Global Bifurcation of Periodic Solutions in Delay Equations with Symmetric Monotone Feedback

We study the periodic solutions of the delay equation $\dot{x}(t)=f(x(t),x(t-1))$, where $f$ scalar is strictly monotone in the delayed component and has even-odd symmetry. We completely describe the global bifurcation structure of periodic solutions via a period map originating from planar ordinary differential equations. Moreover, we prove that the first derivative of the period map determines the local stability of the periodic orbits. This article builds on the pioneering work of Kaplan and Yorke, who found some symmetric periodic solutions for $f$ with even-odd symmetry. We enhance their results by proving that all periodic solutions are symmetric if $f$ is in addition monotone.