# Unbounded solutions to a system of coupled asymmetric oscillators at
resonance††thanks: Under the auspices of INdAM-GNAMPA, Italy. In particular,
the first author acknowledges the support of the GNAMPA Project 2020 ”Problemi
ai limiti per l’equazione della curvatura media prescritta”.

A. Boscaggin, W. Dambrosio and D. Papini

> Abstract. We deal with the following system of coupled asymmetric
> oscillators
>
>
> $\left\\{\begin{array}[]{l}\ddot{x}_{1}+a_{1}x_{1}^{+}-b_{1}x^{-}_{1}+\phi_{1}(x_{2})=p_{1}(t)\vspace{7pt}\\\
> \ddot{x}_{2}+a_{2}\,x_{2}^{+}-b_{2}\,x^{-}_{2}+\phi_{2}(x_{1})=p_{2}(t),\end{array}\right.$
>
> where $\phi_{i}:\mathbb{R}\to\mathbb{R}$ is locally Lipschitz continuous and
> bounded, $p_{i}:\mathbb{R}\to\mathbb{R}$ is continuous and $2\pi$-periodic
> and the positive real numbers $a_{i},b_{i}$ satisfy
>
> $\dfrac{1}{\sqrt{a_{i}}}+\dfrac{1}{\sqrt{b_{i}}}=\dfrac{2}{n},\quad\mbox{
> for some }n\in\mathbb{N}.$
>
> We define a suitable function $L:\mathbb{T}^{2}\to\mathbb{R}^{2}$, appearing
> as the higher-dimensional generalization of the well known resonance
> function used in the scalar setting, and we show how unbounded solutions to
> the system can be constructed whenever $L$ has zeros with a special
> structure. The proof relies on a careful investigation of the dynamics of
> the associated (four-dimensional) Poincaré map, in action-angle coordinates.
>
> Keywords. Systems of ODEs, asymmetric oscillators, unbounded solutions,
> resonance.
>
> AMS Subject Classification. 34C11, 34C15.

##  1 Introduction

In this paper, we investigate the existence of unbounded solutions for a
system of coupled asymmetric oscillators of the type

$\left\\{\begin{array}[]{l}\ddot{x}_{1}+a_{1}x_{1}^{+}-b_{1}x^{-}_{1}+\phi_{1}(x_{2})=p_{1}(t)\vspace{7pt}\\\
\ddot{x}_{2}+a_{2}\,x_{2}^{+}-b_{2}\,x^{-}_{2}+\phi_{2}(x_{1})=p_{2}(t),\end{array}\right.$
(1.1)

where, as usual, $x^{\pm}=\max\\{\pm x,0\\}$ and, for $i=1,2$,
$\phi_{i}:\mathbb{R}\to\mathbb{R}$ is locally Lipschitz continuous and
bounded, $p_{i}:\mathbb{R}\to\mathbb{R}$ is continuous and $2\pi$-periodic. As
for the positive real numbers $a_{i},b_{i}$, we assume that

$\dfrac{1}{\sqrt{a_{i}}}+\dfrac{1}{\sqrt{b_{i}}}=\dfrac{2}{n},\quad\mbox{ for
some }n\in\mathbb{N},$ (1.2)

thus implying that each oscillator is at resonance with respect to the same
curve of the Fucik spectrum [11].

The study of unbounded solutions for oscillators at resonance is a classical
topic in the qualitative theory of ordinary differential equations and we
refer to [15] for an excellent survey on this subject. In order to motivate
our contribution, the crucial reference to be recalled here is the seminal
paper [2] by Alonso and Ortega. It is proved therein (cf. [2, Theorem 4.1])
that, for the scalar asymmetric oscillator

$\ddot{x}+ax^{+}-bx^{-}=p(t),\qquad x\in\mathbb{R},$ (1.3)

with $1/\sqrt{a}+1/\sqrt{b}=2/n$, all large solutions are unbounded (either in
the past or in the future) whenever the $2\pi$-periodic function

$\Phi(\theta)=\int_{0}^{2\pi}C\left(\frac{\theta}{n}+t\right)p(t)\,dt,\qquad\theta\in\mathbb{R}$
(1.4)

has zeros, all simple (in the above formula, $C$ stands for the asymmetric
cosine function, cf. Section 2.1). The function $\Phi$, sometimes referred to
as resonance function, was previously introduced by Dancer [7] to investigate
the $2\pi$-periodic solvability of equation (1.3). In the linear case
($a=b=n^{2}$), the function $\Phi$ has (simple) zeros if and only if
$\int_{0}^{2\pi}p(t)e^{-\textrm{i}nt}\,dt\neq 0$: in this case, as well known,
all the solutions of $\ddot{x}+n^{2}x=p(t)$ are unbounded; instead,
$2\pi$-periodic and unbounded solutions to (1.3) can coexist in the genuinely
asymmetric case $a\neq b$. The proof of this result was obtained by a careful
investigation of the dynamics of the associated Poincaré map: more precisely,
the zeros of the function $\Phi$ were shown to give rise to invariant sets for
the discrete dynamical system associated with (1.3) and eventually to the
existence of unbounded orbits. Generalization of this approach, requiring the
introduction of suitable resonance functions, were later provided for forced
asymmetric oscillators

$\ddot{x}+ax^{+}-bx^{-}+\phi(x)=p(t),\qquad x\in\mathbb{R},$

with $\phi:\mathbb{R}\to\mathbb{R}$ a bounded function (see [6, 9]) and, more
in general, for planar system of the type

$Jz^{\prime}=\nabla H(z)+R(z)+e(t),\qquad z\in\mathbb{R}^{2},$

where $J$ is the standard symplectic matrix, $H:\mathbb{R}^{2}\to\mathbb{R}$
is positive and positively homoegeneous of degree $2$ and
$R:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is bounded (see [8, 10]). We also refer to
[1, 4, 5, 12, 13, 14, 16, 17] for related results.

In spite of this extensive bibliography, the existence of unbounded solutions
for systems of coupled oscillators seems to be an essentially unexplored
topic. To the best of our knowledge, the only available results are the one
contained in the recent paper [3], dealing however with systems of equations
looking like weakly coupled perturbations of linear oscillators (i.e.
$a_{i}=b_{i}=n_{i}^{2}$ for $i=1,2$) and not being applicable to the more
general setting of (1.1).

The aim of the present paper is to extend the approach of [2] in this higher-
dimensional framework. As expected, this is a quite delicate task, since it
leads to the study of the dynamics of a four-dimensional map; nonetheless, we
will succeed in providing some partial generalizations of the results in [2].
In more details, our strategy and results can be described as follows.

In Section 2 we pass to an appropriate set of action-angle coordinates and we
perform an asymptotic expansion, at infinity, of the Poincaré map associated
with (1.1), cf. (2.31). In doing this, we are led to define a resonance
function defined on the two-dimensional torus,

$L:\mathbb{T}^{2}\to\mathbb{R}^{2},\qquad(\theta_{1},\theta_{2})\mapsto(L_{1}(\theta_{1},\theta_{2}),L_{2}(\theta_{1},\theta_{2}))$

which can be thought as the higher-dimensional generalization of the resonance
function $\Phi$ defined in (1.4), see (2.30)-(2.33). We notice that when
system (1.1) is uncoupled (that is, $\phi_{1}=\phi_{2}=0$), then
$L(\theta_{1},\theta_{2})=(L_{1}(\theta_{1}),L_{2}(\theta_{2}))$ and, up to a
constant, $L_{i}=\Phi$ with $p=p_{i}$.

In Section 3 we investigate the dynamics of this four-dimensional Poincaré map
and we construct invariant sets, giving rise to unbounded orbits. As in the
two-dimensional setting, the zeros of the function $L$ are shown to play a
role; however, due to the coupling terms in system (1.1), we need here to
assume that the Jacobian matrix $JL$ has a special structure at the zeros.
More precisely, we introduce the notion of $\mathcal{D}^{\pm}$-matrix, cf.
Definition 3.1: again, we observe that such a condition is satisfied by
diagonal matrices with concordant sign diagonal entries and, hence, by the
matrix $JL$ when system (1.1) is uncoupled and the functions $L_{i}$ have
simple zeros, as in the main result of [2]. This is a quite technical part of
the proof, involving, among other things, a delicate estimate for the $2$-norm
of a two-parameter family of suitable matrices, which are perturbations of the
identity by $\mathcal{D}^{\pm}$-matrices, cf. Lemma 3.2.

In Section 4 we finally give our main result for the existence of unbounded
solutions to system (1.1), Theorem 4.1. It provides a positive measure set of
initial conditions giving rise to unbounded orbits to (1.1), whenever the
function $L$ has a zero $\omega\in\mathbb{T}^{2}$ such that the Jacobian
matrix $JL(\omega)$ is a $\mathcal{D}^{\pm}$-matrix. Theorem 4.1. Notice that
this can be interpreted as a kind of local version of the main result in [2].
Indeed, we do not claim that every large solution of (1.1) is unbounded: due
to the higher-dimensional setting, obtaining this global information seems to
be a very hard task, even in the case when all the zeros of $L$ are such that
the Jacobian at each zero is a $\mathcal{D}^{\pm}$-matrix. We mention that the
condition for $JL$ to be a $\mathcal{D}^{\pm}$-matrix can be, in general, not
easy to verify. To this end, we discuss some situations in which this can be
done and Theorem 4.1 can thus be applied. The first, quite natural,
possibility that we present is a semi-perturbative result (cf. Corollary 4.3),
dealing with the case in which the $L^{\infty}$-norms of the coupling terms
$\phi_{1},\phi_{2}$ are not too big: it is worth noticing that this provided a
genuinely asymmetric (non-quantitative) generalization of a result obtained in
[3] for coupled linear oscillators. Other results, more global in nature but
focusing on specific choices for the parameters $a_{i},b_{i}$ or the forcing
terms $p_{i}$, are given by Corollary 4.6 and Corollary 4.7. It seems that
various other situations could be treated at the expenses of longer
computations.

We finally mention that it should be possible, with the same approach, to
consider also the more general case of resonance with respect to different
curves of the Fucik spectrum, that is, $1/\sqrt{a_{i}}+1/\sqrt{b_{i}}=2/n_{i}$
with $n_{i}\in\mathbb{N}$. Also, the possibility of coupling more oscillators
in a cyclic way $\phi_{i+1}=\phi_{i}$ could be considered. All these
generalizations, however, seem to require substantial technical modifications
of the proofs and they are thus postponed to future investigations.

Notation. Throughout the paper, the symbol $\|\cdot\|$ will be used for the
Euclidean norm of a vector in the plane. Also, for the index $i=1,2$, we will
adopt the cyclic agreement $i+1=2$ for $i=1$.

##  2 Coupled asymmetric oscillators: some preliminary estimates

In this section, we perform some preliminary estimates for the solutions of
system (1.1), with the final goal of obtaining an asymptotic expansion for its
Poincaré-map in action-angle coordinates (see Section 2.2).

