{4,{\rm res}}(J_{1},{J_{2}},\psi_{1})\,,

(40)

with $\mathbb{H}_{4,{\rm res}}$ defined in (39). At this point it is convenient to introduce the “adimensional Hamiltonian”¹⁰¹⁰10Also $\chi$ is adimensional.

\hat{H}(J_{1},\psi_{1})=\hat{H}_{J_{2}}(J_{1},\psi_{1}):=\frac{1}{\chi{J_{2}^{2}}}\mathbb{H}_{{J_{2}}}(J_{1},\psi_{1})\,,\qquad\mbox{with}\ \ \chi:=\frac{f_{3,1}}{2\sqrt{3}(\sqrt{\omega_{-}})^{3}\sqrt{\omega_{+}}}\neq 0\,,

(41)

and rewrite $\hat{H}_{J_{2}}$ as a function of the “adimensional action”

x:=3J_{1}/{J_{2}}\qquad\mbox{with}\quad 0<x<1\,,

(42)

by (37). We have the following

Lemma 1.

It results that

\hat{H}(J_{1},\psi_{1})=\hat{H}_{J_{2}}(J_{1},\psi_{1})=F(\psi_{1},3J_{1}/{J_{2}};{J_{2}})+a_{0}\,,

(43)

where

$\displaystyle F(\psi,x)=F(\psi,x;{J_{2}})$	$\displaystyle:=$	$\displaystyle a(x;{J_{2}})+b(x)\cos\psi\,,$
$\displaystyle a(x)=a(x;J_{2})$	$\displaystyle:=$	$\displaystyle\frac{1}{2}a_{2}x^{2}+a_{1}x\,,$
$\displaystyle b(x)$	$\displaystyle:=$	$\displaystyle\sqrt{(1-x)^{3}x}>0\,,\qquad 0<x<1\,,$
$\displaystyle a_{0}$	$\displaystyle:=$	$\displaystyle\frac{{\mathtt{G}}_{(2,0),(2,0)}}{\chi}\,,$
$\displaystyle a_{1}$	$\displaystyle=$	$\displaystyle a_{1}({J_{2}}):=-2\frac{{\mathtt{G}}_{(2,0),(2,0)}}{\chi}+\frac{{\mathtt{G}}_{(1,1),(1,1)}}{3\chi}+\frac{\omega_{+}-3\omega_{-}}{3{J_{2}}\chi}\,,$
$\displaystyle a_{2}$	$\displaystyle:=$	$\displaystyle 2\frac{{\mathtt{G}}_{(2,0),(2,0)}}{\chi}-2\frac{{\mathtt{G}}_{(1,1),(1,1)}}{3\chi}+2\frac{{\mathtt{G}}_{(0,2),(0,2)}}{9\chi}\,.$	(44)

proof. Multiplying the right hand side of (43) by $\chi{J_{2}^{2}}$ we have, by (1),

	$\displaystyle\chi{J_{2}^{2}}\,F(\psi_{1},3J_{1}/{J_{2}})+\chi{J_{2}^{2}}\,a_{0}=\chi{J_{2}^{2}}\,a(3J_{1}/{J_{2}})+\chi{J_{2}^{2}}\,a_{0}+\chi{J_{2}^{2}}\,b(3J_{1}/{J_{2}})\cos\psi_{1}$
	$\displaystyle\qquad=\frac{9}{2}\chi a_{2}J_{1}^{2}+3\chi a_{1}J_{1}{J_{2}}+\chi{J_{2}^{2}}\,a_{0}+\chi\sqrt{3({J_{2}}-3J_{1})^{3}J_{1}}\cos\psi_{1}$
	$\displaystyle\qquad={\mathtt{G}}_{(2,0),(2,0)}({J_{2}}-3J_{1})^{2}+{\mathtt{G}}_{(1,1),(1,1)}J_{1}({J_{2}}-3J_{1})+{\mathtt{G}}_{(0,2),(0,2)}{J_{2}^{2}}$
	$\displaystyle\qquad\quad+\sigma J_{1}+\chi\sqrt{3({J_{2}}-3J_{1})^{3}J_{1}}\cos\psi_{1}$
	$\displaystyle\qquad\stackrel{{\scriptstyle\eqref{pollo}}}{{=}}\mathcal{H}_{4}(J_{2}-3J_{1},J_{1})+\sigma J_{1}+\chi\sqrt{3({J_{2}}-3J_{1})^{3}J_{1}}\cos\psi_{1}$
	$\displaystyle\quad\ \stackrel{{\scriptstyle\eqref{connessione},\eqref{pluto}}}{{=}}\mathbb{H}_{4,{\rm res}}(J,\psi_{1})+\sigma J_{1}$
	$\displaystyle\qquad\stackrel{{\scriptstyle\eqref{1DHAM}}}{{=}}\mathbb{H}_{{J_{2}}}(J_{1},\psi_{1})\stackrel{{\scriptstyle\eqref{pluto}}}{{=}}\chi{J_{2}^{2}}\hat{H}_{J_{2}}(J_{1},\psi_{1})\,,$

proving (43). $\Box$

Remark 5.

Note that $F(\psi,x;{J_{2}})$ depends on ${J_{2}}$ only through $a(x;{J_{2}})$ , which depends on ${J_{2}}$ only through $a_{1}({J_{2}})$ . Moreover at the exact resonance $\omega_{+}=3\omega_{-}$ the dependence on ${J_{2}}$ disappears.

Since $\psi_{2}$ does not appear in $\hat{H}$ , the conjugated action ${J_{2}}$ is a constant of motion and $\hat{H}$ is actually a one degree of freedom Hamiltonian system depending on ${J_{2}}$ as a parameter. From now on we consider the one degree of freedom Hamiltonian $\hat{H}(J_{1},\psi_{1})=\hat{H}_{J_{2}}(J_{1},\psi_{1})$ on the phase space $(0,3{J_{2}})\times\mathbb{T}\ni(J_{1},\psi_{1})$ with $\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}$ .

3 The phase portrait

In this section we study the phase portrait of the adimensional Hamiltonian $\hat{H}$ in (41) describing level curves, critical points and extrema. An important remark, that simplifies the treatment, is the fact that, thanks to (43), $\hat{H}$ has, up to the rescalings $x=3J_{1}/J_{2}$ and $F=\hat{H}+a_{0}$ , the same level curves, critical points and extrema as the auxiliary function $F$ in (1). Such objects are studied in Subsections 3.1, 3.2 and 3.3, respectively. As usual, the new action coordinates, that integrate the system, are defined as the areas enclosed by the level curves. In order to evaluate them its important to determine the intersections between the level curves and the lines $\{\psi=0\}$ and $\{\psi=\pi\}$ since they appear as endpoints of the involved integrals. It turns out that such intersections correspond to the real roots of the quartic polynomial $\mathbf{P}(x)$ , see (55) and Figure 25. As the energy $E$ varies, it is necessary to distinguish whether $\mathbf{P}$ has $4,2$ or $0$ real roots¹¹¹¹11Note that, excluding the degenerate case of multiple roots, the number of real roots is even. and whether a root corresponds to an intersection with $\{\psi=0\}$ or $\{\psi=\pi\}$ . Explicit formulae for the roots are given in Subsection 3.5, see Figure 14. In Subsections 3.6 and 3.7 as the parameters vary, six topologically different scenarios appear.

3.1 Critical points, elliptic and hyperbolic zones

We now describe how the critical point of $\hat{H}$ depends on the values of the parameters $a_{2}$ and $a_{1}$ in (1). First we note that $(J_{1},\psi_{1})$ is a critical point of $\hat{H}$ if and only if $(\psi,x):=(\psi_{1},3J_{1}/J_{2})$ is a critical point of the auxiliary function $F(\psi,x)=a(x)+b(x)\cos\psi$ defined on $\mathbb{T}\times(0,1)$ (recall (1)). Moreover the nature of a critical point (maximum, minimum or saddle) is the same for $\hat{H}$ and $F$ . Then in the following we will study critical points of $F$ as the parameters $a_{2}$ and $a_{1}$ vary.

It is immediate to see that, since

\partial_{x}F(\psi,x)=a^{\prime}(x)+b^{\prime}(x)\cos\psi\,,\qquad\partial_{\psi}F(\psi,x)=-b(x)\sin\psi

and $b(x)>0$ , the critical points of $F$ have the form $(x,0)$ with $a^{\prime}(x)+b^{\prime}(x)=0$ or $(x,\pi)$ with $a^{\prime}(x)-b^{\prime}(x)=0$ . Namely

	$\displaystyle\nabla F(0,x)=0\$	$\displaystyle\iff$	$\displaystyle\ -a_{2}x-a_{1}=b^{\prime}(x)\,,$		(45)
	$\displaystyle\nabla F(\pi,x)=0\$	$\displaystyle\iff$	$\displaystyle\ a_{2}x+a_{1}=b^{\prime}(x)\,$		(46)

where

b^{\prime}(x)=\frac{(1-4x)\sqrt{1-x}}{2\sqrt{x}}.

(47)

The number of solutions of equations (45),(46) depends on the parameters $a_{1},a_{2}$ .

Set

g(a_{1}):=\frac{1}{27}\textstyle\big{(}\sqrt{9+4a_{1}^{2}}-2a_{1}\big{)}\big{(}9-4a_{1}^{2}-4a_{1}\sqrt{9+4a_{1}^{2}}\big{)}

(48)

and¹²¹²12Note that $g(a_{1})>-a_{1}$ .

$\displaystyle Z_{10}$	$\displaystyle:=$	$\displaystyle\{(a_{1},a_{2})\ \ :\ \ a_{2}<-g(-a_{1})\}\,,$
$\displaystyle Z_{12}$	$\displaystyle:=$	$\displaystyle\{(a_{1},a_{2})\ \ :\ \ -g(-a_{1})<a_{2}<-a_{1}\}\,,$
$\displaystyle Z_{21}$	$\displaystyle:=$	$\displaystyle\{(a_{1},a_{2})\ \ :\ \ -a_{1}<a_{2}<g(a_{1})\}\,,$
$\displaystyle Z_{01}$	$\displaystyle:=$	$\displaystyle\{(a_{1},a_{2})\ \ :\ \ a_{2}>g(a_{1})\,.$	(49)

In particular the following result holds

Proposition 1.

If $(a_{1},a_{2})\in Z_{ij}$ then $F(0,x)$ has $i$ critical points and $F(\pi,x)$ $j$ critical points. More precisely:

•

If $(a_{1},a_{2})\in Z_{10}$ then $F(0,x)$ has a positive maximum at some $x_{1}^{(0)}$ and $F(\pi,x)$ is strictly decreasing;
•

If $(a_{1},a_{2})\in Z_{01}$ then $F(\pi,x)$ has a negative minimum at some $x_{1}^{(\pi)}$ and $F(0,x)$ is strictly increasing;
•

If $(a_{1},a_{2})\in Z_{12}$ then $F(0,x)$ has a positive maximum at some $x_{1}^{(0)}$ , while $F(\pi,x)$ has a negative minimum at some $x_{1}^{(\pi)}$ and a maximum at some $x_{2}^{(\pi)}$ , with $x_{1}^{(\pi)}<x_{2}^{(\pi)}$ ;
•

If $(a_{1},a_{2})\in Z_{21}$ then $F(0,x)$ has a positive maximum at some $x_{1}^{(0)}$ and a minimum at some $x_{2}^{(0)}$ , with $x_{1}^{(0)}<x_{2}^{(0)}$ , while $F(\pi,x)$ has a negative minimum at some $x_{1}^{(\pi)}$ .

As a corollary, if $(a_{1},a_{2})\in Z_{ij}$ then $F$ has $i$ critical points of the form $(0,x)$ and $j$ critical points of the form $(\pi,x)$ . More precisely:

•

If $(a_{1},a_{2})\in Z_{10}$ then $F$ has a positive maximum at $(0,x_{1}^{(0)})$ ;
•

If $(a_{1},a_{2})\in Z_{01}$ then $F$ has a negative minimum at $(\pi,x_{1}^{(\pi)})$ ;
•

If $(a_{1},a_{2})\in Z_{12}$ then $F$ has a positive maximum at $(0,x_{1}^{(0)})$ , a negative minimum at $(\pi,x_{1}^{(\pi)})$ and a saddle at $(\pi,x_{2}^{(\pi)})$ ;
•

If $(a_{1},a_{2})\in Z_{21}$ then $F$ has a positive maximum at $(0,x_{1}^{(0)})$ , a saddle at $(0,x_{2}^{(0)})$ and a negative minimum at $(\pi,x_{1}^{(\pi)})$ .

proof. See Appendix. $\Box$

We call $Z_{21}$ , $Z_{12}$ hyperbolic zones, since they contain hyperbolic equilibria, and $Z_{01}$ , $Z_{10}$ elliptic zones, since they contain only elliptic equilibria. For any fixed pair $(\tilde{M},\tilde{K})$ , it is possible to identify which wave numbers $(\tilde{k}_{1},\tilde{k}_{2})$ in the Brillouin triangle give rise to resonant normal forms with different phase portraits. In particular if the corresponding values of $a_{1}$ and $a_{2}$ belong to $Z_{21}$ , $Z_{12}$ , then the phase portrait contains one hyperbolic and two elliptic equilibria, while, for $a_{1}$ and $a_{2}$ belonging to $Z_{01}$ , $Z_{10}$ only elliptic equilibria appear (see Figure 11). For brevity we denote by BNF of type $Z_{ij}$ the corresponding Birkhoff Normal Form.

Remark 6.

In the following for simplicity we restrict to the case in which $(a_{1},a_{2})\in Z_{ij}$ for some $0\leq i,j\leq 2$ . This means that we avoid the degenerate cases $a_{2}+a_{1}=0$ , when $x=1$ is a solution of (45)-(46), $a_{2}-g(a_{1})=0$ and $a_{2}+g(-a_{1})=0$ , when two solutions coincide. We will briefly discuss such degenerate cases in Subsection 3.8.

3.2 Extrema

We now discuss the extrema of $F$ in (1) and their dependence on the parameters $a_{1},a_{2}$ . Following the notation of Proposition 1 we set

$\displaystyle E_{\rm max}$	$\displaystyle:=$	$\displaystyle F(0,x_{1}^{(0)})\,,\qquad{\rm if}\ \ (a_{1},a_{2})\in Z_{10},\ Z_{12},\ Z_{21}\,,$
$\displaystyle E_{\rm min}$	$\displaystyle:=$	$\displaystyle F(\pi,x_{1}^{(\pi)})\,,\qquad{\rm if}\ \ (a_{1},a_{2})\in Z_{01},\ Z_{12},\ Z_{21}\,,$
$\displaystyle E_{\rm sad}$	$\displaystyle:=$	$\displaystyle F(0,x_{2}^{(0)})\,,\qquad\,{\rm if}\ \ (a_{1},a_{2})\in Z_{21}\,,$
$\displaystyle E_{\rm sad}$	$\displaystyle:=$	$\displaystyle F(\pi,x_{2}^{(\pi)})\,,\qquad{\rm if}\ \ (a_{1},a_{2})\in Z_{12}\,.$	(50)

Then define

E_{+}:=\sup_{\mathbb{T}\times(0,1)}F\,,\qquad E_{-}:=\inf_{\mathbb{T}\times(0,1)}F\,.

(51)

Since $F(\pi,x)<F(\psi,x)<F(0,x)$ for every $0<x<1$ , $0<\psi<2\pi$ , $\psi\neq\pi$ , we have that

E_{+}:=\sup_{(0,1)}F(0,x)\,,\qquad E_{-}:=\inf_{(0,1)}F(\pi,x)\,.

Note that

E_{-}<0<E_{+}\,,

since $F(0,0)=F(0,\pi)=0$ and $F(\pi,x)<0<F(0,x)$ for $x>0$ small enough since $\lim_{x\to 0^{+}}\partial_{x}F(0,x)=+\infty$ and $\lim_{x\to 0^{+}}\partial_{x}F(\pi,x)=-\infty$ . Note that in the cases $Z_{10},Z_{12}$ we have $E_{+}=E_{\rm max}$ , since the function $x\to F(0,x)$ has only one critical point (a maximum); analogously in the cases $Z_{01},Z_{21}$ we have $E_{-}=E_{\rm min}$ , since the function $x\to F(\pi,x)$ has only one critical point (a minimum). Moreover $E_{+}=a(1)$ in the case $Z_{01}$ ; indeed the function $x\to F(0,x)=a(x)+b(x)$ has no critical points then $E_{+}=\max\{a(0)+b(0),a(1)+b(1)\}=\max\{a(0),a(1)\}$ , moreover $a(x)+b(x)$ is increasing close to zero since $\lim_{x\to 0^{+}}\big{(}a^{\prime}(x)+b^{\prime}(x)\big{)}=+\infty$ . Analogously $E_{-}=a(1)$ in the case $Z_{10}$ . Finally in the case $Z_{21}$ we have $E_{+}=\max\{a(1),E_{\rm max}\}$ , since the function $x\to F(0,x)$ has a maximum at $x_{1}^{(0)}$ and a saddle at $x_{2}^{(0)}$ with $x_{1}^{(0)}<x_{2}^{(0)}$ . Analogously in the case $Z_{12}$ we have $E_{-}=\min\{a(1),E_{\rm min}\}$ .

