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

## TL;DR

This paper proves the uniqueness and asymptotic stability of slowly oscillating periodic solutions to Wright's equation for a range of parameters, advancing the understanding of the equation's dynamics and supporting the Jones' Conjecture.

## Contribution

The paper introduces a branch and bound algorithm combined with eigenvalue analysis to rigorously prove stability and uniqueness of solutions for Wright's equation across a parameter interval.

## Key findings

- All SOPS are asymptotically stable for α in [1.9,6.0]
- Established a method to control Floquet multipliers rigorously
- Supported the Jones' Conjecture through these results

## Abstract

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.