摘要
本文考虑保持。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，如果保持一形式的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的变换，使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，参数保持一形式的空间Ａｂｅｌｉａｎ对称群，则原系统可被约化成一保持（ｎ－ｒ）－形式的（ｎ－ｒ）维向量场．特别ｎ＝４，ｒ＝２时，约化向量场有较简单的形式，于是可具体地讨论该类四维扰动系统的一些重要动力学性质．最后本文以著名的Ｌ－Ｋ模型及ＡＢＣ流为例阐述了本文提出的一般方法的应用。
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.
出处
《应用数学学报》
CSCD
北大核心
2000年第1期108-121,共14页
Acta Mathematicae Applicatae Sinica
基金
国家自然科学基金
云南省科委基金!19972058
关键词
保持n-形式
约化
微分方程
高维系统
李对称群
Preserving n-form, Lie group, reduction, action-angle-angle transformation, invariant torus
