This paper is concerned with the following periodic Hamiltonian
elliptic system
$ \{
-\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)\ \hbox{in }\mathbb{R}^N, \\
-\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)\ \hbox{in }\mathbb{R}^N, \\
\varphi(x)\to 0\ \hbox{and }\psi(x)\to0\ \hbox{as }|x|\to\infty.$
Assuming the potential V is periodic and 0 lies in a gap of
$\sigma(-\Delta+V)$, $G(x,\eta)$ is periodic in x and
asymptotically quadratic in $\eta=(\varphi,\psi)$, existence and
multiplicity of solutions are
obtained via variational approach.