Global dynamics of a differential equation with piecewise constant argument

Several aspects of global dynamics are studied for the scalar differential-difference equation ε over(x, ̇) (t) + x (t) = f (x ([t])), 0 < ε ≪ 1, where [{dot operator}] is the integer part function. The equation is a particular case of the special discretization (discrete version) of the singularly perturbed differential delay equation ε over(x, ̇) (t) + x (t) = f (x (t - 1)). Sufficient conditions for the invariance, global stability of equilibria, existence, stability/instability, and shape of periodic solutions, and the chaotic behavior are derived. The principal analysis is based on the reduction of its dynamics to the one-dimensional map F : x → f (x) + [x - f (x)] e^{- 1 / ε}, many relevant properties of which follow from those of the interval map f.