Unbounded Solutions to a System of Coupled Asymmetric Oscillators at Resonance.

We deal with the following system of coupled asymmetric oscillators 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 ϕ i : R → R is locally Lipschitz continuous and bounded, p i : R → R is continuous and 2 π -periodic and the positive real numbers a i , b i satisfy 1 a i + 1 b i = 2 n , for some n ∈ N. We define a suitable function 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. [ABSTRACT FROM AUTHOR]