The Application Of Averaging Method In The Theory Of The Ergodic

Averaging method appeared as early as in the time of Lagrang and Laplace who used it in celestial mechanics. Starting in 1930's, the averaging methods become an important research area. For the past seventy years, the research got substantial development. This method has solved many problems of dynamical system. The more areas it has been used in, the more people realize it. And then, Brad Lehman and many others developed it further and used it in the engineering. Averaging method is an important tool in solving the differential equation. The method and its idea can be used to find the relationship between the solution of the autonomous system and nonautonomous system. This relation can simplify the differential equation, and then make it easier to solve.In this paper, the averaging method is used into dealing with the ergodic sequence and the quantitative aspect of a differential equation in dynamical system. At first, we give the background of this problem. The background consists of two parts. One is the averaging method, including the development and conclusions of it. The other is the theory of ergodic, including the definition and the properties of it. Based on this, we get a new averaging method. Using the new method we get the definition of ergodic sequence and the properties. On the other hand, we deal with a practical problem using these properties: a quantitative aspect of a differential equation in dynamical system. This is an important application in dynamical system.