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

Limit cycle bifurcations from a nilpotent focus or center of planar systems