Unbounded Solutions to a System of Coupled Asymmetric Oscillators at Resonance

Abstract

We deal with the following system of coupled asymmetric oscillators $$\begin{aligned} \left\{ \begin{array}{l} \ddot{x}_1+a_1x_1^+-b_1x^-_1+\phi _1(x_2)=p_1(t) \\ \ddot{x}_2+a_2\,x_2^+-b_2\,x^-_2+\phi _2(x_1)=p_2(t), \end{array} \right. \end{aligned}$$ x ¨ 1 + a 1 x 1 + - b 1 x 1 - + ϕ 1 ( x 2 ) = p 1 ( t ) x ¨ 2 + a 2 x 2 + - b 2 x 2 - + ϕ 2 ( x 1 ) = p 2 ( t ) , where $$\phi _i: \mathbb {R} \rightarrow \mathbb {R}$$ ϕ i : R → R is locally Lipschitz continuous and bounded, $$p_i: \mathbb {R} \rightarrow \mathbb {R}$$ p i : R → R is continuous and $$2\pi $$ 2 π -periodic and the positive real numbers $$a_i, b_i$$ a i , b i satisfy $$\begin{aligned} \dfrac{1}{\sqrt{a_i}}+\dfrac{1}{\sqrt{b_i}}=\dfrac{2}{n}, \quad \text{ for } \text{ some } n \in \mathbb {N}. \end{aligned}$$ 1 a i + 1 b i = 2 n , for some n ∈ N . We define a suitable function $$L: \mathbb {T}^2 \rightarrow \mathbb {R}^2$$ L : T 2 → 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.