Abstract:

In this talk we answer an open question suggested by Galeotti and Gori in 1987 about the stability of the origin for the system

begin{equation}
left{begin{array}{lll}
dot{x}=y^3-x^{2k+1},\
dot{y}=-x+my^{2s+1}, quad minmathbb{R},quad k,sinmathbb{N}^+.
end{array} ight.end{equation}
In addition, in the case $k=1$ and $s=2$, using the Bendixon-Dulac theorem, we prove that the system has at most one hyperbolic limit cycle and that it can only exist when $min(0.547,0.6)$ improving previous results. We also consider the question of the existence of polycycles. The main interest and difficulty for studying this family is that, although it looks simple, it is not a semi-complete family of rotated vector fields.

This is joint work with Armengol Gasull and Hector Giacomini.

The talk will be in Spanish.