Numerical Solutions Of Soliton Equation And Function Cascade Synchronization Of Chaotic System

With the development of science and technology, nonlinear science has developed rapidly and involved in almost all the scientific fields. Chaos, soliton and fractal consist of the main three branches of nonlinear science. There are many theoretical significance and applicable value to investigate them, such as optical soliton communication, chaos secure communication and the length of the coastline et al.This dissertation investigates two problems of numerical solutions of soliton equations and chaos synchronization. Although different with each other, in fact, they have many common characteristics. For example, both of them are relevant to nonlinear system: soliton is associated with nonlinear partial differential,ordinary differential,differential-integral equation (system) et al and chaos relevant with nonlinear ordinary differential,difference one (ones) et al.In this dissertation, based on symbolic-numeric computation software, the applications of Adomian decomposition method and homotopy perturbation method are extended to a number of nonlinear soliton equations owning important physical significance and some useful general numerical (approximate) solutions are obtained. The chaos synchronization is also further investigated: the function cascade synchronization method is proposed for both continuous-time and discrete-time systems and its automatic reasoning scheme is given; the function cascade synchronization is realized for some chaotic systems including continuous, discrete, with and without unknown parameters systems.It is organized as follows:Chapter 1 briefly reviews the history and progress of soliton, Adomian decomposition method, homoptopy perturbation method as well as chaos and chaos synchronization. Some achievements on these subjects involved in this dissertation are presented at home and abroad.Chapter 2 directly extends the Adomian decomposition method and homotopy perturbation method to study some nonlinear soliton equations with physical significance. These two methods were used for differential equations of integer order traditionally. We investigate and obtain some general numerical solutions of the nonlinear evolution equations with nonlinear terms of any order; a type of nonlinear fractional coupled differential equations and some numerical solutions owning actual physical meaning; the complex KdV equation and the numerical positon,negaton solutions as well as numerical complexiton solutions.Chapter 3 gives the automatic reasoning scheme of the function cascade synchronization method for both continuous-time and discrete-time chaotic system. The function cascade syn- chronization of chaotic system is investigated, which includes the unified chaotic system,Lorenz system with unknown parameters,hyperchaotic Lüsystem,discrete-time generalized Hénon map and so on. Numerical simulations are used to verify the effectiveness of the proposed scheme.Chapter 4 is the summary and outlook of the dissertation.