Analysis of a system of linear delay differential equations

A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. The similarity with the concept of the state transition matrix in linear ordinary differential equations enables the approach to be used for general classes of linear delay differential equations using the matrix form of DDEs. The solution is in the form of an infinite series of modes written in terms of Lambert functions. Stability criteria for the individual modes, free response, and forced response for delay equations in different examples are studied, and the results are presented. The new approach is applied to obtain the stability regions for the individual modes of the linearized chatter problem in turning. The results present a necessary condition to the stability in chatter for the whole system, since it only enables the study of the individual modes, and there are an infinite number of them that contribute to the stability of the system.