Qualitative Analysis Of A Class Of Cubic System

In this paper, we study a class of E3 1 system: where b₁ ≠ 0 . Using method of bifurcation theory and qualitative analysis,we obtain the global structure of the system in certain condition. We got the following conclusion: 1 When b₁ > 0,finite critical point O (0,0) can be a fine focus and the highest order of the fine focus O (0,0) is three .so the system has at least three limit cycles.while b₁ < 0 ,the highest order of the fine focus can be four when the system has only one finite singular point and can be three when the system has three finite critical points,obviously,the system has at least four or three limit cycles. 2 In this paper,we study the connections of the two weak-focus when the parameter b₁ < 0. 3 When O (0,0) is a centre,we obtain all the possible global structures of the system through the analysis of the infinite singular points of the system. 4 In the end ,by the perturbing of little parameter all of the possible global structures of system are obtained as the order of the fine focus O (0,0) is four.