Saddle connections

It is shown that vector fields that are close to a fixed field with the same set of connections form a smooth Banach submanifold. A sufficient condition for the birth of saddle connections in a generic family is presented. The following result is proved: in a perturbation of a monodromic hyperbolic polycycle of $n$ connections in a generic family at least $n$ limit cycles can appear.
Bibliography: 21 titles.