Analysis On Normal Form Near Elliptic Equilibrium Point And Transformation Convergence For Some Analytic Hamiltonian Systems Under The Brjuno Condition

The normal form theory that is important tools to simplify ordinary dif-ferential equations or diffeomorphism, beginning from H.Poincare for hundred years has made great development. Especially in recent years, this theory wide-ly used in the areas of Hilbert16problem, bifurcation theory and research of Hamiltonian systems dynamics.This thesis research of Normal form near elliptic equilibrium point and its transformations Convergence for some analytic Hamiltonian systems under the Brjuno condition, and proves the convergence of its transformations, This thesis is divided into four parts:In chapter1, Introduced normal form theory related to the work and its significance, and then point out the main contentIn Chapter2, Give the prior knowledge, including the Brjuno conditions, elliptic equilibrium point of the Hamiltonian system, etc.In Chapter3, We introduce the method of normalizing Hamiltonian sys-tems by using time—1mapping, and then point out that if the characteristic roots of the Hamiltonian system meet the Brjuno condition and its normal form function has some special form we can find convergent canonical trans-formation;In chapter4, This chapter is divided into three parts, the first part is to construct formal canonical transformations; the second part is to estimate the iteration transformation and the new remainder term, the third part is to prove the convergence of the composition transformation.