Global planar dynamics with star nodes: beyond Hilbert’s 16th problem

This is a complete study of the dynamics of polynomial planar vector fields whose linear
part is a multiple of the identity (the origin is a star node) and whose nonlinear part is
a homogeneous polynomial. It extends previous work by other authors that was mainly
concerned with the existence and number of limit cycles (about Hilbert’s 16th problem).
We show that the behaviour at infinity completely determines the dynamics on the plane.
The general results are also applied to two classes of examples where the nonlinearities
have degrees 2 and 3, for which we provide a complete set of phase portraits.
Joint work with Begoña Alarcón (UFF, Brazil) and Sofia Castro (CMUP and FEP.UP).