A proof of Jones' conjecture

Abstract

In this paper, we prove that Wright's equation $y^{\prime }(t)=-\alpha y(t-1)\{1+y(t)\}$ has a unique slowly oscillating periodic solution for parameter values $\alpha \in (\frac{\pi }{2},1.9]$, up to time translation. This result proves Jones' Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all $\alpha >\frac{\pi }{2}$. Furthermore, there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations.

41 pages, 5 figures, 1 table

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