With Finite-dimensional Reduction And Anti-integrable On Certain Infinite Dimensional Problem

This dissertation focuses on the researches on some infinite-dimensional dynamical problems by the methods of finite-dimensional reduction andAnti-integrability. We organize it along the following lines:In Chapter 2 we discuss the problem of reduction of the typical non-autonomous partial differential equations arising in mathematical physics. We prove the existence of uniform attractors of the nonautonomous Shi鰀inger equation, and give the estimate on their Hansdorff dimension. Then we continue to discuss the existence of uniform attractors of non-autonomous KdV equation, obtain the existence of uniform attractor in weak topology and prove that they are also uniform attractor in strong topology.In Chapter 3 we deal with the existence of wea]dy damped forced KdV equation in 2D thin domains, we give the estimates for the blow ?憉p time of the equation in 2D thin domain, and then obtain the existence of local attractors.In Chapter 4 we present the investigation of the finite-dimensional feedback control of a kind of the infinite-dimensional dynamical system. First we establish a linear finite-dimensional state feedback controller to locally stabilize the zero solution for a kind of infinite-dimensional dynamical system, then obtain a result of the global stabilization of zero solution for some particular systems. Finally numerical simulations of the closed-loop system, for different values of the instability parameter, show the effectiveness of the proposed control method.In Chapter 5 we study the dynamics of the discrete systems of the infinite-dimensional systems using the Anti-integrabi]ity method, and mainly deal with the discrete Nagumo equation. We not only obtain the existence of breathers and various spatiotemporal periodic solutions, but also give a discussion on the disorder property of steady solutions.