A relation between small amplitude and big limit cycles

There are two well-known methods for generating limit cycles for planar systems with a nondegenerate critical point of focus typed the degenerate Hopf bifurcation and the Poincare-Melnikov method; that is, the study of small perturbations of Hamiltonian systems. The first one gives the so-called small amplitude limit cycles, while the second one gives limit cycles which tend to some concrete periodic orbit,., of the Hamiltonian system when the perturbation goes to zero (big limit cycles, for short). The goal of this paper is to relate both methods. In facts in all the families of differential equations that we have studied, both methods generate the same number of limit cycles. The families studied include Lienard systems and systems with homogeneous nonlinearities.