Stable and non-symmetric pitchfork bifurcations

TL;DR

This paper introduces a new topological criterion for identifying pitchfork bifurcations in smooth vector fields, expanding previous results and demonstrating its application to non-symmetric cases.

Contribution

It provides a novel topological criterion for pitchfork bifurcations and applies it to non-symmetric vector field families, broadening the understanding of bifurcation phenomena.

Findings

New criterion for pitchfork bifurcation based on topology

Application to non-symmetric vector fields

Significant extension of previous theoretical results

Abstract

In this paper, we present a criterion for pitchfork bifurcation of smooth vector fields based on a topological argument. Our result expands Rajapakse and Smale's result \cite{RS2} significantly. Based on our criterion, we present a class of families of non-symmetric vector fields undergoing a pitchfork bifurcation.