Center conditions and limit cycles for a class of nilpotent-Poincarè systems

A class of nilpotent-Poincarè   system is discussed in this paper. Center conditions are obtained by methods of inverse integrating factor and theory of rotated vector field. When n>2n>2, we proved that there are 2n+42n+4 small amplitude limit cycles enclosing the origin O(0,0)O(0,0). When n=2n=2, there are 14 limit cycles enclosing the origin O(0,0)O(0,0).