The Solution, Periodic Solution And Hopf Bifurcation Of Degenerate Differential System With Delay

In 1974, Rosenbrock put forward the degenerate system originally and formally during the time when he studied the complex electric network system. Since 1970s, people found increasingly that there were many degenerate phenomena existing extensively in many practical fields, such as engineering, society, economy, biology and so on. And this phenomenon has attracted many scholars' attention and lots of available and important results have been obtained (such as [1-12]). However, because delay is a common phenomenon in the objective world and engineering fields. Thus, since 1850s, scholars began to study the system with delay systematically and also made substantial and comprehensive progress (such as [5-19]). Meanwhile, we notice that,in many practical systems, in order to describe the system more accurately and further to design, analysis and apply it, we must take the influence of delay and being degenerate into consideration altogether. So, it is of important practical meanings to study the degenerate differential system with delay.In recent years, many results have been obtained on the studying of the degenerate differential system with delay(such as [4-11]). But, because it has not taken a long time to study such kind of systems, a complete and systematic theory about it has not been formed, which needs our improvement on studying the system further. And in the paper, the solution, periodic solution and Hopf bifurcation of the degenerate differential systems with delay are discussed and many important results are also given in it.There are four chapters in this paper. In the first chapter, the preliminary knowledge which is necessary in the paper is given. The basic concept of {1}-inverse, matrix measure and Hopf bifurcation are given. In chapter 2,based on {l}-inverse structure, the normalization condition of general degenerate differential equations with delay is given. As for the normalizable linear degenerate differential equations with delay, the general expression of their solutions is also given in it. In chapter 3, the degenerate differential system with delay is discussed. By making use of matrix measure and Krasnoselskii's fixed point theorem, the sufficient condition of the existence of its periodic solution is established, and an example is given in order to illustrate its application. In chapter 4, a general 2-dimensional degenerate differential system with delay is discussed. When r ^ 0, the rang for absolute stability of an equilibrium point of this system is given. Also, being regarded the delay r as a parameter, the conditions under which the system has a Hopf bifurcation can be obtained in this chapter.