Normal Form Theory Of Nonlinear Quasiperiodic Systems And Averaging Principle Of Nonlinear Systems With Jordan Blocks

The quasiperiodic system is an important class of system that depict quasiperiodic motion,which is a common natural phenomenon in nature,it is widely present in problems such as mechanics,astronomy and physics.One of the important problems of quasiperiodic systems is to study the normal form theory of the system near the singularity,which is a key tool for studying the local dynamics of quasiperiodic systems.The averaging principle provides a powerful tool for simplifying nonlinear dynamic systems and is used to obtain approximate solutions of differential equations generated in mechanics,mathematics,physics,control and other fields.At present,research on these aspects mainly involves the case where the matrix corresponding to the linear part of the system is diagonal,and there are few research results on the normal form theory of general nonlinear quasiperiodic systems.In this dissertation,we will study the normal form theory of nonlinear quasiperiodic systems in high-dimensional and special cases,including the normal form theory of nilpotent systems,i.e.,the matrix corresponding to the linear part of the system is Jordan matrix.Using the normal form theory of nonlinear quasiperiodic system to derive the averaging principle of nonlinear systems with Jordan blocks.In Chapter 1,we introduce the background and significance of this dissertation,mainly including the development process of normal form theory and averaging principle,and relevant theoretical achievement in recent years.In addition,we introduce the main results and the framework of the dissertation.In Chapter 2,consider the nonlinear quasiperiodic system and introduce the relevant concepts of the Jordan canonical form,which mainly include the definition of Jordan blocks and the real Jordan canonical form theorem.To discuss the proof of the averaging principle of nonlinear system with Jordan blocks,we give the definitions and related properties of complex polynomials,polynomial vector field,and local Lipschitz continuity.Further,we recall the existence and uniqueness of solutions for initial value problems of differential equations with parameters,and the concepts of Lagrange canonical form and other canonical forms.In Chapter 3,based on the basic idea of normal form theory,we prove the normal form theory of high-dimensional nonlinear quasiperiodic system satisfying appropriate conditions through mathematical induction,where the coefficient matrix corresponding to the linear part of the system is nondegenerate.We study two types of normal forms:a nonlinear quasiperiodic system containing only one Jordan block;a nonlinear quasiperiodic system with multiple Jordan blocks,i.e.,a general nonlinear quasiperiodic system.In Chapter 4,we study the normal form theory of low-dimensional nonlinear quasiperiodic systems.Similarly,under appropriate conditions,the proof of normal form theory is obtained by using mathematical induction.For low-dimensional nonlinear quasiperiodic system,we describe two types of normal forms:eigenvalues of the linear part are zero,which is a nilpotent system;eigenvalues of the linear part are nonzero.In Chapter 5,based on the normal form theory of nonlinear quasiperiodic system and variation of constants in high-dimensional Euler space,we transform the nonlinear system with Jordan blocks into the standard form,where we stress that the only restriction on the spectrum of the operator is that it be purely imaginary.Further,we introduce a complex structure and represent the standard form as a complex structure.Finally,we prove the Krylov-Bogolyubov averaging theorem by averaging of vector fields.