Homoclinic orbits for a class of Hamiltonian systems

Consider the second order Hamiltonian system: where q ∊ ℝ n and V ∊ C 1 (ℝ ×ℝ n ℝ) is T periodic in t . Suppose V q ( t , 0) = 0, 0 is a local maximum for V ( t ,.) and V ( t , x ) | x | → ∞ Under these and some additional technical assumptions we prove that (HS) has a homoclinic orbit q emanating from 0. The orbit q is obtained as the limit as k → ∞ of 2 kT periodic solutions (i.e. subharmonics) q k of (HS). The subharmonics q k are obtained in turn via the Mountain Pass Theorem.