Bifurcation Of Limit Cycles Of Planar Quartic Near-integrable Systems Around A Center Point

In this paper,we study the number of limit cycles generated around the primary center of a class of planar quartic polynomial integrable system under small perturbations by using the method of higher-order analysis of focus values.It is proved that there are at most 17 small amplitude limit cycles near the center of the system by calculating the focus values of ?-order.When all the focus values of ?-order vanish,it is proved that the system can generate at most 20 small amplitude limit cycles around the center point by calculating the focus values of ?~2-order.