Location and numerical preservation of characteristic roots of delay differential equations by LMS methods

The local stability of steady state solutions of differential equations with time delays is determined by the roots of a nonlinear characteristic equation. These characteristic roots can be computed by e.g. the discretization of the solution operator using linear multistep (LMS) methods. Ideally, this numerical procedure ensures that all characteristic roots with real part larger than a given constant are computed accurately. This requires some knowledge on the location of these roots. The reliability of the numerical results depends on the preservation of these roots by the discrete approximation. Here we present theoretical results for both issues. The theoretical foundation obtained in this paper allows effective improvements to the numerical procedure.