On the bifurcation of periodic orbits

TL;DR

This paper surveys recent advances in Bautin's bifurcation theory, focusing on higher-order Hopf bifurcations, return mapping techniques, and explicit domain estimates crucial for applications in biology and physiology.

Contribution

It provides an overview of effective methods for analyzing bifurcations of periodic orbits, emphasizing explicit estimates and higher-order Hopf bifurcation analysis.

Findings

Effective analysis of higher-order Hopf bifurcations using return maps

Explicit estimates of domain sizes for limit cycle control

Applications in biology and physiology

Abstract

This article is a survey on recent contributions to an effective version of Bautin's theory about the bifurcation of periodic orbits (limit cycles). The analysis of Hopf bifurcations of higher order is possible by use of the return mapping. Explicit estimates of the size of the domain on which the number of limit cycles is controlled are important in many applications of bifurcation theory (for instance in Biology and Physiology).