可积系统与混沌系统中若干问题的符号计算研究
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摘要
基于符号计算,本文研究了非线性系统中可积系统与混沌系统中的若干问题,工作主要分以下两个部分：一、分别从延拓结构方法、Riccati型伪势与Bell多项式三个方面研究了非线性发展方程的可积性质：Lax对、自-Backlund变换、守恒律、奇异流形方程与双线性形式等,并开发了计算非线性发展方程双线性形式的一个程序包；二、构造了分数阶的Lorenz标准型与一个新的四维混沌系统,并给出了它们的数值模拟。
第一章,介绍了本文所研究内容的理论背景与发展现状,其中包括非线性系统的可积性、延拓结构理论、符号计算与混沌系统。
第二章,改进了延拓结构理论并将其应用于Qiao方程,得到了该方程的两个势与两个伪势,从中得到了新的反散射谱问题、Lax对与无穷多守恒律。将延拓结构理论扩展到变系数非线性发展方程,并应用于变系数KdV方程,得到了变系数KdV方程的Lax对与Pfaffian形式。
第三章,构造了广义五阶KdV方程的Riccati型伪势,得到了广义五阶KdV方程在该条件下的Lax对与奇异流形方程。在三种条件下,得到了广义五阶KdV方程的新奇异流形方程与自-Backlund变换,其中CDG-SK方程、Lax方程与KK方程分别包含在这三种情况之中。
第四章,基于Bell多项式,构造了得到非线性发展方程的双线性形式的机械化算法,并在Maple上给出了算法实现程序包。该程序包首先将非线性方程进行无维化,然后将无维化后的方程表达成P-多项式的线性组合,从而给出其双线性形式。并以实例验证了该算法的有效性和可靠性。
第五章,构造了分数阶的广义Lorenz标准型与一个新四维混沌系统,分析了其动力学性质,并给出了数值模拟。通过选择不同的参数可以分别得到分数阶的经典Lorenz系统、Chen系统、Lu系统、Shimizu-Morioka系统与双曲型广义Lorenz系统。
第六章,对全文的工作进行了总结和讨论,并对下一步工作进行了展望。
Based on symbolic computation, some problems of integrable systems and chaotic systems in nonlinear systems are investigated. There are two main parts in this disserta-tion:1. Some integrable properties of nonlinear equations are investigated with help of the prolongation structure method, Riccati type pseudopotentials and Bell polynomials. These properties include Lax pairs, auto-Backlund transformations, conservation laws, singularity manifold equations, bilinear forms and so on. A Maple package to obtain the bilinear forms of nonlinear evolution equations is developed; 2. A fractional-order gener-alized Lorenz canonical form and a new four-dimensional chaotic system are constructed and numerically simulated.
In chapter 1, an introduction is devoted to review the background and the current situation related the dissertation, which include integrability of nonlinear system, prolon-gation structure method, symbolic computation and chaotic system.
In chapter 2, the prolongation structure technique is improved and applied to Qiao equation. Two potentials and two pseudopotentials are obtained, from which a new spec-tral problem of inverse scattering transformation, Lax equations and infinite number of conserved laws are obtained. The prolongation structure method is generalized to the variable coefficient nonlinear evolution equations and applied to the variable coefficient KdV equation, from which Lax pairs and Pfaffian forms of variable coefficient KdV equa-tion are obtained.
In chapter 3, the Riccati-type pseudopotentials of the generalized fifth-order KdV equation are derived, from which Lax pairs and singularity manifold equations can be obtained. Especially, new singularity manifold equations and auto-Backlund transforma-tions can be obtained under three conditions, which include CDG-SK equation, Lax equation and KK equation.
In chapter 4, based on the Bell polynomials, a mechanization algorithm is proposed to obtain the bilinear forms of nonlinear evolution equations and the corresponding im-plementation software package in maple is developed. Firstly, the package transforms the equation into a dimensionless equation, which can be expressed in the linear combina-tions of P-polynomials, then the bilinear form of this equation can be obtained directly. The validity and reliability of this algorithm are verified by some examples.
In chapter 5, a fractional-order generalized Lorenz canonical form and a new four-dimensional chaotic system are constructed and the dynamical properties are investigated. In addition, the numerical simulations and interesting figures are performed. By choos- ing different coefficient, the fractional-order system of classic Lorenz system, Lii system, Chen system, Shimizu-Morioka system and hyperbolic-type Lorenz system can be ob-tained.
