On a Differential Equation with State-Dependent Delay A Center-Unstable Manifold Connecting an Equilibrium and a Periodic Orbit

We study a family of scalar differential equations with a single parameter > 0 and delay > 0. In the case of the constant delay = 1 it is known that for parameters 0 < < 1 the trivial solution of this family is asymptotically stable, whereas for > 1 the trivial solution gets unstable, and a global center-unstable manifold connects the trivial solution to a slowly oscillating periodic orbit. Here, we consider a state-dependent delay = (()) > 0 instead of the constant one, and generalize the result on the existence of slowly oscillating periodic solutions for parameters > 1 under modest conditions on the delay function .