Unbounded solutions to a system of coupled asymmetric oscillators at resonance

We deal with the following system of coupled asymmetric oscillators \[ \begin{cases} \ddot{x}_1+a_1x_1^+-b_1x^-_1+\phi_1(x_2)=p_1(t) \\ \ddot{x}_2+a_2\,x_2^+-b_2\,x^-_2+\phi_2(x_1)=p_2(t) \end{cases} \] 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) Poincar\'e map, in action-angle coordinates.


Introduction
In this paper, we investigate the existence of unbounded solutions for a system of coupled asymmetric oscillators of the type ẍ1 + a where, as usual, x ± = max{±x, 0} and, for i = 1, 2, φ i : R → R is locally Lipschitz continuous and bounded, p i : R → R is continuous and 2π-periodic.As for the positive real numbers a i , b i , we assume that 1 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 ẍ + ax + − bx − = p(t), x ∈ R, with 1/ √ a+1/ √ b = 2/n, all large solutions are unbounded (either in the past or in the future) whenever the 2π-periodic function has zeros, all simple (in the above formula, C stands for the asymmetric cosine function, cf.Section 2.1).The function Φ, sometimes referred to as resonance function, was previously introduced by Dancer [7] to investigate the 2π-periodic solvability of equation (1.3).In the linear case (a = b = n 2 ), the function Φ has (simple) zeros if and only if 2π 0 p(t)e −int dt = 0: in this case, as well known, all the solutions of ẍ + n 2 x = p(t) are unbounded; instead, 2π-periodic and unbounded solutions to (1.3) can coexist in the genuinely asymmetric case a = 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 Φ 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 ẍ + ax + − bx − + φ(x) = p(t), x ∈ R, with φ : R → R a bounded function (see [6,9]) and, more in general, for planar system of the type where J is the standard symplectic matrix, H : R 2 → R is positive and positively homoegeneous of degree 2 and R : R 2 → 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 2 i 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.(2.23).In doing this, we are led to define a resonance function defined on the two-dimensional torus, which can be thought as the higher-dimensional generalization of the resonance function Φ defined in (1.4), see (2.22)-(2.25).We notice that when system (1.1) is uncoupled (that is, ) and, up to a constant, L i = Φ 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 D ± -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 D ± -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 ω ∈ T 2 such that the Jacobian matrix JL(ω) is a D ± -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 D ± -matrix.We mention that the condition for JL to be a D ± -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 ∞ -norms of the coupling terms φ 1 , φ 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/ √ a i +1/ √ b i = 2/n i with n i ∈ N. Also, the possibility of coupling more oscillators in a cyclic way φ i+1 = φ 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 • 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.

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 : R → R is continuous and 2π-periodic and the function φ i : R → R is locally Lipschitz continuous and bounded.Furthermore, we also suppose that there exist lim x→±∞ φ i (x) := φ i (±∞); moreover, without loss of generality, (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 ẍi with a i and b i as in (1.2).We recall that, for every i = 1, 2, the function C i is even, τ := 2π/n-periodic and its explicit expression in [−τ /2, τ /2] is (2.2) For future reference, let us observe that C i , i = 1, 2, when a i = b i admits the Fourier series expansion where and, for h ≥ 1, In the next sections we will use the integrals of C i over the sets J ± i+1 defined by where θ i+1 ∈ 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 2π/n-periodic implies that does not change if we replace [0, 2π] in the definition of J ± i+1 by any interval of lenght 2π.In particular, in the computation of the integral of C i on J + i+1 , we can replace J + i+1 by the set thus obtaining that where Λ i : R → R is defined by 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 where α i is given by In particular, we will be interested in the situation where (2.7) has simple solutions; in order to face this problem, let us first concentrate on the range of the function Λ i .Introducing the function Σ i : R → R defined by is it possible to prove the following result.
Proof.Let us first observe that we have where The function C n i is continuous, 2π-periodic, even and strictly decreasing in [0, π); as a consequence, Σ i is continuous, 2π-periodic and odd.As far as the sign of Σ i is concerned, let us observe that Σ i (t) = 0 if and only if t = kπ for some k ∈ Z. Indeed, Σ i (t) = 0 if and only if: Now, since a i+1 > n 2 /4, the first alternative cannot hold, and the second one implies that t = kπ.Therefore Σ i has constant sign in (0, π) and a straightforward argument shows that From the above described properties of Σ 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 (2.7), with i = 1, admits simple solutions if and only if 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 The complete description of the open set R is quite difficult; by means of long computations it is possible to show that the vertical sections R∩{(a * 1 , a 2 ) : , are bounded.On the other hand, the study of the horizontal sections R∩{(a 1 , a * 2 ) : , is much more complicated.However, the following simple result holds true.
