$\alpha \in ( {\tfrac{\pi}{2}}, 1.9]$, thus proving the Jones conjecture.

Furthermore, Theorem \[prop:Jones\] allows us to deduce that there are no isolas of rapidly oscillating periodic solutions. Since the nonlinearity in depends only on $x(t-1)$, in fact any periodic orbit is either a SOPS or rescaling thereof. This rescaling between slowly and rapidly oscillating periodic solutions is given in terms of a solution’s lap number [@mallet1988morse] and its period, as detailed in the following theorem:

\[prop:Rescaling\] Let $ x_0$ be a periodic solution to at parameter $\alpha_0$ with period $L_0$ and lap number $ N $. Then there exists a SOPS $x_1(t) = x_0(r t) $ to at parameter $\alpha_1= r\alpha_0$ where $r := 1- \tfrac{N-1}{2} L_0  $.

Thus, every periodic orbit is on a branch originating from one of the Hopf bifurcations at $ \alpha = {\tfrac{\pi}{2}}+ 2 n \pi$. That is to say, there are no isolas of rapidly oscillating periodic solutions. However, this is not sufficient to show there are no folds in the branches of rapidly oscillating periodic solutions. The proofs for Theorem \[prop:Jones\] and Theorem \[prop:Rescaling\] are presented at the end of Section \[sec:GlobalAlgorithm\], and we discuss future directions in Section \[sec:FutureWork\].

Outline of Proof
