  In this paper, we study the analytical property of the Poincare return map
and the generalized focal values of an analytical planar system with a
nilpotent focus or center. Then we use the focal values and the map to study
the number of limit cycles of this kind of systems with parameters, and obtain
some new results on the lower and upper bounds of the maximal number of limit
cycles near the nilpotent focus or center.
