Analytic properties of the complete formal normal form for the Bogdanov–Takens singularity

W pracy badamy analityczność ostatecznych formalnych postaci normalnych dla osobliwości Bogdanova-Takensa. Okazuje się, że analityczne własności dla tej osobliwości są analogiczne jak dla osobliwości elementarnych.

In previous papers a complete formal normal forms for germs of 2-dimensional holomorphic vector fields with nilpotent singularity was obtained. That classification is quite nontrivial (7 cases), but it can be divided into general types like in the case of the elementary singularities. One could expect that also the analytic properties of the normal forms for the nilpotent singularities are analogous to the case of elementary singularities. This is really true. In the cases analogous to the focus and the node the normal form is analytic. In the case analogous to the nonresonant saddle the normal form is often nonanalytic due to the small divisors phenomenon. In the cases analogous to the resonant saddles (including saddle nodes) the normal form is nonanalytic due to a bad behavior of the homological operators associated with the first nontrivial term in the orbital normal form.