On generic homogeneous vector fields on the plane

Background. For applications of mathematics, it is of interest to study dynamical systems with symmetry. We consider vector fields on the plane whose components are homogeneous functions of natural degree n. Their phase portraits are invariant with respect to the group of extensions of the plane. The aim of this paper is to describe an open and everywhere dense set in a Banach space $HF_n^r$ of homogeneous vector fields of degree n and class $C^{r }$ in $R^2 \{0\}$ ($r \geq 2, n \geq 2$). Materials and methods. We use the methods of the qualitative theory of differential equations, functional analysis, and projective geometry. Results and conclusions. The concept of a structurally stable homogeneous vector field is introduced, the topological structure of the phase portrait of which does not change when passing to a vector field sufficiently close to $X$ in $HF_n^r$. Necessary and sufficient conditions for structural stability are obtained. It is shown that structurally stable homogeneous vector fields are generic: they form an open everywhere dense set in the space $HF_n^r$.