Analysis On Convergence Of The Normalization For Analytic Differential Equations Under The Brjuno Condition

For the general analytic differential equations, it is difficult to find out the solution to discuss or determine its properties. So we thought of a way to solve it. namely, the normal form theory. The normal form theory is an important tool to simplify ordinary differential equations and diffeomorphism. So, here I study analytic differential equations, through a form of variable to transform the system as much as possible into a simple normal form. However, trans-formations and normal forms are divergent generally. I will use the Brjuno conditions to restrict the linear part so that we can prove the convergence of the transformation when the normal form is linear. The thesis is divided into four chapters:In Chapter1, we introduce the development history and research status in the relevant field, introduce main content of the thesis;In chapter2, we present preparatory knowledge, including the normal form theory and the Brjuno condition;Chapter3discusses a linear transformation for the analysed ordinary differential equation, and formulates our main result;Chapter4is divided into three parts. The first part is to construct the iteration transformation. In the second part, we estimate the iteration trans-formation and the new remainder. In the last part, the convergence of trans-formation is demonstrate under the Brjuno condition.