In chapter 6, the summary and discussion of this dissertation are given, as well as the outlook of future work is discussed.
第一章,介绍了本文所研究内容的理论背景与发展现状,其中包括非线性系统的可积性、延拓结构理论、符号计算与混沌系统。
第二章,改进了延拓结构理论并将其应用于Qiao方程,得到了该方程的两个势与两个伪势,从中得到了新的反散射谱问题、Lax对与无穷多守恒律。将延拓结构理论扩展到变系数非线性发展方程,并应用于变系数KdV方程,得到了变系数KdV方程的Lax对与Pfaffian形式。
第三章,构造了广义五阶KdV方程的Riccati型伪势,得到了广义五阶KdV方程在该条件下的Lax对与奇异流形方程。在三种条件下,得到了广义五阶KdV方程的新奇异流形方程与自-Backlund变换,其中CDG-SK方程、Lax方程与KK方程分别包含在这三种情况之中。
第四章,基于Bell多项式,构造了得到非线性发展方程的双线性形式的机械化算法,并在Maple上给出了算法实现程序包。该程序包首先将非线性方程进行无维化,然后将无维化后的方程表达成P-多项式的线性组合,从而给出其双线性形式。并以实例验证了该算法的有效性和可靠性。
第五章,构造了分数阶的广义Lorenz标准型与一个新四维混沌系统,分析了其动力学性质,并给出了数值模拟。通过选择不同的参数可以分别得到分数阶的经典Lorenz系统、Chen系统、Lu系统、Shimizu-Morioka系统与双曲型广义Lorenz系统。
第六章,对全文的工作进行了总结和讨论,并对下一步工作进行了展望。
Based on symbolic computation, some problems of integrable systems and chaotic systems in nonlinear systems are investigated. There are two main parts in this disserta-tion:1. Some integrable properties of nonlinear equations are investigated with help of the prolongation structure method, Riccati type pseudopotentials and Bell polynomials. These properties include Lax pairs, auto-Backlund transformations, conservation laws, singularity manifold equations, bilinear forms and so on. A Maple package to obtain the bilinear forms of nonlinear evolution equations is developed; 2. A fractional-order gener-alized Lorenz canonical form and a new four-dimensional chaotic system are constructed and numerically simulated.
In chapter 1, an introduction is devoted to review the background and the current situation related the dissertation, which include integrability of nonlinear system, prolon-gation structure method, symbolic computation and chaotic system.
In chapter 2, the prolongation structure technique is improved and applied to Qiao equation. Two potentials and two pseudopotentials are obtained, from which a new spec-tral problem of inverse scattering transformation, Lax equations and infinite number of conserved laws are obtained. The prolongation structure method is generalized to the variable coefficient nonlinear evolution equations and applied to the variable coefficient KdV equation, from which Lax pairs and Pfaffian forms of variable coefficient KdV equa-tion are obtained.
In chapter 3, the Riccati-type pseudopotentials of the generalized fifth-order KdV equation are derived, from which Lax pairs and singularity manifold equations can be obtained. Especially, new singularity manifold equations and auto-Backlund transforma-tions can be obtained under three conditions, which include CDG-SK equation, Lax equation and KK equation.
In chapter 4, based on the Bell polynomials, a mechanization algorithm is proposed to obtain the bilinear forms of nonlinear evolution equations and the corresponding im-plementation software package in maple is developed. Firstly, the package transforms the equation into a dimensionless equation, which can be expressed in the linear combina-tions of P-polynomials, then the bilinear form of this equation can be obtained directly. The validity and reliability of this algorithm are verified by some examples.
In chapter 5, a fractional-order generalized Lorenz canonical form and a new four-dimensional chaotic system are constructed and the dynamical properties are investigated. In addition, the numerical simulations and interesting figures are performed. By choos- ing different coefficient, the fractional-order system of classic Lorenz system, Lii system, Chen system, Shimizu-Morioka system and hyperbolic-type Lorenz system can be ob-tained.
In chapter 6, the summary and discussion of this dissertation are given, as well as the outlook of future work is discussed.
引文
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