非半单分叉的Normal Form计算

在文［１］基础上，提出一种仅知道派生线性系统零实部特征值时求解非线性系统非半单分叉ＮｏｒｍａｌＦｏｒｍ的方法．通过适当的分类，将要求解的线性代数方程组分为若干相互独立的方程组．将所求系数向量按字典序列排列后，各独立方程组的系数矩阵是上三角矩阵．在非共振情形，各系数向量可按反字典序列递推求出．在共振情形，根据文中的二个定理，巧妙地由一简单的常数矩阵的最大秩子矩阵，定位其系数矩阵的满秩子矩阵，解决了这类方程组的降维简化．通过消元法，把简化后的方程化成类似于半单分叉ＮｏｒｍａｌＦｏｒｍ求解过程中方程的形式，其解法也类似．该方法非常易于在计算机代数软件平台上程序化．

CALCULATING NORMAL FORMS FOR NONSEMI SIMPLE BIFURCATIONS

This paper presents a new scheme of calculating the Normal forms of a set of nonlinear ordinary differential equations when a nonsemi simple bifurcation occurs, with only the eigenvalues of zero real parts of the linearized differential equations given. It is well known that the classical matrix method enables one to establish the algebraic equations that govern the coefficients in the Normal form of a set of ordinary differential equations. However, it offers neither general technique of reducing the maximal dimension of the algebraic equations to be solved, nor practical algorithm of solving the linear algebraic symbolic equations with singular coefficient matrix. The matrix method, hence, is far from solving the Normal form of nonsemi simple bifurcations of a nonlinear system. The primary aim of this paper is to solve the problem of determining the Normal forms by the matrix method. The analysis in the paper begins with the series expression of the nonlinear vector valued function and treats every coefficient vector as a whole from a new point of view, rather than individuals considered in previous publications. The first advantage of the expression is that one can readily classify all the nonlinear terms so that the coefficient vectors corresponding to different kinds of nonlinear terms are uncoupled with each other. As a result, one can reduce greatly the maximal dimension of the coupled equations in reducing the Normal form by classifying the nonlinear terms appropriately. The second advantage is that the coefficient matrix of each set of coupled equations consists of two kinds of sub matrices, i.e., the linearized matrix of the original system and the identity matrix of the same dimension, following a rule found in the previous study of the first author. This feature enables one to get the key idea of this paper. i.e., the sub matrices of full rank in the coefficient matrix can be located by using the constant matrix with the same structure. These advantages are made use of to study a new scheme and its theoretical background for computing the Normal forms of the Nonsemi simple bifurcation. In section 1, a suitable classification of the nonlinear terms according to their exponent vectors is presented and the recursive formulae for solving the coefficient vector of the nonresonant terms is derived. Section 2, as the kernel of the paper, deals with how to find the form of the Normal form and the solution of the coefficient vector of the resonant terms. Because a full rank sub matrix can be found in the coefficient matrix of the set of matrix equations from the maximal rank full sub matrix of a simple constant matrix through Theorem 1 and 2 proposed and proved in this paper, some of their coefficient vectors corresponding to a kind of resonant terms can be solved out, while the others remain unchanged in a new set of fewer matrix equations. Through elimination, the new set of matrix equations can be casted into the similar matrix equations solved in the case of semi simple bifurcations. In section 3, the scheme efficacy is demonstrated through an example. The scheme can be easily programmed on any platforms of computer algebra according to the flow chart given at the end of the paper.