Metastable periodic patterns in singularly perturbed state-dependent delayed equations

Abstract

We consider the scalar delayed differential equation ǫẋ(t) = −x(t)+f (x(t−r)), where ǫ > 0, r = r(x, ǫ) and f represents either a positive feedback df /dx > 0 or a negative feedback df /dx < 0. When the delay is a constant, i.e. r(x, ǫ) = 1, this equation admits metastable rapidly oscillating solutions that are transients whose duration is of order exp(c/ǫ), for some c > 0. In this paper we investigate whether this metastable behavior persists when the delay r(x, ǫ) depends non trivially on the state variable x. Our conclusion is that for negative feedback, the persistence of the metastable behavior depends only on the way r(x, ǫ) depends on ǫ and not on the feedback f . In contrast, for positive feedback, for metastable solutions to exist it is further required that the feedback f is an odd function and the delay r(x, ǫ) is an even function. Our analysis hinges upon the introduction of state dependent transtion layer equations that describe the profiles of the transient oscillations. One novel result is that state dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant.

Absolute stability and absolute hyperbolicity in systems with discrete time-delays

“…, This hints that the stability of the system can be obtained at a formal level by a discrete dynamical system. There are many publications devoted to relations between the DDE (2.2) and the singular map (2.3), see [37,[61][62][63][64][65][66][67][68][69][70]. In fact, in order to obtain equivalent stability conditions, one should consider an extended singular map (2.4) x(T ) = (iωI − A 0 ) −1 A 1 e iϕ x(T − 1).…”

Periodic Solutions of a Singularly Perturbed Delay Differential Equation with Two State-Dependent Delays

“…In [31] the SOPS of (1.3) are studied in detail and the shape of the solution near the local maxima and minima is determined for 0 < ε 1 as well as the width of the transition layer, and the "super-stability" of the solution. Other work on singularly perturbed state-dependent DDEs includes [14] where they arise from the regularisation of neutral state-dependent DDEs, and also [34] where the metastability of solutions of a singularly perturbed state-dependent DDE is studied in the case where the state-dependency vanishes in the limit as ε → 0.…”