The Studies Of Limit Cycles And Hopf Bifurcation For Several Classes Of Polynomial Differential Autonomous Systems

The Problems about bifurcations of limit cycles for polynomial differential autonomy systems and limit cycles and Hopf bifurcation at positive equilibrium of ecological systems are investigated in this paper. It is composed of seven chapters.In chapter 1, the historical background about bifurcations of limit cycles for planar autonomy differential systems is introduced and the main works of this paper are concluded as well.Chapter 2 introduces definitions, theories and methods which are related by bifurcations of limit cycles and Hopf bifurcation.In chapter 3, using an indirect method to study the problem about bifurcations of limit cycles at infinity for a class of cubic and quintic polynomial differential system, we obtain 6 limit cycles and 9 limit cycles respectively. It is the first time to obtain 9 limit cycles at infinity for a class of quintic system by now.In chapter 4, firstly, using a direct method to study bifurcations of limit cycles at infinity for a class of seven-degree system, we obtain 9 limit cycles. Secondly, we use an indirect method to study bifurcations of limit cycles at infinity for another class of seven-degree system and obtain 10 limit cycles, which is obtained the first time.In chapter 5, the problem about limit cycles for a class of 2n-degree Kolmogorov system is studied. The sufficient conditions of nonexistence and existence of limit cycles in the first quadrant are analyzed as well.In chapter 6, Hopf bifurcation on a multi-dimension ecological system is investigated. The qualitative property and stability of equilibrium points are discussed. The case of the period solution at positive equilibrium is analyzed when a small perturbed term is added into the system. Moreover, the conditions of the existence of Hopf bifurcation for the system are investigated. Finally, the numerical example is given.Last chapter summarizes the whole paper and shows the problems which are still solved in the research and aspects of our investigation in the future.