A proof of Jones' conjecture

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.