On the other hand, if α 2 = n 2 , we have Recalling (2.9), we deduce that Λ

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 , ẋ1 , ẋ2 ) and use the change of variables where the functions C i and S i are defined in Subsection 2.1.
This relation implies that and, thus: since C i and S i are smooth enough.By replacing (2.16) in the last two equations of (2.13) we get As a consequence, we infer that where lim ri,0→+∞ F i,i (θ 0 , r 0 ) = 0 uniformly w.r.t.θ 1 , θ 2 , and r i+1,0 .
For i = 1, 2, let us now denote for every where lim ri,0→+∞ G i,i (θ 0 , r 0 ) = 0 uniformly w.r.t.r i+1,0 and θ 0 , G i,i+1 (θ 0 , I 0 ) = 0 uniformly w.r.t.r i,0 and θ 0 . (2.24) The functions L 1 , L 2 will be meant as the components of the vector valued function (2.25) which we will call resonance function for system (1.1).Notice that, due to the 2π-periodicity in both the variables, we can interpret L as a function defined on the two-dimensional torus T 2 = R 2 /(2πZ) 2 .This function will play a crucial role in the statement of our main result (see Section 4).

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).
Lemma 3.2.Assume that A is a D ± -matrix.Then, there exist a 0 > 0, ǫ 0 > 0 and η > 0 such that Proof.We give the proof in the case of D + -matrix; the other case is analogous.We recall that the matrix norm B ǫ 2 coincides with the square root of the maximum eigenvalue of the matrix C ǫ = B T ǫ B ǫ .Let us first observe that, for every ǫ, the elements on the diagonal of C ǫ are given by where ∆ = a 11 a 22 − a 12 a 21 .Now, let us observe that the matrix C ǫ is positive definite; as a consequence, the maximum eigenvalue of C ǫ is given by From (3.4) and (3.5), by means of simple computations we infer that where and as ǫ → 0. where: Observe that g is a positively homogeneous function of degree since the matrix A is a D + -matrix.Using (3.9) we deduce that there exists η > 0 such that for every ǫ such that ǫ = 1 and By homogeneity, we conclude that g (ǫ) < −2a 0 ǫ , for every ǫ ∈ (0, +∞) 2 satisfying (3.10).From (3.8) and (3.11) we deduce that there exists ǭ > 0 such that for every ǫ ∈ C ǭ,η .Let us now take ǫ 0 = min{ǭ, 1/a 0 }; from (3.12) we immediately conclude that for every ǫ ∈ C ǫ0,η .

Invariant sets and unbounded orbits of discrete maps
In (2.23) we have obtained an estimate for the Poincaré map (θ(0), r(0)) → (θ(2π), r(2π)) associated to the system (2.13) 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 (2.23) define a map (θ, r) → (u, ρ), with θ = (θ 1 , θ 2 ), r = (r 1 , r 2 ), u = (u 1 , u 2 ) and ρ = (ρ 1 , ρ 2 ), such that: where and, moreover, L, G, F are all 2π-periodic w.r.t.θ 1 and θ 2 .We recall that (θ i , r i ) and (u i , ρ i ) are modified polar coordinates in R 2 according to (2.12) 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 } ≥ R > 0.
