Periodic Solutions And Stability Problems Of Degenerate Differential Systems With Delay

In many practical systems, people have found that there were many degenerate phenomena existing extensively, such as economic systems, power systems, engineering systems and so on. And this phenomenon has attracted many scholars' attention and lots of available and important results have been obtained. However, time delay is a common phenomenon in the objective world and engineering fields. We notice that, in many practical systems, in order to describe the system more accurately, we must take the influence of degenerate and delay into consideration altogether. So, it has a practical significance to study the character of solutions of degenerate differential systems with delay.In this paper, periodic solutions and stability problems of degenerate differential systems with delay are discussed. This thesis of Master is composed of six chapters, which mainly studies the general solutions of general degenerate neutral differential systems, the periodic solutions of degenerate differential systems with delay, the stability for degenerate differential system with delays, the global exponential stability of the equilibrium for a class of delayed differential systems.In chapter one, the backgrounds and significance of the problems are given. The main work done in this paper is introduced.In chapter two, we mainly consider the solutions of general degenerate neutral differential systems when the degenerate matrix E is not square. Based on existence conditions of solutions for degenerate ordinary differential systems, by defining solvable matrix pair and fundamental solution and using Laplace transformation, existence conditions of solutions and general solutions of degenerate neutral differential systems are obtained.In chapter three, the problems of the periodic solutions of degenerate neutral differential equations are considered. We give the conditions of existence of the periodic solutions and the algebraic criterion of existence of the periodic solutions for two-dimensional degenerate neutral differential equations. we also discuss the periodic solutions of nonlinear degenerate differential systems with infinite delay, by using matrix measure and Krasnoselskii's fixed point theorem, and the sufficient conditions of the existence of periodic solutions are obtained.In chapter four, the problems of stability for degenerate differential systems with variable delay and degenerate neutral differential systems with variable delay are discussed. Based on variation formula and Gronwall-Bellman integral inequality, the exponential estimation of the solution and the sufficient conditions of stability and exponential asymptotic stability for those two systems are obtained. We also give the criterion to check stability for a class of degenerate differential systems with delay by using Liapunov functional.In chapter five, the all-delay stability of degenerate differential systems with delay is discussed. We give some new criteria to check the all-delay stability of degenerate differential systems with delay and degenerate neutral differential systems with delay, and give an example to illustrate the main results.In the last chapter, the global exponential stability of the zero solution for a class of differential systems with delay is discussed. By constructing Liapunov functionals and using some analytical techniques, we derive the criteria to check exponential stability of the zero solution for differential systems with delay. The results establish a relation between the delay time and the parameters of the systems. Two illustrative examples are also given to show the applicability and effectiveness of the obtained results.