3.3 Level curves

Since $F$ is even with respect to $\psi$ we can reduce to consider the “half phase space” $[0,\pi]\times(0,1)$ . Take an energy $E_{-}<E<E_{+}$ with $E\neq E_{\rm max},E_{\rm min},E_{\rm sad}$ , and consider the level set $\{F=E\}$ . If $(\psi_{0},x_{0})\in\{F=E\}$ , namely $F(\psi_{0},x_{0})=E$ , since $(\psi_{0},x_{0})$ is not a critical point (being $E\neq E_{\rm max},E_{\rm min},E_{\rm sad}$ and recalling Proposition 1 and (3.2)), we can locally¹³¹³13Namely in a sufficiently small neighborood of $(\psi_{0},x_{0})$ . express $\{F=E\}$ as a curve by the implicit function theorem. In particular, in the half phase space $[0,\pi]\times(0,1)$ , we can always express $\psi$ as a function of $x$ , indeed the equation $F(\psi,x)=a(x)+b(x)\cos\psi=E$ has the unique solution

\psi(x)=\psi(x;E;{J_{2}})=\arccos\left(\frac{E-a(x)}{b(x)}\right)\,.

(52)

Since the domain of definition of the $\arccos$ is $[-1,1]$ , the domain of $\psi(x)$ is

D:=\{x\in(0,1)\ |\ -b(x)\leq E-a(x)\leq b(x)\}\,.

We now discuss the structure of $D$ . Consider first the case in which 0 is an accumulation point for $D$ ; then it must be $E=0$ . Indeed, taking the limit for $x\to 0^{+}$ , $x\in D$ in the inequality $-b(x)\leq E-a(x)\leq b(x)$ , we get $E=0$ . Moreover, when $E=0$ ,

\lim_{x\to 0^{+}}\psi(x;0)=\lim_{x\to 0^{+}}\arccos\big{(}-a(x)/b(x)\big{)}=\arccos(0)=\pi/2\,.

(53)

Claim 1.

1 cannot be an accumulation point for $D$ , since we are assuming that $a_{2}+a_{1}\neq 0$ (recall Remark 6).

proof. Indeed assume, by contradiction, that 1 is an accumulation point for $D$ . Then taking the limit for $x\to 1^{-}$ , $x\in D$ , in the inequality $-b(x)\leq E-a(x)\leq b(x)$ we get $E=a(1)$ . Substituting $E=a(1)$ in the above inequality and dividing by $1-x$ we get

-\sqrt{x(1-x)}\leq\frac{a(1)-a(x)}{1-x}\leq\sqrt{x(1-x)}\,,\qquad\forall x\in[x_{0},1)\,.

Taking again the limit for $x\to 1^{-}$ we get $0=a^{\prime}(1)=a_{2}+a_{1}$ , which contradicts the assumption $a_{2}+a_{1}\neq 0$ . $\Box$

As a consequence, assuming $E\neq 0$ , we have that $D$ is a compact set contained in $(0,1)$ ; moreover it is not difficult to see that it is formed by a finite number of closed intervals (possibly isolated points), whose endpoints satisfy one of the equations

a(x)-E=\mp b(x)\,.

(54)

This amounts to find the roots of the quartic polynomial

\mathbf{P}(x)=(a(x)-E)^{2}-(b(x))^{2}=\left(\frac{1}{2}a_{2}x^{2}+a_{1}x-E\right)^{2}-(1-x)^{3}x=0\,,

(55)

with $0<x<1$ .

Lemma 2.

If $E$ is not a critical energy for¹⁴¹⁴14Namely the energy of a critical point of $F$ . $F$ , the roots of the quartic polynomial $\mathbf{P}(x)$ in (55) with $0<x<1$ are simple.

proof. By contradiction, if $0<x_{0}<1$ is a multiple root of $\mathbf{P}$ , then $\mathbf{P}(x_{0})=\mathbf{P}^{\prime}(x_{0})=0$ . Write

\mathbf{P}(x)=\big{(}a(x)-E-b(x)\big{)}\big{(}a(x)-E+b(x)\big{)}\,.

Assume that $a(x_{0})-E-b(x_{0})=0$ , the case $a(x_{0})-E+b(x_{0})=0$ being analogous. By $\mathbf{P}^{\prime}(x_{0})=0$ it follows that

\mathbf{P}^{\prime}(x_{0})=\big{(}a^{\prime}(x_{0})-b^{\prime}(x_{0})\big{)}\big{(}a(x_{0})-E+b(x_{0})\big{)}=0\,.

(56)

Since $a(x_{0})-E-b(x_{0})=0$ and $b(x_{0})>0$ , by (56) we get $a^{\prime}(x_{0})-b^{\prime}(x_{0})=0$ . This means that $(x_{0},\pi)$ is a critical point of $F$ , which is a contradiction since $E$ is a not critical energy. $\Box$

From now on we will assume that $E$ is not a critical energy of $F$ . We denote the roots of $\mathbf{P}(x;E)$ with $0<x<1$ by $x_{i}=x_{i}(E)$ with $i\in\{1,2,3,4\}$ . We label the roots in increasing order, namely $x_{i}<x_{i+1}$ .

3.4 The quartic equation

In studying the solutions of (54) (equivalently of (55)) on $0<x<1$ , it is convenient to consider the real variable $t\in\mathbb{R}$ and make the substitution

x=\frac{t^{2}}{1+t^{2}}\,.

Since

a(x)-E=\frac{1}{(1+t^{2})^{2}}\left[\frac{1}{2}a_{2}t^{4}+a_{1}t^{2}(1+t^{2})-E(1+t^{2})^{2}\right]\,,\qquad b(x)=\frac{|t|}{(1+t^{2})^{2}}

and

\frac{1}{2}a_{2}t^{4}+a_{1}t^{2}(1+t^{2})-E(1+t^{2})^{2}=\left(\frac{1}{2}a_{2}+a_{1}-E\right)t^{4}+(a_{1}-2E)t^{2}-E\,,

the two equations in (54) are equivalent to

\left(\frac{1}{2}a_{2}+a_{1}-E\right)t^{4}+(a_{1}-2E)t^{2}-E=\mp|t|\,.

(57)

Lemma 3.

Let $t_{0}$ be a root of the polynomial

P(t):=\left(\frac{1}{2}a_{2}+a_{1}-E\right)t^{4}+(a_{1}-2E)t^{2}-t-E

(58)

and set

x_{0}:=\frac{t_{0}^{2}}{1+t_{0}^{2}}\,.

(59)

If $t_{0}<0$ , resp. $t_{0}>0$ , then $x_{0}$ solves $F(0,x_{0})=E$ , resp. $F(\pi,x_{0})=E$ . Conversely if $0<x_{0}<1$ solve the equation in (54) with the $\mp$ sign, then $t_{0}:=\mp x_{0}/(1-x_{0}^{2})$ solves (57) with the $\mp$ sign and, therefore, is a root of $P(t)$ .

proof. If $P(t_{0})=0$ for some $t_{0}>0$ , then $t_{0}$ satisfies (57) and, therefore (54), with the plus sign. As a consequence $F(\pi,x_{0})=E$ . The proof in the case $t_{0}<0$ is analogous. $\Box$

When $E=\frac{1}{2}a_{2}+a_{1}$ the polynomial $P(t)$ reduces to $(a_{2}+a_{1})t^{2}-t+a_{1}+a_{2}/2$ , whose two roots are easily evaluated. Then we can reduce to the case $E\neq\frac{1}{2}a_{2}+a_{1}$ and consider the equivalent monic polynomial ${\rm P}(t):=P(t)/(\frac{1}{2}a_{2}+a_{1}-E)$ , namely

{\rm P}(t)=t^{4}+{\rm p}t^{2}+{\rm q}t+{\rm r},\quad{\rm p}:=\frac{a_{1}-2E}{\frac{1}{2}a_{2}+a_{1}-E},\ \ {\rm q}:=\frac{-1}{\frac{1}{2}a_{2}+a_{1}-E},\ \ {\rm r}:=\frac{-E}{\frac{1}{2}a_{2}+a_{1}-E}.

(60)

The above quartic polynomial is called “depressed” since it is monic and its third order coefficient vanishes. Obviously $P(t)$ and ${\rm P}(t)$ have the same roots. An immediate corollary of Lemma 3 is the following

Lemma 4.

Fix $E\neq\frac{1}{2}a_{2}+a_{1}$ . Let $t_{0}$ be a root of ${\rm P}(t)$ in (60). If $t_{0}<0$ , resp. $t_{0}>0$ , then $x_{0}$ in (59) solves $F(0,x_{0})=E$ , resp. $F(\pi,x_{0})=E$ . In particular $x_{0}$ is a root of $\mathbf{P}(x)$ in (55).

Remark 7.

If ${\rm P}(t)$ has four real distinct roots and $E\neq 0$ (so that $t=0$ is not a root), then the number of positive/negative roots depends on the sign of ${\rm r}$ defined in (60). Indeed, since $\lim_{t\to\pm\infty}{\rm P}(t)=+\infty$ , the number of positive/negative roots is even if ${\rm r}>0$ and odd otherwise.

3.5 Finding the roots of the quartic equation

Following [CP23] we find the roots of the quartic polynomial ${\rm P}(t)$ in (60). First set¹⁵¹⁵15Compare formulas (20) and (10) in [CP23].

p_{*}:=-\frac{{\rm p}^{2}+12{\rm r}}{3}\,,\quad q_{*}:=-\frac{2{\rm p}^{3}-72{\rm p}{\rm r}+27{\rm q}^{2}}{27}\,,\quad\Delta:=-4p_{*}^{3}-27q_{*}^{2}\,.

Let us define the positive¹⁶¹⁶16Compare Theorem 8 in [CP23]. number $s_{*}>0$ as

s_{*}:=\left\{\begin{array}[]{ll}\sqrt[3]{-\frac{q_{*}}{2}+\sqrt{-\frac{\Delta}{108}}}+\sqrt[3]{-\frac{q_{*}}{2}-\sqrt{-\frac{\Delta}{108}}}-\frac{2{\rm p}}{3}&{\rm if}\ \ \Delta\leq 0\,,\\ 2\sqrt{-\frac{p_{*}}{3}}\cos\left(\frac{1}{3}\arccos\left(-\frac{q_{*}}{2}\sqrt{\left(-\frac{3}{p_{*}}\right)^{3}}\right)\right)-\frac{2{\rm p}}{3}&{\rm if}\ \ \Delta>0\,.\end{array}\right.

(61)

Then the roots of ${\rm P}(t)$ are given by¹⁷¹⁷17Compare formula (9) in [CP23].

t^{\pm}_{\varsigma}:=\frac{-\varsigma\sqrt{s_{*}}\pm\sqrt{\delta_{\varsigma}}}{2}\,,\qquad\delta_{\varsigma}:=\varsigma 2{\rm q}(s_{*})^{-1/2}-2{\rm p}-s_{*}\,,\qquad\varsigma=\pm\,.

(62)

The number of real roots of ${\rm P}(t)$ is:

4 if $\delta_{\pm}>0$ ,

2 if $\delta_{+}\delta_{-}<0$ ,

0 if $\delta_{\pm}<0$ .

Let us now define

x^{\pm}_{\varsigma}:=\frac{(t^{\pm}_{\varsigma})^{2}}{1+(t^{\pm}_{\varsigma})^{2}}=\frac{|t^{\pm}_{\varsigma}|^{2}}{1+|t^{\pm}_{\varsigma}|^{2}}\,.

(63)

Note that $x^{\pm}_{\varsigma}$ is an increasing function of $|t^{\pm}_{\varsigma}|$ . By Lemma 4 $x^{\pm}_{\varsigma}$ are the roots of $\mathbf{P}(x)$ in (55). We now want to order the real roots $x^{\pm}_{\varsigma}$ in increasing order $x_{1}<x_{2}<\ldots$ . We have different cases (see Figure 14):

$\displaystyle x_{1}$	$\displaystyle:=$	$\displaystyle\left\{\begin{array}[]{ll}\min\{x_{+}^{+},x_{-}^{-}\}&{\rm if}\ \ \delta_{\pm}>0\,,\\ \min\{x_{+}^{+},x_{+}^{-}\}&{\rm if}\ \ \delta_{-}\leq 0\leq\delta_{+}\,,\\ \min\{x_{-}^{+},x_{-}^{-}\}&{\rm if}\ \ \delta_{+}\leq 0\leq\delta_{-}\,,\end{array}\right.$	(67)
$\displaystyle x_{2}$	$\displaystyle:=$	$\displaystyle\left\{\begin{array}[]{ll}\max\{x_{+}^{+},x_{-}^{-}\}&{\rm if}\ \ \delta_{\pm}>0\,,\\ \max\{x_{+}^{+},x_{+}^{-}\}&{\rm if}\ \ \delta_{-}\leq 0\leq\delta_{+}\,,\\ \max\{x_{-}^{+},x_{-}^{-}\}&{\rm if}\ \ \delta_{+}\leq 0\leq\delta_{-}\,,\end{array}\right.$	(71)
$\displaystyle x_{3}$	$\displaystyle:=$	$\displaystyle\ \quad\min\{x_{-}^{+},x_{+}^{-}\}\quad{\rm if}\quad\delta_{\pm}>0\,,$
$\displaystyle x_{4}$	$\displaystyle:=$	$\displaystyle\ \quad\max\{x_{-}^{+},x_{+}^{-}\}\quad{\rm if}\quad\delta_{\pm}>0\,.$	(72)

3.6 The separatrices at the saddle points

Recall the definition of the zones $Z_{ij}$ given in (49). We now consider the curve with zero energy bifurcating from the point $(\pi/2,0)$ (recall (53)) in the “half phase space” $[0,\pi]\times(0,1)$ . In the case $Z_{10}$ such curve “turns left” and touches the line $\{\psi=0\}$ at some point $>x_{1}^{(0)}$ . Analogously in the case $Z_{01}$ such curve “turns right” and touches the line $\{\psi=\pi\}$ at some point $>x_{1}^{(\pi)}$ .
The situation in the cases $Z_{21}$ and $Z_{12}$ is more involved; more precisely it depends on the sign of $E_{\rm sad}$ . In particular for $(i,j)\in\left\{(2,1),(1,2)\right\}$ we set

Z_{ij}^{\pm}:=\{(a_{1},a_{2})\in Z_{ij}\ :\ \pm E_{\rm sad}>0\}\,,\qquad Z_{ij}^{0}:=\{(a_{1},a_{2})\in Z_{ij}\ :\ E_{\rm sad}=0\}\,,

(73)

so that

Z_{ij}=Z_{ij}^{+}\cup Z_{ij}^{-}\cup Z_{ij}^{0}\,.

The next result characterises the sets in (73)

Lemma 5.

Setting

\tilde{g}(a_{1}):=-\frac{2}{27}a_{1}(4a_{1}^{2}+27)

(74)

we have

	$\displaystyle Z_{21}^{+}=Z_{21}\cap\{a_{2}>\tilde{g}(a_{1})\}\,,\quad\quad Z_{12}^{+}=Z_{12}\cap\{a_{2}>\tilde{g}(a_{1})\}\,,$
	$\displaystyle Z_{21}^{-}=Z_{21}\cap\{a_{2}<\tilde{g}(a_{1})\}\,,\qquad Z_{12}^{-}=Z_{12}\cap\{a_{2}<\tilde{g}(a_{1})\}\,,\vskip 12.0pt plus 4.0pt minus 4.0pt$
	$\displaystyle Z_{21}^{0}=Z_{21}\cap\{a_{2}=\tilde{g}(a_{1})\}=\{a_{2}=\tilde{g}(a_{1}),\ \ a_{1}<0\}\,,$
	$\displaystyle Z_{12}^{0}=Z_{12}\cap\{a_{2}=\tilde{g}(a_{1})\}=\{a_{2}=\tilde{g}(a_{1}),\ \ a_{1}>0\}\,.$		(75)

Note that, since $\tilde{g}$ is odd and $\tilde{g}(a_{1})\leq-a_{1}$ for $a_{1}\geq 0$ , by the definition of $Z_{21}$ and $Z_{12}$ it follows that $Z_{21}^{-}\subset\{a_{1}<0\}$ and $Z_{12}^{+}\subset\{a_{1}>0\}$ .

proof. We discuss only the case $Z_{21}$ , the study of $Z_{12}$ being analogous. As we said above, the picture of the phase space in the case $Z_{21}$ strongly depends on the sign of the energy of the saddle point $E_{\rm sad}=F(0,x_{2}^{(0)})$ , where $x_{2}^{(0)}$ is the minimum of the function $x\to F(0,x)$ . In particular we claim that $E_{\rm sad}\lesseqqgtr 0$ if and only if $a_{2}\lesseqqgtr\tilde{g}(a_{1})$ . In particular we note that $x=x_{2}^{(0)}$ satisfies the system

\left\{\begin{array}[]{ll}F(0,x)=\frac{1}{2}a_{2}x^{2}+a_{1}x+b(x)=0\,,\\ \partial_{x}F(0,x)=a_{2}x+a_{1}+b^{\prime}(x)=0\,.\end{array}\right.

By algebraic manipulation we get

\left\{\begin{array}[]{ll}\frac{1}{2}a_{2}x^{2}+xb^{\prime}(x)-b(x)=0\,,\\ a_{1}x+2b(x)-xb^{\prime}(x)=0\,,\end{array}\right.

by which we finally have

a_{2}=2\frac{b(x)-xb^{\prime}(x)}{x^{2}}\,,\qquad\quad a_{1}=\frac{xb^{\prime}(x)-2b(x)}{x}\,.

