Solution of Systems of Linear Delay Differential Equations Via Lambert Functions

Abstract

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 to the concept of the state transition matrix in linear ordinary differential equations enables the approach to be used for general classes of linear DDE’s in matrix form. The solution is in the form of an infinite series of modes written in terms of Lambert functions. Results are presented for stability criteria for the individual modes, free response, and forced response in the context of specific examples. This new approach is also applied to the problem of chatter stability in a machining operation on a lathe. The results, since they are only for individual modes, and there are an infinite number of them, represent a necessary condition for system stability.