非多项式向量场的Poincaré紧化
Poincaré compactification for non-polynomial vector fields
摘要：
在这项工作中的理论框架，应用庞加莱紧化技术的局部Lipschitz连续向量场的发展。证明了这些向量场在n维球面上是紧的，而紧化的向量场在赤道上可以恒为零。此外，对于半球上的固定投影，向量场的所有紧化在赤道上不全同为零都是等价的。同时，得到了紧化向量场赤道不变性的条件。据作者所知，这是首次研究局部Lipschitz连续向量场的Poincaré紧化.这些结果被应用到一些向量场族中，如多项式向量场，定义为齐次函数之和的向量场和由分段线性函数定义的向量场.
英文摘要：
In this work a theorical framework to apply the Poincaré compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincaré compactification of locally Lipschitz continuous vector fields is studied. These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.
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