By using (47)

a_{1}=-\frac{3}{2}\sqrt{\frac{1}{x}-1}\,,\qquad a_{2}=\sqrt{\frac{1}{x}-1}\left(\frac{1}{x}+2\right)\,.

(76)

Note that $a_{1}<0$ . By inverting the first expression in (76) we get $\frac{1}{x}=\frac{4}{9}a_{1}^{2}+1$ ; substituting in the second expression we obtain that $a_{2}=\tilde{g}(a_{1})$ defined in (74). Therefore in $Z_{21}^{0}$ , namely when $a_{2}=\tilde{g}(a_{1})$ , the value of the function $x\to F(0,x)=\frac{1}{2}a_{2}x^{2}+a_{1}x+b(x)$ at its minimum $x_{2}^{(0)}$ is exactly 0. On the other hand in $Z_{21}^{+}$ , namely when $a_{2}>\tilde{g}(a_{1})$ , one has $F(0,x_{2}^{(0)})>0$ . Finally in $Z_{21}^{-}$ it is $F(0,x_{2}^{(0)})<0$ . $\Box$

3.7 Different topologies of the level curves

Let us consider the energy level sets in the phase spase

{\mathcal{P}}:=\mathbb{T}\times(0,1)\,,

which is a cylinder. The points where the level curves $\{F=E\}$ touch the lines $\psi=0$ or $\psi=\pi$ are the solutions of the equation $F(0,x)=a(x)+b(x)=E$ and $F(\pi,x)=a(x)-b(x)=E$ , respectively; equivalently they are the roots of the quartic polynomial in (54).

We note that in the cases $Z_{10}$ , $Z_{01}$ the set $\{F=E\}$ has only one connected component. The same holds in the case $Z_{21}$ except for $E_{\rm sad}<E<\min\{a(1),E_{\rm max}\}$ when $\{F=E\}$ possesses two connected components. Analogously in the case $Z_{12}$ the level set $\{F=E\}$ possesses two connected component for $\max\{a(1),E_{\rm min}\}<E<E_{\rm sad}$ and only one otherwise.

Remark 8.

Up to the energy level corresponding to $E=0$ and to the critical energies¹⁸¹⁸18Namely the energy of critical points of $F$ . In the case $a_{1}+a_{2}=0$ , that we are actually excluding (recall Remark 6), there is also a curve which touches the line $x=1$ ., the level sets are curves of three types:
(i) a homotopically trivial, namely contractible, curve making a loop around the maximum $(0,x_{1}^{(0)})$ intersecting twice the line $\psi=0$ ;
(ii) a curve wrapping on the cylinder; in particular it intersects once the line $\psi=0$ and once the line $\psi=\pi$ ;
(iii) a homotopically trivial curve making a loop around the minimum $(\pi,x_{1}^{(\pi)})$ intersecting twice the line $\psi=\pi$ .

In the following we will always label the roots of the quartic polynomial in (54) so that $x_{i}(E)<x_{i+1}(E)$ . Recall Proposition 1.

Case $Z_{10}$ .

The zero level separatrix actually separates the phase space ${\mathcal{P}}$ into two open connected components ${\mathcal{P}}_{10}^{{\rm I}}$ and ${\mathcal{P}}_{10}^{{\rm I\!I}}$ supporting two different kind of motions¹⁹¹⁹19Where $\{F>0\}:=\{(\psi,x)\in{\mathcal{P}}\ :\ F(\psi,x)>0\}$ .

{\mathcal{P}}_{10}^{{\rm I}}:=\{F>0\}\,,\qquad{\mathcal{P}}_{10}^{{\rm I\!I}}:=\{F<0\}\,,

(77)

with ${\mathcal{P}}={\mathcal{P}}_{10}^{{\rm I}}\cup{\mathcal{P}}_{10}^{{\rm I\!I}}\cup\{F=0\}$ . Indeed in ${\mathcal{P}}_{10}^{{\rm I}}$ the level curves have the form in case (ii) above, while in ${\mathcal{P}}_{10}^{{\rm I\!I}}$ they have the form in case (i). In the present case the quartic polynomial in (54) possesses, for $E\neq 0$ and not critical, only two real roots $x_{1}(E)<x_{2}(E)$ . Note that $x_{1}(E)=x_{1}(E;{J_{2}})$ and $x_{2}(E)=x_{2}(E;{J_{2}})$ . If $E>0$ the $E$ –level curve starts at $(0,x_{1}(E))$ and come back on the line $\psi=0$ at $(0,x_{2}(E))$ , otherwise, for $E<0$ , it joints the line $\psi=\pi$ at $(\pi,x_{1}(E))$ and the line $\psi=0$ at $(\pi,x_{2}(E))$ . Recalling (52), the level curve $\{F=E\}$ can be expressed as a graph over $x_{1}(E)<x<x_{2}(E)$ by the function $\psi(x;{J_{2}})$ .

Case $Z_{01}$ .

We set

{\mathcal{P}}_{01}^{{\rm I\!I}}:=\{F>0\}\,,\qquad{\mathcal{P}}_{01}^{{\rm I\!I\!I}}:=\{F<0\}\,,

(78)

with ${\mathcal{P}}={\mathcal{P}}_{01}^{{\rm I\!I\!I}}\cup{\mathcal{P}}_{01}^{{\rm I\!I}}\cup\{F=0\}$ . Again the zero level separatrix actually separates the two different kind of motions: in ${\mathcal{P}}_{01}^{{\rm I\!I\!I}}$ the level curves have the form in case (iii) above, while in ${\mathcal{P}}_{01}^{{\rm I\!I}}$ they have the form in case (ii).

Case $Z_{21}^{+}$ .

The zero level separatrix and the two separatrices emanating from the saddle point $(0,x_{2}^{(0)})$ with energy²⁰²⁰20Recall Proposition 1 and (3.2). $F(0,x_{2}^{(0)})=E_{\rm sad}>0$ (recall (73)) separate the phase space ${\mathcal{P}}$ into 4 open connected components:

	$\displaystyle\!{\mathcal{P}}_{21}^{+,{\rm I}}:=\{F>E_{\rm sad}\ \mbox{containing}\ (0,x_{1}^{(0)})\}\,,\qquad{\mathcal{P}}_{21}^{+,{\rm I\!I}}:=\{0<F<E_{\rm sad}\}\,,$		(79)
	$\displaystyle\!{\mathcal{P}}_{21}^{+,{\rm I\!I\!I}}:=\{F<0\}\,,\quad{\mathcal{P}}_{21}^{+,{\rm I\!V}}:=\{F>E_{\rm sad}\ \mbox{not containing}\ (x_{1}^{(0)},0)\}$

with ${\mathcal{P}}={\mathcal{P}}_{21}^{+,{\rm I}}\cup{\mathcal{P}}_{21}^{+,{\rm I\!I}}\cup{\mathcal{P}}_{21}^{+,{\rm I\!I\!I}}\cup{\mathcal{P}}_{21}^{+,{\rm I\!V}}\cup\{F=0\}\cup\{F=E_{\rm sad}\}$ . The level curves in ${\mathcal{P}}_{21}^{+,{\rm I\!I}}$ and ${\mathcal{P}}_{21}^{+,{\rm I\!V}}$ have the form case (ii) above, the ones in ${\mathcal{P}}_{21}^{+,{\rm I}}$ have the form in case (i), finally the ones in ${\mathcal{P}}_{21}^{+,{\rm I\!I\!I}}$ are as in (iii). In particular the level curves in ${\mathcal{P}}_{21}^{+,{\rm I}}$ pass through the points $(0,x_{1}(E))$ and $(0,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{21}^{+,{\rm I\!I}}$ through $(0,x_{1}(E))$ and $(\pi,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{21}^{+,{\rm I\!I\!I}}$ through $(\pi,x_{1}(E))$ and $(\pi,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{21}^{+,{\rm I\!V}}$ through $(0,x_{3}(E))$ and $(\pi,x_{4}(E))$ .

Case $Z_{21}^{-}$ .

The zero level separatrix and the two separatrices emanating from the saddle point $(0,x_{2}^{(0)})$ with energy $F(0,x_{2}^{(0)})=E_{\rm sad}<0$ (recall (73)) separate the phase space ${\mathcal{P}}$ into 4 open connected components:

{\mathcal{P}}_{21}^{-,{\rm I}}:=\{F>0\}\,,\qquad{\mathcal{P}}_{21}^{-,{\rm I\!I\!I}}:=\{F<E_{\rm sad}\}\,,

(80)

while ${\mathcal{P}}_{21}^{-,{\rm I\!I}}$ and ${\mathcal{P}}_{21}^{-,{\rm I\!V}}$ are the two open connected components of $\{E_{\rm sad}<F<0\}$ with ${\mathcal{P}}_{21}^{-,{\rm I\!I}}$ containing $(0,\pi)$ in its closure. We immediately see that

{\mathcal{P}}={\mathcal{P}}_{21}^{-,{\rm I}}\cup{\mathcal{P}}_{21}^{-,{\rm I\!I}}\cup{\mathcal{P}}_{21}^{-,{\rm I\!I\!I}}\cup{\mathcal{P}}_{21}^{-,{\rm I\!V}}\cup\{F=0\}\cup\{F=E_{\rm sad}\}\,.

The level curves in ${\mathcal{P}}_{21}^{-,{\rm I\!I}}$ and ${\mathcal{P}}_{21}^{-,{\rm I\!V}}$ have the form in case (ii) above, the ones in ${\mathcal{P}}_{21}^{-,{\rm I}}$ have the form in case (i), finally the ones in ${\mathcal{P}}_{21}^{-,{\rm I\!I\!I}}$ are as in (iii). In particular the level curves in ${\mathcal{P}}_{21}^{-,{\rm I}}$ pass through the points $(0,x_{1}(E))$ and $(0,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{21}^{-,{\rm I\!I}}$ through $(\pi,x_{1}(E))$ and $(0,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{21}^{-,{\rm I\!I\!I}}$ through $(\pi,x_{1}(E))$ and $(\pi,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{21}^{-,{\rm I\!V}}$ through $(0,x_{3}(E))$ and $(\pi,x_{4}(E))$ .

Case $Z_{12}^{+}$ .

The zero level separatrix and the two separatrices emanating from the saddle point $(\pi,x_{2}^{(\pi)})$ with energy $F(\pi,x_{2}^{(\pi)})=E_{\rm sad}>0$ (recall (73)) separate the phase space ${\mathcal{P}}$ into 4 open connected components:

{\mathcal{P}}_{12}^{+,{\rm I}}:=\{F>E_{\rm sad}\}\,,\qquad{\mathcal{P}}_{12}^{+,{\rm I\!I\!I}}:=\{F<0\}\,,

(81)

while ${\mathcal{P}}_{12}^{+,{\rm I\!I}}$ and ${\mathcal{P}}_{12}^{+,{\rm I\!V}}$ are the two open connected components of $\{0<F<E_{\rm sad}\}$ with ${\mathcal{P}}_{12}^{+,{\rm I\!I}}$ containing $(0,0)$ in its closure. We note that

{\mathcal{P}}={\mathcal{P}}_{12}^{+,{\rm I}}\cup{\mathcal{P}}_{12}^{+,{\rm I\!I}}\cup{\mathcal{P}}_{12}^{+,{\rm I\!I\!I}}\cup{\mathcal{P}}_{12}^{+,{\rm I\!V}}\cup\{F=0\}\cup\{F=E_{\rm sad}\}\,.

The level curves in ${\mathcal{P}}_{12}^{+,{\rm I\!I}}$ and ${\mathcal{P}}_{12}^{+,{\rm I\!V}}$ have the form case (ii) above, the ones in ${\mathcal{P}}_{12}^{+,{\rm I}}$ have the form in case (i), finally the ones in ${\mathcal{P}}_{12}^{+,{\rm I\!I\!I}}$ are as in (iii). In particular the level curves in ${\mathcal{P}}_{12}^{+,{\rm I}}$ pass through the points $(0,x_{1}(E))$ and $(0,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{12}^{+,{\rm I\!I}}$ through $(0,x_{1}(E))$ and $(\pi,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{12}^{+,{\rm I\!I\!I}}$ through $(\pi,x_{1}(E))$ and $(\pi,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{12}^{+,{\rm I\!V}}$ through $(\pi,x_{3}(E))$ and $(0,x_{4}(E))$ .

Case $Z_{12}^{-}$ .

The zero level separatrix and the two separatrices emanating from the saddle point $(\pi,x_{2}^{(\pi)})$ with energy $F(\pi,x_{2}^{(\pi)})=E_{\rm sad}<0$ (recall (73)) separate the phase space ${\mathcal{P}}$ into 4 open connected components:

	$\displaystyle{\mathcal{P}}_{12}^{-,{\rm I}}:=\{F>0\}\,,\qquad{\mathcal{P}}_{12}^{-,{\rm I\!I\!I}}:=\{F<E_{\rm sad}\ \mbox{containing}\ (x_{1}^{(\pi)},\pi)\}\,,$
	$\displaystyle{\mathcal{P}}_{12}^{-,{\rm I\!I}}:=\{E_{\rm sad}<F<0\}\,,\quad{\mathcal{P}}_{12}^{-,{\rm I\!V}}:=\{F<E_{\rm sad}\ \mbox{not containing}\ (x_{1}^{(\pi)},\pi)\}\,,$		(82)

with ${\mathcal{P}}={\mathcal{P}}_{12}^{-,{\rm I}}\cup{\mathcal{P}}_{12}^{-,{\rm I\!I}}\cup{\mathcal{P}}_{12}^{-,{\rm I\!I\!I}}\cup{\mathcal{P}}_{12}^{-,{\rm I\!V}}\cup\{F=0\}\cup\{F=E_{\rm sad}\}$ . The level curves in ${\mathcal{P}}_{12}^{-,{\rm I\!I}}$ and ${\mathcal{P}}_{12}^{-,{\rm I\!V}}$ have the form in case (ii) above, the ones in ${\mathcal{P}}_{12}^{-,{\rm I}}$ have the form in case (i), finally the ones in ${\mathcal{P}}_{12}^{-,{\rm I\!I\!I}}$ are as in (iii). In particular the level curves in ${\mathcal{P}}_{12}^{-,{\rm I}}$ pass through the points $(0,x_{1}(E))$ and $(0,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{12}^{-,{\rm I\!I}}$ through $(\pi,x_{1}(E))$ and $(0,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{12}^{-,{\rm I\!I\!I}}$ through $(\pi,x_{1}(E))$ and $(\pi,x_{2}(E))$ ; the ones in ${\mathcal{P}}_{12}^{-,{\rm I\!V}}$ through $(\pi,x_{3}(E))$ and $(0,x_{4}(E))$ .

3.8 Degenerate cases

Recalling Remark 6, we briefly illustrate in Figures 22-24 the degenerate cases: $a_{2}=-a_{1}$ , when $x=1$ is a solution of (45)-(46), $a_{2}=g(a_{1})$ and $a_{2}=-g(-a_{1})$ , when two solutions coincide, finally $a_{2}=\tilde{g}(a_{1})$ , when the separatrix and the stable and unstable manifolds of the saddle point coincide and have zero energy.

4 Explicit formulae of the nonlinear frequencies

In this section we first write the integrating action ${\rm I}_{1}$ as a function of the energy $E$ in terms of integrals in the $x$ variables with endpoints given by the roots of the quartic polynomial $\mathbf{P}(x)$ in (55), studied in the previous section. In addition to energy, these representation formulae depend on the values of the parameters $a_{1}$ and $a_{2}$ , according to the resulting different topologies of the phase space described above.

The final integrated Hamiltonian is the inverse $\mathcal{E}:{\rm I}_{1}\to\mathcal{E}({\rm I}_{1};{\rm I}_{2})$ of the function $E\to{\rm I}_{1}(E;{\rm I}_{2})$ in (88). Its derivatives with respect to ${\rm I}_{1}$ and ${\rm I}_{2}$ are the nonlinear frequencies and can be written in terms of the derivatives of ${\rm I}_{1}(E;{\rm I}_{2})$ with respect to $E$ and ${\rm I}_{2}$ , see (95). These derivatives are expressed in terms of elliptic integrals in Proposition 2. The integrals are explicitly evaluated by means of suitable Moebius transformations in Subsections 4.3 and 4.4, in the case that $\mathbf{P}(x)$ has four or two real roots, respectively. In the last subsection we consider the exact 3:1 resonance case, where the above formulae simplify a bit.

4.1 Construction of the integrating action variables

Since $\hat{H}$ has two independent integrals of motions: the Hamiltonian itself and ${J_{2}}$ , by the Arnold-Liouville theorem the Hamiltonian $\hat{H}$ is integrable. A part from ${\rm I}_{2}:={J_{2}}$ the construction of the other action ${\rm I}_{1}$ as function of the energy $E$ is as follows. ${\rm I}_{1}(E)$ is simply the area enclosed by the level curves of $\hat{H}=E$ divided by $2\pi$ . Such level curves coincide with the ones of $F$ .

Our aim is to find a symplectic map $\Psi:({\rm I},\theta)=({\rm I}_{1},{\rm I}_{2},\theta_{1},\theta_{2})\to(J_{1},{J_{2}},\psi_{1},\psi_{2})$ , fixing

{\rm I}_{2}={J_{2}}\,,

(83)

such that, in the new coordinates, the Hamiltonian $\hat{H}$ is integrated, namely²¹²¹21Recalling (41) note that $\mathcal{E}$ is adimensional.

