Normal forms of lienard type for analytic unfoldings of nilpotent singularities

Abstract

Using the technique of gluing complex manifolds (equipped with vector fields) developed by Loray and the theory of deformation of complex structures developed by Kodaira and Spencer, we find normal forms of Liénard type for analytic unfoldings of planar singularities with a nonradial linear part. In particular, we improve normal forms of Takens for analytic unfoldings of nilpotent singularities and normal forms of De Maesschalck, Dumortier and Roussarie for analytic unfoldings of nilpotent contact points in planar slow-fast systems.