Unbounded solutions to a system of coupled asymmetric oscillators at resonance

Abstract

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.