\hat{H}\circ\Psi=:\mathcal{E}({\rm I})

(84)

depends only on the new actions ${\rm I}=({\rm I}_{1},{\rm I}_{2})$ .

Note that the same transformation $\Psi$ also integrates $\mathbb{H}_{{J_{2}}}=\mathbb{H}_{{\rm I}_{2}}$ in (41) and $\hat{\mathbb{H}}_{\rm res}$ in (38). Indeed

\mathbb{H}_{{\rm I}_{2}}\circ\Psi=\chi\,{\rm I}_{2}^{2}\,\mathcal{E}({\rm I})\qquad\mbox{and}\qquad\hat{\mathbb{H}}_{\rm res}\circ\Psi=\mathbb{E}({\rm I}):=\omega_{-}{\rm I}_{2}+\chi\,{\rm I}_{2}^{2}\,\mathcal{E}({\rm I})\,.

(85)

In the new coordinates, the actions are constants of motion and the angles perform a linear motion $\theta(t)=\theta(0)+\omega t$ with frequencies

\omega=(\omega_{1},\omega_{2}):=(\partial_{{\rm I}_{1}}\mathbb{E},\partial_{{\rm I}_{2}}\mathbb{E})=(\chi\,{\rm I}_{2}^{2}\,\partial_{{\rm I}_{1}}\mathcal{E}\,,\ \ \omega_{-}+2\chi\,{\rm I}_{2}\mathcal{E}+\chi\,{\rm I}_{2}^{2}\,\partial_{{\rm I}_{2}}\mathcal{E})\,.

(86)

The classical construction of the Hamiltonian $\mathcal{E}$ , “the adimensional energy”, is as follows. First one constructs, for every fixed value of ${\rm I}_{2}={J_{2}}$ , the action function ${\rm I}_{1}:E\to{\rm I}_{1}(E;{J_{2}})$ defined as the area enclosed by the level curve $\gamma_{E}:=\{\hat{H}_{J_{2}}=E+a_{0}\}$ normalised by $2\pi$ . Then, since the function ${\rm I}_{1}:E\to{\rm I}_{1}(E;{J_{2}})$ turns out to be monotone (being $|\partial_{E}{\rm I}_{1}(E;{J_{2}})|>0$ ), one defines $\mathcal{E}:{\rm I}_{1}\to\mathcal{E}({\rm I}_{1},{J_{2}})$ as its inverse. Namely, in view of (43),

\mathcal{E}\big{(}{\rm I}_{1}(E;{J_{2}}),{J_{2}}\big{)}=\mathcal{E}\big{(}{\rm I}_{1}(E;{\rm I}_{2}),{\rm I}_{2}\big{)}=E+a_{0}\,.

(87)

So the level curves of $\hat{H}_{J_{2}}$ play a crucial role here. Note that by (41) the level curves of $\hat{H}_{J_{2}}$ are the same as the ones of $\mathbb{H}_{{J_{2}}}$ , moreover by (43) they are simple related to the ones of $F$ .
More precisely the new action is defined as²²²²22Recall (43).

{\rm I}_{1}(E)={\rm I}_{1}(E;{\rm I}_{2}):=\frac{{\rm I}_{2}}{3}{\mathcal{A}}(E;{\rm I}_{2})\,,

(88)

where, recalling the notation introduced in Remark 8, ${\mathcal{A}}$ is the area (normalised by $2\pi$ ) enclosed by the $E$ -level curve in the cases (i) and (iii), and below the level curve in the case (ii). In particular we have four cases indexed by ${\rm I},{\rm I\!I},{\rm I\!I\!I},{\rm I\!V}$ , according if one is in the zones ${\mathcal{P}}_{ij}^{{\rm I}},{\mathcal{P}}_{ij}^{{\rm I\!I}},{\mathcal{P}}_{ij}^{{\rm I\!I\!I}},{\mathcal{P}}_{ij}^{\pm,{\rm I}},{\mathcal{P}}_{ij}^{\pm,{\rm I\!I}},{\mathcal{P}}_{ij}^{\pm,{\rm I\!I\!I}},{\mathcal{P}}_{ij}^{\pm,{\rm I\!V}}$ .
Case ${\rm I}$ . The level curve makes a loop around the maximum $(0,x_{1}^{(0)})$ then²³²³23Recall (52).

{\mathcal{A}}(E)={\mathcal{A}}^{{\rm I}}(E;{\rm I}_{2}):=\frac{\mathrm{Area}}{\pi}=\frac{1}{\pi}\int_{x_{1}(E)}^{x_{2}(E)}\psi(x;E)\,dx\,.

(89)

This holds in the zones: ${\mathcal{P}}_{10}^{{\rm I}},{\mathcal{P}}_{ij}^{\pm,{\rm I}}.$
Case ${\rm I\!I}$ . The level curve wraps on the cylinder

{\mathcal{A}}(E)={\mathcal{A}}^{\rm I\!I}(E;{\rm I}_{2}):=\frac{\mathrm{Area}}{\pi}=\left\{\begin{array}[]{ll}x_{1}(E)+\frac{1}{\pi}\int_{x_{1}(E)}^{x_{2}(E)}\psi(x;E)\,dx\,,&{\rm if}\ \ \psi(x_{1})=\pi\\ x_{2}(E)-\frac{1}{\pi}\int_{x_{1}(E)}^{x_{2}(E)}\psi(x;E)\,dx\,,&{\rm if}\ \ \psi(x_{1})=0\,.\end{array}\right.

(90)

in the cases ${\mathcal{P}}_{ij}^{{\rm I\!I}},{\mathcal{P}}_{ij}^{\pm,{\rm I\!I}}$ .
Case ${\rm I\!I\!I}$ . The level curve makes a loop around the minimum $(\pi,x_{1}^{(\pi)})$

{\mathcal{A}}(E)={\mathcal{A}}^{\rm I\!I\!I}(E;{\rm I}_{2}):=\frac{\mathrm{Area}}{\pi}=\frac{1}{\pi}\int_{x_{1}(E)}^{x_{2}(E)}\big{(}\pi-\psi(x;E)\big{)}\,dx\,.

(91)

This holds in the zones: ${\mathcal{P}}_{01}^{{\rm I\!I\!I}},{\mathcal{P}}_{ij}^{\pm,{\rm I\!I\!I}}$ .
Case ${\rm I\!V}$ . The level curve wraps on the cylinder

{\mathcal{A}}(E)={\mathcal{A}}^{\rm I\!V}(E;{\rm I}_{2}):=\frac{\mathrm{Area}}{\pi}=\left\{\begin{array}[]{ll}x_{3}(E)+\frac{1}{\pi}\int_{x_{3}(E)}^{x_{4}(E)}\psi(x;E)\,dx\,,&{\rm if}\ \ \psi(x_{3})=\pi\\ x_{4}(E)-\frac{1}{\pi}\int_{x_{3}(E)}^{x_{4}(E)}\psi(x;E)\,dx\,,&{\rm if}\ \ \psi(x_{3})=0\,.\end{array}\right.

(92)

in the cases ${\mathcal{P}}_{ij}^{\pm,{\rm I\!V}}$ .

Remark 9 (KAM Theory).

The above integrating construction holds for the truncated Hamiltonian $\hat{\mathbb{H}}_{\rm res}$ in (38) but it does not work for the complete Hamiltonian. In fact the complete system is genuinely two dimensional and, therefore, not integrable. In particular ${J_{2}}$ is not more a constant of motion. One might wonder whether, for $\epsilon$ small enough, the invariant structures, both NNMs and stable and unstable manifolds, that exist for the truncated Hamiltonian survive, slightly deformed, for the full Hamiltonian. The answer is substantially positive thanks to KAM Theory. More precisely, the hyperbolic periodic orbit and its (local) stable and unstable manifolds survive as can be demonstrated following, e.g., [Graff] and [Val]. The conservation of two dimensional invariant tori is ensured when the frequencies are strongly rationally independent. This implies that the majority of invariant tori still exist in the complete system, whereas a minority is destroyed. However we note that, in this resonant case, the application of KAM Theory is not straightforward. In fact the standard KAM theory only regards the persistence of the so called primary tori, namely tori that are graphs over the angles. However, as we have already shown, in the resonant case also the so called secondary tori appear (the blue and the yellow tori in Figure 5). All our analysis can bee seen as a necessary preparatory step in view of the application of KAM techniques, since it integrates the resonant BNF up to order four. This means that, in the final action angle variables, the invariant tori are graphs over the angles and KAM methods can be applied. For a KAM result in presence of resonances and the persistence of secondary tori see [MNT].
Finally we note that, since the complete system is, in general, not integrable, KAM tori do not completely fill the phase space but some gaps appear between them. In these gaps chaotic behaviour may occur. However one has to notice that, since we are in two degrees of freedom, every orbit is perpetually stable in the sense that the solutions exist for all times and the values of the action variables remain close to the initial ones forever. The argument is standard in KAM Theory: the orbits evolve on the three dimensional energy surface we have two cases. 1) If on orbit starts on a KAM torus, then it remains on it forever, since the torus is invariant for the Hamiltonian flow. 2) If an orbit starts in a gap between two KAM tori then, since the tori are invariant and bidimensional and the energy surface is three dimensional, the orbit cannot cross them and it remains trapped between them forever.

4.2 Evaluation of the nonlinear frequencies as functions of the energy

In evaluating the new frequencies in (86), it is convenient to use $(E;{\rm I}_{2})$ as independent variables, rather than $({\rm I}_{1},{\rm I}_{2})$ . In particular, we have to evaluate $\partial_{{\rm I}_{1}}\mathcal{E}\big{(}{\rm I}_{1}(E;{\rm I}_{2}),{\rm I}_{2}\big{)}$ and $\partial_{{\rm I}_{2}}\mathcal{E}\big{(}{\rm I}_{1}(E;{\rm I}_{2}),{\rm I}_{2}\big{)}$ . Deriving (87) with respect to $E$ we get

\partial_{{\rm I}_{1}}\mathcal{E}\big{(}{\rm I}_{1}(E;{\rm I}_{2}),{\rm I}_{2}\big{)}\partial_{E}{\rm I}_{1}(E;{\rm I}_{2})=1\,.

Then

\partial_{{\rm I}_{1}}\mathcal{E}\big{(}{\rm I}_{1}(E;{\rm I}_{2}),{\rm I}_{2}\big{)}=\frac{1}{\partial_{E}{\rm I}_{1}(E;{\rm I}_{2})}\,.

(93)

Analogously, deriving (87) with respect to ${\rm I}_{2}$ , we get

\partial_{{\rm I}_{1}}\mathcal{E}\big{(}{\rm I}_{1}(E;{\rm I}_{2}),{\rm I}_{2}\big{)}\partial_{{\rm I}_{2}}{\rm I}_{1}(E;{\rm I}_{2})+\partial_{{\rm I}_{2}}\mathcal{E}\big{(}{\rm I}_{1}(E;{\rm I}_{2}),{\rm I}_{2}\big{)}=0\,,

and, therefore,

\partial_{{\rm I}_{2}}\mathcal{E}\big{(}{\rm I}_{1}(E;{\rm I}_{2}),{\rm I}_{2}\big{)}=-\partial_{{\rm I}_{1}}\mathcal{E}\big{(}{\rm I}_{1}(E;{\rm I}_{2}),{\rm I}_{2}\big{)}\partial_{{\rm I}_{2}}{\rm I}_{1}(E;{\rm I}_{2})\stackrel{{\scriptstyle\eqref{peperosa}}}{{=}}-\frac{\partial_{{\rm I}_{2}}{\rm I}_{1}(E;{\rm I}_{2})}{\partial_{E}{\rm I}_{1}(E;{\rm I}_{2})}\,.

(94)

Then, using (87), we rewrite (86) as

\omega_{1}(E,{\rm I}_{2})=\chi\,{\rm I}_{2}^{2}\,\frac{1}{\partial_{E}{\rm I}_{1}(E;{\rm I}_{2})}\,,\qquad\omega_{2}(E,{\rm I}_{2})=\omega_{-}+2\chi\,{\rm I}_{2}(E+a_{0})-\chi\,{\rm I}_{2}^{2}\,\frac{\partial_{{\rm I}_{2}}{\rm I}_{1}(E;{\rm I}_{2})}{\partial_{E}{\rm I}_{1}(E;{\rm I}_{2})}\,,

namely, recalling (88),

\omega_{1}(E,{\rm I}_{2})=3\chi\,{\rm I}_{2}\,\frac{1}{\partial_{E}{\mathcal{A}}(E;{\rm I}_{2})}\,,\qquad\omega_{2}(E,{\rm I}_{2})=\omega_{-}+2\chi\,{\rm I}_{2}(E+a_{0})-\chi\,{\rm I}_{2}^{2}\,\frac{\partial_{{\rm I}_{2}}{\mathcal{A}}(E;{\rm I}_{2})}{\partial_{E}{\mathcal{A}}(E;{\rm I}_{2})}\,.

(95)

As a final symplectic change of variables we consider the inverse of the map in (36), namely the map $\tilde{\Phi}:(\tilde{I},\tilde{\varphi})\to({\rm I},\theta)$

\left\{\begin{array}[]{l}{\rm I}_{1}=\tilde{I}_{2}\\ {\rm I}_{2}=\tilde{I}_{1}+3\tilde{I}_{2}\,,\end{array}\right.\qquad\left\{\begin{array}[]{l}\theta_{1}=\tilde{\varphi}_{2}-3\tilde{\varphi}_{1}\\ \theta_{2}=\tilde{\varphi}_{1}\,.\end{array}\right.

(96)

Applying the above map to the Hamiltonian $\mathbb{E}$ in (85) we get $\tilde{\mathbb{E}}:=\mathbb{E}\circ\tilde{\Phi}$ , namely

\tilde{\mathbb{E}}(\tilde{I})=\mathbb{E}(\tilde{I}_{2},\tilde{I}_{1}+3\tilde{I}_{2})\,.

(97)

In order to describe the frequencies of $\tilde{\mathbb{E}}(\tilde{I})$ it is convenient to use $(E,{\rm I}_{2})$ as variables instead of $(\tilde{I}_{1},\tilde{I}_{2})$ . The (invertible) relation between the two set of variable is the following

E=\frac{\mathbb{E}(\tilde{I}_{2},\tilde{I}_{1}+3\tilde{I}_{2})-\omega_{-}(\tilde{I}_{1}+3\tilde{I}_{2})}{\chi\,(\tilde{I}_{1}+3\tilde{I}_{2})^{2}}-a_{0}\,,\qquad{\rm I}_{2}=\tilde{I}_{1}+3\tilde{I}_{2}\,,

(98)

(recalling (85), (87)). We are now able to evaluate the final nonlinear frequencies, namely the partial derivatives of $\tilde{\mathbb{E}}(\tilde{I})$ in (97), namely

\omega_{-}^{\rm nlr}:=\partial_{\tilde{I}_{1}}\tilde{\mathbb{E}}=\omega_{2}\,,\qquad\omega_{+}^{\rm nlr}:=\partial_{\tilde{I}_{2}}\tilde{\mathbb{E}}=\omega_{1}+3\omega_{2}\,.

(99)

Indeed, recalling (86) and (95), we have

	$\displaystyle\omega_{-}^{\rm nlr}(E,{\rm I}_{2})$	$\displaystyle:=$	$\displaystyle\omega_{-}+\chi\,{\rm I}_{2}\left(2(E+a_{0})-{\rm I}_{2}\,\frac{\partial_{{\rm I}_{2}}{\mathcal{A}}(E;{\rm I}_{2})}{\partial_{E}{\mathcal{A}}(E;{\rm I}_{2})}\right)\,,$
	$\displaystyle\omega_{+}^{\rm nlr}(E,{\rm I}_{2})$	$\displaystyle:=$	$\displaystyle 3\omega_{-}+3\chi\,{\rm I}_{2}\left(2(E+a_{0})+\frac{1}{\partial_{E}{\mathcal{A}}(E;{\rm I}_{2})}-{\rm I}_{2}\,\frac{\partial_{{\rm I}_{2}}{\mathcal{A}}(E;{\rm I}_{2})}{\partial_{E}{\mathcal{A}}(E;{\rm I}_{2})}\right)\,.$		(100)

It remains to evaluate $\partial_{E}{\mathcal{A}}(E;{\rm I}_{2})$ and $\partial_{{\rm I}_{2}}{\mathcal{A}}(E;{\rm I}_{2})$ .

Proposition 2.

Set²⁴²⁴24 $\mathbf{P}(x)$ was defined in (55). Note that $\mathbf{P}(x_{i})=0$ for $i=1,2,3,4$ .

W(x,E,{\rm I}_{2}):=\frac{1}{\pi\sqrt{(1-x)^{3}x-\big{(}E-\frac{1}{2}a_{2}x^{2}-a_{1}x\big{)}^{2}}}=\frac{1}{\pi\sqrt{-\mathbf{P}(x)}}\,.

