保持n-形式系统的Lie对称群约化及应用

本文考虑保持ｎ－形式（面积、体积的高维推广概念）的ｎ维向量场,应用Ｌｉｅ群方法对其约化问题进行了系统性研究,得到了下列结果,第一,如何保持ｎ－ｇａａａ的ｎ维向量场具有一个单参数保持ｎ－形式的空间对称群,则可具体地构造出一个与向量场无关变换,便得原向量场约化掉一维,并且该约化向量场保持相应的（ｎ－１）形式,特别ｎ＝３时可直接得到「１」中的重要结果,第二,上述ｎ维向量场如果具有一个ｒ参数保持ｎ－ｇａａ

LIE SYMMETRY GROUP REDUCTION OF THE SYSTEMS PRESERVING n-FORM AND ITS APPLICATIONS

in this paper, n-dimensional vectorfields preserving n-form (a high dimensional generalization of area, volume) are considered. By the mean of Lie group the reduction of this kind of vectorfields is studied systematically3 and the following results are obtained. Firstly if a n-dimensional vectorfiels preserving "form admits a one-parameter symmetry group that is spatial and preserving n-form, then it can be reduced into a (n-1)-dimensional vectorfield preserving the corresponding (n-1) form by constructing a concrete transformation independent of the vectorfield. Especially) when n-3, the important results in 1] can be obtained directly. Secondly, if the above n-dimensional vectorfield admits a r-parameter symmetry group that is spatial, Abelian and preserving n-form, then it can be reduced into a (n-r)-dimensional vectorfield preserving (n-r)-form. In particular, when n=4 and r = 2, some important dynamical behavior in this kind of four-dimensional perturbed systems can be discussed in detail. Finally the applications of the method proposed in this paper are illustrated by two examples: the famous L-K model and ABC flow.