Homoclinic orbits in a first order superquadratic hamiltonian system: Convergence of subharmonic orbits

Abstract

We consider the existence of homoclinic orbits for a first order Hamiltonian system z ̇ = JH z(t, z) . We assume H( t, z) is of form H(t, z) = 1 2 (Az, z) + W(t, z) , where A is a symmetric matrix with δ( JA)∩ i R = ∅ and W( t, z) is 2π-periodic in t and has superquadratic growth in z. We prove the existence of a nontrivial homoclinic solution z ∞( t) and subharmonic solutions ( z T ( t)) TϵN (i.e., 2π T-periodic solutions) of (HS) such that Z T ( t) → Z ∞( t) in C loc 1( R , R 2 N ) as T → ∞.