Connecting Orbits near the Adiabatic Limit of Lagrangian Systems with Turning Points

We consider a natural Lagrangian system defined on a complete Riemannian manifold being subjected to action of a time-periodic force field with potential $U(q,t, \varepsilon) = f(\varepsilon t)V(q)$ depending slowly on time. It is assumed that the factor $f(\tau)$ is periodic and vanishes at least at one point on the period. Let $X_{c}$ denote a set of isolated critical points of $V(x)$ at which $V(x)$ distinguishes its maximum or minimum. In the adiabatic limit $\varepsilon \to 0$ we prove the existence of a set $\mathcal{E}_{h}$ such that the system possesses a rich class of doubly asymptotic trajectories connecting points of $X_{c}$  for $\varepsilon \in \mathcal{E}_{h}$.