(101)

In the zones labelled by ${\rm I},{\rm I\!I},{\rm I\!I\!I}$

	$\displaystyle\partial_{E}{\mathcal{A}}(E;{\rm I}_{2})$	$\displaystyle=$	$\displaystyle\pm\int_{x_{1}(E,{\rm I}_{2})}^{x_{2}(E,{\rm I}_{2})}W(x,E,{\rm I}_{2})\,dx\,,$
	$\displaystyle\partial_{{\rm I}_{2}}{\mathcal{A}}(E;{\rm I}_{2})$	$\displaystyle=$	$\displaystyle\pm\frac{\sigma}{3\chi{\rm I}_{2}^{2}}\int_{x_{1}(E,{\rm I}_{2})}^{x_{2}(E,{\rm I}_{2})}x\,W(x,E,{\rm I}_{2})\,dx\,,$		(102)

where the $+$ sign holds in the zones labelled by ${\rm I\!I\!I}$ and ${\mathcal{P}}_{01}^{{\rm I\!I}},{\mathcal{P}}_{21}^{+,{\rm I\!I}},{\mathcal{P}}_{12}^{+,{\rm I\!I}}$ , while the $-$ sign in the zones labelled by ${\rm I}$ and ${\mathcal{P}}_{10}^{{\rm I\!I}},{\mathcal{P}}_{21}^{-,{\rm I\!I}},{\mathcal{P}}_{12}^{-,{\rm I\!I}}$ . Finally

	$\displaystyle\partial_{E}{\mathcal{A}}^{{\rm I\!V}}(E;{\rm I}_{2})$	$\displaystyle=$	$\displaystyle\pm\int_{x_{3}(E,{\rm I}_{2})}^{x_{4}(E,{\rm I}_{2})}W(x,E,{\rm I}_{2})\,dx\,,$
	$\displaystyle\partial_{{\rm I}_{2}}{\mathcal{A}}^{{\rm I\!V}}(E;{\rm I}_{2})$	$\displaystyle=$	$\displaystyle\pm\frac{\sigma}{3\chi{\rm I}_{2}^{2}}\int_{x_{3}(E,{\rm I}_{2})}^{x_{4}(E,{\rm I}_{2})}x\,W(x,E,{\rm I}_{2})\,dx\,.$		(103)

where the $+$ sign holds in the zones ${\mathcal{P}}_{21}^{\pm,{\rm I\!V}}$ and the $-$ one in ${\mathcal{P}}_{12}^{\pm,{\rm I\!V}}$ .

proof. First note that from (52) and (1) we get

	$\displaystyle\partial_{E}\psi(x,{\rm I}_{2})$	$\displaystyle=$	$\displaystyle-\frac{1}{b(x)}\cdot\frac{1}{\sqrt{1-\left(\displaystyle\frac{E-a(x)}{b(x)}\right)^{2}}}=-\frac{1}{\sqrt{b(x)^{2}-(E-a(x))^{2}}}=-W(x,E,{\rm I}_{2})\,,$
	$\displaystyle\partial_{{\rm I}_{2}}\psi(x,{\rm I}_{2})$	$\displaystyle=$	$\displaystyle\frac{3\omega_{-}-\omega_{+}}{3\chi{\rm I}_{2}^{2}}\,x\,W(x,E,{\rm I}_{2})\,.$		(104)

Case ${\rm I}$ . Since $\psi(x_{2}(E,{\rm I}_{2});E,{\rm I}_{2})=\psi(x_{1}(E,{\rm I}_{2});E,{\rm I}_{2})=0$ , we have²⁵²⁵25For brevity we omit to write the dependence on ${\rm I}_{2}$ .

	$\displaystyle\partial_{E}{\mathcal{A}}^{\rm I}(E)$	$\displaystyle=$	$\displaystyle\frac{1}{\pi}\left[\psi(x_{2}(E);E)\partial_{E}x_{2}(E)-\psi(x_{1}(E);E)\partial_{E}x_{1}(E)\right]+\frac{1}{\pi}\int_{x_{1}(E)}^{x_{2}(E)}\partial_{E}\psi(x;E)\,dx$
		$\displaystyle=$	$\displaystyle\frac{1}{\pi}\int_{x_{1}(E)}^{x_{2}(E)}\partial_{E}\psi(x;E)\,dx\stackrel{{\scriptstyle\eqref{derivopsi(x)}}}{{=}}-\int_{x_{1}(E)}^{x_{2}(E)}W(x,E,{\rm I}_{2})\,dx\,,\,,$

and, analogously,

	$\displaystyle\partial_{{\rm I}_{2}}{\mathcal{A}}^{\rm I}(E)$	$\displaystyle=$	$\displaystyle\frac{1}{\pi}\left[\psi(x_{2}(E);E)\partial_{{\rm I}_{2}}x_{2}(E)-\psi(x_{1}(E);E)\partial_{{\rm I}_{2}}x_{1}(E)\right]+\frac{1}{\pi}\int_{x_{1}(E)}^{x_{2}(E)}\partial_{{\rm I}_{2}}\psi(x;E)\,dx$
		$\displaystyle=$	$\displaystyle\frac{1}{\pi}\int_{x_{1}(E)}^{x_{2}(E)}\partial_{{\rm I}_{2}}\psi(x;E)\,dx\,.$

Then (2) follows by (4.2).
Case ${\rm I\!I}$ . We have two sub-cases: $\psi(x_{2}(E,{\rm I}_{2});E,{\rm I}_{2})=0\,,\psi(x_{1}(E,{\rm I}_{2});E,{\rm I}_{2})=\pi$ or $\psi(x_{2}(E,{\rm I}_{2});E,{\rm I}_{2})=\pi\,,\psi(x_{1}(E,{\rm I}_{2});E,{\rm I}_{2})=0$ . In the first sub-case by the first formula in (90) we have²⁶²⁶26For brevity we omit to write the dependence on $E$ and ${\rm I}_{2}$ .

	$\displaystyle\partial_{E}{\mathcal{A}}^{\rm I\!I}(E)$	$\displaystyle=$	$\displaystyle\partial_{E}x_{1}+\frac{1}{\pi}\left[\psi(x_{2})\partial_{E}x_{2}-\psi(x_{1})\partial_{E}x_{1}\right]+\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{E}\psi\,dx$
		$\displaystyle=$	$\displaystyle\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{E}\psi\,dx\stackrel{{\scriptstyle\eqref{derivopsi(x)}}}{{=}}-\int_{x_{1}}^{x_{2}}W(x)\,dx\,,\,,$

and, analogously,

	$\displaystyle\partial_{{\rm I}_{2}}{\mathcal{A}}^{\rm I\!I}(E)$	$\displaystyle=$	$\displaystyle\partial_{{\rm I}_{2}}x_{1}+\frac{1}{\pi}\left[\psi(x_{2})\partial_{{\rm I}_{2}}x_{2}-\psi(x_{1})\partial_{{\rm I}_{2}}x_{1}\right]+\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{{\rm I}_{2}}\psi(x)\,dx$
		$\displaystyle=$	$\displaystyle\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{{\rm I}_{2}}\psi(x)\,dx\,.$

In the second sub-case by the second formula in (90) we have

	$\displaystyle\partial_{E}{\mathcal{A}}^{\rm I\!I}(E)$	$\displaystyle=$	$\displaystyle\partial_{E}x_{2}-\frac{1}{\pi}\left[\psi(x_{2})\partial_{E}x_{2}-\psi(x_{1})\partial_{E}x_{1}\right]-\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{E}\psi\,dx$
		$\displaystyle=$	$\displaystyle-\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{E}\psi\,dx\stackrel{{\scriptstyle\eqref{derivopsi(x)}}}{{=}}\int_{x_{1}}^{x_{2}}W(x)\,dx\,,\,,$

and, analogously,

	$\displaystyle\partial_{{\rm I}_{2}}{\mathcal{A}}^{\rm I\!I}(E)$	$\displaystyle=$	$\displaystyle\partial_{{\rm I}_{2}}x_{1}-\frac{1}{\pi}\left[\psi(x_{2})\partial_{{\rm I}_{2}}x_{2}-\psi(x_{1})\partial_{{\rm I}_{2}}x_{1}\right]-\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{{\rm I}_{2}}\psi(x)\,dx$
		$\displaystyle=$	$\displaystyle-\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{{\rm I}_{2}}\psi(x)\,dx\,.$

We conclude by (4.2).
Case ${\rm I\!I\!I}$ . Since $\psi(x_{2}(E,{\rm I}_{2});E,{\rm I}_{2})=\psi(x_{1}(E,{\rm I}_{2});E,{\rm I}_{2})=\pi$ , by (91) we have

	$\displaystyle\partial_{E}{\mathcal{A}}^{\rm I\!I\!I}(E)$	$\displaystyle=$	$\displaystyle\frac{1}{\pi}\left[\big{(}\pi-\psi(x_{2})\big{)}\partial_{E}x_{2}-\big{(}\pi-\psi(x_{1})\big{)}\partial_{E}x_{1}\right]-\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{E}\psi(x)\,dx$
		$\displaystyle=$	$\displaystyle-\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{E}\psi(x)\,dx\stackrel{{\scriptstyle\eqref{derivopsi(x)}}}{{=}}\int_{x_{1}}^{x_{2}}W(x)\,dx\,,\,,$

and, analogously,

	$\displaystyle\partial_{{\rm I}_{2}}{\mathcal{A}}^{\rm I\!I\!I}(E)$	$\displaystyle=$	$\displaystyle\frac{1}{\pi}\left[\big{(}\pi-\psi(x_{2})\big{)}\partial_{{\rm I}_{2}}x_{2}-\big{(}\pi-\psi(x_{1})\big{)}\partial_{{\rm I}_{2}}x_{1}\right]-\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{{\rm I}_{2}}\psi(x)\,dx$
		$\displaystyle=$	$\displaystyle-\frac{1}{\pi}\int_{x_{1}}^{x_{2}}\partial_{{\rm I}_{2}}\psi(x)\,dx\,.$

Again we conclude by (4.2).
Case ${\rm I\!V}$ is analogous to case ${\rm I\!I}$ sending $1\to 3$ and $2\to 4$ . $\Box$

Remark 10.

In the case of exact 3:1 resonance, namely when $\omega_{+}=3\omega_{-}$ the functions $F$ and $a$ in (1), $\psi$ in (52), $\mathbf{P}$ in (55) with its roots $x_{i}$ , do not depend on ${\rm I}_{2}$ . As a consequence the functions $W$ and ${\mathcal{A}}$ in Proposition 2 do not depend on ${\rm I}_{2}$ . In particular $\partial_{{\rm I}_{2}}{\mathcal{A}}(E;{\rm I}_{2})=0$ , ${\mathcal{A}}(E;{\rm I}_{2})={\mathcal{A}}(E)$ and formula (100) simplifies

	$\displaystyle\omega_{-}^{\rm nlr}(E,{\rm I}_{2})$	$\displaystyle:=$	$\displaystyle\omega_{-}+\chi\,{\rm I}_{2}\left(2(E+a_{0})\right)\,,$
	$\displaystyle\omega_{+}^{\rm nlr}(E,{\rm I}_{2})$	$\displaystyle:=$	$\displaystyle\omega_{+}+3\chi\,{\rm I}_{2}\left(2(E+a_{0})+\frac{1}{\partial_{E}{\mathcal{A}}(E)}\right)\,.$		(105)

Let us now practically evaluate the elliptic integrals²⁷²⁷27For a wide treatment of elliptic integrals see, e.g., [Elliptic]. $\int W(x)dx$ and $\int xW(x)dx$ in (2). Assume that the polynomial $\mathbf{P}$ in (55) has 4 distinct roots: $x_{1},x_{2},x_{3},x_{4}$ , namely²⁸²⁸28 $(1+a_{2}^{2}/4)$ is the coefficient of the fourth order term of $\mathbf{P}(x)$ .

\mathbf{P}(x)=\left(1+\frac{a_{2}^{2}}{4}\right)(x-x_{1})(x-x_{2})(x-x_{3})(x-x_{4})\,.

(106)

We have two cases:
i) the four roots are real, $x_{1}<x_{2}<x_{3}<x_{4}$ ;
ii) we have two real roots, $x_{1}<x_{2}$ and two complex conjugated roots $x_{3}=\bar{x}_{4}$ .

4.3 Elliptic integrals: the case of four real roots

Let us define the cross ratio²⁹²⁹29Note that $\lambda\neq 0,1,\infty$ , since $x_{j}$ , $j=1,2,3,4,$ are distinct.:

\lambda:=\frac{(x_{2}-x_{1})(x_{4}-x_{3})}{(x_{3}-x_{1})(x_{4}-x_{2})}\,.

(107)

Note that $0<\lambda<1$ . Define the elliptic modulus:

{\mathtt{k}}:=\frac{1-\sqrt{\lambda}}{1+\sqrt{\lambda}}\,.

(108)

Note that $0<{\mathtt{k}}<1$ . We now construct a change of variable $x={\mathtt{T}}(z)$ given by a Möbius transformation

{\mathtt{T}}(z):=\frac{{\mathtt{A}}z+{\mathtt{B}}}{{\mathtt{C}}z+{\mathtt{D}}}\,,

(109)

such that³⁰³⁰30As is well known the cross ratio is invariant under Möbius transformations. Then, by (110), we get $\lambda=\frac{({\mathtt{k}}-1)^{2}}{({\mathtt{k}}+1)^{2}}$ , which is consistent with (108). See Lemma 2.3 and Exercise 2.4 of [Elliptic].

{\mathtt{T}}(-1/{\mathtt{k}})=x_{4}\,,\quad{\mathtt{T}}(-1)=x_{3}\,,\quad{\mathtt{T}}(1)=x_{2}\,,\quad{\mathtt{T}}(1/{\mathtt{k}})=x_{1}\,.

(110)

It is simple to show (see formula (2.7) of [Elliptic]) that the transformation $x={\mathtt{T}}(z)$ can be construct as the solution of equation

\frac{(x-x_{1})(x_{3}-x_{4})}{(x-x_{4})(x_{3}-x_{1})}=\frac{(z-1/{\mathtt{k}})(-1+1/{\mathtt{k}})}{(z+1/{\mathtt{k}})(-1-1/{\mathtt{k}})}\,.

(111)

Then the (real) coefficients of ${\mathtt{T}}$ are given by

$\displaystyle{\mathtt{A}}$	$\displaystyle:=$	$\displaystyle-{\mathtt{k}}x_{1}x_{3}-{\mathtt{k}}^{2}x_{1}x_{3}+2{\mathtt{k}}x_{1}x_{4}-{\mathtt{k}}x_{3}x_{4}+{\mathtt{k}}^{2}x_{3}x_{4}\,,$
$\displaystyle{\mathtt{B}}$	$\displaystyle:=$	$\displaystyle-x_{1}x_{3}-{\mathtt{k}}x_{1}x_{3}+2{\mathtt{k}}x_{1}x_{4}+x_{3}x_{4}-{\mathtt{k}}x_{3}x_{4}\,,$
$\displaystyle{\mathtt{C}}$	$\displaystyle:=$	$\displaystyle{\mathtt{k}}x_{1}-{\mathtt{k}}^{2}x_{1}-2{\mathtt{k}}x_{3}+{\mathtt{k}}x_{4}+{\mathtt{k}}^{2}x_{4}\,,$
$\displaystyle{\mathtt{D}}$	$\displaystyle:=$	$\displaystyle-x_{1}+{\mathtt{k}}x_{1}-2{\mathtt{k}}x_{3}+x_{4}+{\mathtt{k}}x_{4}\,.$	(112)

Note that, since ${\mathtt{k}}>0$ and $x_{1}<x_{2}<x_{3}<x_{4}$ we have³¹³¹31Indeed $x_{1}-{\mathtt{k}}x_{1}-2x_{3}+x_{4}+{\mathtt{k}}x_{4}=0$ implies, by (108), that $\sqrt{\lambda}x_{1}-(1+\sqrt{\lambda})x_{3}+x_{4}=0$ , namely $\sqrt{\lambda}=(x_{4}-x_{3})/(x_{3}-x_{1})$ . Squaring, by (107), we get the right hand side of (113).

{\mathtt{C}}=0\qquad\iff\qquad(x_{2}-x_{1})(x_{3}-x_{1})=(x_{4}-x_{2})(x_{4}-x_{3})\,.

(113)

Note also that ${\mathtt{T}}$ is invertible (on the Riemann sphere $\mathbb{C}\cup\{\infty\}$ ) and ${\mathtt{T}}(\mathbb{R})=\mathbb{R}$ . Note that, since $x_{1}<x_{2}<x_{3}<x_{4}$ and $0<{\mathtt{k}}<1$ , then

{\mathtt{A}}{\mathtt{D}}-{\mathtt{B}}{\mathtt{C}}=2{\mathtt{k}}(1-{\mathtt{k}}^{2})(x_{1}-x_{3})(x_{1}-x_{4})(x_{3}-x_{4})<0\,.

(114)

We have

\frac{d{\mathtt{T}}}{dz}(z)=\frac{{\mathtt{A}}{\mathtt{D}}-{\mathtt{B}}{\mathtt{C}}}{({\mathtt{C}}z+{\mathtt{D}})^{2}}<0\,.

