Symmetry and the Buchanan-Lillo conjecture: A resolution of the mixed feedback case

Buchanan and Lillo both conjectured that oscillatory solutions of the first-order delay differential equation with positive feedback $x^{\prime }\left(t\right)=p\left(t\right)x\left(\tau \right(t\left)\right)$, $t\ge 0$, where $0\le p\left(t\right)\le 1$, $0\le t-\tau \left(t\right)\le 2.75+\ln 2,t\in R$, are asymptotic to a shifted multiple of a unique periodic solution. This special solution can also be described from the more general perspective of the mixed feedback case (sign-changing p), thanks to its symmetry (antiperiodicity). The analogue of this conjecture for negative feedback, $p\left(t\right)\le 0$, was resolved by Lillo, and the mixed feedback analog was recently set as an open problem. In this paper, we resolve the case of mixed feedback, obtaining results in support of the conjecture of Buchanan and Lillo, underlining its link to the symmetry of the periodic solution. In particular, we obtain and describe the optimal estimates on the necessary delay for existence of periodic (more generally, nonvanishing) solutions, with respect to the period (oscillation speed). These apply to almost any first-order delay system, as we consider the general nonautonomous case, under minimal assumptions of measurability of the parameters. We furthermore discuss and elucidate the relations between the periodic and the nonautonomous case.