Hopf bifurcation at infinity and dissipative vector fields of the plane

TL;DR

This paper investigates families of planar vector fields exhibiting Hopf bifurcation at infinity, focusing on spectral properties and perturbations without relying on standard compactification methods.

Contribution

It introduces a novel approach to analyze Hopf bifurcation at infinity in non-differentiable vector fields using spectral conditions and perturbations, bypassing Poincaré compactification.

Findings

Identified conditions for Hopf bifurcation at infinity in dissipative vector fields.

Developed a method to analyze perturbations of systems with a global period annulus.

Extended bifurcation analysis to non-differentiable vector fields with spectral constraints.

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 Poincar\'e compactification. Moreover, the perturbation of planar systems with a global period annulus is also considered.