Quasi-homogeneous normal forms

In this paper, we extend the concepts of the normal form theory for vector fields that are expanded in quasi-homogeneous components of a fixed type (these expansions have been used by some authors in the analysis of the determinacy of a given singularity). Also, the use of reparametrizations in the time are considered. Namely, beyond the use of C∞-conjugation to determine normal forms, we present a method useful to determine how much a vector field can be simplified by using C∞-equivalence. The results obtained are applied in the case of the Bogdanov–Takens singularity, firstly using C∞-conjugation and later, showing the improvements provided by the C∞-equivalence.