Abstract:

In this talk we consider the family of planar differential systems given by
begin{equation*}label{nLoud}
(mathcal L_{n,mu})qquadleft{
begin{array}{l}
dot x=-y+Bx^{n-1}y, \ dot y=x+Dx^n+Fx^{n-2}y^2,
end{array}
ight.
end{equation*}
with $mu!:=(B,F,D)inmathbb{R}^3$ and an odd natural number $nge 3$. This family is an extension to higher degree of the Loud's systems. The origin is a nondegenerate center for all values of the parameter and we are interested in the qualitative properties of its period function. By analyzing the ideal generated
by the period constants, in a previous work we proved that $(mathcal L_{n,mu})$ has at most one (respectively, two) critical periods bifurcating from the origin for $n$ odd (respectively, even). Expecting this local bound to be global, we tackle the case $n$ odd. This case has the additional advantage that the phase portrait of $(mathcal L_{n,mu})$ is symmetric with respect to both coordinate axes and this simplifies the computations that we must carry through. In this talk we will focus on the bifurcations occurring at the polycycle that bounds the period annulus of the center.

This is a joint work with David Marín and the talk will be in catalan.