Global Dynamics and Periodic Solutions in a Singular Differential Delay Equation

Abstract

Differential delay equation $\varepsilon \left [x^{\,{\prime}}(t) + cx^{\,{\prime}}(t - 1)\right ] + x(t) = f(x(t - 1))$ is considered where $\varepsilon> 0$ and c ∈ R are parameters, and f: R → R is piece-wise continuous. For small values of the parameter $\varepsilon$ a connection is made to the continuous time difference equation $x(t) = f(x(t - 1)),$ which is further linked to the one-dimensional dynamical system x ↦ f(x). Two cases of the nonlinearity f are treated: when it is continuous and of the negative feedback with respect to a unique equilibrium, and when it is of the so-called Farrey-type with a single jump-discontinuity. Several properties are studied, such as continuous dependence of solutions on the singular parameter $\varepsilon$ and the existence of periodic solutions. Open problems and conjectures are stated for the case of genuinely neutral equation, when c ≠ 0.