seminar: Tangential Center problem for Darboux Centers
speaker: Pavao Mardesic (Université de Bourgogne, Francia)abstract: I will present a joint work with Colin Christopher (U. Plymouth UK). It concerns planar polynomial vector fields.
A center of a planar vector field is a singularity surrounded by a continuous family of periodic orbits. The center problem (Poincaré), asks for a characterization of polynomial vector fields having a center. It is solved in a satisfactory way only for quadratic vector fields (Dulac, Kapteyn around 1900). We note that all quadratic vector fields with a center have a Darboux first integral. That is a first integral of the form F(x,y)=\prod{j=1}k fj(x,y){lj}, with fj(x,y) polynomials and lj real numbers.
In our problem, we start with a system having a center and a Darboux first integral F. We call it a Darboux center. We study polynomial deformations of the given system. The infinitesimal center problem asks to identify all deformations preserving a center. The tangential center problem is the first order version of this problem. It amounts to characterize the vanishing of pseudo-abelian integrals; that is integrals of rational differential one-forms along cycles D(h)\subset F{-1}(h).
We solve the tangential center problem for Darboux centers under some conditions on the first integral F. Our results generalizes analogous results of Ilyashenko (1969) on abelian integrals in the case when the first integral is polynomial
timetable:
Wed 15 Apr, 10:00 - 11:00, Sala Conferenze Centro De Giorgi
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