Limit Cycle Bifurcation And Local Critical Period Bifurcation In Some Planar Polynomial Systems

This thesis is devoted to two kinds of bifurcation problems. One is the bifurcation of limit cycles of several planar vector fields and the other is critical period bifurcation by perturbing a reversible rigidly isochronous center with multiple parameters. It is divided into four chapters.In Chapter one§we introduce the research backgrounds about the topic, and we also give the theory of bifurcation of limit cycles and the critical period, research methods!research status!main conclusions and innovation points.In Chapter two, First, we consider Hopf cyclicity of two kinds of Li′enard systems.The main results can be proved according to the conclusions of Professor Han about a generalized Li′enard system. In fact, we use his another conclusion in another paper, that is, the small positive zeros of F(α(x))- F(x) ≡ 0 corresponds to the limit cycles of the researched system. Thus we prove the obtained conclusions from the more general case.Then, some problems about the multiplicity of the smooth(or analytic) functions are introduced in Section 2.2. Further, Hopf cyclicity of another special Li′enard system is considered. By the known theorem we calculate the cyclicity of the system for the cases/1 ≤ m, n ≤ 4, m = n.0In Chapter three, we consider a planar system whose unperturbed system has multiple factors(1- y)m. By calculating the Melnikov function, we get the lower bound of the number of the limit cycles bifurcating from the origin for some cases.In Chapter four, we investigate the local bifurcation of critical period of the reversible system, which has rigidly isochronous center with multiple parameters. By concrete calculations, the integral expressions of the period bifurcation functions are obtained. Then, by applying a new method, we study the number of critical points of the period bifurcation function. The number of critical periods can be obtained for some special systems.