Connecting orbits of Lagrangian systems in a nonstationary force field

We study connecting orbits of a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q, t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t→±∞ and vanishes at a unique point t ₀ ∈ ?. Let X ₊, X _? denote the sets of isolated critical points of V (x) at which U(x, t) as a function of x distinguishes its maximum for any fixed t > t ₀ and t < t ₀, respectively. Under nondegeneracy conditions on points of X _± we prove the existence of infinitely many doubly asymptotic trajectories connecting X _? and X ₊.