Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation

TL;DR

This paper uses validated continuation to rigorously analyze the global structure of slowly oscillating periodic solutions in Wright's delay equation, providing partial evidence towards the conjecture of their uniqueness for all parameter values.

Contribution

The paper reformulates the longstanding conjecture and applies validated continuation to compute a global branch of solutions, showing it lacks fold points or secondary bifurcations.

Findings

Part of the solution branch has no fold points.

Part of the solution branch has no secondary bifurcations.

Provides partial confirmation of the conjecture for certain parameters.

Abstract

An old conjecture in delay equations states that Wright's equation \[ y'(t)= - \alpha y(t-1) [ 1+y(t)], \alpha \in \mathbb{R} \] has a unique slowly oscillating periodic solution (SOPS) for every parameter value $\alpha>\pi/2$. We reformulate this conjecture and we use a method called validated continuation to rigorously compute a global continuous branch of SOPS of Wright's equation. Using this method, we show that a part of this branch does not have any fold point nor does it undergo any secondary bifurcation, partially answering the new reformulated conjecture.