Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations

We study the singularly perturbed state-dependent delay-differential equation(⁎)εx˙(t)=−x(t)−kx(t−r),r=r(x(t))=1+x(t), which is a special case of the equationεx˙(t)=g(x(t),x(t−r)),r=r(x(t)). One knows that for every sufficiently small ε>0, Eq. (⁎) possesses at least one so-called slowly oscillating periodic solution, and moreover, the graph of every such solution approaches a specific sawtooth-like shape as ε→0. In this paper we obtain higher-order asymptotics of the sawtooth, including the location of the minimum and maximum of the solution with the form of the solution near these turning points, and as well an asymptotic formula for the period. Using these and other asymptotic formulas, we further show that the solution enjoys the property of superstability, namely, the nontrivial characteristic multipliers are of size O(ε) for small ε. This stability property implies that this solution is unique among all slowly oscillating periodic solutions, again for small ε.