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

We study the singularly perturbed state-dependent delay-differential equation εẋ(t) = −x(t)− kx(t− r), r = r(x(t)) = 1 + x(t), (∗) which is representative of a broader class of such equations of the form εẋ(t) = g(x(t), x(t− r)), r = r(x(t)). It is known that for every sufficiently small ε > 0, equation (∗) 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 a higher-order asymptotic description 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 a property which we term superstability, namely, that the nontrivial characteristic multipliers are of size O(ε) for small ε. Additionally, this stability property implies that this solution is unique among all slowly oscillating periodic solutions of (∗), again for small ε.