Periodic solutions of a discretized differential delay equation

Several aspects of dynamics are addressed for the differential-difference equation ∈ẋ(t) + x(t) = f(x([t - k + 1])),0 < ∈ ≪ 1, where [̇] is the integer part function, k is a positive integer. The equation can be viewed as a special discretization (discrete version) of the singularly perturbed differential delay equation ∈ẋ(t) + x(t) = f(x(t - k)). Sufficient conditions for the invariance, existence, stability and shape of periodic solutions are derived. The principal analysis is based on reduction to special multi-dimensional maps whose relevant properties follow from those of 1D map f.