Limit Cycles For Some Planar Differential Systems

In this thesis,we mainly study the bifurcation of limit cycles for several kinds of planar differential systems,a class of planar near-Hamiltonian systems and two kinds of planar piecewise smooth systems with invariant lines parallel or perpendicular to the switching line.We mainly use Melnikov function method and averaging method to study limit cycles in this paper.This thesis is divided into five chapters as follows.Chapter 1 is devoted to the introduction concerning the background,the development,the research methods and the present situation of the research.We also put forward the research work and the innovation of this paper.Chapter 2 is concerned with a class of near-Hamiltonian systems with nilpotent saddle points.By using Chebyshev criterion we obtain an upper bound of the number of limit cycles bifurcating from the period annulus of the unperturbed system,and the upper bound can be achieved by using Melnikov function and its expansions.In addition,we discuss the Chebyshev property of the related hyper-elliptic integrals and correct some results in reference [60].A lot of auxiliary calculations are made by Maple 20 in this chapter.Chapter 3 is concerned with the number of limit cycles bifurcating from the period annulus for some planar piecewise smooth non-Hamiltonian systems.We construct a planar piecewise quadratic system with multiple parameters,obtain its lower bound for the maximum number of limit cycles by using Melnikov function method.More limit cycles are obtained by using the multi-parameter perturbation method.Chapter 4 is concerned with limit cycles bifurcating from planar piecewise integrable systems,whose unperturbed systems have a center at the origin and have invariant lines perpendicular to the switching line.By using the first order averaging method,we present an estimate of the maximum number of crossing limit cycles bifurcating from the periodic orbits around the center.Chapter 5 is the summary and prospect.