The problem of general center-focus and bifurcation of limit cycles for a planar system of nine degrees

Our researches are concerned with a class of planar general equivariant system of nine degrees. By making two appropriate transformations of system and calculating focal values carefully, we obtain the conditions that the infinity and three elementary focuses (−12,0),(−12,12),(−12,−12) become four general centers at the same time. Moreover, 20 limit cycles including 15 small limit cycles from three elementary foci and 5 large limit cycles from the infinity can occur under a certain condition. What is worth mentioning is that similar conclusions are hardly seen in published papers up till now and our work is completely new.