Global Existence of Homoclinic and Periodic Orbits for a Class of Autonomous Hamiltonian Systems

When nonlinear, reversible fourth order Hamiltonian systems have a saddle-focus at a hyperbolic equilibrium of zero energy, geometric conditions on the Hamiltonian are given to ensure the existence of a symmetric homoclinic orbit which is the limit of certain specific zero-energy periodic orbits uniformly on compact time intervals. When the equilibrium is a centre. the existence of a large amplitude, zero-energy, periodic orbit which is not given by Lyapunov’s Centre Theorem is proved.