(115)

Since

{\mathtt{T}}(z)-{\mathtt{T}}(\zeta)=\big{(}{\mathtt{A}}-{\mathtt{C}}\,{\mathtt{T}}(\zeta)\big{)}\frac{z-\zeta}{{\mathtt{C}}z+{\mathtt{D}}}\,,

recalling (106) and (110), the substitution $x={\mathtt{T}}(z)$ gives

	$\displaystyle\mathbf{P}({\mathtt{T}}(z))$	$\displaystyle=$	$\displaystyle(1+\frac{a_{2}^{2}}{4})[{\mathtt{T}}(z)-{\mathtt{T}}(1/{\mathtt{k}})][{\mathtt{T}}(z)-{\mathtt{T}}(1)][{\mathtt{T}}(z)-{\mathtt{T}}(-1)][{\mathtt{T}}(z)-{\mathtt{T}}(-1/{\mathtt{k}})]$		(116)
		$\displaystyle=$	$\displaystyle{\mathtt{c}}\frac{p_{\mathtt{k}}(z)}{({\mathtt{C}}z+{\mathtt{D}})^{4}}$		(116)

where

p_{\mathtt{k}}(z):=(1-z^{2})(1-{\mathtt{k}}^{2}z^{2})\,,\qquad{\mathtt{c}}:=(1+a_{2}^{2}/4){\mathtt{k}}^{-2}\prod_{1\leq j\leq 4}\big{(}{\mathtt{A}}-{\mathtt{C}}\,x_{j}\big{)}\,.

(117)

Note that

	$\displaystyle\prod_{1\leq j\leq 4}\big{(}{\mathtt{A}}-{\mathtt{C}}\,x_{j}\big{)}$	$\displaystyle=$	$\displaystyle 16\sqrt{\lambda}\big{(}1+\sqrt{\lambda}\big{)}^{-7}\big{(}-1+\sqrt{\lambda}\big{)}^{4}(x_{1}-x_{3})^{2}(x_{1}-x_{4})^{2}(x_{3}-x_{4})^{2}\cdot$
			$\displaystyle\cdot\Big{(}(x_{1}-x_{2})(x_{3}-x_{4})+(x_{1}-x_{3})(x_{2}-x_{4})\sqrt{\lambda}\Big{)}\,>\,0\,,$

which implies that ${\mathtt{c}}>0$ .

By (101), (115), (110) and (116) we get

\int_{x_{1}}^{x_{2}}W(x)dx=\frac{{\mathtt{B}}{\mathtt{C}}-{\mathtt{A}}{\mathtt{D}}}{\pi\sqrt{\mathtt{c}}}\int_{1}^{1/{\mathtt{k}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}=\frac{{\mathtt{B}}{\mathtt{C}}-{\mathtt{A}}{\mathtt{D}}}{\pi\sqrt{\mathtt{c}}}\int_{-1/{\mathtt{k}}}^{-1}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}=\int_{x_{3}}^{x_{4}}W(x)dx\,,

(118)

where the second equality holds since $p_{\mathtt{k}}(z)$ is even. It remains to evaluate $\int_{1}^{1/{\mathtt{k}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}$ , which is an elliptic integral. We get the complete elliptic integral of the first kind³²³²32Note that ${\tt EllipticK}:(-\infty,1)\to(0,+\infty)$ is an analytic strictly increasing function with $\lim_{x\to-\infty}{\tt EllipticK}(x)=0$ and $\lim_{x\to 1^{-}}{\tt EllipticK}(x)=+\infty$ .

\int_{1}^{1/{\mathtt{k}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}=\int_{0}^{1}\frac{ds}{\sqrt{(1-s^{2})(1-m_{1}s^{2})}}=:{\tt EllipticK}(m_{1})\,,\qquad m_{1}:=1-{\mathtt{k}}^{2}\,,

(119)

by the change of variable $z=\frac{1}{\sqrt{1-m_{1}s^{2}}}$ . Note that, since $0<{\mathtt{k}}<1$ we have $0<m_{1}<1$ . By (118) and (119) we get

\int_{x_{1}}^{x_{2}}W(x)dx=\int_{x_{3}}^{x_{4}}W(x)dx=\frac{{\mathtt{B}}{\mathtt{C}}-{\mathtt{A}}{\mathtt{D}}}{\pi\sqrt{\mathtt{c}}}{\tt EllipticK}(1-{\mathtt{k}}^{2})\,.

(120)

Similarly

\int_{x_{1}}^{x_{2}}x\,W(x)dx=\frac{{\mathtt{B}}{\mathtt{C}}-{\mathtt{A}}{\mathtt{D}}}{\pi\sqrt{\mathtt{c}}}\int_{1}^{1/{\mathtt{k}}}\frac{{\mathtt{A}}z+{\mathtt{B}}}{{\mathtt{C}}z+{\mathtt{D}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}\,.

(121)

We have two cases: ${\mathtt{C}}\neq 0$ and ${\mathtt{C}}=0$ . In the first case setting

{\mathtt{a}}:=\frac{{\mathtt{D}}}{{\mathtt{C}}}\,,\qquad{\mathtt{b}}:=\frac{{\mathtt{B}}{\mathtt{C}}-{\mathtt{A}}{\mathtt{D}}}{{\mathtt{C}}^{2}}\,,

(122)

we have

\int_{1}^{1/{\mathtt{k}}}\frac{{\mathtt{A}}z+{\mathtt{B}}}{{\mathtt{C}}z+{\mathtt{D}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}=\frac{{\mathtt{A}}}{{\mathtt{C}}}\int_{1}^{1/{\mathtt{k}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}+{\mathtt{b}}\int_{1}^{1/{\mathtt{k}}}\frac{1}{z+{\mathtt{a}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}\,.

(123)

Note that the real number ${\mathtt{a}}$ satisfies $|{\mathtt{a}}|>1/{\mathtt{k}}$ . Otherwise, by contradiction, assume that $|{\mathtt{a}}|\leq 1/{\mathtt{k}}$ . Since ${\mathtt{T}}(-{\mathtt{a}})=\infty$ (in the Riemann sphere), by (110) we have that $|{\mathtt{a}}|=|-{\mathtt{a}}|<1/{\mathtt{k}}$ . Since the real function ${\mathtt{T}}(z)$ has a vertical asymptote at $z=-{\mathtt{a}}$ , has ${\mathtt{A}}/{\mathtt{C}}$ as horizontal asymptote and is decreasing (recall (115)) in the intervals $(-\infty,-{\mathtt{a}})$ and $(-{\mathtt{a}},+\infty)$ , we have that ${\mathtt{T}}(-1/{\mathtt{k}})<{\mathtt{A}}/{\mathtt{C}}<{\mathtt{T}}(1/{\mathtt{k}})$ . Then by (110) we obtain $x_{4}<x_{1}$ , which is a contradiction. We conclude that $|{\mathtt{a}}|>1/{\mathtt{k}}$ .

The first integral on the right hand side of (123) has been evaluated in (119). Regarding the second one we have

\int_{1}^{1/{\mathtt{k}}}\frac{1}{z+{\mathtt{a}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}=\int_{1}^{1/{\mathtt{k}}}\frac{z}{z^{2}-{\mathtt{a}}^{2}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}-\int_{1}^{1/{\mathtt{k}}}\frac{{\mathtt{a}}}{z^{2}-{\mathtt{a}}^{2}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}\,.

(124)

We have

\int_{1}^{1/{\mathtt{k}}}\frac{z}{z^{2}-{\mathtt{a}}^{2}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}\stackrel{{\scriptstyle z^{2}=t}}{{=}}\frac{1}{2}\int_{1}^{1/{\mathtt{k}}^{2}}\frac{1}{t-{\mathtt{a}}^{2}}\frac{dt}{\sqrt{(t-1)(1-{\mathtt{k}}^{2}t)}}=-\frac{\pi}{2\sqrt{({\mathtt{a}}^{2}-1)({\mathtt{a}}^{2}{\mathtt{k}}^{2}-1)}}

(125)

and, by the change of variable $z=\frac{1}{\sqrt{1-m_{1}s^{2}}}$ , we obtain

$\displaystyle\int_{1}^{1/{\mathtt{k}}}\frac{{\mathtt{a}}}{z^{2}-{\mathtt{a}}^{2}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}$	$\displaystyle=$	$\displaystyle{\mathtt{a}}\int_{0}^{1}\frac{1-m_{1}s^{2}}{(1-{\mathtt{a}}^{2}+m_{1}{\mathtt{a}}^{2}s^{2})\sqrt{(1-s^{2})(1-m_{1}s^{2})}}\,ds$	(126)
	$\displaystyle=$	$\displaystyle-\frac{1}{{\mathtt{a}}}\int_{0}^{1}\frac{1}{\sqrt{(1-s^{2})(1-m_{1}s^{2})}}\,ds$	(128)
		$\displaystyle+\frac{1}{m_{1}{\mathtt{a}}^{3}}\int_{0}^{1}\frac{1}{\frac{1}{n_{1}}-s^{2}}\frac{1}{\sqrt{(1-s^{2})(1-m_{1}s^{2})}}\,ds\,,$	(128)

where

0<n_{1}:=\frac{m_{1}{\mathtt{a}}^{2}}{{\mathtt{a}}^{2}-1}\stackrel{{\scriptstyle\eqref{EllipticK}}}{{=}}\frac{{\mathtt{a}}^{2}-{\mathtt{k}}^{2}{\mathtt{a}}^{2}}{{\mathtt{a}}^{2}-1}<1\,,

since $|{\mathtt{a}}|>1/{\mathtt{k}}>1$ . Recalling that

\mathtt{EllipticPi}(n_{1},m_{1}):=\int_{0}^{1}\frac{1}{1-n_{1}s^{2}}\frac{1}{\sqrt{(1-s^{2})(1-m_{1}s^{2})}}\,ds

(129)

is the complete elliptic integral of the third kind, by (119) and (121)-(128) and noting that $\frac{{\mathtt{A}}}{{\mathtt{C}}}+\frac{{\mathtt{b}}}{{\mathtt{a}}}=\frac{{\mathtt{B}}}{{\mathtt{D}}},$ we get

	$\displaystyle\int_{x_{1}}^{x_{2}}x\,W(x)dx=$		(130)
	$\displaystyle\frac{{\mathtt{B}}{\mathtt{C}}-{\mathtt{A}}{\mathtt{D}}}{\pi\sqrt{\mathtt{c}}}\left(\frac{{\mathtt{B}}}{{\mathtt{D}}}{\tt EllipticK}(m_{1})-\frac{\pi{\mathtt{b}}}{2\sqrt{({\mathtt{a}}^{2}-1)({\mathtt{a}}^{2}{\mathtt{k}}^{2}-1)}}-\frac{n_{1}{\mathtt{b}}}{m_{1}{\mathtt{a}}^{3}}\mathtt{EllipticPi}(n_{1},m_{1})\right)\,.$

Let us now consider the case ${\mathtt{C}}=0$ . By (121) and since

\int_{1}^{1/{\mathtt{k}}}\frac{z\,dz}{\sqrt{-p_{\mathtt{k}}(z)}}\stackrel{{\scriptstyle z^{2}=t}}{{=}}\frac{1}{2}\int_{1}^{1/{\mathtt{k}}^{2}}\frac{dt}{\sqrt{(t-1)(1-{\mathtt{k}}^{2}t)}}=\frac{\pi}{2{\mathtt{k}}}\,,

we have

	$\displaystyle\int_{x_{1}}^{x_{2}}x\,W(x)dx$	$\displaystyle=$	$\displaystyle-\frac{{\mathtt{A}}{\mathtt{B}}}{\pi\sqrt{\mathtt{c}}}{\tt EllipticK}(m_{1})-\frac{{\mathtt{A}}^{2}}{\pi\sqrt{\mathtt{c}}}\int_{1}^{1/{\mathtt{k}}}\frac{z\,dz}{\sqrt{-p_{\mathtt{k}}(z)}}$		(131)
		$\displaystyle=$	$\displaystyle-\frac{{\mathtt{A}}{\mathtt{B}}}{\pi\sqrt{\mathtt{c}}}{\tt EllipticK}(m_{1})-\frac{{\mathtt{A}}^{2}}{2{\mathtt{k}}\sqrt{\mathtt{c}}}\,,$		(131)

which is exactly the limit for ${\mathtt{C}}\to 0$ of (130).

Let us now evaluate

\int_{x_{3}}^{x_{4}}x\,W(x)dx=\frac{{\mathtt{B}}{\mathtt{C}}-{\mathtt{A}}{\mathtt{D}}}{\pi\sqrt{\mathtt{c}}}\int_{-1/{\mathtt{k}}}^{-1}\frac{{\mathtt{A}}z+{\mathtt{B}}}{{\mathtt{C}}z+{\mathtt{D}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}\,.

(132)

Changing variable $z\to-z$ we get

\int_{-1/{\mathtt{k}}}^{-1}\frac{{\mathtt{A}}z+{\mathtt{B}}}{{\mathtt{C}}z+{\mathtt{D}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}=\int_{1}^{1/{\mathtt{k}}}\frac{{\mathtt{A}}z-{\mathtt{B}}}{{\mathtt{C}}z-{\mathtt{D}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}

When ${\mathtt{C}}\neq 0$ , recalling (122), we have

\int_{1}^{1/{\mathtt{k}}}\frac{{\mathtt{A}}z+{\mathtt{B}}}{{\mathtt{C}}z+{\mathtt{D}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}=\frac{{\mathtt{A}}}{{\mathtt{C}}}\int_{1}^{1/{\mathtt{k}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}-{\mathtt{b}}\int_{1}^{1/{\mathtt{k}}}\frac{1}{z-{\mathtt{a}}}\frac{dz}{\sqrt{-p_{\mathtt{k}}(z)}}\,.

(133)

Reasoning as in derivation of (130) we get

	$\displaystyle\int_{x_{3}}^{x_{4}}x\,W(x)dx=$		(134)
	$\displaystyle\frac{{\mathtt{B}}{\mathtt{C}}-{\mathtt{A}}{\mathtt{D}}}{\pi\sqrt{\mathtt{c}}}\left(\frac{{\mathtt{B}}}{{\mathtt{D}}}{\tt EllipticK}(m_{1})+\frac{\pi{\mathtt{b}}}{2\sqrt{({\mathtt{a}}^{2}-1)({\mathtt{a}}^{2}{\mathtt{k}}^{2}-1)}}-\frac{n_{1}{\mathtt{b}}}{m_{1}{\mathtt{a}}^{3}}\mathtt{EllipticPi}(n_{1},m_{1})\right)\,.$

The case ${\mathtt{C}}=0$ can be obtained taking the limit for ${\mathtt{C}}\to 0$ of (134).

4.4 Elliptic integrals: the case of two real roots

In this case define the cross ratio and the elliptic modulus as:³³³³33Setting $w:=(x_{1}-x_{4})(x_{2}-x_{3})$ we have that $\lambda_{*}=\bar{w}/w$ since $x_{1},x_{2}\in\mathbb{R}$ and $\bar{x}_{3}=x_{4}$ . Then $\sqrt{\lambda}_{*}:=|w|/w$ satisfies $(\sqrt{\lambda}_{*})^{2}=|w|^{2}/w^{2}=\lambda_{*}$ .

\lambda_{*}:=\frac{(x_{1}-x_{3})(x_{2}-x_{4})}{(x_{1}-x_{4})(x_{2}-x_{3})}\,,\quad{\mathtt{k}}_{*}:=\frac{1-\sqrt{\lambda}_{*}}{1+\sqrt{\lambda}_{*}}\,,\quad\sqrt{\lambda}_{*}:=\frac{|x_{1}-x_{4}||x_{2}-x_{3}|}{(x_{1}-x_{4})(x_{2}-x_{3})}\,.

(135)

Since $|\sqrt{\lambda}_{*}|=1$ there exists a real $\theta$ such that $\sqrt{\lambda}_{*}=e^{\mathrm{i}\theta}$ , so that ${\mathtt{k}}_{*}=-\mathrm{i}\tan(\theta/2)$ , namely ${\mathtt{k}}_{*}$ is purely imaginary and ${\mathtt{k}}_{*}^{2}<0$ (see page 40 of [Elliptic] for details). We now construct a Möbius transformation

{\mathtt{T}}_{*}(z):=\frac{{\mathtt{A}}_{*}z+{\mathtt{B}}_{*}}{{\mathtt{C}}_{*}z+{\mathtt{D}}_{*}}\,,

(136)

such that

{\mathtt{T}}_{*}(-1/{\mathtt{k}}_{*})=x_{4}\,,\quad{\mathtt{T}}_{*}(-1)=x_{2}\,,\quad{\mathtt{T}}_{*}(1)=x_{1}\,,\quad{\mathtt{T}}_{*}(1/{\mathtt{k}}_{*})=x_{3}\,.

(137)

It is simple to show (see formula (2.7) of [Elliptic]) that the transformation $x={\mathtt{T}}_{*}(z)$ can be construct as the solution of equation

\frac{(x-x_{3})(x_{2}-x_{4})}{(x-x_{4})(x_{2}-x_{3})}=\frac{(z-1/{\mathtt{k}}_{*})(-1+1/{\mathtt{k}}_{*})}{(z+1/{\mathtt{k}}_{*})(-1-1/{\mathtt{k}}_{*})}\,.

