Stability and Uniqueness
of Slowly Oscillating Periodic Solutions to Wright’s
Equation
Jonathan Jaquette,
Jean-Philippe Lessard and Konstantin Mischaikow.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 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 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.