On the dynamics of a class of difference equations with continuous arguments and its singular perturbation

Abstract

The dynamical properties of a class of difference equations with continuous arguments of the form x(t)=g(x(t-r1),x(t-r2)) and its singularly perturbed counterpart ∊dxdt=-x(t)+g(x(t-r1),x(t-r2)) are investigated here. We discuss the effect of the time delays r1 and r2 on the qualitative behavior of the considered dynamical systems. The local stability of the fixed points is studied. It is proved that the systems exhibit Hopf bifurcation which means that periodic orbits can be created from a fixed point by varying the delays. We compare the results of the singularly perturbed equation with those of the associated difference equation with continuous arguments when the perturbation parameter ∊⟶0 and with those of the corresponding delay differential equation when ∊⟶1. By letting the perturbation parameter ∊⟶0, we find that the singularly perturbed equation exhibits the same qualitative behavior as its corresponding difference equation. Furthermore, the singularly perturbed equation behaves qualitatively the same as its corresponding delay differential equation when ∊⟶1. Finally, we discuss that how this work can be generalized in the fractional differential calculus sense.