The Local And Global Bifurcations Of Plane Polynomial Vector Fields

The bifurcation theory of the plane polynomial vector fields is one important research domain of the theory of ordinary differential equation. The bifurcations of limit cycles which have important theory values and apply values are generally acknowledged as the hot areas of the research work. The second part of the famous Hilbert 16th problem is about the maximal number and relative dispositions of limit cycles of the planar polynomial vector field.The local and global bifurcations of the planar polynomial vector fields are considered. The main aim of this dissertation is to study the exist of limit cycles of plane nonlinear dynamic system of high degree and the number and the configurations of limit cycles. This dissertation also discusses the bifurcation theory's application on the nonlinear dynamic system. The major works of this dissertation mainly are as follows:(1) The development history, research development, achievements and the apply background in engineering of the local and global bifurcations theory of the plane polynomial vector fields are summarized;(2) The concepts of limit cycles, the bifurcations of plane polynomial vector fields and Zq- equivariant vector fields are introduced. The methods for studying the local and global bifurcations of plane polynomial vector fields are also introduced.(3) The local and global bifurcations of a Z 2-equivariant perturbed polynomial Hamiltonian plane polynomial vector fields of degree 5 are considered. Utilizing the bifurcation theory of nonlinear dynamical system, the existence of multiple limit cycles and the numbers and relative position of the limit cycles of this system are obtained.(4) A generic nonlinear system with unperturbed parameters is also discussed. The curves decided by the unperturbed parameters partition the (a, b)-plane into 15 different regions. And the fifteen different phase portraits are also given. The number of limit cycles and the configuration under different control conditions are given. The bifurcations of multiple limit cycles of the system with only one perturbed term are also discussed. Finally, we discussed the application of the bifurcation theory on nonlinear mechanical system. A parametrically and externally excited mechanical...