Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Homoclinic orbits for a class of Hamiltonian systems

Paul H. Rabinowitza1

a1 Mathematics Department and Center for the Mathematical Sciences, University of Wisconsin-Madison, Madison, Wisconsin 53705, U.S.A.

Synopsis

Consider the second order Hamiltonian system:

S0308210500024240_eqnU1

where q ∊ ℝn and VC1 (ℝ ×ℝn ℝ) is T periodic in t. Suppose Vq (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 2kT periodic solutions (i.e. subharmonics) qk of (HS). The subharmonics qk are obtained in turn via the Mountain Pass Theorem.

(Received June 05 1989)