On stability of LMS methods and characteristic roots of delay differential equations

We investigate the use of linear multistep (LMS) methods for computing characteristic roots of systems of (linear) delay differential equations (DDEs) with multiple fixed discrete delays. These roots are important in the context of stability and bifurcation analysis. We prove convergence orders for the characteristic root approximations and analyze under what condition for the steplength the discrete integration scheme retains certain delay-independent stability properties of the original equations. Unlike existing results, we concentrate on the recovery of both stability and instability. We illustrate our findings with a number of numerical test results.