Rigorous verification of Hopf bifurcations via desingularization and continuation

TL;DR

This paper introduces a rigorous computational method combining analytic estimates and computer-assisted calculations to validate Hopf and saddle-node bifurcations of periodic orbits in ODE systems, extending validated continuation techniques.

Contribution

It presents a novel approach that uses blowup analysis and validated continuation to rigorously verify bifurcations in dynamical systems, enhancing the mathematical rigor of bifurcation analysis.

Findings

Successfully validated Hopf bifurcations in example systems

Rigorously continued solution curves through bifurcations

Extended applicability of validated continuation methods

Abstract

In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems.