坐标变换下平面解析系统单值轨道的不变性

Invariance of monodromic orbits of planar analytic system under coordinate transformation

为了研究平面解析系统在何种坐标变换下单值轨道具有不变性,首先给出平面解析系统的轨线沿固定方向进入奇点的两个定义,并证明了它们是等价的;其次引入判别轨线沿固定方向进入奇点的一个充要条件,得到平面解析系统在非正则变换下系统轨道可以具有不同的单值性;最后通过对平面解析系统做正则变换,证明了变换前后的系统轨线具有相同的单值性。该结果对研究平面解析系统单值轨道的不变性具有参考价值。

In order to study the invariance of monodromic orbits under which coordinate transformation of planar analytic system,firstly,two definitions of orbits of planar analytic system entering singular point along a fixed direction are given,and they are proved to be equivalent.Secondly,a necessary and sufficient condition for discriminating that orbits enter singular point along a fixed direction is introduced,and it is obtained that the orbits for the planar analytic system and the associated system under non-regular coordinate transformation have a different monodromy.Finally,by making a regular coordinate transformation on the planar analytic system,it is proved that the orbits for the planar analytic system and the associated system have the same monodromy.The result can provide a reference for studying the invariance of monodomic orbits of planar analytic system.