The second issue is that (3.13) defines a lifting of the actual dynamical system that, indeed, acts on T 2 × R 2 + , where, as usual, T 2 = R 2 /(2πZ) 2 denotes the two-dimensional torus.More precisely, the coordinates (θ, r) and (u, ρ) should be projected to T 2 × R 2 + to determine the correct behavior of the dynamical system, but computations are more easily performed on the "flat" covering space R 2 × R 2 + .To this aim, we denote by θi the equivalence class of θ i in T 1 = R/2πZ and, thus, we will have θ = ( θ1 , θ2 ) ∈ T 2 for each θ = (θ 1 , θ 2 ) ∈ R 2 ; the group metrics in T 1 and T 2 are respectively defined by It will be clear from the context when | • | and • are meant on either R and R 2 or T 1 and T 2 , respectively.In particular, we observe that The invariant sets we obtain are built around a fixed ω ∈ T 2 and depend of four other parameters as follows: where R > 0, 0 < Θ < π, λ > 0 and 0 < η < λ.We will denote by f : E R,Θ,λ,η → T 2 × R 2 + the map which has (3.13) as a lifting.We remark that all different choices of n 1 , n 2 ∈ Z in (3.13) 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 ω ∈ R 2 such that L(ω) = 0 and suppose that the Jacobian JL(ω) is a D + -matrix.Moreover, assume that Then, there exist R > 0, Θ ∈ ]0, π[, λ > 0 and η ∈ ]0, λ[ such that: Proof.We divide the proof into three parts.Part 1. Choice of the constants R, Θ, λ and η.Let let η, ǫ 0 > 0 be as in Lemma 3.2 and let R 0 = 1/ǫ 0 .Since JL(ω) is a D + -matrix we deduce that there exist Θ 0 ∈ ]0, π[ and γ i > 0, i = 1, 2, such that Moreover, according to assumption (3.15), let R 1 ≥ R 0 such that By the continuity of JL(θ) in θ = ω, a simple computation shows that there exists Θ Moreover, using again assumption (3.15), we deduce that there exists R 2 ≥ R 1 such that for i = 1, 2 and θ ∈ R 2 , with lim with α(θ) := (α 1 (θ), α 2 (θ)).Then, we choose Θ ∈ ]0, Θ 1 ] such that where a 0 is given in Lemma 3.2.
Let us now define and consider the set E R,Θ,λ,η 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 (θ, r) such that ( θ, r) ∈ E and consider ρ = (ρ 1 , ρ 2 ) given by (3.13).From conditions (3.16) and (3.17) we immediately deduce that On the other hand, we have r 1 ≤ (λ + η) r 2 and, then, we infer that Let us now observe that (3.25) implies that r 2 − ∂ 2 L 2 (θ) + F 2 (θ, r) > 0 in E; moreover, from the first relations in (3.18) and (3.19), we deduce that From Part 3. Invariance with respect to the angular components.We have to show that, if ( θ, r) ∈ E then ū − ω ≤ Θ, where u is given in (3.13).By the definition of the metric on T 2 in (3.14) and the choice Θ < π, it is enough to work on the covering space and to prove that for a suitable lifting (3.13) we have u − ω ≤ Θ, with θ ∈ R 2 such that θ − ω ≤ Θ, where these last two norms are Euclidean in the covering space R 2 of T 2 .As already announced just before the statement of the theorem, the choice n 1 = n 2 = 0 in (3.13) will work here.
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(ω) is a D − -matrix; indeed, in this situation there exist R > 0, 0 < Θ < π, λ > 0 and 0 < η < λ such that: Then, for every (θ 0 , r 0 ) ∈ E R,Θ,λ,η it is possible to define for every n ≤ 0, and we have lim 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 We recall that, for i = 1, 2, we are assuming the resonance condition Moreover, the function p i : R → R is continuous and 2π-periodic and the function φ i : R → R is locally Lipschitz continuous and bounded, with In this setting, and recalling the definition of the function L given in (2.22)-(2.25), the following result holds true.
Remark 4.2.According to Remark 3.5, an analogous result for t → −∞ can be proved when JL(ω) is a D − -matrix.
Proof.The result follows from an application of Theorem 3.4, taking into account the fact that, from (2.23), the Poincaré map associated with (4.1) is of the form (3.13), with (2.24) implying (3.15).More precisely, let E ⊂ R 2 × R 2 be the set corresponding, via action-angle coordinates, to the set E R,Θ,ω,λ,η given in the statement of Theorem 3.4 and let x be a solution of (4.1) such that (x(0), x ′ (0)) ∈ E.Then, from Theorem 3.4 we infer that lim k→+∞ The thesis (4.4) 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 φ 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 Φ i , i = 1, 2, defined in (2.22).
The first situation we deal with is the one in which both Φ 1 and Φ 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 Under this assumption, the following result holds true.
