Limit Cycles And Center Problems For A Class Of Homogeneous Systems

In this paper, we mainly investigate center problems and limit cycle bifurcations in a class of quasi-homogeneous systems.In Chapter one, we introduce the background of our research and main topics that we will study in the following chapters.In Chapter two, we use an algorithm to obtain various forms of a class of quasihomogeneous polynomial di?erential systems of degree 5. Further we classify all centers of a class of quasi-homogeneous polynomial di?erential systems of degree 5, which have four types of forms.In Chapter three, first of all, we extend this kind of systems to a generalized polynomial di?erential system. Then, we extend the results of Chapter two to such systems. The method is for transformation into Lienard system, using Lienard system has known results,then get the necessary and su?cient conditions for which to have a center at the origin.In Chapter four, we mainly focus on a class of quasi-homogeneous polynomial center containing small perturbations of limit cycles bifurcation, is to study the number of zeros of the first order Melnikov function. The method is Melnikov function for processing,and finally we get that Melnikov function is a combination with monomials of h up to a fractional power, further we get Poincare cyclicity.