Unbounded Solutions to a System of Coupled Asymmetric Oscillators at Resonance

We deal with the following system of coupled asymmetric oscillators {x¨1+a1x1+-b1x1-+ϕ1(x2)=p1(t)x¨2+a2x2+-b2x2-+ϕ2(x1)=p2(t),where ϕi: R→ R is locally Lipschitz continuous and bounded, pi: R→ R is continuous and 2 π-periodic and the positive real numbers ai, bi satisfy 1ai+1bi=2n,forsomen∈N.We define a suitable function L: T2→ R2, 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.