Stability and Uniqueness of Slowly Oscillating Periodic Solutions to Wright's Equation

In this paper, we prove that Wright's equation $y'(t) = - \alpha y(t-1) \{1 + y(t)\}$ has a unique slowly oscillating periodic solution (SOPS) for all parameter values $\alpha \in [ 1.9,6.0]$, up to time translation. Our proof is based on a same strategy employed earlier by Xie [27]; show that every SOPS is asymptotically stable. We first introduce a branch and bound algorithm to control all SOPS using bounding functions at all parameter values $\alpha \in [ 1.9,6.0]$. Once the bounding functions are constructed, we then control the Floquet multipliers of all possible SOPS by solving rigorously an eigenvalue problem, again using a formulation introduced by Xie. Using these two main steps, we prove that all SOPS of Wright's equation are asymptotically stable for $\alpha \in [ 1.9,6.0]$, and the proof follows. This result is a step toward the proof of the Jones' Conjecture formulated in 1962.