Computing Simplest Normal Forms for the Takens–Bogdanov Singularity

A computational method to obtain simplest normal forms, which we refer as hypernormal forms, for vector fields having a linear degeneracy corresponding to a double zero eigenvalue with geometric multiplicity one, is presented. The procedure of simplifying the classical normal forms requires some hypothesis on the nonlinear terms of the vector field. Once the hypothesis are established, different hypernormal forms are obtained. Several examples are included in order to show the applicability of our approach in concrete cases.