Bifurcations Of Multiple Limit Cycles Of Planar 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 fields.In this paper, the bifurcations of multiple limit cycles of planar polynomial vector fields are considered by using the bifurcation theory of planar dynamical system and the method of detection function.In the third chart, the bifurcations of multiple limit cycles of a Z2-equivariant plane polynomial vector fields of degree 5 are considered and the number and the relative position of limit cycles under one parameter control condition group are obtained. In the fourth chart, the bifurcations of multiple limit cycles of a Z2-equivariant plane polynomial vector fields of degree 7 are considered. The system is discussed in four different cases. For each case, the number and the relative position of limit cycles under one parameter control condition group are obtained. Finally, an important result based on the analysis on the four cases is obtained.