From now on, as in the Introduction we will always assume that, for $i=1,2$,
the positive real numbers $a_{i},b_{i}$ satisfy (1.2), the function
$p_{i}:\mathbb{R}\to\mathbb{R}$ is continuous and $2\pi$-periodic and the
function $\phi_{i}:\mathbb{R}\to\mathbb{R}$ is locally Lipschitz continuous
and bounded. Furthermore, we also suppose that there exist

$\lim_{x\to\pm\infty}\phi_{i}(x):=\phi_{i}(\pm\infty);$ (2.5)

moreover, without loss of generality,

$\phi_{i}(-\infty)=-\phi_{i}(+\infty).$ (2.6)

###  2.1 Remarks on the asymmetric cosine and related functions

We collect here some results on various functions related to the asymmetric
cosine function $C_{i}$, $i=1,2$, which is defined as the solution of

$\left\\{\begin{array}[]{l}\ddot{x}_{i}+a_{i}x_{i}^{+}-b_{i}x^{-}_{i}=0\vspace{7pt}\\\
x_{i}(0)=1,\ \dot{x}_{i}(0)=0,\end{array}\right.$ (2.7)

with $a_{i}$ and $b_{i}$ as in (1.2). We recall that, for every $i=1,2$, the
function $C_{i}$ is even, $\tau:=2\pi/n$-periodic and its explicit expression
in $[-\tau/2,\tau/2]$ is

$C_{i}(t)=\left\\{\begin{array}[]{ll}\cos\sqrt{a_{i}}t&\mbox{if
}|t|\leq\pi/2\sqrt{a_{i}}\\\ &\\\
\displaystyle{-\sqrt{\dfrac{a_{i}}{b_{i}}}\sin\sqrt{b_{i}}\left(|t|-\dfrac{\pi}{2\sqrt{a_{i}}}\right)}&\mbox{if
}\pi/2\sqrt{a_{i}}\leq|t|\leq\tau/2;\end{array}\right.$ (2.8)

For future reference, let us observe that $C_{i}$, $i=1,2$, when $a_{i}\neq
b_{i}$ admits the Fourier series expansion

$C_{i}(t)=\sum_{h=0}^{+\infty}c_{h,i}\cos hnt,\quad t\in{\mathbb{R}},$ (2.9)

where

$c_{0,i}=\dfrac{2}{\tau}\,\dfrac{b_{i}-a_{i}}{b_{i}\sqrt{a_{i}}}$

and, for $h\geq 1$,

$c_{h,i}=\left\\{\begin{array}[]{ll}\dfrac{4}{\tau}\,\dfrac{b_{i}-a_{i}}{b_{i}-h^{2}n^{2}}\,\dfrac{\sqrt{a_{i}}}{a_{i}-h^{2}n^{2}}\,\cos\dfrac{hn\pi}{2\sqrt{a_{i}}}&\text{if
}a_{i}\neq h^{2}n^{2},\,b_{i}\neq h^{2}n^{2},\vspace{4pt}\\\
\dfrac{1}{2h}&\text{otherwise}\end{array}\right.$ (2.10)

(see [2, Lemma 4.2]).

In the next sections we will use the integrals of $C_{i}$ over the sets
$J^{\pm}_{i+1}$ defined by

$\begin{array}[]{l}\displaystyle{J^{+}_{i+1}=\left\\{t\in[0,2\pi]:\
C_{i+1}\left(\dfrac{\theta_{i+1}}{n}+t\right)>0\right\\},\quad
J^{-}_{i+1}=\left\\{t\in[0,2\pi]:\
C_{i+1}\left(\dfrac{\theta_{i+1}}{n}+t\right)<0\right\\}}\end{array}$ (2.11)

where $\theta_{i+1}\in{\mathbb{R}}$ and we have used the notation $i+1=1$ for
$i=2$. It is immediate to observe that the fact that $C_{i}$ and $C_{i+1}$ are
both $2\pi/n$-periodic implies that

$\int_{J^{\pm}_{i+1}}C_{i}\left(\dfrac{\theta_{i}}{n}+t\right)\,dt,\quad\theta_{i}\in{\mathbb{R}},$

does not change if we replace $[0,2\pi]$ in the definition of $J^{\pm}_{i+1}$
by any interval of lenght $2\pi$. In particular, in the computation of the
integral of $C_{i}$ on $J_{i+1}^{+}$, we can replace $J^{+}_{i+1}$ by the set

$\bigcup_{k=0}^{n-1}\left(-\dfrac{\pi}{2\sqrt{a_{i}}}-\dfrac{\theta_{i+1}}{n}+2k\dfrac{\pi}{n},\dfrac{\pi}{2\sqrt{a_{i}}}-\dfrac{\theta_{i+1}}{n}+2k\dfrac{\pi}{n}\right),$

thus obtaining that

$\int_{J^{+}_{i+1}}C_{i}\left(\dfrac{\theta_{i}}{n}+t\right)\,dt=n\Lambda_{i}(\theta_{i}-\theta_{i+1}),\quad\forall\
\theta_{i},\theta_{i+1}\in{\mathbb{R}},\quad i=1,2,$ (2.12)

where $\Lambda_{i}:{\mathbb{R}}\to{\mathbb{R}}$ is defined by

$\Lambda_{i}(t)=K_{i}\left(\dfrac{t}{n}+\dfrac{\pi}{2\sqrt{a_{i+1}}}\right)-K_{i}\left(\dfrac{t}{n}-\dfrac{\pi}{2\sqrt{a_{i+1}}}\right),\quad\forall\
t\in{\mathbb{R}},$ (2.13)

being $K_{i}$ the primitive of $C_{i}$ such that $K_{i}(0)=0$.

A crucial point in our analysis will be the study ot the resolubility of the
equation

$\Lambda_{i}(t)=\alpha_{i},$ (2.14)

where $\alpha_{i}$ is given by

$\alpha_{i}=\dfrac{1}{\sqrt{a_{i}}}-\dfrac{\sqrt{a_{i}}}{b_{i}},\quad i=1,2,$
(2.15)

which is is related to $C_{i}$ by

$\int_{0}^{2\pi}C_{i}\left(t\right)\,dt=2n\alpha_{i},\quad i=1,2.$ (2.16)

In particular, we will be interested in the situation where (2.14) has simple
solutions; in order to face this problem, let us first concentrate on the
range of the function $\Lambda_{i}$. Introducing the function
$\Sigma_{i}:{\mathbb{R}}\to{\mathbb{R}}$ defined by

$\Sigma_{i}(t)=n\Lambda^{\prime}_{i}(t),\quad\forall\ t\in{\mathbb{R}},$
(2.17)

is it possible to prove the following result.

######  Lemma 2.1.

The function $\Lambda_{i}$ given in (2.13) is even, $2\pi$-periodic,
decreasing in $(0,\pi)$ and increasing in $(-\pi,0)$.

###### Proof.

Let us first observe that we have

$\begin{array}[]{l}\Sigma_{i}(t)=C_{i}\left(\dfrac{t}{n}+\dfrac{\pi}{2\sqrt{a_{i+1}}}\right)-C_{i}\left(\dfrac{t}{n}-\dfrac{\pi}{2\sqrt{a_{i+1}}}\right)=\\\
\\\
=C_{i}^{n}\left(t+\dfrac{n\pi}{2\sqrt{a_{i+1}}}\right)-C_{i}^{n}\left(t-\dfrac{n\pi}{2\sqrt{a_{i+1}}}\right),\quad\forall\
t\in{\mathbb{R}},\end{array}$ (2.18)

where $C_{i}^{n}:{\mathbb{R}}\to{\mathbb{R}}$ is defined by

$C^{n}_{i}(t)=C_{i}\left(\dfrac{t}{n}\right),\quad\forall\ t\in{\mathbb{R}}.$

The function $C^{n}_{i}$ is continuous, $2\pi$-periodic, even and strictly
decreasing in $[0,\pi)$; as a consequence, $\Sigma_{i}$ is continuous,
$2\pi$-periodic and odd. As far as the sign of $\Sigma_{i}$ is concerned, let
us observe that $\Sigma_{i}(t)=0$ if and only if $t=k\pi$ for some
$k\in{\mathbb{Z}}$. Indeed, $\Sigma_{i}(t)=0$ if and only if:

$\text{either
}t+\frac{n\pi}{2\sqrt{a_{i+1}}}=t-\frac{n\pi}{2\sqrt{a_{i+1}}}+2k\pi\quad\text{or}\quad
t+\frac{n\pi}{2\sqrt{a_{i+1}}}=-t+\frac{n\pi}{2\sqrt{a_{i+1}}}+2k\pi\quad\text{for
some }k\in{\mathbb{Z}}.$

Now, since $a_{i+1}>n^{2}/4$, the first alternative cannot hold, and the
second one implies that $t=k\pi$. Therefore $\Sigma_{i}$ has constant sign in
$(0,\pi)$ and a straightforward argument shows that

$\Sigma_{i}\left(\pi-\frac{n\pi}{2\sqrt{a_{i+1}}}\right)<0.$

From the above described properties of $\Sigma_{i}$ we immediately deduce the
thesis. ∎

From now on, in order to simplify the notation, let us continue with the case
$i=1$; the case $i=2$ is completely analogous.

From Lemma 2.1 we deduce that equation (2.14), with $i=1$, admits simple
solutions if and only if

$\Lambda_{1}(\pi)<\alpha_{1}<\Lambda_{1}(0);$

in general, the validity of this condition depends on the original pairs
$(a_{1},b_{1})$ and $(a_{2},b_{2})$. Hence, let us define the resolubility set

$\mathcal{R}=\left\\{(a_{1},a_{2})\in\left(\dfrac{n^{2}}{4},+\infty\right)\times\left(\dfrac{n^{2}}{4},+\infty\right)\subset\mathbb{R}^{2}:\
\Lambda_{1}(\pi)<\alpha_{1}<\Lambda_{1}(0)\right\\}.$ (2.19)

The complete description of the open set $\mathcal{R}$ is quite difficult; by
means of long computations it is possible to show that the vertical sections
$\mathcal{R}\cap\\{(a_{1}^{*},a_{2}):\ a_{2}>n^{2}/4\\}$, with
$a_{1}^{*}>n^{2}/4,\ a_{1}^{*}\neq n^{2}$, are bounded. On the other hand, the
study of the horizontal sections $\mathcal{R}\cap\\{(a_{1},a_{2}^{*}):\
a_{1}>n^{2}/4\\}$, with $a_{2}^{*}>n^{2}/4,\ a_{2}^{*}\neq n^{2}$, is much
more complicated. However, the following simple result holds true.

######  Lemma 2.2.

The set $\mathcal{R}$ contains the half-lines
$\\{(n^{2},a_{2}):a_{2}>n^{2}/4\\}$ and $\\{(a_{1},n^{2}):a_{1}>n^{2}/4\\}$.

###### Proof.

Let us first assume that $a_{1}=n^{2}$ (and, thus, $b_{1}=n^{2}$) and fix
$\sqrt{a_{2}}>n/2$; we then have

$\displaystyle C_{1}(t)=\cos(nt),\quad
K_{1}(t)=\frac{1}{n}\sin(nt),\quad\Lambda_{1}(t)=\frac{2}{n}\sin\frac{\pi
n}{2\sqrt{a_{2}}}\cos t,\quad\forall t\in{\mathbb{R}}.$

A simple computation proves that $\Lambda_{1}(\pi)<0<\Lambda_{1}(0)$; noticing
that $\alpha_{1}=0$, by (2.15), this shows that $(n^{2},a_{2})\in\mathcal{R}$.

On the other hand, if $\alpha_{2}=n^{2}$, we have

$\displaystyle\Lambda_{1}(0)=2\int_{0}^{\pi/2n}C_{1}(t)dt,\quad\Lambda_{1}(\pi)=2\int_{\pi/2n}^{\pi/n}C_{1}(t)dt.$

Recalling (2.16), we deduce that
$\Lambda_{1}(\pi)=2\alpha_{1}-\Lambda_{1}(0)<\Lambda_{1}(0)$ and, thus,
$\alpha_{1}<\Lambda_{1}(0)$. From these relations we also obtain
$\Lambda_{1}(\pi)=2\alpha_{1}-\Lambda_{1}(0)<\alpha_{1}$, proving that
$(a_{1},n^{2})\in\mathcal{R}$. ∎

###  2.2 Asymptotic analysis

We now perform an asymptotic expansion of the Poincaré map associated to
(1.1). We adapt the argument of the proof of [2, Theorem 4.1] to our case: we
write (1.1) as a first order system in
$(x_{1},x_{2},y_{1},y_{2})=(x_{1},x_{2},\dot{x}_{1},\dot{x}_{2})$ and use the
change of variables

$\left\\{\begin{aligned}
x_{i}&=\gamma_{i}r_{i}C_{i}\left(\dfrac{\theta_{i}}{n}\right)\\\
y_{i}&=\gamma_{i}r_{i}S_{i}\left(\dfrac{\theta_{i}}{n}\right)\end{aligned}\right.\quad\text{with
}\gamma_{i}=\sqrt{2n/a_{i}},$ (2.20)

where the functions $C_{i}$ and $S_{i}$ are defined in Subsection 2.1.