Remark 4.4.A dual result, ensuring the existence of solutions unbounded in the past, could be proved when (4.5) is replaced by We omit the details for briefness.Let us now focus on the situation where the function Φ 1 (or Φ 2 ) is identically zero, i.e.For the sake of brevity and clarity, we present here just a couple of corollaries in which (4.8) is assumed.In the first we suppose that a 2 is such that c r,2 = 0 for some r ∈ N, ( and that with µ > 0. Corollary 4.6.Let a i , b i > 0 satisfy, for i = 1, 2, assumption (4.2); moreover, suppose that where R is defined in (2.11), and that (4.10) is fulfilled.Finally, assume that conditions (4.3), (4.8) and (4.11) are satisfied.Then, for every φ 1 (+∞) = 0 and for every φ 2 (+∞) ∈ R there exists µ * > 0 such that for every µ > µ * there exist two infinite measure sets E ± ⊂ R 2 × R 2 such that: • for every solution x of (4.1) such that (x(0), • for every solution x of (4.1) such that (x(0), 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 (4.12) holds if (a 1 , a 2 ) is close to (n 2 , n 2 ).This happens, for instance if a 1 satisfies (4.9) with s = 2k and k large enough, and if √ a 2 is irrational and close to n.With these choices (4.8) holds with p 1 (t) = cos(2knt), while (4.10) is trivially satisfied (see (2.4)).
Proof.Let us first notice that, from (2.22) and (4.11), recalling the Fourier expansion of C 2 given in (2.3), we obtain As a consequence, recalling (4.8), we obtain for every θ ∈ R 2 .Now, let us look for solutions of L(θ) = 0; from the relation L 1 (θ) = 0, recalling that φ 1 (+∞) = 0, we deduce From Lemma 2.1, taking into account (4.12), we infer that there exists Λ In particular, we choose m = 0; then, from (4.14) and (4.15) we obtain Replacing the last equality in the expression of L 2 in (4.13) and recalling that Λ 2 is even and 2π-periodic, the equation L 2 (θ) = 0 reduces to then, for every µ > μ the equation (4.16) can be solved and we obtain Choosing h = 0, we then conclude that, for every µ > μ, the equation L(θ) = 0 has the four solutions .
In order to apply Theorem 4.1, we claim that one of the above four solutions, to be named ω + , is such that JL(ω + ) is a D + -matrix and another one, to be named ω − , is such that JL(ω − ) is a D − -matrix.
To do this, recalling (4.13) and the fact that Λ ′ i = Σ i /n is 2π-periodic and odd, we observe that for i = 1, 2. Now, since φ 1 (+∞) = 0 and recalling (4.15), we have moreover, there exists μ ≥ μ such that for every µ > μ we have Hence, for µ > μ, the choice of ω ±,i µ has to be made according to the signs of φ 1 (+∞) and c r,2 .For the sake of briefness, we discuss the case φ 1 (+∞) > 0 and c r,2 > 0, the other ones being similar.We set ω + = ω −,1 µ and ω − = ω +,2 µ ; hence, by construction, JL(ω + ) and JL(ω − ) satisfy the sign conditions on the diagonal coefficients in order to be a D ± -matrix.As far as the third condition in Definition 3.1 is concerned, we have that Hence, there exists µ * ≥ μ such that for every µ > µ * the third condition in Definition 3.1 is satisfied; hence the values ω ± are such that JL(ω ± ) is a D ± 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 (4.8) holds true; as already observed, this is equivalent to the assumption p 1,n = 0, (4.17) where p 1,n is as in (4.7).Let us observe that this situation is not covered by the results in [3].with p 2,n as in (4.7), there exist two infinite measure sets E ± ⊂ R 2 × R 2 such that: • for every solution x of (4.1) such that (x(0), x ′ (0)) ∈ E + , lim t→+∞ • for every solution x of (4.1) such that (x(0), x ′ (0)) Recalling that φ 1 (+∞) = 0, we can solve the equation L 1 (θ) = 0 to obtain We now observe that assumption (4.18) implies that p 2,n = 0; hence, from (4.22) we infer that Choosing in particular m = h = 0, we conclude that the equation L(θ) = 0 has the four solutions (4.23) We now claim that one of the above four solutions, to be named ω + , is such that JL(ω + ) is a D + -matrix and another one, to be named ω − , is such that JL(ω − ) is a D − -matrix.
22 + |a 12 a 22 + a 11 a 21 | .8)Incidentally, let us observe that in the linear symmetric case a 1 = b 1 = n 2 assumption (4.8) corresponds to the case when the number p 1,n in (4.7) is zero.Instead, in the asymmetric case a 1 = b 1 , condition (4.8) is more tricky to be checked.However, some examples in which it holds can be provided.For instance, if a 1 satisfies Fourier coefficient c s,1 of C 1 vanishes (see (2.4)), and (4.8) holds when p 1 (t) = cos snt.