Bifurcation at Infinity in Polynomial Vector Fields

We study here the appearance of limit cycles from the equator in polynomial vector fields with no singular points at infinity: this bifurcation is a generalized Hopf bifurcation from the point at infinity. We start with the general theory and then specialize to the particular case of cubic polynomial systems for which we study the simultaneous bifurcation of limit cycles from the origin and from the equator. We finally discuss the transformation of a weak focus at infinity into a finite weak focus.