It is straightforward to see that (1.1) is formally equivalent to

$\left\\{\begin{aligned}
\dot{\theta}_{1}&=n-\dfrac{\gamma_{1}}{2r_{1}}C_{1}\left(\dfrac{\theta_{1}}{n}\right)\left[p_{1}(t)-\phi_{1}\left(\gamma_{2}r_{2}C_{2}\left(\dfrac{\theta_{2}}{n}\right)\right)\right]\\\
\dot{\theta}_{2}&=n-\dfrac{\gamma_{2}}{2r_{2}}C_{2}\left(\dfrac{\theta_{2}}{n}\right)\left[p_{2}(t)-\phi_{2}\left(\gamma_{1}r_{1}C_{1}\left(\dfrac{\theta_{1}}{n}\right)\right)\right]\\\
\dot{r}_{1}&=\dfrac{\gamma_{1}}{2n}S_{1}\left(\dfrac{\theta_{1}}{n}\right)\left[p_{1}(t)-\phi_{1}\left(\gamma_{2}r_{2}C_{2}\left(\dfrac{\theta_{2}}{n}\right)\right)\right]\\\
\dot{r}_{2}&=\dfrac{\gamma_{2}}{2n}S_{2}\left(\dfrac{\theta_{2}}{n}\right)\left[p_{2}(t)-\phi_{2}\left(\gamma_{1}r_{1}C_{1}\left(\dfrac{\theta_{1}}{n}\right)\right)\right]\end{aligned}\right.$
(2.21)

We denote by $(\theta_{1},\theta_{2},r_{1},r_{2})$ the solution of (2.21)
satisfying
$(\theta_{1},\theta_{2},r_{1},r_{2})(0)=(\theta_{1,0},\theta_{2,0},r_{1,0},r_{2,0})$
and study the behavior of $(\theta_{1},\theta_{2},r_{1},r_{2})(2\pi)$ as
$\min\\{r_{1,0},r_{2,0}\\}\to+\infty$. We also set
$\theta_{0}=(\theta_{1,0},\theta_{2,0})$, $r_{0}=(r_{1,0},r_{2,0})$ and remark
that $\theta_{0}\in{\mathbb{R}}^{2}$ and $r_{1,0},r_{2,0}>0$.

The boundedness of $p_{i}$ and $\phi_{i}$ implies that $\dot{r}_{i}$ is
uniformly bounded and, hence, we have

$r_{i}=r_{i,0}+O(1)\qquad\text{and}\qquad
r_{i}^{-1}=r_{i,0}^{-1}+O(r_{i,0}^{-2})\qquad\text{as }r_{i,0}\to+\infty,$
(2.22)

where these and all the following estimates hold uniformly w.r.t.
$t\in[0,2\pi]$, $\theta_{1,0}$, $\theta_{2,0}$ and $r_{i+1,0}$. We deduce that

$\begin{aligned}
\dot{\theta}_{i}&=n-\dfrac{\gamma_{i}}{2}C_{i}\left(\dfrac{\theta_{i}}{n}\right)\left(\dfrac{1}{r_{i,0}}+O(r_{i,0}^{-2})\right)\left[p_{i}(t)-\phi_{i}\left(\gamma_{i+1}r_{i+1}C_{i+1}\left(\dfrac{\theta_{i+1}}{n}\right)\right)\right]\\\
&=n-\dfrac{\gamma_{i}}{2r_{i,0}}C_{i}\left(\dfrac{\theta_{i}}{n}\right)\left[p_{i}(t)-\phi_{i}\left(\gamma_{i+1}r_{i+1}C_{i+1}\left(\dfrac{\theta_{i+1}}{n}\right)\right)\right]+O(r_{i,0}^{-2}),\end{aligned}\qquad\text{as
}r_{i,0}\to+\infty.$ (2.23)

This relation implies that

$\dfrac{\theta_{i}}{n}=\dfrac{\theta_{i,0}}{n}+t+O(r_{i,0}^{-1}),\qquad\text{as
}r_{i,0}\to+\infty;$

and, thus:

$\begin{aligned}
C_{i}\left(\dfrac{\theta_{i}}{n}\right)&=C_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)+O(r_{i,0}^{-1})\\\
S_{i}\left(\dfrac{\theta_{i}}{n}\right)&=S_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)+O(r_{i,0}^{-1})\end{aligned}\qquad\text{as
}r_{i,0}\to+\infty,$ (2.24)

since $C_{i}$ and $S_{i}$ are smooth enough. By replacing (2.24) in the last
two equations of (2.21) we get

$\dot{r}_{i}=\dfrac{\gamma_{i}}{2n}S_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)\left[p_{i}(t)-\phi_{i}\left(\gamma_{i+1}r_{i+1}C_{i+1}\left(\frac{\theta_{i+1}}{n}\right)\right)\right]+O(r_{i,0}^{-1})\qquad\text{as
}r_{i,0}\to+\infty.$

As a consequence, we infer that

$\displaystyle r_{i}(2\pi)$
$\displaystyle=r_{i,0}+\dfrac{\gamma_{i}}{2n}\int_{0}^{2\pi}S_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)p_{i}(t)\,dt$
(2.25)
$\displaystyle\hphantom{=}-\dfrac{\gamma_{i}}{2n}\int_{0}^{2\pi}S_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)\phi_{i}\left(\gamma_{i+1}r_{i+1}C_{i+1}\left(\frac{\theta_{i+1}}{n}\right)\right)dt+F_{i,i}(\theta_{0},r_{0})$

where

$\lim_{r_{i,0}\to+\infty}F_{i,i}(\theta_{0},r_{0})=0\quad\text{uniformly
w.r.t. }\theta_{1},\theta_{2},\text{ and }r_{i+1,0}.$

Now, we deduce from (2.24) that $C_{i+1}(\theta_{i+1}(t)/n)\to
C_{i+1}(\theta_{i+1,0}/n+t)$ uniformly w.r.t. $t\in[0,2\pi]$, $\theta_{1,0}$,
$\theta_{2,0}$ and $r_{i,0}$, as $r_{i+1,0}\to+\infty$ and, setting

$\displaystyle J^{+}_{i+1,0}=\left\\{t\in[0,2\pi]:\
C_{i+1}\left(\frac{\theta_{i+1,0}}{n}+t\right)>0\right\\}$ $\displaystyle
J^{-}_{i+1,0}=\left\\{t\in[0,2\pi]:\
C_{i+1}\left(\frac{\theta_{i+1,0}}{n}+t\right)<0\right\\},$

we have that

$t\in
J^{\pm}_{i+1,0}\implies\lim_{r_{i+1,0}\to+\infty}\phi_{i}\left(\gamma_{i+1}r_{i+1}C_{i+1}\left(\frac{\theta_{i+1}}{n}\right)\right)=\phi_{i}(\pm\infty).$

where these two limits are not uniform w.r.t. $t\in[0,2\pi]$. However, using
that $\phi_{i}$ is bounded and $C_{i+1}(\theta_{i+1}(t)/n)$ converges
uniformly, it is possible to show that:

$\lim_{r_{i+1,0}\to+\infty}\int_{J^{\pm}_{i+1,0}}S_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)\left[\phi_{i}(\pm\infty)-\phi_{i}\left(\gamma_{i+1}r_{i+1}C_{i+1}\left(\frac{\theta_{i+1}}{n}\right)\right)\right]dt=0$

uniformly w.r.t. $\theta_{1,0}$, $\theta_{2,0}$ and $r_{i,0}$. Therefore, we
can write equation (2.25) in the following way:

$\displaystyle r_{i}(2\pi)$
$\displaystyle=r_{i,0}+\dfrac{\gamma_{i}}{2n}\int_{0}^{2\pi}S_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)p_{i}(t)\,dt$
(2.26)
$\displaystyle\hphantom{=}-\dfrac{\gamma_{i}}{2n}\left(\phi_{i}(+\infty)\int_{J^{+}_{i+1,0}}S_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)\,dt+\phi_{i}(-\infty)\int_{J^{-}_{i+1,0}}S_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)\,dt\right)$
$\displaystyle\hphantom{=}+F_{i,i}(\theta_{0},r_{0})+F_{i,i+1}(\theta_{0},r_{0}),$

where also

$\lim_{r_{i+1,0}\to+\infty}F_{i,i+1}(\theta_{0},r_{0})=0\qquad\text{uniformly
w.r.t. }\theta_{1,0},\theta_{2,0}\text{ and }r_{i,0}.$ (2.27)

We now substitute (2.22) and (2.24) in (2.23), obtaining

$\dot{\theta}_{i}=n-\dfrac{\gamma_{i}}{2r_{i,0}}C_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)\left[p_{i}(t)-\phi_{i}\left(\gamma_{i+1}r_{i+1}C_{i+1}\left(\dfrac{\theta_{i+1}}{n}\right)\right)\right]+O(r_{i,0}^{-2}),\qquad\text{as
}r_{i,0}\to+\infty.$

Integrating on $[0,2\pi]$ and making similar considerations as done for
$r_{i}(2\pi)$, we deduce that

$\displaystyle\theta_{i}(2\pi)$
$\displaystyle=\theta_{i,0}+2n\pi-\dfrac{\gamma_{i}}{2r_{i,0}}\int_{0}^{2\pi}C_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)p_{i}(t)\,dt$
(2.28)
$\displaystyle\hphantom{=}+\dfrac{\gamma_{i}}{2r_{i,0}}\left(\phi_{i}(+\infty)\int_{J^{+}_{i+1,0}}C_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)\,dt+\phi_{i}(-\infty)\int_{J^{-}_{i+1,0}}C_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)\,dt\right)$
$\displaystyle\hphantom{=}+\dfrac{1}{r_{i,0}}\left(G_{i,i}(\theta_{0},r_{0})+G_{i,i+1}(\theta_{0},r_{0})\right),$

where

$\lim_{r_{i,0}\to+\infty}G_{i,i}(\theta_{0},r_{0})=\lim_{r_{i+1,0}\to+\infty}G_{i,i+1}(\theta_{0},r_{0})=0$
(2.29)

uniformly w.r.t. the other variables.

Recalling (2.6), we observe that (2.12) and (2.16) imply that
$\theta_{i}(2\pi)$, $i=1,2$, can be written as

$\displaystyle\theta_{i}(2\pi)$
$\displaystyle=\theta_{i,0}+2n\pi-\dfrac{\gamma_{i}}{2r_{i,0}}\int_{0}^{2\pi}C_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)p_{i}(t)\,dt+\frac{1}{r_{i,0}}\gamma_{i}n\phi_{i}(+\infty)(\Lambda_{i}(\theta_{i,0}-\theta_{i+1,0})-\alpha_{i})$
$\displaystyle\hphantom{=}+\dfrac{1}{r_{i,0}}\left(G_{i,i}(\theta_{0},r_{0})+G_{i,i+1}(\theta_{0},r_{0})\right),$

where $\alpha_{i},\Lambda_{i}$ are defined in (2.15), (2.13).

For $i=1,2$, let us now denote

$\displaystyle\Phi_{i}(\theta_{i,0})$
$\displaystyle=-\dfrac{\gamma_{i}}{2}\int_{0}^{2\pi}C_{i}\left(\dfrac{\theta_{i,0}}{n}+t\right)p_{i}(t)\,dt,$
(2.30) $\displaystyle L_{i}(\theta_{0})$
$\displaystyle=\Phi_{i}(\theta_{i,0})+\gamma_{i}n\phi_{i}(+\infty)(\Lambda_{i}(\theta_{i,0}-\theta_{i+1,0})-\alpha_{i}),$

for every $\theta_{0}=(\theta_{1,0},\theta_{2,0})\in{\mathbb{R}}^{2}$. Then,
we can summarize (2.26), (2.27), (2.28) and (2.29) as follows:

$\left\\{\begin{aligned} {\theta_{i}}(2\pi)&=\theta_{i,0}+2\pi
n+\dfrac{1}{r_{i,0}}\left[L_{i}(\theta_{0})+G_{i,i}(\theta_{0},r_{0})+G_{i,i+1}(\theta_{0},r_{0})\right]\\\
r_{i}(2\pi)&=r_{i,0}-\dfrac{\partial
L_{i}}{\partial\theta_{i,0}}(\theta)+F_{i,i}(\theta_{0},r_{0})+F_{i,i+1}(\theta_{0},r_{0})\end{aligned}\right.\qquad\text{for
}i=1,2,$ (2.31)

where

$\displaystyle\lim_{r_{i,0}\to+\infty}F_{i,i}(\theta_{0},r_{0})=\lim_{r_{i,0}\to+\infty}G_{i,i}(\theta_{0},r_{0})=0$
$\displaystyle\text{uniformly w.r.t. }r_{i+1,0}\text{ and }\theta_{0},$ (2.32)
$\displaystyle\lim_{r_{i+1,0}\to+\infty}F_{i,i+1}(\theta_{0},r_{0})=\lim_{r_{i+1,0}\to+\infty}G_{i,i+1}(\theta_{0},I_{0})=0$
$\displaystyle\text{uniformly w.r.t. }r_{i,0}\text{ and }\theta_{0}.$

The functions $L_{1},L_{2}$ will be meant as the components of the vector
valued function

$L(\theta_{0})=(L_{1}(\theta_{0}),L_{2}(\theta_{0})),\qquad\theta_{0}=(\theta_{1,0},\theta_{2,0})\in{\mathbb{R}}^{2}.$
(2.33)

which we will call _resonance function_ for system (1.1). Notice that, due to
the $2\pi$-periodicity in both the variables, we can interpret $L$ as a
function defined on the two-dimensional torus
$\mathbb{T}^{2}={\mathbb{R}}^{2}/(2\pi\mathbb{Z})^{2}$. This function will
play a crucial role in the statement of our main result (see Section 4).

