双线性方法与混沌同步算法研究
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摘要
非线性科学中主要包含孤立子、混沌、分形三个部分。本文主要以符号计算系统Maple为工具,研究了非线性发展方程孤子解的求解以及混沌同步的算法。
第一章主要回顾了孤立子研究的历史与发展以及非线性偏微分方程精确解的若干构造性方法,同时介绍了一些关于该学科领域的国内外学者所取得的成果。同时介绍了混沌同步的研究历史与现状。
第二章主要以Hirota双线性方法为理论基础,介绍了用双线性方法求解偏微分方程孤子解的具体步骤和一些具体格式,同时给出一些常见方程的双线性形式。并且研究了KP方程,给出了KP方程的Wronskian解,并且证明了其解最后等价于普吕克关系。最后以双线性方法所能得到的解为形式,用待定系数法求得并证明了Zakharov–Kuznetsov(ZK)方程的N孤子解,其中的双线性形式由双线性算子线性部分和非线性部分组成。
第三章首先介绍了广义Lorenz系统及其规范式,并且介绍几种常见的属于该规范式的几种系统,并分析给出这些系统变换到规范式的过程以及参数关系。通过Q-S混沌同步以及自动推理格式,在符号计算软件Maple的帮助下,得到了广义Lorenz系统规范式与R?ssler系统的Q-S同步。
Nonlinear Science consists mainly of three parts: soliton, chaos and fractal. With the aid of computer algebra system Maple, this dissertation mainly studies the soliton solutions of nonlinear evolution equations and the algorithm of chaos synchronization.
Chapter 1 of this dissertation is devoted to reviewing the history and development of the soliton theory and the construction of the nonlinear partial differential equation. In addition, some domestic achievements and abroad ones on the subject are presented. At the same time, the history and progress of the chaos synchronization are briefly introduced.
Chapter 2 , based on Hirota bilinear method, introduces the concrete scheme of solving the soliton solution of the nonlinear partial differential equation. In addition, some bilinear forms of equations are given. By using Hirota bilinear method and Wronskian technique to consider the Wronskian solution of KP equation, it is shows that the bilinear equation is nothing but the Plücker relation for determinants. At last, combing the form of solution obtained by Hirota bilinear method with the unified structure method, we investigate the N-soliton solution of the Zakharov–Kuznetsov(ZK) equation, in which the Hirota bilinear operator form consists of linear and nonlinear part.
Chap 3 introduces the generalized Lorenz canonical form at the beginning. Some systems which belong to the canonical form are given, and the relation between the parameter of these systems and the parameter of the canonical form are shown. Then, Q-S synchronization and the automatic reasoning for Q-S synchronization backstepping scheme and its algorithm are investigated. Based on the symbolic-numeric computation software Maple, the Q-S synchronization between the generalized Lorenz canonical form and the R?ssler system is obtained.
第一章主要回顾了孤立子研究的历史与发展以及非线性偏微分方程精确解的若干构造性方法,同时介绍了一些关于该学科领域的国内外学者所取得的成果。同时介绍了混沌同步的研究历史与现状。
第二章主要以Hirota双线性方法为理论基础,介绍了用双线性方法求解偏微分方程孤子解的具体步骤和一些具体格式,同时给出一些常见方程的双线性形式。并且研究了KP方程,给出了KP方程的Wronskian解,并且证明了其解最后等价于普吕克关系。最后以双线性方法所能得到的解为形式,用待定系数法求得并证明了Zakharov–Kuznetsov(ZK)方程的N孤子解,其中的双线性形式由双线性算子线性部分和非线性部分组成。
第三章首先介绍了广义Lorenz系统及其规范式,并且介绍几种常见的属于该规范式的几种系统,并分析给出这些系统变换到规范式的过程以及参数关系。通过Q-S混沌同步以及自动推理格式,在符号计算软件Maple的帮助下,得到了广义Lorenz系统规范式与R?ssler系统的Q-S同步。
Nonlinear Science consists mainly of three parts: soliton, chaos and fractal. With the aid of computer algebra system Maple, this dissertation mainly studies the soliton solutions of nonlinear evolution equations and the algorithm of chaos synchronization.
Chapter 1 of this dissertation is devoted to reviewing the history and development of the soliton theory and the construction of the nonlinear partial differential equation. In addition, some domestic achievements and abroad ones on the subject are presented. At the same time, the history and progress of the chaos synchronization are briefly introduced.
Chapter 2 , based on Hirota bilinear method, introduces the concrete scheme of solving the soliton solution of the nonlinear partial differential equation. In addition, some bilinear forms of equations are given. By using Hirota bilinear method and Wronskian technique to consider the Wronskian solution of KP equation, it is shows that the bilinear equation is nothing but the Plücker relation for determinants. At last, combing the form of solution obtained by Hirota bilinear method with the unified structure method, we investigate the N-soliton solution of the Zakharov–Kuznetsov(ZK) equation, in which the Hirota bilinear operator form consists of linear and nonlinear part.
Chap 3 introduces the generalized Lorenz canonical form at the beginning. Some systems which belong to the canonical form are given, and the relation between the parameter of these systems and the parameter of the canonical form are shown. Then, Q-S synchronization and the automatic reasoning for Q-S synchronization backstepping scheme and its algorithm are investigated. Based on the symbolic-numeric computation software Maple, the Q-S synchronization between the generalized Lorenz canonical form and the R?ssler system is obtained.
引文
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