Bifurcations Of Limit Cycles For Some Perturbed Polynomial Systems

Finding the lowest upper bound for the number of zeros of Abelian integrals is an important problem in bifurcation theory. It is closed related to determining the number of limit cycles of polynomial perturbations on the plane Hamiltonian vector fields. This paper studies the number of Abelian integrals and limit cycle bifurcations of Hamiltonian perturbed systems.This paper includes four chapters.The first chapter is overview. It introduces the history, development and some main results of bifurcation theory.In the second chapter, we study the Abelian integrals of a class of reversible Hamiltonian system under cubic perturbations, and we get this system has 2 limit cycles at most.In the third chapter, a class of quasi-reversible Hamiltonian system is investigated. We get the conclusion that this system has at most 3 limit cycles. So the result of Zhao Yulin [43] is extended.In the four chapter, we consider the number of limit cycles of a class of Hamiltonian system with parameter, and get the number of the zeros of Abelian integrals of this system at most 12. So we extend the result of Zhang Tonghua and Chen Wencheng[38].