##  3 Dynamics of discrete maps

In this section, we establish the abstract result that will be used to prove
the existence of unbounded solutions to system (1.1).

###  3.1 $\mathcal{D}^{\pm}$-matrices

We consider $2\times 2$-matrices $A=(a_{ij})$, $i,j=1,2$.

######  Definition 3.1.

A $2\times 2$-matrix $A$ is said to be a $\mathcal{D}^{+}$-matrix if

$a_{11}<0,\qquad a_{22}<0,\qquad|a_{12}a_{22}+a_{11}a_{21}|<2a_{11}a_{22}.$
(3.34)

Analogously, a $2\times 2$-matrix $A$ is said to be a $\mathcal{D}^{-}$-matrix
if

$a_{11}>0,\qquad a_{22}>0,\qquad|a_{12}a_{22}+a_{11}a_{21}|<2a_{11}a_{22}.$
(3.35)

Notice that a diagonal matrix with negative entries (resp., positive entries)
is a $\mathcal{D}^{+}$ matrix (resp., $\mathcal{D}^{-}$ matrix). Given a
$\mathcal{D}^{\pm}$-matrix $A$ and
$\epsilon=(\epsilon_{1},\epsilon_{2})\in(0,+\infty)^{2}$, let us define

$B_{\epsilon}=\begin{pmatrix}1+\epsilon_{1}a_{11}&\epsilon_{1}a_{12}\\\ &\\\
\epsilon_{2}a_{21}&1+\epsilon_{2}a_{22}\end{pmatrix}.$ (3.36)

Moreover, for every $\epsilon_{0}>0$ and $\eta>0$ let us define

$C_{\epsilon_{0},\eta}=\left\\{\epsilon=(\epsilon_{1},\epsilon_{2})\in(0,+\infty)^{2}:\
\dfrac{a_{11}}{a_{22}}-\eta\leq\dfrac{\epsilon_{2}}{\epsilon_{1}}\leq\dfrac{a_{11}}{a_{22}}+\eta,\quad\|\epsilon\|\leq\epsilon_{0}\right\\}.$
(3.37)

We prove the following result.

######  Lemma 3.2.

Assume that $A$ is a $\mathcal{D}^{\pm}$-matrix. Then, there exist $a_{0}>0$,
$\epsilon_{0}>0$ and $\eta>0$ such that

$\|B_{\epsilon}\|_{2}\leq 1-\dfrac{1}{2}a_{0}\|\epsilon\|,\quad\forall\
\epsilon\in C_{\epsilon_{0},\eta}.$ (3.38)

###### Proof.

We give the proof in the case of $\mathcal{D}^{+}$-matrix; the other case is
analogous. We recall that the matrix norm $\|B_{\epsilon}\|_{2}$ coincides
with the square root of the maximum eigenvalue of the matrix
$C_{\epsilon}=B_{\epsilon}^{T}\,B_{\epsilon}$.

Let us first observe that, for every $\epsilon$, the elements on the diagonal
of $C_{\epsilon}$ are given by

$(1+\epsilon_{1}a_{11})^{2}+\epsilon_{2}^{2}a_{21}^{2},\quad(1+\epsilon_{2}a_{22})^{2}+\epsilon_{1}^{2}a_{12}^{2};$

hence, we have

${\rm
tr}(C_{\epsilon})=2+2(a_{11}\epsilon_{1}+a_{22}\epsilon_{2})+(a_{11}^{2}+a_{12}^{2})\epsilon_{1}^{2}+(a_{22}^{2}+a_{21}^{2})\epsilon_{2}^{2}.$
(3.39)

Hence, a simple computation shows that

$\begin{array}[]{ll}\left({\rm
tr}(C_{\epsilon})\right)^{2}=&4+8(a_{11}\epsilon_{1}+a_{22}\epsilon_{2})+4(a_{11}\epsilon_{1}+a_{22}\epsilon_{2})^{2}+4(a_{11}^{2}+a_{12}^{2})\epsilon_{1}^{2}+4(a_{22}^{2}+a_{21}^{2})\epsilon_{2}^{2}+\\\
&\\\
&+4(a_{11}\epsilon_{1}+a_{22}\epsilon_{2})((a_{11}^{2}+a_{12}^{2})\epsilon_{1}^{2}+(a_{22}^{2}+a_{21}^{2})\epsilon_{2}^{2})+((a_{11}^{2}+a_{12}^{2})\epsilon_{1}^{2}+(a_{22}^{2}+a_{21}^{2})\epsilon_{2}^{2})^{2}.\end{array}$
(3.40)

On the other hand, we have

$\begin{array}[]{ll}{\det}(C_{\epsilon})=(\det
B_{\epsilon})^{2}=&1+2(a_{11}\epsilon_{1}+a_{22}\epsilon_{2})+(a_{11}\epsilon_{1}+a_{22}\epsilon_{2})^{2}+2\Delta\epsilon_{1}\epsilon_{2}+\\\
&\\\
&+2(a_{11}\epsilon_{1}+a_{22}\epsilon_{2})\Delta\epsilon_{1}\epsilon_{2}+\Delta^{2}\epsilon_{1}^{2}\epsilon_{2}^{2},\end{array}$
(3.41)

where $\Delta=a_{11}a_{22}-a_{12}a_{21}$.

Now, let us observe that the matrix $C_{\epsilon}$ is positive definite; as a
consequence, the maximum eigenvalue of $C_{\epsilon}$ is given by

$\lambda_{+}(\epsilon)=\dfrac{{\rm tr}(C_{\epsilon})+\sqrt{({\rm
tr}(C_{\epsilon}))^{2}-4\det(C_{\epsilon})}}{2}.$ (3.42)

From (3.40) and (3.41), by means of simple computations we infer that

$\sqrt{({\rm
tr}(C_{\epsilon}))^{2}-4\det(C_{\epsilon})}=2\sqrt{(1+a_{11}\epsilon_{1}+a_{22}\epsilon_{2})d_{2}(\epsilon)+P_{4}(\epsilon)},$
(3.43)

where

$d_{2}(\epsilon)=(a_{11}^{2}+a_{12}^{2})\epsilon_{1}^{2}-2\Delta\epsilon_{1}\epsilon_{2}+(a_{21}^{2}+a_{22}^{2})\epsilon_{2}^{2}=(a_{11}\epsilon_{1}-a_{22}\epsilon_{2})^{2}+(a_{12}\epsilon_{1}+a_{21}\epsilon_{2})^{2}$
(3.44)

and

$\displaystyle P_{4}(\epsilon)$
$\displaystyle=\dfrac{1}{4}\left[(a_{11}^{2}+a_{12}^{2})\epsilon_{1}^{2}+(a_{22}^{2}+a_{21}^{2})\epsilon_{2}^{2}\right]^{2}-\Delta^{2}\epsilon_{1}^{2}\epsilon_{2}^{2}.$
(3.45)
$\displaystyle=\dfrac{1}{4}d_{2}(\epsilon)\left[(a_{11}^{2}+a_{12}^{2})\epsilon_{1}^{2}+(a_{22}^{2}+a_{21}^{2})\epsilon_{2}^{2}+2\Delta\epsilon_{1}\epsilon_{2}\right]$
$\displaystyle=d_{2}(\epsilon)\cdot O(\|\epsilon\|^{2})\qquad\text{as
}\epsilon\to 0.$

Using (3.39) and (3.42)–(3.45), we can estimate:

$\displaystyle\lambda_{+}(\epsilon)$
$\displaystyle=1+a_{11}\epsilon_{1}+a_{22}\epsilon_{2}+O(\|\epsilon\|^{2})+\sqrt{d_{2}(\epsilon)}\,\sqrt{1+a_{11}\epsilon_{1}+a_{22}\epsilon_{2}+O(\|\epsilon\|^{2})}$
(3.46) $\displaystyle=1+g(\epsilon)+O(\|\epsilon\|^{2})\qquad\text{as
}\epsilon\to 0,$

where:

$g(\epsilon)=a_{11}\epsilon_{1}+a_{22}\epsilon_{2}+\sqrt{d_{2}(\epsilon)}.$

Observe that $g$ is a positively homogeneous function of degree $1$. A simple
computation shows that

$g\left(-\dfrac{a_{22}}{\sqrt{a_{11}^{2}+a_{22}^{2}}},-\dfrac{a_{11}}{\sqrt{a_{11}^{2}+a_{22}^{2}}}\right)=\dfrac{-2a_{11}a_{22}+|a_{12}a_{22}+a_{11}a_{21}|}{a_{11}^{2}+a_{22}^{2}}:=-4a_{0}<0,$
(3.47)

since the matrix $A$ is a $\mathcal{D}^{+}$-matrix. Using (3.47) we deduce
that there exists $\eta>0$ such that

$g\left(\epsilon\right)<-2a_{0},$ (3.48)

for every $\epsilon$ such that $\|\epsilon\|=1$ and

$0<\dfrac{a_{11}}{a_{22}}-\eta\leq\dfrac{\epsilon_{2}}{\epsilon_{1}}\leq\dfrac{a_{11}}{a_{22}}+\eta.$
(3.49)

By homogeneity, we conclude that

$g\left(\epsilon\right)<-2a_{0}\|\epsilon\|,$ (3.50)

for every $\epsilon\in(0,+\infty)^{2}$ satisfying (3.49).

From (3.46) and (3.50) we deduce that there exists $\bar{\epsilon}>0$ such
that

$\lambda_{+}(\epsilon)\leq 1-a_{0}\|\epsilon\|,$ (3.51)

for every $\epsilon\in C_{{\bar{\epsilon}},\eta}$. Let us now take
$\epsilon_{0}=\min\\{{\bar{\epsilon}},1/a_{0}\\}$; from (3.51) we immediately
conclude that

$\sqrt{\lambda_{+}(\epsilon)}\leq 1-\dfrac{1}{2}\,a_{0}\|\epsilon\|,$

for every $\epsilon\in C_{{\epsilon}_{0},\eta}$. ∎

###  3.2 Invariant sets and unbounded orbits of discrete maps

In (2.31) we have obtained an estimate for the Poincaré map
$(\theta(0),r(0))\mapsto(\theta(2\pi),r(2\pi))$ associated to the system
(2.21) when both components $r_{1,0}$ and $r_{2,0}$ of $r(0)$ are large. Here
we provide sufficient conditions under which the discrete dynamical systems
generated by similar maps possess invariant sets that contain unbounded
trajectories.

Few words are in order to clarify the setting in which the dynamical system is
defined and represented. Equations (2.31) define a map
$(\theta,r)\mapsto(u,\rho)$, with $\theta=(\theta_{1},\theta_{2})$,
$r=(r_{1},r_{2})$, $u=(u_{1},u_{2})$ and $\rho=(\rho_{1},\rho_{2})$, such
that:

$\left\\{\begin{aligned} u&=\theta+\left[\begin{array}[]{c}2\pi n_{1}\\\ 2\pi
n_{2}\end{array}\right]+\left[\begin{array}[]{c}L_{1}(\theta)/r_{1}\\\
L_{2}(\theta)/r_{2}\end{array}\right]+\left[\begin{array}[]{c}G_{1}(\theta,r)/r_{1}\\\
G_{2}(\theta,r)/r_{2}\end{array}\right]\\\
\rho&=r-\left[\begin{array}[]{c}\partial_{1}L_{1}(\theta)\\\
\partial_{2}L_{2}(\theta)\end{array}\right]+F(\theta,r),\end{aligned}\right.$
(3.52)

where $n_{1},n_{2}\in\mathbb{N}$,
$G(\theta,r)=(G_{1}(\theta,r),G_{2}(\theta,r))$ and
$F(\theta,r)=(F_{1}(\theta,r),F_{2}(\theta,r))$ are continuous,
$L(\theta)=(L_{1}(\theta),L_{2}(\theta))$ is a $C^{1}$-function with
$\partial_{j}L_{i}=\partial L_{i}/\partial\theta_{j}$, and, moreover, $L,G,F$
are all $2\pi$-periodic w.r.t. $\theta_{1}$ and $\theta_{2}$. We recall that
$(\theta_{i},r_{i})$ and $(u_{i},\rho_{i})$ are modified polar coordinates in
${\mathbb{R}}^{2}$ according to (2.20) and, hence, there is a couple of well
known issues to take into account.

