积分因子在定性理论中的性质及其应用

在微分方程定性理论中.一次奇点的分类.高次奇点的分类,极限环的稳定性等,都是需要研究的重要问题,并且是用不同的方法来加以解决的.而高次奇点中焦点与中心的区分,至今还是一个未解决的问题.在本文中,我们从理论上阐明了.所有上述问题都可利用积分因子的概念而统一地加以处理.此外,我们并给出了判别中心与焦点的方法,这一方法对于一次奇点与高次奇点都是同样适用的.从而解决了关于高次奇点的中心与焦点的区分问题.

The Properties and Applications of the Integrating Factor in the Qualitative Theory of Ordinary Differential Equations

In this article, we indicated that all the problems, such as the classification of the singular point and the determination of the stability of limit cycle, can be solved by the application of the integral factor. Especially we gave a criterion for deciding the center and the focus, which is appropriate for the singular point of the first order as well as that of the higher order.In the qualitative theory of ordinary differential equations,all the problems, such as the classification of singular point of the first order and higher order, and the determination of stability of limit cycle, are important problems to be solved by different ways, the distinction between a focus and a center of singular point of the higher order is an unsolved problem. In this paper, we show that all the problems mentioned above can be solved by the use of the integrating factor. A criterion is given to decide the center and the focus, and this criterion is applicable to the singular point of the first order as well as the higher order.