Hopf bifurcation at infinity and dissipative vector fields of the plane

Alarcón, Begoña; Rabanal, Roland

doi:10.1090/proc/13462

Hopf bifurcation at infinity and dissipative vector fields of the plane
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Proc. Amer. Math. Soc. 145 (2017), 3033-3046 Request permission

Abstract:

This work deals with one–parameter families of differentiable (not necessarily $C^1$) 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. It is worth noting that the adopted approach is not restricted to consider vector fields which are strongly dominated by the linear part. Moreover, the Poincaré compactification is not applied in this paper.

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