Global Dynamics and Periodic Solutions in a Singular Differential Delay Equation

Differential delay equation 𝜀 x ′ ( t ) + c x ′ ( t − 1 ) + x ( t ) = f ( x ( t − 1 ) ) $$\varepsilon \left [x^{\,{\prime}}(t) + cx^{\,{\prime}}(t - 1)\right ] + x(t) = f(x(t - 1))$$ is considered where 𝜀 > 0 $$\varepsilon> 0$$ and c ∈ R are parameters, and f: R → R is piece-wise continuous. For small values of the parameter 𝜀 $$\varepsilon$$ a connection is made to the continuous time difference equation x ( t ) = f ( x ( t − 1 ) ) , $$x(t) = f(x(t - 1)),$$ 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 𝜀 $$\varepsilon$$ and the existence of periodic solutions. Open problems and conjectures are stated for the case of genuinely neutral equation, when c ≠ 0.