Random Specification Of Some Of The Manifold Method

Normal Form theory is a useful tool for the research of dynamical system, its main idea is to study the equations of the system and transform the equations into simple forms. In this paper, we introduce the develop history of Normal Form theory and methods, which include some basic concepts and theorems. We discuss and further more, proof the rule of deterministic near identity transformations (Chapter 1). Then we study the stochastic case, which is in its beginning period. Proof the rule of stochastic near identity transformations, these are important for the research of Normal form. We also provide a method to calculate the Stochastic Normal Form of Noisy Van der Pol-Duffing oscillator. We got 104 homological equations and an optimal result (Chapter 2). Basing on the work of Zhang and Leung, we present two new concepts: stochastic Averaging Normal Forms and High order stochastic Averaging Normal Forms. The fore is based on the work of Sri and Leung; we use the last method calculating Duffing oscillator system, the result we got is prior to that of the former (Chapter 3).