General center conditions and bifurcation of limit cycles for a quasi-symmetric seventh degree system

In this paper, we study a class of quasi-symmetric seventh degree system. By making two appropriate transformations of system (3) and calculating general focal values carefully, we obtain the conditions that the infinity and the elementary focus (−12,0) become centers at the same time. Moreover, 12 limit cycles including 6 small limit cycles from the elementary focus and 6 large limit cycles from the infinity can occur under a certain condition. What is worth mentioning is that similar conclusions are less and our work is new in terms of research about quasi-symmetric systems up till now.