---
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 for parameter values $\alpha \in (\tfrac{\pi}{2},  1.9]$, up to time translation. This result proves Jones’ Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all $ \alpha > \tfrac{\pi}{2}$. Furthermore, there are no isolas of periodic solutions to Wright’s equation; all periodic orbits arise from Hopf bifurcations.'
author:
- 'Jonathan Jaquette [^1] [^2]'
bibliography:
- 'BibWright.bib'
title: 'A proof of Jones’ conjecture'
---

[**Key words.**]{} [ Wright’s Equation $\cdot$ Jones’ Conjecture $\cdot$ Delay Differential Equations\
Computer-Assisted Proofs $\cdot$ Branch and Bound $\cdot$ Krawczyk method]{}

Introduction
============

An often studied class of delay differential equations are negative feedback systems of the form: $$x'(t) = - \alpha f(x(t-1)) 
\label{eq:MNF}$$ where $ xf(x) > 0 $ for $x \neq 0$ and $ f'(0) >0$. One particularly well studied example of is when $f(x) =e^x -1$, better known as Wright’s equation, which after making the change of variables $ y =e^x -1 $ can be written in the following form: $$y'(t) = - \alpha \,y(t-1) \left[  1+ y(t)  \right].
\label{eq:Wright}$$

In [@jones1962existence], Jones proved 