Invariant Tori Of Nearly Integrable Systems And Reducibility Of Quasi-periodic Systems

This paper primarily takes advantage of KAM theory, Brouwer degree theory and Banach fixed point theory to study the persistence of invariant tori with given frequency direction for nearly integrable hamiltonian system, the existence of quasi-periodic solution for a nonlinear schrodinger equation with a prescribed potential under Dirichlet condition, the existence of periodic solution for a class of perturbed periodic systems with degenerate equilibrium, and the reducibility for a two-dimensional linear quasi-periodic system without any non-degeneracy condition. The main contents are arranged as follows:Chapter 1 is devoted to introducing elementary concepts and vital theory on hamiltonian systems and on symplectic geometry, and to presenting a concrete statement of KAM theory. Furthermore, this paper also introduces the development of research on the reducibility of per-turbed periodic and quasi-periodic systems. At the end of the first chapter, an outline of this paper about our major work is given, also encompassing innovative keys.Chapter 2 studies the existence of Lagangian invariant tori with the prearranged direc-tion of a given Diophantine frequencies for nearly integrable hamiltonian systems under Bruno non-degeneracy condition. Moreover, the obtained whole family constitutes an one-parameter analytic family.Chapter 3 is concerned with the persistence of elliptic lower dimensional tori for the per-turbed real analytic hamiltonian systems without any non-degeneracy condition. Assume the frequencies ω0 and Ω0 satisfy the following non-resonance condition Here Aκ= 1+|κ|Ï„, τ≥ n+1. Suppose the Brouwer degree of the frequency mapping ω(ξ) at ω0 ∈ D never vanishes. In this paper, an external parameter is introduced to adjust the shift of the frequencies during KAM iteration and to cope with the resonance between tangential frequencies and normal frequencies. At last, the assumption on the Brouwer degree ensures that the result for the parameterized system can be transferred to the original system.Chapter 4 considers a class of nonlinear schrodinger equation with a given analytic po-tential under Dirichlet condition. This paper employs the localization property of eigenvalues of Sturm-Liouville operator and certain techniques to obtain Birkhoff normal form to param-eterize the infinite dimensional hamiltonian systems. Ultimately, KAM theory guarantees the parameterized system possesses analytic, linearly stable quasi-periodic solutions.Chapter 5 deals with the reducibility of higher dimensional periodic systems with degenerate equilibrium. Consider the perturbation of n-dimensional real analytic periodic system where f(x,t) is a small perturbation, N(x)= (x12l1+1,...xn2ln+1)T with l1,...,ln being non-negative integers. h= (h1,..., hn)T denotes high order terms and satisfies hj= O(xj2lj+2), xâ†' 0, j=1,...,n. This paper introduces external parameter and applies Banach fixed point theory to transform the parameterized system into one with zero equilibrium. Finally, a periodic solution for the original system is obtained through Brouwer degree theory.Chapter 6 studies the reducibility of a class of two-dimensional linear quasi-periodic systems without any non-degeneracy condition. Let λ1 and A2 be the eigenvalues of A. Suppose (ω, λ1,λ2) satisfy the following non-resonance condition where α0> 0, Ï„> Ï„-1. Then for all ε∈ (0, ε0) the above system is reducible; or there exists a non-empty Cantor set E C (0, ε0) with positive Lebesgue measure such that for ε∈ E the system is reducible. Moreover, this paper presents the reducibility of a certain n-dimensional linear quasi-periodic system and of a certain n-dimensional nonlinear quasi-periodic system with eigenvalues satisfying several restrictions, serving as interesting examples and applications of two-dimensional reducibility result.