Homoclinic Bifurcation from Perturbing a Cubic Integrable and Non-Hamiltonian System

In this paper, we examine homoclinic bifurcation in a planar cubic integrable system, incorporating cubic polynomial perturbations, by employing first-order and second-order Melnikov functions. We derive the simplest expressions for these two Melnikov functions and their asymptotic expansions in proximity to the homoclinic loop. By utilizing the coefficients from these expansions, we identify four and six limit cycles within a small neighborhood of the homoclinic loop using the first-order and second-order Melnikov functions, respectively. Our findings surpass existing results for cubic near-Hamiltonian systems by yielding more limit cycles through the application of both first-order and second-order Melnikov functions.