\(C^\infty\)-normal forms of local vector fields
It is well known that a resonant normal form and a normalizing transformation are not uniquely determined. This fact can be used for a further simplification of a normal form (the so-called ``secondary'' normalization). Here, a direct definition of a secondary normalization is given. In contrast with a traditional normalization, a secondary normalization turns out to be uniquely determined for formally equivalent systems. As follows from examples treated in the paper, at least in the case of low dimensions the secondary normalization leads to the most simple formal system a given system can be brought to. As a rule, even for analytic systems formal normalizing transformations diverge. However, as shown in the paper, these transformations in certain cases correspond to \(C^{\infty}\)-smooth changes of variables.
scientific article; zbMATH DE number 1747773