Triple equivalence of the oscillatory behavior for scalar delay differential equations

We study the oscillation of a first-order delay equation with negative feedback at the critical threshold $1/e$. We apply a novel center manifold method, proving that the oscillation of the delay equation is equivalent to the oscillation of a $2$-dimensional system of ordinary differential equations (ODEs) on the center manifold. It is well known that the delay equation oscillation is equivalent to the oscillation of a certain second-order ODE, and we furthermore show that the center manifold system is asymptotically equivalent to this same second-order ODE. In addition, the center manifold method has the advantage of being applicable to the case where the parameters oscillate around the critical value $1/e$, thereby extending and refining previous results in this case.