幂零奇点的局部可积性及其分类

平面多项式微分系统的可积问题与退化奇点的完全分类问题是常微分方程定性理论中的 2 个重要问题. 目前, 几乎所有可积问题的工作都集中于讨论中心焦点和 $p:-q$ 共振中心上, 而退化奇点的完全分类问题的结果很少. 考虑带幂零奇点的平面实多项式微分系统, 给出了相应的局部可积的理论与方法, 并在可积的条件下讨论了幂零奇点的完全分类问题. 进一步地, 对相应的 2 次系统与 1 类 3 次系统给出了可积的充要条件以及可积条件下幂零奇点的完整分类.

Local integrability and classification of nilpotent critical points

The integrability of planar polynomial differential systems and the complete classification of degenerate critical points are both important problems in the qualitative theory of ordinary differential equations. Currently, almost all results on the local integrability are related to a center, focus on real polynomial systems or a $p:-q$ resonant center for polynomial systems in a complex plane. The complete classification of a degenerate critical point is also very difficult, and there are few related studies on this. In this study, by considering polynomial systems with a nilpotent critical point (0, 0), the corresponding theory is established for local integrability and present a method for the complete classification of nilpotent critical points under each integrable condition. Moreover, the necessary and sufficient condition is obtained for the integrability of the quadratic system and a kind of cubic system, and then the nilpotent critical points are completely classified under each integrable condition.