A twist condition and periodic solutions of Hamiltonian systems

In this paper, we investigate existence of nontrivial periodic solutions to the Hamiltonian systemequation(HS)−Jz˙=H′(t,z),z∈R2N. Under a general twist condition for the Hamiltonian function in terms of the difference of the Conley–Zehnder index at the origin and at infinity we establish existence of nontrivial periodic solutions. Compared with the existing work in the literature, our results do not require the Hamiltonian function to have linearization at infinity. Our results allow interactions at infinity between the Hamiltonian and the linear spectra. The general twist condition raised here seems to resemble more the spirit of Poincaré's last geometric theorem.