Limit cycles near a homoclinic loop by perturbing a class of integrable systems

In this paper, the problem of limit cycle bifurcation is investigated by perturbing a class of integrable systems with a homoclinic loop. Under the assumption that the homoclinic loop passes through a degenerate singular point at the origin, the asymptotic expansion of the Melnikov function along the level curves of the first integral inside the homoclinic loop is studied near the loop. Meanwhile, the formulas for the first coefficients in the expansion are given, which can be used to study the number of limit cycles near the homoclinic loop. Finally, an example is provided to demonstrate the obtained results.