Characterization of the generic unfolding of a weak focus
The author considers analytic families of planar vector fields depending on a real parameter \(\mu\) and having a singular point at the origin with eigenvalues \(\alpha(\mu)\pm\beta(\mu)\,i\), with \(\alpha(0)=0\) and \(\beta(0)\neq0\), which are generic (\(\alpha'(0)\neq0\)) and when \(\mu=0\) have a weak focus of first order (the first Lyapunov constant is non-zero) at this point. He proves that their orbital class is determined by the conjugacy class of their associated Poincaré monodromy and vice versa. The natural approach to these questions is via complexification of time, coordinates and parameter. The used tools include quasiconformal surgery and normal form theory.
scientific article; zbMATH DE number 6069298
scientific article