Some remarks on Puiseux Series for multiple imaginary characteristic roots of linear time-delay systems

Obtaining the Puiseux Series expansion of multiple imaginary (characteristic) roots (MIRs) is a fundamental issue in the stability analysis of time-delay systems. However, this issue has not been fully investigated up to date. This work focuses on the Puiseux Series expansion of MIRs for a class of linear time-delay systems including commensurate delays. For an MIR of any multiplicity, we propose an algorithm for defining the structure of the Puiseux Series, as well as the explicit computation of the corresponding coefficients. By using the proposed method, we can find all the Puiseux Series corresponding to all the root trajectories. Some illustrative examples complete the work.