The application of Puiseux-Newton diagram on the asymptotic analysis of multiple imaginary characteristic roots of LTI delay systems

The paper presents an application of the Puiseux-Newton diagram associated with the Reduction Theorem (a simple case of Weierstrass Preparation Theorem) to the study of linear time-invariant delay systems, focusing on the asymptotic behavior of critical characteristic roots on the imaginary axis which is necessary for the stability analysis. With the systems given in quasi-polynomials, we characterize the asymptotic behaviors of the characteristic roots of such systems in an algebraic way and determine whether the imaginary roots cross from one half plane into another or only touch the imaginary axis. An analogue of Weierstrass Preparation Theorem has been proposed to reduce the characteristic equation to the algebraic equation whose explicit expression is also given in this paper, from which the classic Puiseux-Newton diagram can be used to obtain the asymptotic expansions directly. Some illustrative examples complete the paper.