The first one concerns the singularity of polar coordinates whenever the
radius vanishes and will be easily dealt with since the invariant sets we are
going to define will be contained in a region where $\min\\{r_{1},r_{2}\\}\geq
R>0$.

The second issue is that (3.52) defines a lifting of the actual dynamical
system that, indeed, acts on $\mathbb{T}^{2}\times{\mathbb{R}}_{+}^{2}$,
where, as usual, $\mathbb{T}^{2}={\mathbb{R}}^{2}/(2\pi\mathbb{Z})^{2}$
denotes the two-dimensional torus. More precisely, the coordinates
$(\theta,r)$ and $(u,\rho)$ should be projected to
$\mathbb{T}^{2}\times{\mathbb{R}}_{+}^{2}$ to determine the correct behavior
of the dynamical system, but computations are more easily performed on the
“flat” covering space ${\mathbb{R}}^{2}\times{\mathbb{R}}_{+}^{2}$. To this
aim, we denote by $\bar{\theta}_{i}$ the equivalence class of $\theta_{i}$ in
$\mathbb{T}^{1}={\mathbb{R}}/2\pi\mathbb{Z}$ and, thus, we will have
$\bar{\theta}=(\bar{\theta}_{1},\bar{\theta}_{2})\in\mathbb{T}^{2}$ for each
$\theta=(\theta_{1},\theta_{2})\in{\mathbb{R}}^{2}$; the group metrics in
$\mathbb{T}^{1}$ and $\mathbb{T}^{2}$ are respectively defined by

$|\bar{\theta}_{i}-\bar{u}_{i}|=\min\\{|\theta_{i}-u_{i}+2n\pi|:n\in\mathbb{Z}\\}\quad\text{and}\quad\|\bar{\theta}-\bar{u}\|=\sqrt{|\bar{\theta}_{1}-\bar{u}_{1}|^{2}+|\bar{\theta}_{2}-\bar{u}_{2}|^{2}}.$
(3.53)

It will be clear from the context when $|\cdot|$ and $\|\cdot\|$ are meant on
either ${\mathbb{R}}$ and ${\mathbb{R}}^{2}$ or $\mathbb{T}^{1}$ and
$\mathbb{T}^{2}$, respectively. In particular, we observe that
$|\bar{\theta}_{i}-\bar{u}_{i}|=|\theta_{i}-u_{i}|$ if and only if
$|\theta_{i}-u_{i}|\leq\pi$.

The invariant sets we obtain are built around a fixed
$\bar{\omega}\in\mathbb{T}^{2}$ and depend of four other parameters as
follows:

$E_{R,\Theta,\lambda,\eta}=\left\\{(\bar{\theta},r)\in\mathbb{T}^{2}\times{\mathbb{R}}_{+}^{2}:r_{1}\geq
R,\ r_{2}\geq R,\ \lambda-\eta\leq\dfrac{r_{1}}{r_{2}}\leq\lambda+\eta,\
\|\bar{\theta}-\bar{\omega}\|\leq\Theta\right\\}.$ (3.54)

where $R>0$, $0<\Theta<\pi$, $\lambda>0$ and $0<\eta<\lambda$. We will denote
by $f:E_{R,\Theta,\lambda,\eta}\to\mathbb{T}^{2}\times{\mathbb{R}}_{+}^{2}$
the map which has (3.52) as a lifting. We remark that _all different choices
of $n_{1},n_{2}\in\mathbb{Z}$ in (3.52) define good liftings of the map $f$:_
we will use the choice $n_{1}=n_{2}=0$ in the proof of the next result.

######  Theorem 3.3.

In the above setting, let us assume that there exists
$\omega\in{\mathbb{R}}^{2}$ such that $L(\omega)=0$ and suppose that the
Jacobian $JL(\omega)$ is a $\mathcal{D}^{+}$-matrix. Moreover, assume that

$\lim_{\begin{subarray}{c}r_{i}\to+\infty\\\
i=1,2\end{subarray}}G(\theta,r)=0\quad\text{and}\quad\lim_{\begin{subarray}{c}r_{i}\to+\infty\\\
i=1,2\end{subarray}}F(\theta,r)=0\qquad\text{uniformly w.r.t. }\theta.$ (3.55)

Then, there exist $R>0$, $\Theta\in\left]0,\pi\right[$, $\lambda>0$ and
$\eta\in\left]0,\lambda\right[$ such that:

$f(E_{R,\Theta,\lambda,\eta})\subset E_{R,\Theta,\lambda,\eta}.$ (3.56)

###### Proof.

We divide the proof into three parts.

Part 1. Choice of the constants $R,\Theta,\lambda$ and $\eta$. Let

$\lambda=\dfrac{\partial_{1}L_{1}(\omega)}{\partial_{2}L_{2}(\omega)}>0,$
(3.57)

let $\eta,\epsilon_{0}>0$ be as in Lemma 3.2 and let $R_{0}=1/\epsilon_{0}$.
Since $JL(\omega)$ is a $\mathcal{D}^{+}$-matrix we deduce that there exist
$\Theta_{0}\in\left]0,\pi\right[$ and $\gamma_{i}>0$, $i=1,2$, such that

$\partial_{i}L_{i}(\theta)\leq-\gamma_{i}<0,\quad\text{for }i=1,2\text{ and
}\forall\theta\in{{\mathbb{R}}}^{2}:\|\bar{\theta}-\bar{\omega}\|\leq\Theta_{0}.$
(3.58)

Moreover, according to assumption (3.55), let $R_{1}\geq R_{0}$ such that

$F_{i}(\theta,r)\geq-\dfrac{\gamma_{i}}{2},\quad\text{for }i=1,2\text{ and
}\forall\ \theta\in{{\mathbb{R}}}^{2},\ r_{1}\geq R_{1},\ r_{2}\geq R_{1}.$
(3.59)

By the continuity of $JL(\theta)$ in $\theta=\omega$, a simple computation
shows that there exists $\Theta_{1}\in\left]0,\Theta_{0}\right]$ such that

$\begin{aligned}
\dfrac{\partial_{1}L_{1}(\theta)}{\lambda+\eta}-\partial_{2}L_{2}(\theta)&\geq\dfrac{-\partial_{2}L_{2}(\omega)\eta}{2(\lambda+\eta)}\\\
\partial_{2}L_{2}(\theta)-\dfrac{\partial_{1}L_{1}(\theta)}{\lambda-\eta}&\geq\dfrac{-\partial_{2}L_{2}(\omega)\eta}{2(\lambda-\eta)}\end{aligned}\qquad\forall\theta:\|\bar{\theta}-\bar{\omega}\|\leq\Theta_{1}.$
(3.60)

Moreover, using again assumption (3.55), we deduce that there exists
$R_{2}\geq R_{1}$ such that

$\begin{aligned}
\left|F_{2}(\theta,r)-\dfrac{F_{1}(\theta,r)}{\lambda+\eta}\right|&<\dfrac{-\partial_{2}L_{2}(\omega)\eta}{2(\lambda+\eta)}\\\
\displaystyle\left|\dfrac{F_{1}(\theta,r)}{\lambda-\eta}-F_{2}(\theta,r)\right|&<\dfrac{-\partial_{2}L_{2}(\omega)\eta}{2(\lambda-\eta)}\end{aligned}\qquad\text{for
each }r_{1}\geq R_{2},\ r_{2}\geq R_{2}\text{ and }\theta\in{\mathbb{R}}^{2}.$
(3.61)

Now, let us write

$L_{i}(\theta)=\langle\nabla
L_{i}(\omega),\theta-\omega\rangle+\alpha_{i}(\theta)\|\theta-\omega\|,$
(3.62)

for $i=1,2$ and $\theta\in{{\mathbb{R}}}^{2}$, with

$\lim_{\theta\to\omega}\alpha(\theta)=0,\quad i=1,2.$

with $\alpha(\theta):=(\alpha_{1}(\theta),\alpha_{2}(\theta))$. Then, we
choose $\Theta\in\left]0,\Theta_{1}\right]$ such that

$\|(\alpha_{1}(\theta),\alpha_{2}(\theta))\|\leq\dfrac{a_{0}}{4},\qquad\text{if
}\|\theta-\omega\|\leq\Theta,$ (3.63)

where $a_{0}$ is given in Lemma 3.2.

Let us now define

$L^{*}=\max\\{\|L(\theta)\|:\|\theta-\omega\|\leq\Theta/2\\};$ (3.64)

according to assumption (3.55), let $R_{3}\geq R_{2}$ be such that

$\displaystyle\|G(\theta,r)\|<\min\left\\{L^{*},\dfrac{a_{0}\,\Theta}{8}\,\right\\}\quad\text{for
every }\theta\in{{\mathbb{R}}}^{2}\text{ and }r_{1},r_{2}\geq R_{3}.$ (3.65)

Finally, let us fix

$R\geq\max\left\\{R_{3},\dfrac{4L^{*}}{\Theta}\right\\}$ (3.66)

and consider the set $E_{R,\Theta,\lambda,\eta}$ corresponding to the chosen
constants. From now on, we will simply denote this set by $E$.

Part 2. Invariance of $E$ with respect to the radial components. Let us fix
$(\theta,r)$ such that $(\bar{\theta},r)\in E$ and consider
$\rho=(\rho_{1},\rho_{2})$ given by (3.52). From conditions (3.58) and (3.59)
we immediately deduce that

$\rho_{i}\geq r_{i}+\dfrac{\gamma_{i}}{2}>r_{i},\quad\text{for }i=1,2.$ (3.67)

On the other hand, we have $r_{1}\leq\left(\lambda+\eta\right)r_{2}$ and,
then, we infer that

$\displaystyle\dfrac{\rho_{1}}{\rho_{2}}$
$\displaystyle=\dfrac{r_{1}-\partial_{1}L_{1}(\theta)+F_{1}(\theta,r)}{r_{2}-\partial_{2}L_{2}(\theta)+F_{2}(\theta,r)}$
(3.68)
$\displaystyle\leq(\lambda+\eta)\dfrac{r_{2}-\dfrac{\partial_{1}L_{1}(\theta)}{\lambda+\eta}+\dfrac{F_{1}(\theta,r)}{\lambda+\eta}}{r_{2}-\partial_{2}L_{2}(\theta)+F_{2}(\theta,r)}$
$\displaystyle=(\lambda+\eta)\left(1-\dfrac{\dfrac{\partial_{1}L_{1}(\theta)}{\lambda+\eta}-\partial_{2}L_{2}(\theta)+F_{2}(\theta,R)-\dfrac{F_{1}(\theta,r)}{\lambda+\eta}}{r_{2}-\partial_{2}L_{2}(\theta)+F_{2}(\theta,r)}\right).$

Let us now observe that (3.67) implies that
$r_{2}-\partial_{2}L_{2}(\theta)+F_{2}(\theta,r)>0$ in $E$; moreover, from the
first relations in (3.60) and (3.61), we deduce that

$\dfrac{\partial_{1}L_{1}(\theta)}{\lambda+\eta}-\partial_{2}L_{2}(\theta)+F_{2}(\theta,R)-\dfrac{F_{1}(\theta,r)}{\lambda+\eta}>0.$
(3.69)

From (3.68) and (3.68) we thus conclude that

$\dfrac{\rho_{1}}{\rho_{2}}\leq\lambda+\eta.$ (3.70)

In an analogous way, taking into account the second relations in (3.60) and
(3.61), it is possible to prove that

$\dfrac{\rho_{1}}{\rho_{2}}\geq\lambda-\eta.$ (3.71)

From (3.67), (3.70) and (3.71) we deduce the invariance of the set $E$ with
respect to the radial components.
Part 3. Invariance with respect to the angular components. We have to show
that, if $(\bar{\theta},r)\in E$ then $\|\bar{u}-\bar{\omega}\|\leq\Theta$,
where $u$ is given in (3.52). By the definition of the metric on
$\mathbb{T}^{2}$ in (3.53) and the choice $\Theta<\pi$, it is enough to work
on the covering space and to prove that for a suitable lifting (3.52) we have
$\|u-\omega\|\leq\Theta$, with $\theta\in{\mathbb{R}}^{2}$ such that
$\|\theta-\omega\|\leq\Theta$, where these last two norms are Euclidean in the
covering space ${\mathbb{R}}^{2}$ of $\mathbb{T}^{2}$. As already announced
just before the statement of the theorem, the choice $n_{1}=n_{2}=0$ in (3.52)
will work here.