(138)

Note that ${\mathtt{T}}_{*}$ is invertible (on the Riemann sphere $\mathbb{C}\cup\{\infty\}$ ) and ${\mathtt{T}}_{*}(\mathbb{R})=\mathbb{R}$ . Indeed the last claim is equivalent to show that if $x\in\mathbb{R}$ in (138) then also $z\in\mathbb{R}$ . This can be proven taking the complex conjugate of (138) and inverting both sides³⁴³⁴34More precisely denoting by $\ell$ and $r$ , respectively, the left and right hand side of (138), we have that, if $x\in\mathbb{R}$ then $\ell=1/\bar{\ell}$ (recall $\bar{x}_{3}=x_{4}$ ), which implies $r=1/\bar{r}$ (recall $\bar{\mathtt{k}}_{*}=-{\mathtt{k}}_{*}$ ), namely $\frac{z-a}{z+a}=\frac{\bar{z}-a}{\bar{z}+a}$ denoting for brevity $a:=1/{\mathtt{k}}_{*}$ . Then $z=\bar{z}$ , namely $z\in\mathbb{R}$ . . The coefficients of ${\mathtt{T}}_{*}$ , which are given by

$\displaystyle{\mathtt{A}}_{*}$	$\displaystyle:=$	$\displaystyle-x_{2}(x_{3}+x_{4})+{\mathtt{k}}_{*}x_{2}(x_{4}-x_{3})+2x_{3}x_{4}\,,$
$\displaystyle{\mathtt{B}}_{*}$	$\displaystyle:=$	$\displaystyle-x_{2}(x_{3}+x_{4})+x_{2}(x_{4}-x_{3})/{\mathtt{k}}_{*}+2x_{3}x_{4}\,,$
$\displaystyle{\mathtt{C}}_{*}$	$\displaystyle:=$	$\displaystyle-2x_{2}+x_{3}+x_{4}+{\mathtt{k}}_{*}(x_{4}-x_{3})\,,$
$\displaystyle{\mathtt{D}}_{*}$	$\displaystyle:=$	$\displaystyle-2x_{2}+x_{3}+x_{4}+(x_{4}-x_{3})/{\mathtt{k}}_{*}\,,$	(139)

are real since, $x_{2}\in\mathbb{R},$ $\bar{x}_{3}=x_{4}$ and ${\mathtt{k}}_{*}$ is purely imaginary. We have that

\frac{d{\mathtt{T}}_{*}}{dz}(z)=\frac{{\mathtt{A}}_{*}{\mathtt{D}}_{*}-{\mathtt{B}}_{*}{\mathtt{C}}_{*}}{({\mathtt{C}}_{*}z+{\mathtt{D}}_{*})^{2}}<0\qquad\mbox{for}\qquad z\in\mathbb{R}\,,

(140)

since ${\mathtt{T}}_{*}(-1)=x_{2}>{\mathtt{T}}_{*}(1)=x_{1}$ by (137). It follows that

{\mathtt{A}}_{*}{\mathtt{D}}_{*}-{\mathtt{B}}_{*}{\mathtt{C}}_{*}=2({\mathtt{k}}_{*}^{2}-1)(x_{2}-x_{3})(x_{2}-x_{4})(x_{3}-x_{4})/{\mathtt{k}}_{*}<0\,.

(141)

Arguing as in (116), the substitution $x={\mathtt{T}}_{*}(z)$ gives

\mathbf{P}({\mathtt{T}}_{*}(z))=-{\mathtt{c}}_{*}\frac{p_{{\mathtt{k}}_{*}}(z)}{({\mathtt{C}}_{*}z+{\mathtt{D}}_{*})^{4}}

(142)

where $p_{{\mathtt{k}}_{*}}(z):=(1-z^{2})(1-{\mathtt{k}}_{*}^{2}z^{2})$ and

{\mathtt{c}}_{*}:=-(1+a_{2}^{2}/4){\mathtt{k}}_{*}^{-2}\prod_{1\leq j\leq 4}\big{(}{\mathtt{A}}_{*}-{\mathtt{C}}_{*}\,x_{j}\big{)}\,.

(143)

Note that ${\mathtt{c}}_{*}>0$ ; indeed ${\mathtt{k}}_{*}^{-2}<0$ , $({\mathtt{A}}_{*}-{\mathtt{C}}_{*}x_{3})({\mathtt{A}}_{*}-{\mathtt{C}}_{*}x_{4})=|{\mathtt{A}}_{*}-{\mathtt{C}}_{*}x_{3}|^{2}>0$ (since³⁵³⁵35Note that ${\mathtt{A}}_{*}-{\mathtt{C}}_{*}x_{j}\neq 0$ for $j=1,2,3,4$ , since ${\mathtt{A}}_{*}-{\mathtt{C}}_{*}x_{j}=0$ implies ${\mathtt{T}}_{*}^{-1}(x_{j})=\infty$ that contradicts (137). ${\mathtt{A}}_{*},{\mathtt{C}}_{*}\in\mathbb{R}$ and $\bar{x}_{3}=x_{4}$ ), finally, denoting for brevity $w:=(x_{1}-x_{4})(x_{2}-x_{3})$ , we have

	$\displaystyle({\mathtt{A}}_{}-{\mathtt{C}}_{}x_{1})({\mathtt{A}}_{}-{\mathtt{C}}_{}x_{2})=4(x_{2}-x_{3})^{2}(x_{1}-x_{4})(x_{2}-x_{4})\frac{1+\bar{w}/\|w\|}{1+w/\|w\|}$
	$\displaystyle=4\|x_{2}-x_{3}\|^{2}w\frac{1+\bar{w}/\|w\|}{1+w/\|w\|}=4\|x_{2}-x_{3}\|^{2}\|w\|>0\,.$

By (101), (140), (137) and (142) we get

\int_{x_{1}}^{x_{2}}W(x)dx=\frac{{\mathtt{B}}_{*}{\mathtt{C}}_{*}-{\mathtt{A}}_{*}{\mathtt{D}}_{*}}{\pi\sqrt{{\mathtt{c}}_{*}}}\int_{-1}^{1}\frac{dz}{\sqrt{p_{\mathtt{k}}(z)}}=2\frac{{\mathtt{B}}_{*}{\mathtt{C}}_{*}-{\mathtt{A}}_{*}{\mathtt{D}}_{*}}{\pi\sqrt{{\mathtt{c}}_{*}}}\int_{0}^{1}\frac{dz}{\sqrt{p_{{\mathtt{k}}_{*}}(z)}}\,,

(144)

since $p_{{\mathtt{k}}_{*}}(z)$ is even; in particular

\int_{0}^{1}\frac{dz}{\sqrt{p_{{\mathtt{k}}_{*}}(z)}}=\int_{0}^{1}\frac{dz}{\sqrt{(1-z^{2})(1-{\mathtt{k}}_{*}^{2}z^{2})}}=:{\tt EllipticK}({\mathtt{k}}_{*}^{2})\,.

(145)

By (144) and (145) we get

\int_{x_{1}}^{x_{2}}W(x)dx=2\frac{{\mathtt{B}}_{*}{\mathtt{C}}_{*}-{\mathtt{A}}_{*}{\mathtt{D}}_{*}}{\pi\sqrt{{\mathtt{c}}_{*}}}{\tt EllipticK}({\mathtt{k}}_{*}^{2})\,.

(146)

Arguing as in (144) and recalling the definition of $p_{{\mathtt{k}}_{*}}(z)$ in (117) we obtain

\int_{x_{1}}^{x_{2}}xW(x)dx=\frac{{\mathtt{B}}_{*}{\mathtt{C}}_{*}-{\mathtt{A}}_{*}{\mathtt{D}}_{*}}{\pi\sqrt{{\mathtt{c}}_{*}}}\int_{-1}^{1}\frac{{\mathtt{A}}_{*}z+{\mathtt{B}}_{*}}{{\mathtt{C}}_{*}z+{\mathtt{D}}_{*}}\frac{dz}{\sqrt{p_{{\mathtt{k}}_{*}}(z)}}=\frac{{\mathtt{B}}_{*}{\mathtt{C}}_{*}-{\mathtt{A}}_{*}{\mathtt{D}}_{*}}{\pi\sqrt{{\mathtt{c}}_{*}}}\int_{-1}^{1}{\mathtt{T}}_{*}(z)\frac{dz}{\sqrt{p_{{\mathtt{k}}_{*}}(z)}}\,,

(147)

Since the last integration interval is symmetric and $p_{{\mathtt{k}}_{*}}(z)$ is an even function we can substitute ${\mathtt{T}}_{*}(z)$ with its even part, namely

\frac{1}{2}\big{(}{\mathtt{T}}_{*}(z)+{\mathtt{T}}_{*}(-z)\big{)}=\frac{{\mathtt{A}}_{*}{\mathtt{C}}_{*}z^{2}-{\mathtt{B}}_{*}{\mathtt{D}}_{*}}{{\mathtt{C}}_{*}^{2}z^{2}-{\mathtt{D}}_{*}^{2}}=\frac{{\mathtt{A}}_{*}}{{\mathtt{C}}_{*}}+\frac{{\mathtt{B}}_{*}{\mathtt{C}}_{*}-{\mathtt{A}}_{*}{\mathtt{D}}_{*}}{{\mathtt{C}}_{*}{\mathtt{D}}_{*}}\frac{1}{1-({\mathtt{C}}_{*}/{\mathtt{D}}_{*})^{2}z^{2}}\,,

obtaining (since the integrands are even)

\int_{-1}^{1}{\mathtt{T}}_{*}(z)\frac{dz}{\sqrt{p_{{\mathtt{k}}_{*}}(z)}}=2\frac{{\mathtt{A}}_{*}}{{\mathtt{C}}_{*}}\int_{0}^{1}\frac{dz}{\sqrt{p_{{\mathtt{k}}_{*}}(z)}}+2\frac{{\mathtt{B}}_{*}{\mathtt{C}}_{*}-{\mathtt{A}}_{*}{\mathtt{D}}_{*}}{{\mathtt{C}}_{*}{\mathtt{D}}_{*}}\int_{0}^{1}\frac{1}{1-({\mathtt{C}}_{*}/{\mathtt{D}}_{*})^{2}z^{2}}\frac{dz}{\sqrt{p_{{\mathtt{k}}_{*}}(z)}}\,.

Then, by (119) and (129)

\int_{-1}^{1}{\mathtt{T}}_{*}(z)\frac{dz}{\sqrt{p_{{\mathtt{k}}_{*}}(z)}}=2\frac{{\mathtt{A}}_{*}}{{\mathtt{C}}_{*}}{\tt EllipticK}({\mathtt{k}}_{*}^{2})+2\frac{{\mathtt{B}}_{*}{\mathtt{C}}_{*}-{\mathtt{A}}_{*}{\mathtt{D}}_{*}}{{\mathtt{C}}_{*}{\mathtt{D}}_{*}}\mathtt{EllipticPi}({\mathtt{C}}_{*}^{2}{\mathtt{D}}_{*}^{-2},{\mathtt{k}}_{*}^{2})\,.

(148)

Recalling (2), (146), (147), (148), in the case of two real roots, the last term in (100) writes

\chi\,{\rm I}_{2}^{2}\,\frac{\partial_{{\rm I}_{2}}{\mathcal{A}}(E;{\rm I}_{2})}{\partial_{E}{\mathcal{A}}(E;{\rm I}_{2})}=\frac{\omega_{+}-3\omega_{-}}{3}\left(\frac{{\mathtt{A}}_{*}}{{\mathtt{C}}_{*}}+\frac{{\mathtt{B}}_{*}{\mathtt{C}}_{*}-{\mathtt{A}}_{*}{\mathtt{D}}_{*}}{{\mathtt{C}}_{*}{\mathtt{D}}_{*}}\frac{\mathtt{EllipticPi}({\mathtt{C}}_{*}^{2}{\mathtt{D}}_{*}^{-2},{\mathtt{k}}_{*}^{2})}{{\tt EllipticK}({\mathtt{k}}_{*}^{2})}\right)\,.

(149)

4.5 Explicit expression of the nonlinear frequencies for the exact 3:1 resonance

In this subsection we consider only the case of exact 3:1 resonance, namely when $\omega_{+}=3\omega_{-}$ . Let the energy $E$ be such that the polynomial $\mathbf{P}$ in (55) has 4 distinct roots: $x_{1}(E),x_{2}(E),$ $x_{3}(E),x_{4}(E).$ Recalling the definitions of ${\mathtt{k}}$ in (108), of ${\mathtt{A}},{\mathtt{B}},{\mathtt{C}},{\mathtt{D}}$ in (112), of ${\mathtt{c}}$ in (117), of ${\mathtt{k}}_{*}$ in (135), of ${\mathtt{A}}_{*},{\mathtt{B}}_{*},{\mathtt{C}}_{*},{\mathtt{D}}_{*}$ in (139), of ${\mathtt{c}}_{*}$ in (143), note that all these quantities depend on $E$ . Recalling Proposition 2, (2), (2), (120) and (146), formula (105) in Remark 10 becomes

	$\displaystyle\omega_{-}^{\rm nlr}(E,{\rm I}_{2})$	$\displaystyle:=$	$\displaystyle\omega_{-}+2\chi\,{\rm I}_{2}(E+a_{0})\,,$
	$\displaystyle\omega_{+}^{\rm nlr}(E,{\rm I}_{2})$	$\displaystyle:=$	$\displaystyle\omega_{+}+6\chi\,{\rm I}_{2}\left(E+a_{0}+V(E)\right)\,,$		(150)

where the function $V(E)$ is defined as follows:

V(E):=\pm\frac{\pi\sqrt{{\mathtt{c}}_{*}}}{2({\mathtt{B}}_{*}{\mathtt{C}}_{*}-{\mathtt{A}}_{*}{\mathtt{D}}_{*}){\tt EllipticK}({\mathtt{k}}_{*}^{2})}

(151)

with the $+$ sign in the zones ${\mathcal{P}}_{01},$ ${\mathcal{P}}_{21}^{+,{\rm I\!I}}$ , ${\mathcal{P}}_{21}^{+,{\rm I\!I\!I}}$ , ${\mathcal{P}}_{21}^{-,{\rm I\!I\!I}}$ , ${\mathcal{P}}_{12}^{+,{\rm I\!I\!I}}$ , and with $-$ sign in the zones ${\mathcal{P}}_{10},$ ${\mathcal{P}}_{21}^{-,{\rm I}}$ , ${\mathcal{P}}_{12}^{+,{\rm I}}$ , ${\mathcal{P}}_{12}^{-,{\rm I}}$ , ${\mathcal{P}}_{12}^{-,{\rm I\!I}}$ , moreover

V(E):=\pm\frac{\pi\sqrt{\mathtt{c}}}{({\mathtt{B}}{\mathtt{C}}-{\mathtt{A}}{\mathtt{D}}){\tt EllipticK}(1-{\mathtt{k}}^{2})}

(152)

with the $+$ sign in the zones ${\mathcal{P}}_{21}^{+,{\rm I\!V}}$ , ${\mathcal{P}}_{21}^{-,{\rm I\!V}}$ , ${\mathcal{P}}_{12}^{+,{\rm I\!I}}$ , ${\mathcal{P}}_{12}^{-,{\rm I\!I\!I}}$ , and with $-$ sign in the zones ${\mathcal{P}}_{21}^{+,{\rm I}}$ , ${\mathcal{P}}_{21}^{-,{\rm I\!I}}$ , ${\mathcal{P}}_{12}^{+,{\rm I\!V}}$ , ${\mathcal{P}}_{12}^{-,{\rm I\!V}}$ .
Note that, recalling (1), in (150) we have that

\chi a_{0}={\mathtt{G}}_{(2,0),(2,0)}\,.

Then we can rewrite (150) as

	$\displaystyle\omega_{-}^{\rm nlr}(E,{\rm I}_{2})$	$\displaystyle:=$	$\displaystyle\omega_{-}+2{\rm I}_{2}(\chi E+{\mathtt{G}}_{(2,0),(2,0)})\,,$
	$\displaystyle\omega_{+}^{\rm nlr}(E,{\rm I}_{2})$	$\displaystyle:=$	$\displaystyle\omega_{+}+6{\rm I}_{2}\left(\chi E+{\mathtt{G}}_{(2,0),(2,0)}+\chi V(E)\right)\,.$		(153)

Finally we can see the nonlinear resonant frequencies as functions of the initial amplitudes $a_{-}$ and $a_{+}$ . By (35) and (36) we get

J_{1}(0)=\frac{1}{2}\omega_{+}a_{+}^{2}\,,\qquad J_{2}(0)=\frac{1}{2}(\omega_{-}a_{-}^{2}+3\omega_{+}a_{+}^{2})\,,\qquad\psi_{1}(0)=\psi_{2}(0)=0\,.

(154)

By (83) we have

{\rm I}_{2}={\rm I}_{2}(0)=\frac{1}{2}(\omega_{-}a_{-}^{2}+3\omega_{+}a_{+}^{2})

(155)

and by (43) and (1) we get

E=F\Big{(}3J_{1}(0)/{J_{2}}(0),\psi_{1}(0);{J_{2}}(0)\Big{)}=a(x_{\dagger};{\rm I}_{2})+b(x_{\dagger})\,,\quad\mbox{with}\quad x_{\dagger}:=\frac{3\omega_{+}a_{+}^{2}}{\omega_{-}a_{-}^{2}+3\omega_{+}a_{+}^{2}}\,.

(156)

5 Nonlinear bandgap for the honeycomb metamaterial

In this section we present some outcomes of our analysis and discuss its application to the honeycomb metamaterial described in the introduction. In particular we investigate the effect of nonlinearity on the bandgap size, highlighting the differences between the resonant and non resonant cases. First, we briefly recall what we proved in [DL].

