Georgia Institute of Technology College of Sciences

A conjecture of Buchanan and Lillo states that all nontrivial oscillatory solutions of
\begin{equation*}
x'(t)=p(t)x(t-\tau(t)),
\end{equation*}
 with $0\leq p(t)\leq 1,0\leq \tau(t)\leq 2.75+\ln2 \approx 3.44$ tend to a known function $\varpi$, which is antiperiodic:
 \begin{equation*}
 \varpi(t+T/2)\equiv - \varpi(t)
 \end{equation*}
 where $T$ is its minimal period. We discuss recent developments on this question, focusing on the periodic solutions characterizing the threshold case. We consider the case of positive feedback ($0\leq p(t)\leq 1$) with $\sup\tau(t)= 2.75+\ln2$, the well-known $3/2$-criterion corresponding to negative feedback ($0\leq -p(t)\leq 1$) with $\sup\tau(t)=1.5$, as well as higher order equations. 

 We investigate the behavior of the threshold periodic solutions under perturbation together with the symmetry (antiperiodicity) which characterizes them. This problem is set within the broader background of delay effects on stability for autonomous and nonautonomous equations, taking into account the fundamental relation between oscillation speed and dynamics of delay equations. We highlight the crucial role of symmetry in both the intuitions behind this vein of research, as well as the relevant combinatorial-variational problems.