Monodromy problem for the degenerate critical points

Abstract

For the polynomial planar vector fields with a hyperbolic or nilpotent critical point at the origin, the monodromy problem has been solved, but for the strongly degenerate critical points this problem is still open. When the critical point is monodromic, the stability problem or the center- focus problem is an open problem too. In this paper we will consider the polynomial planar vector fields with a degenerate critical point at the origin. At first we give some normal form for the systems which has no characteristic directions. Then we consider the systems with some characteristic directions at which the origin is still a monodromic critical point and we give a monodromy criterion. Finally we clarify our work by some examples.