Solving Nonlinear Differential Equation And Chaos Synchronization

In this dissertation, under the guidance of mathematics mechanization and by means of symbolic-numeric computation software, some problems in the theory of soliton, fractional dif-ferential equation and chaotic system are discussed as follows:1. the realization of mechanization for constructing the exact solutions for nonlinear evo-lution equations;2. constructing the numerical solutions for nonlinear fractional differential equations;3. chaos synchronization and automatic reasoning scheme developed for them.Chapter 1 is devoted to reviewing the history and development of the mathematics mech-anization, soliton theory, fractional calculous and chaos synchronization, with an emphasis on some achievements on the subjects involved in this dissertation are presented at home and abroad.Chapter 2 introduces Wu-Ritt differential elimination theory, some basic notations and properties on fractional calculous, basic theories on AC=BD as well as the construction of exact solutions of nonlinear evolution equation(s) under the instruction of this theory.Based on the idea of solving nonlinear evolution equations, algebraic method, algorithm reality, mechanization, Chapter 3 firstly presents the rational expansion method to uniformly construct the exact solutions for nonlinear evolution equations, and then by means of Wu elimina-tion theory and symbolic computation software, considers some concrete forms and applications of the rational expansion method:(1) the high dimension coupled Burgers equation is considered by a new auxiliary equation method and some new complexiton solutions are found;(2) based on the elliptic functions, the elliptic function rational expansion method is pre-sented and some new rational formal elliptic function solutions of (2+1)-dimensional dispersive long wave equation are found;(3) Riccati equation rational expansion method is presented based on the Riccati equa-tion, and some new exact solutions of (2+1)-dimensional Broer-Kaup-Kupershmidt equation are obtained;(4) Whitham-Broer-Kaup equation is studied by using the generalized elliptic rational expansion method and some new rational formal exact solutions are found; (5) the Riccati equation rational expansion method is extended to study the (2+1)-dimensiona Burgers equation;(6) multiple auxiliary equations rational expansion method and its application to construct new complexiton solutions;(7) generalized Riccati equation rational expansion method and some non-travelling wave solutions of nonlinear evolution equations;(8) for stochastic differential equations, the Riccati equation rational expansion method is further improved and some new stochastic exact solutions are obtained;(9) the rational expansion method is further improved for differential-difference equations and some new exact solutions of two Toda equations are found.In chapter 4, the Adomian decomposition method and homotopy perturbation method which traditionally developed for differential equations of integer order are directly extended to derive numerical solutions of the nonlinear fractional differential equations. Some nonlinear fractional differential equations with physical significance, such as fractional KdV-mKdV equa-tion, fractional Boussinesq equation, fractional KdV-Burgers equation and fractional Kuramoto-Sivashinsky equation are investigated. Some available numerical solutions are firstly obtained. A simple theorem is also given to determine convergence of these methods.Chapter 5 first improves the automatic reasoning scheme for generalized Q-S (lag, com-plete and anticipated) synchronization of chaotic and hyperchaotic system. Based on the symbolic-numeric computation software, the generalized Q-S synchronization between two iden-tical Chua's circuit with different initial values, hyperchaotic Rossler system and hyperchaotic Tamasevicius-Namajunas-Cenys system, two identical generalized Henon map with different ini-tial values, Henon-like map and generalized Henon map are obtained. Numerical simulations verify the effectiveness of the proposed scheme. Then we present the definition of bidirectional partial generalized (lag, complete and anticipated) synchronization and an automatic reason-ing scheme to obtain it. The Rossler system, a new unified Lorenz-Chen-Lu system as well as the hyperchaotic Tamasevicius-Namajunas-Cenys system are chosen to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme.