Distributions Of The Eigenvalues Of The Differential Equations With Time Delays

In recent years,the research of the equation with time delay has attracted wide attention of the academic circles.Compared with the differential equation,the delay differential equation have widely used in the research of demography,natural ecology and radar warning system,vibration control,neural network.For the stability,asymptotic and periodic synchronization of the analysis system equilibrium,the distribution of the system characteristic toot needs to be given.However,in general,the delay differential equation has an infinite number of characteristic roots,and it is impossible to find out these characteristic roots precisely.This paper mainly considers the distribution of the characteristic toot of the feedback system with small amplitude delay state,and provide the research idea for the analysis of the local dynamics of the system.This paper first introduces the development course and related basic knowledge of the delay differential equation.For example,the definition of stability,gradual stability of two kinds of delay differential and its distribution characteristics of characteristic toot,delay differential equation and differential equation and its particularly.In this paper,we mainly study the characteristic curves of the delay differential equation with weak time delay:And<<0,we analyze the distribution of the system characteristic,and the main characteristics are estimated by Taylor expansion and perturbation method.The article also studies the delay equation with small perturbation ?1,?2 analyze the distribution of its characteristic roots,and estimate its main characteristics.