Research On Normal Forms Of Some Non-autonomous Systems And Related Problems

This paper mainly studies the normal forms of some non-autonomous system.This thesis is mainly divided into two parts:The first part studies the normal forms for non-autonomous difference systems under the condition of non-uniform dichotomy spectrum;In the second part,we study the normal forms for non-autonomous differential system and difference systems on R1 under the condition of uniform dichotomy spectrum,and the specific form of normal form is given.In the first part of this paper,we study the normal forms for non-autonomous difference systems under the condition of non-uniform dichotomy spectrum.The specific way is based on the relevant knowledge of strong non-uniform exponential dichotomy,the non-resonance con-dition defined by the endpoint of spectrum interval of non-uniform dichotomy spectrum,and the method of classical normal form theory,taking the nonlinear part of the non-autonomous difference system satisfies certain conditions as the prerequisite,the finite jet normal form of this system under the condition of non-uniform dichotomy spectrum can be obtained by a trans-formation of nearly identical polynomialIn the second part,we study the normal forms for non-autonomous differential systems and difference systems on R1 under the condition of uniform dichotomy spectrum.The specific way is to use the resonance conditions defined by the endpoint of spectrum interval of uniform dichotomy spectrum and we simplify this system according to the different value of the spec-trum interval endpoint,obtain the normal forms for non-autonomous systems on R1,and the specific form of normal form is given.