Connecting orbits near the adiabatic limit of Lagrangian systems with turning points (Q1695066)

The author considers a natural Lagrangian system on a complete Riemannian manifold influenced by the action of a time-periodic force field with potential $U(q, t, e) = f(\varepsilon t) V(q)$ depending slowly on time. Under the condition that the factor $f(t)$ is periodic and vanishes at least at one point on the period, the author, in the adiabatic limit $\varepsilon\rightarrow0$, proves the existence of a set $\epsilon_h$ such that the system possesses a rich class of doubly asymptotic trajectories connecting points of $X_c$ for $\varepsilon \in \epsilon_h$, where $X_c$ denote a set of isolated critical points of $V(x)$ at which $V(x)$ distinguishes its maximum or minimum.