The Number Of Limit Cycles For Planar Polynomial Perturbations Of Hamiltonian Systems Near Heteroclinic Loops

In the study of limit cycles for near-Hamiltonian systems,the first-order Melnikov function plays an important role.Suppose there is a heteroclinic loop defined by the equation H(x,y)= hs,the form of expansion of Melnikov function at h = hshas been obtained.It is very difficult to get more coefficients of the expansion of Melnikov functions near a heteroclinic loop.In this paper,we study heteroclinic bifurcations of limit cycles in perturbed planar Hamiltonian systems.We present a method to derive more coefficients of the expansion of Melnikov functions at h = hs.Then by using those coefficients,more limit cycles would be found around heteroclinic loops.As an application,we consider polynomial perturbations of degree 5,7 and 9 for a type of cubic Hamiltonian systems with a heteroclinic loop.The first chapter mainly introduces the historical background and current situation of the research.The second chapter mainly introduces the main results and proof of this paper.The third chapter is the application of the conclusion.how to calculate by our method in the specific system and the verification of the main conclusion.