A contribution to the topological classification of homogeneous vector fields in R3

Abstract

In this paper we study the topological classification problem for homogeneous polynomial vector fields of degree k > 2 in three variables. In 1960 Markus [7] (see also [6]) classified topologically the quadratic homogeneous vector fields in two variables. Subsequent works relative to the differentiable classification of these vector fields can be found in Date [4], Newton [lo], Samardzija [13], Vulpe and Sibirskii [14]. Let us now consider a homogeneous vector field Q = (Q,, Q2, Q,), of degree k > 2 in R3 with an isolated singularity at 0 E R3, i.e., all the Qi, i = 1, 2, 3, are homogeneous polynomials of degree k, with an isolated common zero at 0 E R3. The vector field defined as Qr(.?) = Q(i) (X, Qi(.%) + X2Q2(X) +X3 Q&f)). X, X = (2, ,X2, X3) E S2, is tangent to the 2-sphere S2, and for any regular orbit y of Qr the set