Global dynamics of a differential equation with piecewise constant argument Original Research Article

Several aspects of global dynamics are studied for the scalar differential-difference equation View the MathML sourceεẋ(t)+x(t)=f(x([t])),0<ε≪1, where [⋅][⋅] is the integer part function. The equation is a particular case of the special discretization (discrete version) of the singularly perturbed differential delay equation View the MathML sourceεẋ(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 View the MathML sourceF:x→f(x)+[x−f(x)]e−1/ε, many relevant properties of which follow from those of the interval map ff.