Reducibility Of Quasi-periodic Systems Under Weaker Non-degeneracy Conditions

In this thesis,by KAM theory we mainly study the reducibility of some classes of linear real quasi-periodic systems whose coefficient matrix depends on a small parameter and closes to constant.Without any non-degeneracy assumption,we first consider 2-dimensional and 3-dimensional quasi-periodic linear systems whose coefficient matrix depends on a small parameter Cm-smoothly.We also consider a 3-dimensional quasi-periodic linear systems whose coefficient matrix analytically depends on a small parameters.Moreover,we study the existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions.This thesis is divided into 7 chapters as follows:In the first chapter,we introduce the background of hamiltonian systems briefly,and the classical KAM results and the idea of the KAM theory.Furthermore,we also introduce the background and the development of the research on the reducibility of quasi-periodic systems.Finally we give the main work and the innovative points of this paper.In the second chapter,we study the following linear quasi-periodic hamiltonian systems with small parameters where A is a 2-order real Hamiltonian matrix,? is a small parameter and Q is analytic quasi-periodic in t and depends on ? Cm-smoothly.Under some non-resonance conditions about the basic frequencies and the eigenvalues of A and without any non-degeneracy assumption with respect to the small parameter,we prove the following two conclusions:(1)For m=1,there exists a sufficiently small ?0 and a subset E?(0,?0)with positive Lebesgue measure,such that for ??E the system is reducible.(2)For m=0,there exists a sufficiently small ?0 and a subset E?(0,?0)with cardinality number of continuum,such that for ??E the system is reducible.In Chapter 3,we extend the results of Chapter 2 to the following 3-dimensional anti-symmetric quasi-periodic systems:where A is an antisymmetric constant matrix and Q is an antisymmetric analytic quasi-periodic matrix with a Cm-smooth small parameter ?.Under some non-resonance conditions about the basic frequencies and the eigenvalues of A and without any non-degeneracy as-sumption with respect to the small parameter,we prove that the system(0.1)is reducible for many of sufficiently small parameters.In Chapter 4,we study the following three-dimensional system:where A is a real constant 3-order matrix and possesses a pair of complex eigenvalues ?±i?with ??0 and a real eigenvalue ? with ???.Q is a real analytic quasi-periodic matrix and analytically depends on a small parameter ?.Under some non-resonance conditions about the basic frequencies and the eigenvalues of A and without any non-degeneracy assumption with respect to the small parameter,we prove that system(0.2)is reducible for most of sufficiently small parameters in the sense of the Lebesgue measure.In Chapter 5,we still consider the three-dimensional systems(0.2),where A is a real constant matrix and possesses a pair of complex eigenvalues ?±i? with ??0 and a real eigenvalue ?.Q is a real analytic quasi-periodic matrix and depends Cm-smoothly on a small parameter ?.Under some non-resonance conditions about the basic frequencies and the eigenvalues of A and without any non-degeneracy assumption with respect to the small parameter,we prove that system(0.2)is reducible for many of sufficiently small parameters.In Chapter 6,we consider an area-preserving mapping as follows:where f and g are real analytic in(x,y)on T×[a,b]and depend Cm-smoothly on a parameter ?.The average of g with respect to x is 0.Without imposing on any non-degeneracy assumption,we first prove a formal KAM theorem for the mappings(0.3).Then by application of this formal KAM theorem we obtain many previous KAM-type results under some non-degeneracy conditions.Moreover,by this formal KAM theorem we can also obtain some KAM-type results under some weaker non-degeneracy conditions.In Chapter 7,we give the prospects and the plan for future research.