A proof of Jones' conjecture

TL;DR

This paper proves Jones' Conjecture by establishing the uniqueness of a slowly oscillating periodic solution for Wright's equation within a specific parameter range, confirming the conjecture for all relevant parameters.

Contribution

The paper provides a rigorous proof that Wright's equation has a unique slowly oscillating periodic solution for all parameters greater than π/2, confirming Jones' Conjecture from 1962.

Findings

Uniqueness of the slowly oscillating periodic solution for α in (π/2, 1.9]

No isolas of periodic solutions exist for Wright's equation

All periodic orbits originate from Hopf bifurcations

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.