The Stability And Boundedness Of Non-linear High-order Differential Equation With Delays

In 1982,mathematician and dynamicist Lyapunov in Russia introduce the strict exact mathematical definition and general methods on motion stability in his D. thesis titled "general probems of motion stability", which established the basis of motion stability. With the development of science and technology, the stability theory of Lyapunov is widely applied and developed in many fields, such as physics, engineer technology, ecological system, hereditary problems and neural networks, etc .In this paper, the stability and boundedness of second-order and third-order nonlinear differential equations is studied by means of analogy method.This paper consists of four parts.In the preface, the research actuality of higher-order and the significance of this paper is introduced.In chapter one, the basic definitions and necessary lemmas are posed.In chapter two, the boundedness of solution of a class of second-order retarded nonlinear differential system is discussed which extends the results of document[20],Moreover, the bounded problems of non-ordinary second-order retarded non-linear differentialsystem is solved.In chapter three, the global stability of a class of third-order retarded non-linear differential system is discussed and the results generalizes the results of literatures [21 -22].