On generic homogeneous vector fields

Abstract

Background. The study of dynamical systems with symmetry is of both theoretical and applied interest. Phase portraits of dynamical systems defined by homogeneous vector fields are invariant with respect to the phase space dilation group. First of all, it is natural to describe generic homogeneous vector fields. The structure of the phase portraits of generic homogeneous vector fields on the projective plane is known. The purpose of this work is to obtain some generic properties of homogeneous vector fields in a phase space of three or more dimensions. Materials and methods. Methods of the qualitative theory of differential equations and functional analysis are applied. Results. We consider homogeneous vector fields in an m-dimensional arithmetic space of degree n > 0, which have continuous derivatives everywhere, except perhaps at the origin. The trajectories of homogeneous vector fields extend naturally to the space compactification, the Poincaré ball. It is shown that vector fields that have only hyperbolic singular points and closed trajectories in the Poincaré ball with a punctured origin are generic: they form a massive set in the Banach space of homogeneous vector fields. The concept of a weakly structurally stable homogeneous vector field is introduced. Necessary and sufficient conditions for weak structural stability at m = 3 are obtained. It is shown that weakly structurally stable vector fields in the Banach space of homogeneous vector fields form an open everywhere dense set. Conclusions. Weak structural stability of homogeneous vector fields in three-dimensional space is a generic property.

Keywords

• homogeneous vector field
• compactification of the phase space
• hyperbolic singular points and closed trajectories
• generality
• weak structural stability