The Stability of Difference Formulas for Delay Differential Equations

Abstract

A new simple geometric technique is presented for analyzing the stability of difference formulas for the model delay differential equation \[ y'(t) = py(t) + qy(t - \delta ), \] where p and q are complex constants, and the delay $\delta $ is a positive constant. The technique is based on the argument principle and directly relates the region of absolute stability for ordinary differential equations corresponding to the $py(t)$ term with the region corresponding to the delay term $qy(t - \delta )$. A sufficient condition for stability is that these regions be disjoint. The technique is used to show that for each A-stable, $A(\alpha )$-stable, or stiffly stable linear multistep formula for ordinary differential equations, there is a corresponding linear multistep formula for delay differential equations with analogous stability properties. The analogy does not extend, however, to A-stable one-step formulas.