On the existence of periodic and homoclinic orbits for first order superquadratic Hamiltonian systems

In this paper we consider the following Hamiltonian system $$Jdot u + B(t)u +nabla W(t,u)=0.quadquad (HS)$$ Under a new superquadratic assumption on the potential, we prove that (HS) has a sequence of subharmonics. This will be done using a minimax result in critical point theory. Also, we study the asymptotic behavior of these subharmonics and we establish the existence of a homoclinic orbit for (HS). Previous results in the topic, mainly those due to Rabinowitz and Tanaka, are significantly improved.