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, ε) = f(εt)V(q) depending slowly on time. It is assumed that the factor f(τ) 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 ε → 0 we prove the existence of a set E _h such that the system possesses a rich class of doubly asymptotic trajectories connecting points of X _c for ε ∈ E _h .