Limit cycle bifurcations near double homoclinic and heteroclinic loops of a class of cubic Hamiltonian systems

This paper studies the double homoclinic and heteroclinic bifurcations by perturbing a cubic Hamiltonian system with polynomial perturbations of degree n. It is proved that $5\left[\frac{n-1}{2}\right],n\ge 3$ and $2\left[\frac{n-1}{2}\right]$ limit cycles can be bifurcated from the period annuli near the double homoclicic loop and the heteroclinic loop, respectively. This result improves the lower bound on the number of the bifurcated limit cycles comparing with the known results for the related problems. To achieve our results we develop the techniques on calculating the base and the relative relations of the elements in the base, formed partly by curve integral functions along ovals of level sets of the Hamiltonian function, which appear in the expansions of the first order Melnikov functions.