Some Properties Of A Class Of Singular Analytical Curve

We use the classical conclusion of frequency analysis-Routh Hurwitz criterion, to determine the stability of the retarded time-delay system by analysis the spectrum of retarded time-delay system’s characteristics equation. Due to the characteristics equation of the time-delay systems is transcendental equation, this equation in the complex plane corresponds to infinite isolated eigenvalues, and compute the precise position of each root is an arduous task. So, using the method of partition the parameter space of time-delay system, It should be point out that it’s not easy to compute the number of positive real eigenvalue in a certain period of time delay interval. So we analysis the across direction of imaginary characteristic roots in adjacent delay interval, If we know the number of unstable characteristic roots in a certain interval of time delay, we can compute the number of unstable characteristic roots in the subsequent intervals by recursive calculation, and some the complete stability of time-delay systems can be relieazed.The result about the simple imaginary root had been researched, our main research on multiple imaginary roots.Due to the background of Automatic control, we describe our main work in mathematics, namely to analysis the properties near the singular point of singular analysis curve, we mainly analyze the properties near the singular point from the view of algebraic. We found that multiple imaginary roots correspond several groups of Puiseux series, each Puiseux series correspond to a conjugacy class of Puiseux series. At first, we discusses a invariance ANU of a conjugacy class of Puiseux series, which react stability of system, and we found that Each conjugate Puiseux series have this property. Then we can analyze the complete stability of the time-delay system.