Cubic homogeneous polynomial centers

Agraïments: The first author is partially supported by NSFC-11271027 and NSFC- 11171267.First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all cubic homogeneous polynomial differential systems having a center. After using the averaging method of first order we show that we can obtain at most 1 limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all cubic polynomial differential systems. Moreover, there are examples with 1 limit cycles

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

AbstractIn 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