Bifurcation Of Limit Cycles For Several Classes Of Piecewise Smooth Near-hamiltonian Systems

In this dissertation,we mainly study the bifurcation of limit cycles of several classes of piecewise smooth near-Hamiltonian systems in a plane,which is divided into five chapters.In the first chapter,we introduce the research background,research methods and main results of this thesis.In the second chapter,we study the bifurcation of limit cycles for planar piecewise smooth near-Hamiltonian systems with polynomials of order n.A piecewise smooth linear differential system with two centers can be constructed in two ways.One is that there is a central fold point at the origin.The other is that there is a central fold point at the origin,and another unique center point exists.We first explore expressions for the first order Melnikov function in both cases.Then,by using the Melnikov function method,we give estimations of the number of limit cycles bifurcating from periodic annuluses.For the latter case,the simultaneous occurrence of limit cycles near both sides of the homoclinic loop is partially addressed.Finally,we give a general conjecture.In the third chapter,we discuss a class of planar piecewise smooth near-Hamiltonian systems with polynomial perturbation of order n.Two ways of composing a piecewise smooth linear differential system with two centers,one is that both centers are on the same side of the origin,the other is that the two centers are distributed on either side of the origin.We first give the expressions of the first order Melnikov function,and then estimate the number of limit cycles bifurcated from periodic annuluses by Melnikov function method.In addition,we discuss the number of limit cycles that can appear simultaneously on both sides of generalized homoclinic loop or generalized double homoclinic loop,and give their distributions.In the fourth chapter,we explore the bifurcation of limit cycles for a class of piecewise smooth near-Hamiltonian systems with linear center.The two sides of the dividing line of a piecewise smooth linear differential system are a group of periodic orbits composed of a center and a saddle point respectively.We discuss the distribution of center point and saddle point on the same side and on both sides of the origin,respectively.We give the expressions of first order Melnikov function,and estimate the number of limit cycles bifurcating from periodic annuluses by Melnikov function method.In the fifth chapter,summary and prospect.