Homoclinic solutions to invariant tori in a center manifold

We consider the Lagrangian \[ L(y_1, \dot{y}_1, y_2, \dot{y}_2, q, \dot{q}) = 1/2(\dot{y}_1^2 - omega_1^2 y_1^2) + 1/2(\dot{y}_2^2 - omega_2^2 y_2^2) + 1/2 \dot{q}^2 + (1 + \delta(y_1, y_2))V(q), \] where $V$ is non-negative, periodic in $q$ and such that $V(0) = V'(0) = 0$. We prove, using critical point theory, the existence of infinitely many solutions of the corresponding Euler-Lagrange equations which are asymptotic, as $t \to \pm \infinity$, to invariant tori in the center manifold of the origin, that is, to solutions of the form $q(t) = 0$, $y_1(t) = R \cos(\omega_1 t + \varphi_1)$, $y_2 (t) = R \cos(\omega_2 t + \varphi_2)$.