Qualitative Analysis Of A Class Of Cubic System With Planar Polynomial

This paper studies a class of cubic system with planar polynomial:We use the qualitative theory of ordinary differential equations method of analysis, reached the following conclusions of four parts:First, we analyze the quality of all critical points of the system, and draw the conditions of existence of singular integrals for system (E), in front based on the analysis, using Pioncare form series method to calculate the focal value of O(0,0), it is C0=D, C1=-3/4(l+bc), C2=5/8abc. Finally, we also study the behavior of critical points at infinity.The second part, the inexistence of limit cycle around O(0,0) about the singularity of the system is perfectly studied with limit cycle theory. We obtain that:System (E) has no limit cycle around O(0,0) with the time c=0,a=0,b≠0,D≥l/b2or c=0,b2=4a,b≠0,D≥4l/b2well as O(0,0) is two order of fine focus. When c≠0, we get some conclusions of inexistence.Thirdly, with limit cycle theory and Hopf bifurcation theory, we can obtain when abc>0(<0), l+bc>0(<0),|l+bc|<<1, as well D>0(<0),|D|<|l+bc|<<1, it bifurcates at least two limit cycle around O(0,0). System (E) has at most one limit cycle around O(0,0) when for the one order of fine focus.Fourth, we study the global structure of the system. We discuss the global structure of the system when O(0,0) is the center as well as one or two order of the fine focus．Finally numerical simulation method is verified the accuracy of thecalculation.