平面系统中心与焦点判定问题的若干注释

Remarks on the center and focus problem for planar systems

常微分方程理论是数学的一个十分重要的学科,其主要任务是研究解的性态,其中平面系统中心与焦点的判定问题是常微分方程定性理论的重要内容之一.对于高维(包括无穷维)系统,在一定条件下可以通过中心流形定理降维至二维自治系统,因此,平面系统中心与焦点的判定问题是最基本的内容.微分方程定性理论著作,都会不同程度地论过这一问题.针对这一问题进行总结、思考和研究,对已有概念做一些引伸,对已有结果给出新的认识与证明,提出一些新的结论.这些内容都很难在现有文献中找到.

The center and focus problem is one of the main topics of the qualitative theory of ordinary differential equations. Systems in higher dimensional spaces can be reduced to systems in the plane with the help of the center manifold theorem. Therefore, it is fundamental to study the center and focus problem for planar systems. One can find the materials in related text books for graduate students of mathematics. In this article,we give a survey and present some further studies on the problem.