一种构造高维Nilpotent正规型的方法

AN APPROACH FOR CONSTRUCTING THE NILPOTENT NORMAL FORMS OF HIGH DIMENSIONAL SYSTEMS

正规型方法是一种有效的简化一类非线性方程的方法．今提出了一种简便的代数方法去构造高维非线性系统的Ｎｉｌｐｏｔｅｎｔ范式．通过引入一系列简单的变换和代数运算，而无需求解任何偏微分方程，即可得到高维非线性系统的Ｎｉｌｐｏｔｅｎｔ范式．以四维非线性系统为例介绍这个方法，该方法完全适用于分析高于四维的非线性系统的Ｎｉｌｐｏｔｅｎｔ范式．

It is well known that the normal form theory is one of the powerful tools for simplifying ordinary differential equations. In general, if the linear part of a nonlinear differential equation is in a diagonal form, the corresponding normal form consists of resonant monomials, which is easy to obtain. However, if the linear part of a differential equation is not in diagonal form, the associated normal form is very difficult to obtain. In order to construct the formal normal forms, one needs to solve a series of partial differential equations. This is not an easy task, especially when the dimension of the discussed system is higher than 3. In this paper, an algebraic approach is presented for determining normal forms of ordinary differential equations with a nilpotent linear part. Certain transformations introduced in this paper result in a simplified procedure for the calculation of Nilpotent normal forms. One does not need to solve any partial differential equations and algebraic calculations are sufficient. The approach is introduced with regard to a four dimensional system, but the procedure can be applied to higher order dimensional systems as well. The necessary formulations associated with the construction of Nilpotent normal forms of high dimensional systems are presented. All the results presented in this paper can be obtained by the symbolic calculation in seconds. It is noted that any matrix can be transformed into Jordan canonical form through a linear transformation; therefore, the linear part of a given system can always be transformed into Jordan canonical form which, in turn, can be regarded to be composed of a diagonal and a nilpotent component. The normal forms associated with each of these cases can readily be combined to obtain the normal form of the general case. In this paper, two examples are presented to illustrate some of the advantages of the new approach. With the aid of a symbolic calculation, like MAPLE, the Nilpotent normal forms are obtained within one second in both cases.