Hopf bifurcation at infinity for planar vector fields

We study, from a new point of view, families of planar vector fields without singularities $ \{ X_{\mu}$  &nbsp:&nbsp  $-\varepsilon < \mu < \varepsilon\} $ defined on the complement of an open ball centered at the origin such that, at $\mu=0$, infinity changes from repellor to attractor, or vice versa. We also study a sort of local stability of some $C^1$ planar vector fields around infinity.