Subharmonic bifurcations in a perturbed nonlinear oscillation

In this paper we discuss subharmonic bifurcations inside the homoclinic orbits and outside for a general form of perturbed two-dimensional Hamiltonian systems with a pair of real parameters λ,μ and a pair of function parameters g1,f1. By approximating periodic orbits to homoclinic orbits, we determine bifurcation curves for (λ,μ) in R2 and bifurcation manifolds for (g1,f1) in the corresponding space of functions. Furthermore, we determine the codimensions of those bifurcation manifolds. All results on subharmonic bifurcations are given with conditions on homoclinic orbits. Our results generalize Hale and Spezamiglio's in Nonlinear Anal. 9 (1985) 181.