We present one–parameter families of differentiable planar vector fields for which the infinity reverses its stability as the parameter goes through zero. These vector fields are defined on the complement of some compact ball centered at the origin and have isolated singularities. They may be considered as linear perturbations at infinity of a vector field with some spectral property, for instance, dissipativity. We also address the case concerning linear perturbations of planar systems with a global period annulus. We strongly focus on the change of the sign of an index defined at infinity. Joint work with Roland Rabanal.
[AR17] B. Alarcon and R. Rabanal. Hopf bifurcation at infinity and dissipative vector fields of the plane, Proc. Amer. Math. Soc. 145 (2017), 3033-3046.