R^3中一次齐次向量场的全局拓扑结构

THE GLOBAL TOPOLOGICAL STRUCTURE OF HOMOGENEOUS VECTOR FIELDS OF DEGREE ONE IN R￣3

证明了Ｒ３中一次齐次向量场４ｘ延拓到射影空间Ｐ３仍然是一次齐次向量场，当且仅当Ａ相似于对角型矩阵．此外，证明了ＡＸ在Ｒ３中无穷远的流拓扑等价于其切向量场ＡＴ（ｕ１，ｕ２，ｕ３）在单位球面Ｓ２上的流；Ａｘ有２６种不同的全局拓扑相图．

In this paper, the global phase portraits of 26 types of different topological structure are given for one-degree homogeneous vector fields (IDHVF) in Euclidean space R3. They are also phase portraits for the projective space P3 extended from lDHVF in R3. The approach in the paper includes the investigation of behaviour of the vector fields both at infinity and in the neighbourhood of the origin of R3.