Periodic orbit of the pendulum with a small nonlinear damping

We study the pendulum with a small nonlinear damping, which can be expressed by a Hamiltonian system with a small perturbation. We prove that a unique periodic orbit exists for any initial position between the equilibrium point and the heteroclinic orbit of the unperturbed system, depending on the choice of the bifurcation parameter in the damping. The main tools are bifurcation theory and Abelian integral technique, as well as the Zhang''s uniqueness theorem on Li''enard equations.