Bifurcations Of Limit Cycles For Several Special Kinds Of Differential Systems

In this thesis,we study the bifurcations of limit cycles for two special kinds of planar differential systems,a class of the Li′enard system which is transformed from the generalized Burgers-Fisher equation by the traveling wave transformation,a kind of planar piecewise smooth near-Hamiltonian systems with multiple switching curves,respectively.For the former,we mainly obtain the traveling wave solution bifurcation of the generalized Burgers-Fisher equation by studying Hopf bifurcation,Poincar′e bifurcation,homoclinic loop bifurcation and heteroclinic loop bifurcation of a related Li′enard system.For the latter,we mainly obtain a formula of the first order Melnikov function in general case and give an application example.The first chapter is an introduction,which is devoted to introducing the research background and methods used in this paper.In chapter 2,we deal with the traveling wave solutions(including periodic wave solutions,solitary wave solutions and kink or anti-kink solutions)of a generalized Burgers-Fisher equation via studying the dynamics of a related Li′enard system.We obtain necessary and sufficient conditions for the Li′enard system to have a unique limit cycle,a homoclinic loop and a heteroclinic loop respectively.In chapter 3,we are devoted to the study of limit cycle bifurcations in piecewise smooth near-Hamiltonian systems with multiple switching curves,obtaining a formula of the first order Melnikov function in general case.As an application,we give lower bounds of the number of limit cycles for a piecewise smooth near-Hamiltonian system with a closed switching curve passing through the origin under piecewise polynomial perturbations.