Global bifurcations in generic one-parameter families with a parabolic cycle on $S^2$

TL;DR

This paper classifies global bifurcations in generic one-parameter families of vector fields on the sphere with a parabolic cycle, revealing differences from classical bifurcation theory and establishing structural stability.

Contribution

It provides a new classification of bifurcations involving parabolic cycles on $S^2$, contrasting with classical results, and proves the structural stability of these families.

Findings

New bifurcation classification for vector fields with parabolic cycles

Demonstration of structural stability in generic families

Distinct from classical bifurcation results

Abstract

We classify global bifurcations in generic one-parameter local families of \vfs on $S^2$ with a parabolic cycle. The classification is quite different from the classical results presented in monographs on the bifurcation theory. As a by product we prove that generic families described above are structurally stable.