Normal Forms Of Some Nonlinear Systems And Related Reduction Problems

This paper mainly studies the normal forms of some nonlinear systems and related reduction problems.This thesis is mainly divided into three parts: The first part studies the smooth autonomization and analytic polynomial normal forms of periodic difference systems;The second part studies the real analytic linearization of real analytic differential homeomorphisms;The third part studies the formal normal forms of quasi-periodic differential systems.In the first part,we study the smooth autonomization and analytic polynomial normal forms of periodic difference systems.For the study of the smooth autonomization of periodic difference systems,we use the knowledge of matrix and series convergence to construct a matrix satisfying the weak non-resonance condition,thus the conditions for the autonomization of periodic difference systems are further obtained.For the study of the analytic polynomial normal forms of periodic difference systems,we prove the convergence of conjugate equivalent transformation by using the method of classical normal form,and then the analytic polynomial normal forms of the systems are obtained.In the second part,we discuss the real analytic linearization of real analytic differential homeomorphisms.With the help of the Jordan normal form theory of linear algebra and the classical normal form theory,the variable transformation of the real analytic linearization of the system is given.In the third part,we study the formal normal forms of quasi-periodic differential systems.By using Fourier-Taylor expansion of function and the classical normal form theory,formal autonomization and linearization of quasi-periodic differential systems are studied under frequency non-resonance and non-resonance conditions.