Let us split the set $E$ into the following two subsets

$E_{1}=\left\\{(\bar{\theta},r)\in E:\
\|\bar{\theta}-\bar{\omega}\|\leq\dfrac{\Theta}{2}\right\\},\quad
E_{2}=\left\\{(\bar{\theta},r)\in E:\
\dfrac{\Theta}{2}\leq\|\bar{\theta}-\bar{\omega}\|\leq\Theta\right\\}.$

If $(\bar{\theta},r)\in E_{1}$, then, using the first equation in (3.52), with
$n_{1}=n_{2}=0$, and also (3.64), (3.65) and (3.66), we deduce that

$\|u-\omega\|\leq\|\theta-\omega\|+\frac{1}{R}\|L(\theta)\|+\dfrac{1}{R}\|G(\theta,r)\|\leq\|\theta-\omega\|+\frac{1}{R}L^{*}+\frac{1}{R}L^{*}\leq\frac{\Theta}{2}+\frac{2L^{*}}{R}\leq\Theta.$
(3.72)

On the other hand, if $(\bar{\theta},r)\in E_{2}$, we use (3.62) and write:

$\displaystyle u-\omega$
$\displaystyle=B(\theta-\omega)+\left[\begin{array}[]{c}\dfrac{\alpha_{1}(\theta)}{r_{1}}+\dfrac{G_{1}(\theta,r)}{r_{1}\|\theta-\omega\|}\vspace{4pt}\\\
\dfrac{\alpha_{2}(\theta)}{r_{2}}+\dfrac{G_{2}(\theta,r)}{r_{2}\|\theta-\omega\|}\end{array}\right]\|\theta-\omega\|,$

where the matrix $B$ is given by

$B=\left(\begin{array}[]{cc}1+\partial_{1}L_{1}(\omega)/r_{1}&\partial_{2}L_{1}(\omega)/r_{1}\vspace{4pt}\\\
\partial_{1}L_{2}(\omega)/r_{2}&1+\partial_{2}L_{2}(\omega)/r_{2}\end{array}\right)$

and has the form (3.36) with $\epsilon=(1/r_{1},1/r_{2})$. Using (3.63) and
(3.65) we deduce that

$\|u-\omega\|\leq\|B\|_{2}\|\theta-\omega\|+\left(\|\alpha(\theta)\|\|\epsilon\|+\frac{2\|G(\theta,r)\|}{\Theta}\|\epsilon\|\right)\|\theta-\omega\|\leq\left(\|B\|_{2}+\frac{a_{0}}{2}\|\epsilon\|\right)\Theta$

Now, $(\bar{\theta},r)\in E$ implies that $\epsilon=(1/r_{1},1/r_{2})\in
C_{\epsilon_{0},\eta}$, see (3.37), and we can use Lemma 3.2 to obtain that
$\|B\|_{2}\leq(1-a_{0}\|\epsilon\|/2)$ and conclude that
$\|u-\omega\|\leq\Theta$. ∎

Now, let $(\theta_{0},r_{0})\in E_{R,\Theta,\lambda,\eta}$, with
$E_{R,\Theta,\lambda,\eta}$ given by Theorem 3.3; since
$E_{R,\Theta,\lambda,\eta}$ is positively invariant, we can recursively define

$(\theta_{n+1},r_{n+1})=f(\theta_{n},r_{n})\in
E_{R,\Theta,\lambda,\eta},\quad\forall\ n\geq 0.$

From (3.67) we know that

$(r_{1})_{i}\geq(r_{0})_{i}+\dfrac{\gamma_{i}}{2},\quad i=1,2,$

and iterating we infer that

$(r_{n})_{i}\geq(r_{0})_{i}+n\dfrac{\gamma_{i}}{2},\quad i=1,2,\quad n\geq 1.$

This relation is sufficient to prove the final result of this section.

######  Theorem 3.4.

In the same setting of Theorem 3.3, for every $(\theta_{0},r_{0})\in
E_{R,\Theta,\lambda,\eta}$ we have

$\lim_{n\to+\infty}(r_{n})_{i}=+\infty,\quad i=1,2,$

where $(\theta_{n+1},r_{n+1})=f(\theta_{n},r_{n})$, for every $n\geq 0$.

######  Remark 3.5.

We observe that, in the case of a one-to-one map $f$ as above, an analogous
result can be proved when $JL(\omega)$ is a $\mathcal{D}^{-}$-matrix; indeed,
in this situation there exist $R>0$, $0<\Theta<\pi$, $\lambda>0$ and
$0<\eta<\lambda$ such that:

$f^{-1}(E_{R,\Theta,\lambda,\eta})\subset E_{R,\Theta,\lambda,\eta},$

Then, for every $(\theta_{0},r_{0})\in E_{R,\Theta,\lambda,\eta}$ it is
possible to define

$(\theta_{n-1},r_{n-1})=f(\theta_{n},r_{n}),$

for every $n\leq 0$, and we have

$\lim_{n\to-\infty}(r_{n})_{i}=+\infty,\quad i=1,2.$

##  4 The main result and some corollaries

In this section we apply the theory developed in Section 3 in order to prove
our main result, dealing with the existence of unbounded solutions to the
system

$\left\\{\begin{array}[]{l}\ddot{x}_{1}+a_{1}x_{1}^{+}-b_{1}x^{-}_{1}+\phi_{1}(x_{2})=p_{1}(t)\vspace{7pt}\\\
\ddot{x}_{2}+a_{2}\,x_{2}^{+}-b_{2}\,x^{-}_{2}+\phi_{2}(x_{1})=p_{2}(t).\end{array}\right.$
(4.73)

We recall that, for $i=1,2$, we are assuming the resonance condition

$\dfrac{1}{\sqrt{a_{i}}}+\dfrac{1}{\sqrt{b_{i}}}=\dfrac{2}{n},\quad\mbox{ for
some }n\in\mathbb{N},$ (4.74)

Moreover, the function $p_{i}:\mathbb{R}\to\mathbb{R}$ is continuous and
$2\pi$-periodic and the function $\phi_{i}:\mathbb{R}\to\mathbb{R}$ is locally
Lipschitz continuous and bounded, with

$\phi_{i}(-\infty)=-\phi_{i}(+\infty).$ (4.75)

In this setting, and recalling the definition of the function $L$ given in
(2.30)-(2.33), the following result holds true.

######  Theorem 4.1.

Assume conditions (4.74) and (4.75); moreover, suppose that there exists
$\omega\in\mathbb{R}^{2}$ such that $L(\omega)=0$ and $JL(\omega)$ is a
$\mathcal{D}^{+}$-matrix. Then, there exists an infinite measure set
$E\subset{{\mathbb{R}}}^{2}\times{{\mathbb{R}}}^{2}$ such that

$\lim_{t\to+\infty}(|x_{i}(t)|^{2}+|x^{\prime}_{i}(t)|^{2})=+\infty,\qquad
i=1,2,$ (4.76)

for every solution $x$ of (4.73) such that $(x(0),x^{\prime}(0))\in E$.

######  Remark 4.2.

According to Remark 3.5, an analogous result for $t\to-\infty$ can be proved
when $JL(\omega)$ is a $\mathcal{D}^{-}$-matrix.

###### Proof.

The result follows from an application of Theorem 3.4, taking into account the
fact that, from (2.31), the Poincaré map associated with (4.73) is of the form
(3.52), with (2.32) implying (3.55).

More precisely, let $E\subset{{\mathbb{R}}}^{2}\times{{\mathbb{R}}}^{2}$ be
the set corresponding, via action-angle coordinates, to the set
$E_{R,\Theta,\omega,\lambda,\eta}$ given in the statement of Theorem 3.4 and
let $x$ be a solution of (4.73) such that $(x(0),x^{\prime}(0))\in E$. Then,
from Theorem 3.4 we infer that

$\lim_{k\to+\infty}(|x_{i}(2k\pi)|^{2}+|x^{\prime}_{i}(2k\pi)|^{2})=+\infty.$

The thesis (4.76) follows from this relation and an application of Gronwall’s
lemma (see e.g. [2, Proof of Th. 41]), taking into account the boundedness of
$\phi_{i}$, for $i=1,2$. ∎

In the rest of the section, we discuss some concrete situations in which the
abstract condition on the zeros of the function $L$ is verified, thus
providing more explicit corollaries of Theorem 4.1, depending on the structure
of the set of zeroes of the functions $\Phi_{i}$, $i=1,2$, defined in (2.30).

The first situation we deal with is the one in which both $\Phi_{1}$ and
$\Phi_{2}$ have a simple zero (in the scalar setting, this situation was the
one treated by [2, Th. 4.1]). More precisely, we assume that there exists
$\omega^{*}=(\omega^{*}_{1},\omega^{*}_{2})\in\mathbb{R}^{2}$ such that

$\Phi_{1}(\omega^{*}_{1})=\Phi_{2}(\omega^{*}_{2})=0,\qquad\Phi^{\prime}_{i}(\omega^{*}_{i})<0,\quad
i=1,2.$ (4.77)

Under this assumption, the following result holds true.

######  Corollary 4.3.

Assume conditions (4.74), (4.75) and (4.77). Then, there exists
$\phi^{*}=\phi^{*}(a_{1},a_{2},p_{1},p_{2})>0$ such that, for every functions
$\phi_{i}$ with $|\phi_{i}(+\infty)|<\phi^{*}$ ($i=1,2$), there exists an
infinite measure set $E\subset{{\mathbb{R}}}^{2}\times{{\mathbb{R}}}^{2}$ such
that

$\lim_{t\to+\infty}(|x_{i}(t)|^{2}+|x^{\prime}_{i}(t)|^{2})=+\infty,\qquad
i=1,2,$

for every solution $x$ of (4.73) such that $(x(0),x^{\prime}(0))\in E$.

###### Proof.

Let us observe that, in view of Theorem 4.1 it is sufficient to prove that,
under the given assumptions, there exist $\omega\in{{\mathbb{R}}}^{2}$ such
that $L(\omega)=0$ and $JL(\omega)$ is a $\mathcal{D}^{+}$-matrix.

Let us first recall, from (2.30), that we have

$L_{i}(\theta)=\Phi_{i}(\theta_{i})+\gamma_{i}n\phi_{i}(+\infty)(\Lambda_{i}(\theta_{i}-\theta_{i+1})-\alpha_{i}),\quad\forall\
\theta\in{{\mathbb{R}}}^{2},$ (4.78)

where $\Lambda_{i}$ is defined in (2.13). Let us define
$H:{{\mathbb{R}}}^{2}\times{{\mathbb{R}}}^{2}\to{{\mathbb{R}}}^{2}$ by

$H(\theta,v)=(\Phi_{1}(\theta_{1})+\gamma_{1}nv_{1}(\Lambda_{1}(\theta_{1}-\theta_{2})-\alpha_{1}),\Phi_{2}(\theta_{2})+\gamma_{2}nv_{2}(\Lambda_{2}(\theta_{2}-\theta_{1})-\alpha_{2})),\quad\forall\
\theta\in{{\mathbb{R}}}^{2},\ v\in{{\mathbb{R}}}^{2}.$ (4.79)

From (4.77) we immediately infer that

$H(\omega^{*}_{1},\omega^{*}_{2},0,0)=0$

and

$J_{\theta}H(\omega^{*}_{1},\omega^{*}_{2},0,0)=\left(\begin{array}[]{cc}\Phi^{\prime}_{1}(\omega^{*}_{1})&0\\\
&\\\ 0&\Phi^{\prime}_{2}(\omega^{*}_{2})\end{array}\right)\neq 0.$

Hence, by the implicit function theorem, we deduce that there exists
$\hat{\phi}>0$ such that for every
$(\phi_{1}(+\infty),\phi_{2}(+\infty))\in{{\mathbb{R}}}^{2}$ with
$|\phi_{i}(+\infty)|<\hat{\phi}$, $i=1,2$, there exists
$\omega=\omega(\phi_{1}(+\infty),\phi_{2}(+\infty))\in\mathbb{R}^{2}$ near
$\omega^{*}$ such that

