Dissipativity and Exponential Stability of $θ$-Methods for Singularly Perturbed Delay Differential Equations with a Bounded Lag (Journal of Computational Mathematics, 2021)
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 $ε>0$. We will study the numerical solution defined by the linear $θ-$method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small $ε>0$ if and only if $θ=1$.