Stability and uniqueness of slowly oscillating periodic solutions to Wright's equation

Abstract

In this paper, we prove that Wright's equation y′(t)=−αy(t−1){1+y(t)} has a unique slowly oscillating periodic solution (SOPS) for all parameter values α∈[1.9,6.0], up to time translation. Our proof is based on the 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 α∈[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 α∈[1.9,6.0], and the proof follows. This result is a step toward the proof of the Jones' Conjecture formulated in 1962.