Bifurcation of limit cycles near polycycles with n vertices

In the study of Hilbert 16th problem the most difficult part is to find the maximal number of limit cycles appearing near a polycycle by perturbations. In this paper we study the bifurcation of limit cycles near a polycycle with n hyperbolic saddle points. We obtain a sufficient condition for the polycycle to generate at least n limit cycles. We also establish a necessary and sufficient condition for the existence of a separatrix connecting any two saddle points.