Asymptotics for the Characteristic Roots of Delayed Dynamic Systems

Delayed dynamical systems appear in many areas of science and engineering. Analysis of general nonlinear delayed systems often begins with the linearized delay differential equation (DDE). The study of these linearized constant coefficient DDEs involves transcendental characteristic equations, which have infinitely many complex roots not obtainable in closed form. Here, after motivating our study with a well-known delayed dynamical system model for tool vibrations in metal cutting, we obtain asymptotic expressions for the large characteristic roots of several delayed systems. These include first- and second-order DDEs with single delays, and a first-order DDE with distributed as well as multiple incommensurate delays. For reasonable magnitudes of the coefficients of the DDEs, the approximations in each case are very good. Subsequently, a fourth delayed system involving coefficients of disparate magnitude is analyzed using an alternative asymptotic strategy. Finally, the large root asymptotics are complemented with calculations using Padé approximants to find all the roots of these systems.