一类平面解析系统强退化奇点的可积性  

The integrability of strongly degenerate singularity for a class of planar analytical systems

作　　者：梅世明 黄土森[1] MEI Shiming;HUANG Tusen(School of Science,Zhejiang Sci-Tech University,Hangzhou 310018,China)

机构地区：[1]浙江理工大学理学院,杭州310018

出　　处：《浙江理工大学学报（自然科学版）》2024年第2期255-264,共10页Journal of Zhejiang Sci-Tech University(Natural Sciences)

基　　金：国家自然科学基金项目(11671359,11672270)。

摘　　要：利用正规型理论研究一类平面解析系统强退化奇点的可积性问题。首先证明,任意一个可微系统,在可微同胚的变量变换以及时间尺度变换下,可积性保持不变。其次,对于二次齐次的微分系统,可以通过仿射变换及时间尺度变换化为4种标准形;针对其中的一种标准形,结合正规型理论,利用主系统是哈密尔顿的,且所对应的哈密尔顿函数的分解式是单因式的形式微分系统可积的充要条件,给出了相应的非线性系统可积的条件。该结果为研究平面系统的相图和局部定性结构提供了理论依据。The integrability problems of strong degenerate singularities for a type of planar analytical systems are studied by means of the normal form theory.Firstly,it is proved that the integrability of any differentiable system remains unchanged under the variable transformation of the differentiable homeomorphism and time scale transformation.Secondly,the quadratic homogeneous differential system can be transformed into four canonical forms by affine transformation and time scale transformation.Finally,as for one of the canonical forms,the conditions of the integrability for the corresponding nonlinear analytical systems are given by using the normal form theory together with the necessary and sufficient conditions that the main system is Hamiltonian system and that the factorization of the corresponding Hamiltonian function only has simple factors.The results provide a theoretical basis for studying the phase diagram and local qualitative structure of planar systems.

关 键 词：平面解析系统 强退化奇点 可积性 正规型 时间尺度变换 

分 类 号：O175.14[理学—数学]