Abstract: 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.

Key words: variational principle, numerical manifold method, nonlinear analysis, plastical flow