高维非线性动力系统最简规范形的计算  

General program of calculating the simplest normal forms for high-dimensional nonlinear dynamical systems

作　　者：张琪昌[1] 田瑞兰[1] 李小涛[1] 

机构地区：[1]天津大学机械学院力学系,天津300072

出　　处：《振动工程学报》2008年第5期436-440,共5页Journal of Vibration Engineering

基　　金：国家自然科学基金(10872141)资助项目;教育部博士点基金(20060056004)资助项目;高等学校博士学科点专项科研基金资助项目

摘　　要：运用可逆线性变换和近恒同变换,研究了不经计算传统规范形,直接计算高维非线性动力系统的最简规范形。引进可逆线性变换,将非线性动力系统的线性矩阵拓扑等价于符合实际研究需求的分块对角线矩阵:相伴矩阵分布在对角线上,其余元素均为0。利用低阶项来化简高阶项,得到了高维非线性动力系统的最简规范形。在该最简规范形中,对应于每一个相伴矩阵的非线性系数矩阵,只有最后一行含有非0元素,其余各行元素均为0。借助M athem atica语言,编制了计算任意高维非线性动力系统的最简规范形的通用程序。运行该程序,分别计算了4维、6维和7维非线性动力系统的直到4阶的最简规范形。Applying a reversible linear transformation and a near-identity transformation, the simplest normal forms for high-dimensional nonlinear dynamical system is studied without the calculating of its traditional normal form. Using a reversible linear transformation, the matrix of the linear part for the nonlinear dynamical system is topologically equivalent to the block diagonal matrix that adapts to the demand of the practical research： companion matrixes distribute on the diagonal line and the remaining elements are zero. In order to obtain the simplest normal form, lower order nonlinear terms are used in the normal form for the simplifications of higher order terms. In the simplest normal form, the nonlinear coefficient matrix contains non- zero elements only in the row corresponding to the last row of each companion matrix and zero elements in the remaining rows. The general program with the Mathematica language is provided, which can compute the simplest normal form of an arbitrary nonlinear dynamical system easily. For nonlinear dynamical systems of 4-dimensional, 6-dimensional and 7-dimension, the simplest normal forms up to order 4 are discussed by executing the program.

关 键 词：非线性变换 最简规范形 高维非线性动力系统 符号运算 

分 类 号：O322[理学—一般力学与力学基础]