$L(\omega)=0.$

Now, let us observe that

$JL(\omega)=\left(\begin{array}[]{cc}\Phi^{\prime}_{1}(\omega_{1})+\gamma_{1}\phi_{1}(+\infty)\Sigma_{1}(\omega_{1}-\omega_{2})&-\gamma_{1}\phi_{1}(+\infty)\Sigma_{1}(\omega_{1}-\omega_{2})\\\
&\\\
-\gamma_{2}\phi_{2}(+\infty)\Sigma_{2}(\omega_{2}-\omega_{1})&\Phi^{\prime}_{2}(\omega_{2})+\gamma_{2}\phi_{2}(+\infty)\Sigma_{2}(\omega_{2}-\omega_{1})\end{array}\right),$

where $\Sigma_{i}$ is given in (2.17). The continuity of $\omega$ as function
of $(\phi_{1}(+\infty),\phi_{2}(+\infty))$, ensured by the implicit function
theorem, implies that

$\lim_{|(\phi_{1}(+\infty),\phi_{2}(+\infty))|\to
0^{+}}JL(\omega)=\left(\begin{array}[]{cc}\Phi^{\prime}_{1}(\omega^{*}_{1})&0\\\
&\\\ 0&\Phi^{\prime}_{2}(\omega^{*}_{2})\end{array}\right);$

by (4.77) the limit matrix is a $\mathcal{D}^{+}$-matrix. As a consequence,
there exists $\phi^{*}\in(0,\hat{\phi})$ such that for every
$(\phi_{1}(+\infty),\phi_{2}(+\infty))\in{{\mathbb{R}}}^{2}$ with
$|\phi_{i}(+\infty)|<\phi^{*}$ the matrix $JL(\omega)$ is a
$\mathcal{D}^{+}$-matrix, as well. The result is then proved. ∎

######  Remark 4.4.

A dual result, ensuring the existence of solutions unbounded in the past,
could be proved when (4.77) is replaced by

$\Phi_{1}(\omega^{*}_{1})=\Phi_{2}(\omega^{*}_{2})=0,\qquad\Phi^{\prime}_{i}(\omega^{*}_{i})>0,\quad
i=1,2.$

We omit the details for briefness.

######  Remark 4.5.

Let us analyze the result of Corollary 4.3 in the symmetric linear case
$a_{i}=b_{i}=n^{2}$, $i=1,2$. In this situation, in the recent paper [3] the
existence of unbounded solutions has been proved under the assumption

$4|\phi_{i}(+\infty)|<|{\widehat{p}}_{i,n}|^{2},\quad i=1,2,$ (4.80)

where

${\widehat{p}}_{i,n}=\int_{0}^{2\pi}p_{i}(t)e^{int}\,dt$ (4.81)

(see Theorem 3.1 in [3]). The assumption $|\phi_{i}(+\infty)|<\phi^{*}$
($i=1,2$), with $\phi^{*}=\phi^{*}(a_{1},b_{1},p_{1},p_{2})$, in Corollary 4.3
is then on the same spirit of (4.80).

Let us now focus on the situation where the function $\Phi_{1}$ (or
$\Phi_{2}$) is identically zero, i.e.

$\Phi_{1}(\theta_{1})=0,\quad\forall\ \theta_{1}\in{\mathbb{R}}.$ (4.82)

Incidentally, let us observe that in the linear symmetric case
$a_{1}=b_{1}=n^{2}$ assumption (4.82) corresponds to the case when the number
${\widehat{p}}_{1,n}$ in (4.81) is zero. Instead, in the asymmetric case
$a_{1}\neq b_{1}$, condition (4.82) is more tricky to be checked. However,
some examples in which it holds can be provided. For instance, if $a_{1}$
satisfies

$\frac{\sqrt{a_{1}}}{n}=\frac{s}{1+2k}\qquad\text{for some
}s,k\in\mathbb{N}\text{ and }s>k,$ (4.83)

then the Fourier coefficient $c_{s,1}$ of $C_{1}$ vanishes (see (2.10)), and
(4.82) holds when $p_{1}(t)=\cos snt$.

For the sake of brevity and clarity, we present here just a couple of
corollaries in which (4.82) is assumed. In the first we suppose that $a_{2}$
is such that

$c_{r,2}\neq 0\quad\text{for some }r\in\mathbb{N},$ (4.84)

and that

$p_{2}(t)=\mu\cos rnt,\quad\forall\ t\in{\mathbb{R}},$ (4.85)

with $\mu>0$.

######  Corollary 4.6.

Let $a_{i},b_{i}>0$ satisfy, for $i=1,2$, assumption (4.74); moreover, suppose
that

$(a_{1},a_{2})\in\mathcal{R},$ (4.86)

where $\mathcal{R}$ is defined in (2.19), and that (4.84) is fulfilled.
Finally, assume that conditions (4.75), (4.82) and (4.85) are satisfied. Then,
for every $\phi_{1}(+\infty)\neq 0$ and for every
$\phi_{2}(+\infty)\in{\mathbb{R}}$ there exists $\mu^{*}>0$ such that for
every $\mu>\mu^{*}$ there exist two infinite measure sets
$E^{\pm}\subset{{\mathbb{R}}}^{2}\times{{\mathbb{R}}}^{2}$ such that:

  * •

for every solution $x$ of (4.73) such that $(x(0),x^{\prime}(0))\in E^{+}$,

$\lim_{t\to+\infty}(|x_{i}(t)|^{2}+|x^{\prime}_{i}(t)|^{2})=+\infty,\qquad
i=1,2,$

  * •

for every solution $x$ of (4.73) such that $(x(0),x^{\prime}(0))\in E^{-}$

$\lim_{t\to-\infty}(|x_{i}(t)|^{2}+|x^{\prime}_{i}(t)|^{2})=+\infty,\qquad
i=1,2.$

We observe that it is possible to find situations in which Corollary 4.6
applies. Indeed, let us first notice that Lemma 2.2 implies that (4.86) holds
if $(a_{1},a_{2})$ is close to $(n^{2},n^{2})$. This happens, for instance if
$a_{1}$ satisfies (4.83) with $s=2k$ and $k$ large enough, and if
$\sqrt{a_{2}}$ is irrational and close to $n$. With these choices (4.82) holds
with $p_{1}(t)=\cos(2knt)$, while (4.84) is trivially satisfied (see (2.10)).

###### Proof.

Let us first notice that, from (2.30) and (4.85), recalling the Fourier
expansion of $C_{2}$ given in (2.9), we obtain

$\Phi_{2}(\theta_{2})=-\dfrac{\gamma_{2}}{2}\pi\mu c_{r,2}\cos
r\theta_{2},\quad\forall\ \theta_{2}\in{\mathbb{R}}.$ (4.87)

As a consequence, recalling (4.82), we obtain

$\begin{array}[]{l}\displaystyle
L_{1}(\theta)=\gamma_{1}n\phi_{1}(+\infty)(\Lambda_{1}(\theta_{1}-\theta_{2})-\alpha_{1})\\\
\\\ \displaystyle L_{2}(\theta)=-\dfrac{\gamma_{2}}{2}\pi\mu c_{r,2}\cos
r\theta_{2}+\gamma_{2}n\phi_{2}(+\infty)(\Lambda_{2}(\theta_{2}-\theta_{1})-\alpha_{2}),\end{array}$
(4.88)

for every $\theta\in{{\mathbb{R}}}^{2}$.

Now, let us look for solutions of $L(\theta)=0$; from the relation
$L_{1}(\theta)=0$, recalling that $\phi_{1}(+\infty)\neq 0$, we deduce

$\Lambda_{1}(\theta_{1}-\theta_{2})=\alpha_{1}.$ (4.89)

From Lemma 2.1, taking into account (4.86), we infer that there exists
$\Lambda_{1}^{*}\in(0,\pi)$ such that

$\begin{array}[]{l}\Lambda_{1}(t)=\alpha_{1}\quad\Longleftrightarrow\quad
t=\pm\Lambda_{1}^{*}+2m\pi,\ m\in\mathbb{Z}\\\ \\\
\textnormal{sgn}(\Lambda^{\prime}_{1}(\pm\Lambda_{1}^{*}))=\mp 1.\end{array}$
(4.90)

In particular, we choose $m=0$; then, from (4.89) and (4.90) we obtain

$\theta_{1}-\theta_{2}=\pm\Lambda_{1}^{*}.$ (4.91)

Replacing the last equality in the expression of $L_{2}$ in (4.88) and
recalling that $\Lambda_{2}$ is even and $2\pi$-periodic, the equation
$L_{2}(\theta)=0$ reduces to

$-\pi\mu c_{r,2}\cos
r\theta_{2}+2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})=0,$
(4.92)

i.e.

$\cos
r\theta_{2}=\dfrac{2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})}{\pi\mu
c_{r,2}}.$ (4.93)

Let now set

$\hat{\mu}=\left|\dfrac{2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})}{\pi
c_{r,2}}\right|;$ (4.94)

then, for every $\mu>\hat{\mu}$ the equation (4.93) can be solved and we
obtain

$\theta_{2}=\dfrac{1}{r}\left(\pm\arccos\dfrac{2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})}{\pi\mu
c_{r,2}}+2h\pi\right),\quad h\in\mathbb{Z}.$ (4.95)

Choosing $h=0$, we then conclude that, for every $\mu>\hat{\mu}$, the equation
$L(\theta)=0$ has the four solutions

$\omega^{\pm,1}_{\mu}=\left(\Lambda_{1}^{*}+\dfrac{1}{r}\left(\pm\arccos\dfrac{2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})}{\pi\mu
c_{r,2}}\right),\dfrac{1}{r}\left(\pm\arccos\dfrac{2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})}{\pi\mu
c_{r,2}}\right)\right)$

and

$\omega^{\pm,2}_{\mu}=\left(-\Lambda_{1}^{*}+\dfrac{1}{r}\left(\pm\arccos\dfrac{2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})}{\pi\mu
c_{r,2}}\right),\dfrac{1}{r}\left(\pm\arccos\dfrac{2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})}{\pi\mu
c_{r,2}}\right)\right).$

In order to apply Theorem 4.1, we claim that one of the above four solutions,
to be named $\omega^{+}$, is such that $JL(\omega^{+})$ is a
$\mathcal{D}^{+}$-matrix and another one, to be named $\omega^{-}$, is such
that $JL(\omega^{-})$ is a $\mathcal{D}^{-}$-matrix. To do this, recalling
(4.88) and the fact that $\Lambda^{\prime}_{i}=\Sigma_{i}/n$ is
$2\pi$-periodic and odd, we observe that

$\begin{array}[]{ll}\displaystyle\partial_{1}L_{1}(\omega^{\pm,i}_{\mu})=&\displaystyle(-1)^{i+1}\gamma_{1}\phi_{1}(+\infty)\Sigma_{1}(\Lambda_{1}^{*})\\\
&\\\
\displaystyle\partial_{2}L_{1}(\omega^{\pm,i}_{\mu})=&\displaystyle(-1)^{i}\gamma_{1}\phi_{1}(+\infty)\Sigma_{1}(\Lambda_{1}^{*})\\\
&\\\
\displaystyle\partial_{1}L_{2}(\omega^{\pm,i}_{\mu})=&\displaystyle(-1)^{i}\gamma_{2}\phi_{2}(+\infty)\Sigma_{2}(\Lambda_{1}^{*})\\\
&\\\
\displaystyle\partial_{2}L_{2}(\omega^{\pm,i}_{\mu})=&\displaystyle\pm\dfrac{\gamma_{2}}{2}\pi\mu
c_{r,2}r\sin\arccos\dfrac{2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})}{\pi\mu
c_{r,2}}+\displaystyle(-1)^{i+1}\gamma_{2}\phi_{2}(+\infty)\Sigma_{2}(\Lambda_{1}^{*}),\end{array}$
(4.96)

for $i=1,2$. Now, since $\phi_{1}(+\infty)\neq 0$ and recalling (4.90), we
have

$\operatorname{sgn}(\partial_{1}L_{1}(\omega^{\pm,i}_{\mu}))=(-1)^{i}\operatorname{sgn}(\phi_{1}(+\infty));$
(4.97)

moreover, there exists $\check{\mu}\geq\hat{\mu}$ such that for every
$\mu>\check{\mu}$ we have

$\operatorname{sgn}(\partial_{2}L_{2}(\omega^{\pm,i}_{\mu}))=\pm\operatorname{sgn}(c_{r,2}).$
(4.98)

Hence, for $\mu>\check{\mu}$, the choice of $\omega^{\pm,i}_{\mu}$ has to be
made according to the signs of $\phi_{1}(+\infty)$ and $c_{r,2}$.

