Multibump solutions homoclinic to periodic orbits of large energy in a center manifold

We consider the Lagrangian $\frac{1}{2}(\dot{x}^2-\omega^2 x^2) + \frac{1}{2}\dot{q}^2 + (1+\delta(x))V(q)$ , $V$ $2 \pi$-periodic and $\delta$ bounded. The corresponding Euler–Lagrange equations have as the origin a saddle-centre stationary point whose (globally defined) centre manifold is foliated in periodic orbits. We prove that for $\omega$ small enough there exist multi-bump solutions, of large energy and heteroclinic to periodic orbits in the centre manifold.