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

摘要

We study a family of scalar differential equations with a single parameter a > 0 and delay r > 0. In the case of the constant delay r = 1 it is known that for parameters 0 < a < 1 the trivial solution of this family is asymptotically stable, whereas for a > 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 r = r(x(t)) > 0 instead of the constant one, and generalize the result on the existence of slowly oscillating periodic solutions for parameters a > 1 under modest conditions on the delay function r.