Hopf bifurcation at infinity and dissipative vector fields of the plane

Abstract

We describe some families of differentiable vector fields with the Hopf bifurcation at infinity, without assuming the continuous differentiability. These vector fields have isolated singular points on the plane, and the initial families are obtained by special perturbations at infinity of a vector field with some spectral property, for instance the dissipativity. The strong domination imposed by the spectral condition in the differentiable vector field is used, and then we do not apply the standard Poincaré compactification. Moreover, the perturbation of planar systems with a global period annulus is also considered.