For a given pair $(\tilde{M},\tilde{K})$ , the bandgap is defined as the interval between the maximum of the acoustic frequency and the minimum of the optical frequency as the wave numbers run over the Brillouin triangle. In the linear case, since the gradients of $\omega_{-}$ and $\omega_{+}$ (with respect to $(\tilde{k}_{1},\tilde{k}_{2})$ ) never vanish in the interior of $\triangle$ , maxima and minima are attained on the boundary $\partial\triangle$ . In particular, for every pair $(\tilde{M},\tilde{K})$ , the maximum of the linear acoustic frequency is attained at ${\bf X}$ , while the minimum of the linear optical frequency is attained at ${\bf\Gamma}$ . We anticipate that, in evaluating the nonlinear bandgap, the point ${\bf X}$ plays a crucial role, more important than ${\bf\Gamma}$ . Indeed, typically, in the set of parameters we are considering, namely the rectangle $[0.05,0.3]\times[1,20]$ in the $(\tilde{M},\tilde{K})$ -plane, the displacement of the maximum of the acoustic frequency due to the nonlinearity is more relevant than that of the minimum of the optical frequency.

Resonant parameters

As in [DL], within the reference rectangle $[0.05,0.3]\times[1,20]$ , we identify the curve $\mathcal{R}$ formed by the pairs $(\tilde{M},\tilde{K})$ such that the linear acoustic and optical frequencies evaluated at $(\tilde{k}_{1},\tilde{k}_{2})={\bf X}$ are in 3:1 resonance, namely satisfy $3\omega_{-}=\omega_{+}$ . $\mathcal{R}$ is shown in Figure 26. In [DL], we identify the set of nonresonant pairs $(\tilde{M},\tilde{K})$ within the rectangle $[0.05,0.3]\times[1,20]$ (represented by the light yellow region in Figure 26 (left)), for which the maximum/minimum of the nonlinear acoustic/optical frequencies on the boundary of the Brillouin triangle are attained at non resonant wave numbers $(\tilde{k}_{1},\tilde{k}_{2})$ , i.e. at points where the quantity $|3\omega_{-}-\omega_{+}|$ is not small.

Formula (2.1) is valid in this nonresonant set, allowing us to directly evaluate the bandgap in [DL]. In contrast, in the complementary light purple zone in Figure 26, formula (2.1) is not applicable due to resonances and one has to use (153) as we will show here.

The final result of our analysis is presented in Figure 27, where the maximum percentage increment between the nonlinear and linear bandgap³⁶³⁶36Namely $100\times(W^{\rm nl}/W-1)$ , where $W^{\rm nl}$ and $W$ denote the width of the nonlinear and linear bandgap, respectively. is plotted as the pair $(\tilde{M},\tilde{K})$ varies over the rectangle $[0.05,0.3]\times[1,20]$ in the softening case ( $N_{3}=-10^{4}$ ). We emphasize that, while in [DL] we derived Figure 27 using (2.1) only for the pairs $(\tilde{M},\tilde{K})$ belonging to the light yellow set in Figure 26, in this section, we show how to derive it in the light purple set by (153).

Let us first recall how in [DL] we identified the two regions in Figure 26 (left). Given a pair $(\tilde{M},\tilde{K})$ , we define a set in the $(\tilde{k}_{1},\tilde{k}_{2})$ -plane as resonant if every point in the set satisfies the 3:1 resonance condition $3\omega_{-}(\tilde{M},\tilde{K},\tilde{k}_{1},\tilde{k}_{2})=\omega_{+}(\tilde{M},\tilde{K},\tilde{k}_{1},\tilde{k}_{2})$ . For a fixed pair $(\tilde{M},\tilde{K})$ within the rectangle $[0.05,0.3]\times[1,20]$ (see Figure 26, (right)) there are always one or two resonant curves in the $(\tilde{k}_{1},\tilde{k}_{2})$ -plane, that intersect the Brillouin triangle $\triangle$ (see Figure 28). The curve $\mathcal{R}$ divides the rectangle $[0.05,0.3]\times[1,20]$ into two regions: the one above and the one below $\mathcal{R}$ , corresponding to the green region and the blue region in Figure 26 (right), respectively. For every fixed pair $(\tilde{M},\tilde{K})$ in the green region, there is only one resonant curve in the plane of wave numbers $(\tilde{k}_{1},\tilde{k}_{2})$ , that intersects the Brillouin triangle (the green curve in Figure 28). Conversely, for every fixed pair $(\tilde{M},\tilde{K})$ in the blue region, there are two resonant curves in the plane of wave numbers $(\tilde{k}_{1},\tilde{k}_{2})$ , that intersect the Brillouin triangle (the blue curves in Figure 28). Finally, in the limit case when the pair $(\tilde{M},\tilde{K})$ belongs to the curve $\mathcal{R}$ , there are two resonant curves in the $(\tilde{k}_{1},\tilde{k}_{2})$ -plane, that intersect the Brillouin triangle, but one intersects $\triangle$ only at ${\bf X}$ (see the red curves in Figure 28).

Admissible amplitudes

Both in formula (2.1) and in formula (153), (recall also (155) and (156)), the nonlinear corrections to the frequencies are essentially proportional to the squares of the amplitudes $a_{+}$ and $a_{-}$ . Thus, the larger the amplitudes $a_{\pm}$ , the greater the displacement of the nonlinear bandgap relative to the linear one. On the other hand, (2.1) and (153) are perturbative in nature, as they are derived from the non resonant and resonant BNF, respectively. Therefore, $a_{\pm}$ must be sufficiently small for the formulae to remain valid. As shown in [DL], where they are analytically evaluated, the “admissible” amplitudes are smaller in the nonresonant case than in the resonant one. Indeed, since the nonresonant BNF cancels more terms, it is “stronger” than the resonant one. In particular the admissible amplitudes in the nonresonant case approach zero as the quantity $|3\omega_{-}-\omega_{+}|$ vanishes. For example, when taking $(\tilde{k}_{1},\tilde{k}_{2})=\mathbf{X}$ , the admissible amplitudes vanish for parameters values $(\tilde{M},\tilde{K})$ on the curve $\mathcal{R}$ . This is not the case of the admissible amplitudes in the resonant case, namely the ones appearing in formulae (153), (155) and (156)). Indeed they are bounded away from zero on the resonances.

Shifting perspective, we can fix $(\tilde{M},\tilde{K})$ and observe at the variation of $a_{\pm}$ in the nonresonant case, as the wave numbers $(\tilde{k}_{1},\tilde{k}_{2})$ vary along the boundary $\partial\triangle$ of the Brillouin triangle $\triangle$ . Notably, $a_{\pm}$ decreases to zero at certain resonant points, denoted ${\bf R}_{i}$ . These points correspond to the intersections of the boundary of the Brillouin triangle with the resonant curves plotted in Figure 28. Formula (2.1) loses validity in the vicinity of any point ${\bf R}_{i}$ . The values of the admissible initial amplitude $a_{+}$ (in the nonresonant case) as $(\tilde{k}_{1},\tilde{k}_{2})$ traverses $\partial\triangle$ are shown in Figure 29 for three different pairs of $(\tilde{M},\tilde{K})$ .

In conclusion, due to the presence of the 3:1 resonance, formula (2.1) becomes invalid in the vicinity of the points ${\bf R}_{i}$ , when the parameters are resonant or nearly resonant. Specifically, this occurs when they give rise to an exact, or nearly exact, 3:1 resonance between acoustic and optical frequencies. In this resonant case the correct expression for the nonlinear frequencies is $\omega_{\pm}^{\rm nlr}$ , as given by (153).

Nonlinear bandgap

Let us consider the softening case; the hardening case can be treated analogously, leading to a general decrement of the bandgap. We note that, since we are considering pairs $(\tilde{M},\tilde{K})$ belonging to the rectangle $[0.05,0.3]\times[1,20]$ , the point ${\bf\Gamma}$ , where the minimum of the linear acoustic frequency is attained, is always far from being resonant. Therefore, in the following discussion, we will focus on the maximum of acoustic frequency because it undergoes the most significant displacements and may be resonant. It turns out that, for the calculation of the nonlinear bandgap, there are essentially three cases:
i) the maximum of the acoustic frequency and the minimum of the optical frequency are attained away from resonant points,
ii) $\mathbf{X}$ is resonant or nearly resonant,
iii) $\omega_{-}$ has an almost flat maximum, so that, even if ${\bf X}$ is away from resonance, the nonlinear acoustic frequency may attain its maximum at some resonant (or nearly resonant) point away from ${\bf X}$ .
Note that case i) corresponds to the light yellow region in Figure 26, while cases ii) and iii) correspond to the light purple one. These three cases are shown in Figure 30.

To summarize, one applies formula (2.1) in case (i), as we did in [DL], and formula (153) in cases (ii) and (iii), as we do here. Using the expressions for the admissible amplitudes evaluated in [DL], we are able to compute the bandgap, thereby obtaining Figure 27 in its entirety.

6 Conclusions

In this study, we investigated a broad range of structural engineering models by analyzing a general system of two coupled harmonic oscillators with cubic nonlinearity. Our examination revealed that, in the absence of damping, the system exhibits Hamiltonian dynamics, with an elliptic equilibrium at the origin characterized by two distinct linear frequencies. In particular, we focused on the resonant or nearly resonant case, specifically when the two frequencies are close to a 3:1 resonance.
Our investigation involved employing Hamiltonian Perturbation Theory to transform the system into (resonant) Birkhoff Normal Form up to order 4. This transformation provided a new set of symplectic action-angle variables, on which the Hamiltonian, up to six-order terms, depends only on the actions and the slow angle. Notably, our analysis highlighted the dependency of the construction on the system’s physical parameters, necessitating a meticulous case analysis of the phase portrait in the 3:1 resonant case. We found that the system can exhibit up to six topologically different behaviors, depending on the values of the physical parameters. In each of these configurations, we described the nonlinear normal modes (elliptic/hyperbolic periodic orbits, invariant tori) and their stable and unstable manifolds of the truncated Hamiltonian (neglecting order six or higher terms). This is a fundamental step for proving the persistence of the majority of these structures for the complete Hamiltonian by KAM Theory.

By using elliptic integrals, we derived explicit analytic formulas for the nonlinear frequencies. While this analytic expression was already known away from resonances, it is, as far as we know, new in this context for the resonant or nearly resonant case.

As an application of our findings, we explored wave propagation in metamaterial honeycombs equipped with periodically distributed nonlinear resonators. Our investigation allowed us to examine the bandgap phenomenon in the presence of resonance. We found that while nonlinear effects far from resonances can significantly alter the bandgap, in the resonant case, the nonlinear frequencies, especially the acoustic one, closely align with the linear ones, resulting in a less pronounced variation in the bandgap.

7 Appendix

7.1 Proof of Proposition 1

We first count the solutions of equation (46), namely the intersections between the line $\ell(x):=a_{2}x+a_{1}$ and the function $b^{\prime}(x)$ in (47). We note that, since $b^{\prime}$ is strictly convex, if $\ell(1)>0$ there is only one intersection. Note that condition $\ell(1)=a_{2}+a_{1}>0$ is equivalent to $a_{2}>-a_{1}$ . Since $g(a_{1})>-a_{1}$ , condition $a_{2}>-a_{1}$ implies that we are in the zones $Z_{01}$ or $Z_{21}$ in which we have, indeed, one intersection that we call $x_{1}^{(\pi)}$ .
Moreover, in this case $a_{2}>-a_{1}$ , the function $x\to F(\pi,x)=a(x)-b(x)$ with $x\in(0,1)$ has only one critical point, which is exactly $x_{1}^{(\pi)}$ . This critical point is a minimum since $\lim_{x\to 0^{+}}\partial_{x}F(\pi,x)=\lim_{x\to 0^{+}}a^{\prime}(x)-b^{\prime}(x)=-\infty$ and $\lim_{x\to 1^{-}}\partial_{x}F(\pi,x)=\lim_{x\to 1^{-}}a^{\prime}(x)-b^{\prime}(x)=a_{2}+a_{1}>0$ .
Assume now that $a_{2}<-a_{1}$ . Note that for every fixed $a_{1}\in\mathbb{R}$ there exists a unique $a_{2}=h(a_{1})$ such that $a_{2}x+a_{1}$ is tangent to $b^{\prime}(x)$ at some point $0<x_{0}<1$ . In order to evaluate the function $h(a_{1})$ above let us consider the tangent $r(x)$ in a point $x_{0}$ to $b^{\prime}(x)$ ; namely:

r(x)=b^{\prime}(x_{0})+b^{\prime\prime}(x_{0})(x-x_{0})\,.

Since we want that $r(x)=a_{2}x+a_{1}$ we have to impose $r(0)=a_{1}$ and $b^{\prime\prime}(x_{0})=a_{2}$ . Since

b^{\prime\prime}(x)=\frac{8x^{2}-4x-1}{4x^{3/2}\sqrt{1-x}}\,,

(157)

imposing $r(0)=a_{1}$ we have

$\displaystyle a_{1}=r(0)$	$\displaystyle=$	$\displaystyle b^{\prime}(x_{0})-b^{\prime\prime}(x_{0})x_{0}=\frac{(1-4x_{0})\sqrt{1-x_{0}}}{2\sqrt{x_{0}}}-\frac{8x_{0}^{2}-4x_{0}-1}{4x_{0}^{3/2}\sqrt{1-x_{0}}}x_{0}$	(158)
	$\displaystyle=$	$\displaystyle\frac{2(1-4x_{0})(1-x_{0})-(8x_{0}^{2}-4x_{0}-1)}{4\sqrt{1-x_{0}}\sqrt{x_{0}}}$
	$\displaystyle=$	$\displaystyle\frac{2-8x_{0}-2x_{0}+8x_{0}^{2}-8x_{0}^{2}+4x_{0}+1}{4\sqrt{1-x_{0}}\sqrt{x_{0}}}$
	$\displaystyle=$	$\displaystyle\frac{-6x_{0}+3}{4\sqrt{1-x_{0}}\sqrt{x_{0}}}=\frac{3(1-2x_{0})}{4\sqrt{1-x_{0}}\sqrt{x_{0}}}\,.$

Note that:

a_{1}>0,<0,=0\qquad\implies\qquad x_{0}<\frac{1}{2},>\frac{1}{2},=\frac{1}{2}\,.

(159)

Squaring we get

a_{1}^{2}=\frac{9(1+4x_{0}^{2}-4x_{0})}{16(1-x_{0})x_{0}}

namely

(36+16a_{1}^{2})x_{0}^{2}-(36+16a_{1}^{2})x_{0}+9=0\,.

The solutions of the above second order equation are

x_{0}=\frac{1}{2}\pm\frac{a_{1}}{\sqrt{9+4a_{1}^{2}}}\,,

but by (159) we have to choose the minus sign. Since by (158) we have

\frac{1}{4\sqrt{1-x_{0}}\sqrt{x_{0}}}=\frac{a_{1}}{3(1-2x_{0})}

by (157) and denoting for brevity $s:=\sqrt{4a_{1}^{2}+9}$ , we get³⁷³⁷37Note that $2x_{0}-1=-2a_{1}/\sqrt{4a_{1}^{2}+9}=-2a_{1}/s$ .

$\displaystyle a_{2}$	$\displaystyle=$	$\displaystyle b^{\prime\prime}(x_{0})=\frac{8x_{0}^{2}-4x_{0}-1}{x_{0}}\frac{a_{1}}{3(1-2x_{0})}=\frac{2(2x_{0}-1)^{2}+2(2x_{0}-1)-1}{x_{0}}\frac{a_{1}}{3(1-2x_{0})}$	(160)
	$\displaystyle=$	$\displaystyle\frac{8a_{1}^{2}-4a_{1}s-s^{2}}{3(s-2a_{1})}=\frac{8a_{1}^{2}-4a_{1}s-s^{2}}{27}(s+2a_{1})$
	$\displaystyle=$	$\displaystyle\frac{1}{27}\textstyle(4a_{1}^{2}-4a_{1}\sqrt{4a_{1}^{2}+9}-9)(2a_{1}+\sqrt{4a_{1}^{2}+9})=:h(a_{1})\,.$

Note that $h(a_{1})=-g(-a_{1})<-a_{1}$ . Since we are in the case $a_{2}<-a_{1}$ and we have proved that the line $h(a_{1})x+a_{1}$ is tangent to $b^{\prime}(x)$ , we have that for $a_{2}<h(a_{1})$ there are not intersections (zone $Z_{10}$ ) while for $h(a_{1})<a_{2}<-a_{1}$ there are two intersections (zone $Z_{12}$ ), that we call $0<x_{1}^{(\pi)}<x_{2}^{(\pi)}<1$ .
In this last case, the function $x\to F(\pi,x)=a(x)-b(x)$ with $x\in(0,1)$ has two critical points, which are exactly $x_{1}^{(\pi)}$ and $x_{2}^{(\pi)}$ . Since $\lim_{x\to 0^{+}}\partial_{x}F(\pi,x)=\lim_{x\to 0^{+}}a^{\prime}(x)-b^{\prime}(x)=-\infty$ and $\lim_{x\to 1^{-}}\partial_{x}F(\pi,x)=\lim_{x\to 1^{-}}a^{\prime}(x)-b^{\prime}(x)=a_{2}+a_{1}<0$ , $x_{1}^{(\pi)}$ must be a minimum and $x_{2}^{(\pi)}$ a maximum.
Finally the case of equation (45) and the critical points of the function $F(0,x)$ can be studied in the same way sending $a_{2}\to-a_{2}$ and $a_{1}\to-a_{1}$ . ∎

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