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

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.