幂零奇点的局部可积性及其分类

doi:10.12066/j.issn.1007-2861.2200

Abstract

Abstract:

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.

Key words: integrability, nilpotent critical point, Darboux factor, cofactor

CLC Number:

O175.1

WANG Yinzi, HU Zhaoping. Local integrability and classification of nilpotent critical points[J]. Journal of Shanghai University（Natural Science Edition）, 2021, 27(5): 891-906.

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Local integrability and classification of nilpotent critical points

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