Dissipativity and exponential stability of θ-methods for singularly perturbed delay differential equations with a bounded lag

This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small epsilon > 0. We will study the numerical solution defined by the linear theta-method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small epsilon > 0 if and only if theta = 1.