For the sake of briefness, we discuss the case $\phi_{1}(+\infty)>0$ and
$c_{r,2}>0$, the other ones being similar. We set
$\omega^{+}=\omega^{-,1}_{\mu}$ and $\omega^{-}=\omega^{+,2}_{\mu}$; hence, by
construction, $JL(\omega^{+})$ and $JL(\omega^{-})$ satisfy the sign
conditions on the diagonal coefficients in order to be a
$\mathcal{D}^{\pm}$-matrix. As far as the third condition in Definition 3.1 is
concerned, we have that

$\begin{array}[]{l}\displaystyle\partial_{1}L_{1}(\omega^{\pm})\,\partial_{1}L_{2}(\omega^{\pm})+\partial_{2}L_{1}(\omega^{\pm})\,\partial_{2}L_{2}(\omega^{\pm})=\displaystyle-2\gamma_{1}\gamma_{2}\phi_{1}(+\infty)\phi_{2}(+\infty)\Sigma_{1}(\Lambda_{1}^{*})\Sigma_{2}(\Lambda_{1}^{*})\\\
\\\ \qquad\displaystyle+\gamma_{1}\dfrac{\gamma_{2}}{2}\pi\mu
c_{r,2}rn\sin\arccos\dfrac{2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})}{\pi\mu
c_{r,2}}\phi_{1}(+\infty)\Sigma_{1}(\Lambda_{1}^{*})\end{array}$ (4.99)

and

$\begin{array}[]{l}\displaystyle
2\partial_{1}L_{1}(\omega^{\pm})\,\partial_{2}L_{2}(\omega^{\pm})=\displaystyle
2\gamma_{1}\gamma_{2}\phi_{1}(+\infty)\phi_{2}(+\infty)\Sigma_{1}(\Lambda_{1}^{*})\Sigma_{2}(\Lambda_{1}^{*})+\\\
\\\ \qquad\displaystyle-\gamma_{1}\gamma_{2}\pi\mu
c_{r,2}rn\sin\arccos\dfrac{2n\phi_{2}(+\infty)(\Lambda_{2}(\Lambda_{1}^{*})-\alpha_{2})}{\pi\mu
c_{r,2}}\phi_{1}(+\infty)\Sigma_{1}(\Lambda_{1}^{*}).\end{array}$ (4.100)

Hence, there exists $\mu^{*}\geq\check{\mu}$ such that for every $\mu>\mu^{*}$
the third condition in Definition 3.1 is satisfied; hence the values
$\omega^{\pm}$ are such that $JL(\omega^{\pm})$ is a $\mathcal{D}^{\pm}$
matrix. The thesis then follows from an application of Theorem 4.1. ∎

As a last application, we discuss the case when the oscillators are symmetric,
i.e. $a_{i}=b_{i}=n^{2}$ for $i=1,2$, and (4.82) holds true; as already
observed, this is equivalent to the assumption

$\widehat{p}_{1,n}=0,$ (4.101)

where $\widehat{p}_{1,n}$ is as in (4.81). Let us observe that this situation
is not covered by the results in [3].

######  Corollary 4.7.

Let $a_{i}=b_{i}=n^{2}$, for $i=1,2$, and suppose that conditions (4.75) and
(4.101) are satisfied.

Then, for every $\phi_{1}(+\infty)\neq 0$ and for every
$\phi_{2}(+\infty)\in{\mathbb{R}}$ such that

$|\phi_{2}(+\infty)|<\dfrac{3}{16}|\widehat{p}_{2,n}|,$ (4.102)

with ${\widehat{p}}_{2,n}$ as in (4.81), there exist two infinite measure sets
$E^{\pm}\subset{{\mathbb{R}}}^{2}\times{{\mathbb{R}}}^{2}$ such that:

  * •

for every solution $x$ of (4.73) such that $(x(0),x^{\prime}(0))\in E^{+}$,

$\lim_{t\to+\infty}(|x_{i}(t)|^{2}+|x^{\prime}_{i}(t)|^{2})=+\infty,\qquad
i=1,2,$

  * •

for every solution $x$ of (4.73) such that $(x(0),x^{\prime}(0))\in E^{-}$

$\lim_{t\to-\infty}(|x_{i}(t)|^{2}+|x^{\prime}_{i}(t)|^{2})=+\infty,\qquad
i=1,2.$

###### Proof.

First of all, let us observe that in this situation the functions $C_{i}$ and
$\Lambda_{i}$ in (2.8) and (2.13) are given by

$C_{i}(t)=\cos nt,\quad\Lambda_{i}(t)=\dfrac{2}{n}\cos t,\quad\forall\
t\in{\mathbb{R}},$ (4.103)

respectively, while the number $\alpha_{i}$ in (2.15) is zero. Moreover, the
function $\Phi_{2}$ in (2.30) becomes

$\Phi_{2}(u)=-\dfrac{1}{\sqrt{2n}}\int_{0}^{2\pi}\cos(nt+u)p_{2}(t)dt,\quad\forall\
u\in{\mathbb{R}};$ (4.104)

this expression can be written as

$\Phi_{2}(u)=-\dfrac{1}{\sqrt{2n}}\,|\widehat{p}_{2,n}|\cos(u+\psi_{2}),\quad\forall\
u\in{\mathbb{R}},$ (4.105)

for some $\psi_{2}\in{\mathbb{R}}$.

From (4.103), (4.104), (4.105) and the assumption on $\widehat{p}_{1,n}$ we
deduce that

$\begin{array}[]{ll}\displaystyle
L_{1}(\theta)=2\sqrt{\frac{2}{n}}\phi_{1}(+\infty)\cos(\theta_{1}-\theta_{2})\\\
\\\ \displaystyle
L_{2}(\theta)=-\dfrac{1}{\sqrt{2n}}\,|\widehat{p}_{2,n}|\cos(\theta_{2}+\psi_{2})+2\sqrt{\frac{2}{n}}\phi_{2}(+\infty)\cos(\theta_{2}-\theta_{1}),\end{array}$
(4.106)

for every $\theta\in{{\mathbb{R}}}^{2}$.

Recalling that $\phi_{1}(+\infty)\neq 0$, we can solve the equation
$L_{1}(\theta)=0$ to obtain

$\theta_{1}=\theta_{2}\pm\dfrac{\pi}{2}+2m\pi,\quad m\in\mathbb{Z};$ (4.107)

as a consequence the equation $L_{2}(\theta)=0$ reduces to

$|\widehat{p}_{2,n}|\cos(\theta_{2}+\psi_{2})=0.$ (4.108)

We now observe that assumption (4.102) implies that $\widehat{p}_{2,n}\neq 0$;
hence, from (4.108) we infer that

$\theta_{2}=-\psi_{2}\pm\dfrac{\pi}{2}+2h\pi,\quad h\in\mathbb{Z}.$ (4.109)

Choosing in particular $m=h=0$, we conclude that the equation $L(\theta)=0$
has the four solutions

$\begin{array}[]{l}\displaystyle\omega^{\pm,1}=\left(-\psi_{2}+\dfrac{\pi}{2}\pm\dfrac{\pi}{2},-\psi_{2}\pm\dfrac{\pi}{2}\right)\in\mathbb{R}^{2},\\\
\\\
\displaystyle\omega^{\pm,2}=\left(-\psi_{2}-\dfrac{\pi}{2}\pm\dfrac{\pi}{2},-\psi_{2}\pm\dfrac{\pi}{2}\right)\in\mathbb{R}^{2}.\end{array}$
(4.110)

We now claim that one of the above four solutions, to be named $\omega^{+}$,
is such that $JL(\omega^{+})$ is a $\mathcal{D}^{+}$-matrix and another one,
to be named $\omega^{-}$, is such that $JL(\omega^{-})$ is a
$\mathcal{D}^{-}$-matrix.

To see this, let us observe that, from (4.110),

$\begin{array}[]{ll}\displaystyle\partial_{1}L_{1}(\omega^{\pm,i})&=\displaystyle\partial_{1}L_{1}(\theta)_{|\theta=\omega^{\pm,i}}=\displaystyle-2\sqrt{\frac{2}{n}}\phi_{1}(+\infty)\sin(\theta_{1}-\theta_{2})_{|\theta=\omega^{\pm,i}}=\displaystyle(-1)^{i}2\sqrt{\frac{2}{n}}\phi_{1}(+\infty)\\\
\\\
\displaystyle\partial_{2}L_{1}(\omega^{\pm,i})&=\displaystyle\partial_{2}L_{1}(\theta)_{|\theta=\omega^{\pm,i}}=2\sqrt{\frac{2}{n}}\phi_{1}(+\infty)\sin(\theta_{1}-\theta_{2})_{|\theta=\omega^{\pm,i}}=(-1)^{i-1}2\sqrt{\frac{2}{n}}\phi_{1}(+\infty)\\\
\\\
\displaystyle\partial_{1}L_{2}(\omega^{\pm,i})&=\displaystyle\partial_{1}L_{2}(\theta)_{|\theta=\omega^{\pm,i}}=2\sqrt{\frac{2}{n}}\phi_{2}(+\infty)\sin(\theta_{2}-\theta_{1})_{|\theta=\omega^{\pm,i}}=(-1)^{i}2\sqrt{\frac{2}{n}}\phi_{2}(+\infty)\\\
\\\
\displaystyle\partial_{2}L_{2}(\omega^{\pm,i})&=\displaystyle\partial_{2}L_{2}(\theta)_{|\theta=\omega^{\pm,i}}=\dfrac{1}{\sqrt{2n}}|\widehat{p}_{2,n}|\sin(\theta_{2}+\psi_{2})-2\sqrt{\frac{2}{n}}\phi_{2}(+\infty)\sin(\theta_{2}-\theta_{1})_{|\theta=\omega^{\pm,i}}\\\
&=\displaystyle\pm\dfrac{1}{\sqrt{2n}}|\widehat{p}_{2,n}|+(-1)^{i-1}2\sqrt{\frac{2}{n}}\phi_{2}(+\infty),\end{array}$
(4.111)

for $i=1,2$. Focusing for the sake of briefness on the case
$\phi_{1}(+\infty)>0$, we obtain from (4.102) that

$\begin{array}[]{l}\operatorname{sgn}(\partial_{1}L_{1}(\omega^{-,1}))\cdot\operatorname{sgn}(\partial_{2}L_{2}(\omega^{-,1}))>0\\\
\\\
\operatorname{sgn}(\partial_{1}L_{1}(\omega^{+,2}))\cdot\operatorname{sgn}(\partial_{2}L_{2}(\omega^{+,2}))>0.\end{array}$
(4.112)

Setting $\omega^{+}=\omega^{+,2}$ and $\omega^{-}=\omega^{-,1}$, since

$\begin{array}[]{l}\displaystyle\partial_{1}L_{1}(\omega^{\pm})\,\partial_{1}L_{2}(\omega^{\pm})+\partial_{2}L_{1}(\omega^{\pm})\,\partial_{2}L_{2}(\omega^{\pm})=\displaystyle\frac{1}{n}\left(16\phi_{1}(+\infty)\phi_{2}(+\infty)-2|\widehat{p}_{2,n}|\phi_{1}(+\infty)\right)\\\
\\\ \displaystyle\end{array}$ (4.113)

and

$\begin{array}[]{l}\displaystyle
2\partial_{1}L_{1}(\omega^{\pm})\,\partial_{2}L_{2}(\omega^{\pm})=\frac{1}{n}\left(-16\phi_{1}(+\infty)\phi_{2}(+\infty)+4|\widehat{p}_{2,n}|\phi_{1}(+\infty)\right),\end{array}$
(4.114)

from the same assumption (4.102) we deduce that

$\begin{array}[]{l}|\partial_{1}L_{1}(\omega^{\pm})\,\partial_{1}L_{2}(\omega^{\pm})+\partial_{2}L_{1}(\omega^{\pm})\,\partial_{2}L_{2}(\omega^{\pm})|<2\partial_{1}L_{1}(\omega^{\pm})\,\partial_{2}L_{2}(\omega^{\pm})\end{array}$
(4.115)

as well. From (4.112) and (4.115) we conclude that $JL(\omega^{\pm})$ is a
$\mathcal{D}^{\pm}$-matrix. The thesis then follows from an application of
Theorem 4.1. ∎

Acknowledgments. The authors are grateful to Rafael Ortega for having proposed
the subject of this investigation and for his enduring encouragement.

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