ON BIFURCATIONS OF HOMOGENEOUS POLINOMIAL VECTOR FIELDS ON THE PLANE

Author(s):  V.Sh. Roitenberg, candidate of Sciences, associate Professor, Yaroslavl State Technical University, Yaroslavl, Russia, vroitenberg@mail.ru

Issue:  Volume 51, № 2

Rubric:  Mathematics

Annotation:  The paper considers vector fields on the plane whose components are homogeneous polynomials of degree n. The set HPn of such vector fields is identified with the 2n + 2-dimensional space of coefficients of these polynomials. Phase portraits of vector fields are viewed on the projective plane. Structurally stable vector fields X, for which the topological structure of the phase portrait does not change when passing to a vector field close enough to X in HPn, form an open everywhere dense set in HPn. In this paper, we describe an open everywhere dense set in the subspace of structurally unstable vector fields in HPn. It is an analytic submanifold of codimension one in HPn and consists of vector fields X of the first degree of structural instability, for which the topological structure of the phase portrait does not change when passing to a structural unstable vector field close enough to X in HPn . We describe bifurcations for vector fields of the first degree structurally instability.

Keywords:  homogeneous polynomial vector field on the plane, structural stability, first degree of structural instability, bifurcation manifold, singular point

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