  We deal with a weakly coupled system of ODEs of the type $$ x_j'' + n_j^2
\,x_j + h_j(x_1,\ldots,x_d) = p_j(t), \qquad j=1,\ldots,d, $$ with $h_j$
locally Lipschitz continuous and bounded, $p_j$ continuous and $2\pi$-periodic,
$n_j \in \mathbb{N}$ (so that the system is at resonance). By means of a
Lyapunov function approach for discrete dynamical systems, we prove the
existence of unbounded solutions, when either global or asymptotic conditions
on the coupling terms $h_1,\ldots,h_d$ are assumed.
