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 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.