| In this paper,we mainly study the asymptotic behavior library of time-delay systems,and determine the asymptotic behavior characteristics triad(S,g,n)and asymptotic behavior rules of time-delay systems.The asymptotic behavior of time-delay systems is classified based on the asymptotic behavior characteristics traid(S,g,n),and the asymptotic behavior of time-delay systems is obtained when the topological invariants(S,g,n)are determined.At present,the study of time-delay systems is to analyze the stability.With the continuous development of big data,the complexity of time-delay systems in the future can be predicted.In this paper,an analytic curve method is introduced to analyze the curve information of time-delay systems.Through the analysis of the literature,it is found that the asymptotic behavior of the time-delay system has some rules,and the asymptotic behavior of the time-delay system is analyzed numerically.It is obviously difficult to cover all cases,so the concept of topology is introduced.The analytic curves of time-delay systems are analyzed by introducing the Puiseux series.Because of the equivalence relation between the Puiseux series and the dual Puiseux series,the dual Puiseux series is introduced.The time-delay systems are divided into singular types and non-singular types,the time-delay systems of non-singular types are analyzed and the topological equivalence relations are determined.For the time-delay system of singular type,by introducing the Blowing-up method in topology and the concept of Puiseux pair,a set of resolution method for asymptotic behavior characteristics triad(S,g,n)is established,and the information matrix is established.Because the time-delay systems with the same asymptotic behavior characteristics triad(S,g,n)have the same resolution information matrix,the asymptotic behavior characteristics triad is determined to be a topological invariant.The asymptotic behavior of time-delay systems is classified based on the asymptotic behavior characteristics triad(S,g,n).By analyzing the asymptotic behavior of time delay systems of non-singular types,a method for analyzing time delay systems of singular types is established.Based on the case that the Puiseux series in the first term is non-degenerate(S=n+g-1),the asymptotic behavior of the time-delay system is studied in the case that the system with time delay is degenerate(S>n+g-1)and the Puiseux series terms are continuous,the asymptotic behavior of the time-delay system is determined.In this paper,topological invariants of asymptotic behavior for time-delay systems are determined by introducing the concept of topology in mathematics.Based on the study of the asymptotic behavior for time-delay systems,the asymptotic behavior rules of time-delay systems are obtained. |
