Topological equivalences for one-parameter bifurcations of maps

TL;DR

This paper constructs homeomorphisms to establish topological equivalences between one-parameter families of maps experiencing the same bifurcation, solving a longstanding classical problem in bifurcation theory.

Contribution

It provides a novel method for proving topological equivalences in bifurcation theory, addressing a problem unsolved for over thirty years.

Findings

Constructed homeomorphisms for topological equivalences

Solved a classical, longstanding problem in bifurcation theory

Established a framework applicable to various bifurcation scenarios

Abstract

Homeomorphisms allowing us to prove topological equivalences between one-parameter families of maps undergoing the same bifurcation are constructed in this paper. This provides a solution for a classical problem in bifurcation theory that was set out three decades ago and remained unexpectedly unpublished until now