Hypernormal forms for equilibria of vector fields. Codimension one linear degeneracies (Q1293279)

The Poincaré-Dulac-Birkhoff theorem determines how much a vector field can be simplified near an equilibrium point, depending uniquely on its linear part. The authors show how it is possible to obtain further simplifications in the classical normal form taking into account nonlinear terms. This procedure leads to hypernormal forms which are the simplest that can be achieved using \(C^\infty\)-conjugation. The hypernormal forms are characterized using the theory of transformations based on Lie transforms. Since the computation of a hypernormal form requires the solution of some nonlinear equations, the authors define the pseudohypernormal form, which is not as general as the hypernormal form, but its computation involves only linear equations. As illustration of the method, two cases of codimension one linear degeneracies are treated: saddle-node and Hopf singularities. In both examples, additional simplifications are considered that can be achieved by using \(C^\infty\)-equivalence.