相关论文: Vector fields with a given set of singular points

We consider manifolds with isolated singularities, i.e., topological spaces which are manifolds (say, $C^\infty$--) outside discrete subsets (sets of singular points). For (germs of) manifolds with, so called, cone--like singularities, a…

We discuss different generalizations of the classical notion of the index of a singular point of a vector field to the case of vector fields or 1-forms on singular varieties, describe relations between them and formulae for their…

We define two types of local indices of a vector field at an isolated zero on the boundary, and prove Poincare-Hopf-type index theorems for certain vector fields on a compact smooth manifold which have only isolated zeros.

We describe all possible topological structures of codimension one gradient vector fields on the shpere with at most ten singular points. To describe structures, we use a graph whose edges are one-dimensional stable manifolds. The…

This work establishes a strong uniqueness property for a class of planar locally integrable vector fields. A result on pointwise convergence to the boundary value is also proved for bounded solutions.

Indices of vector fields on (complex analytic) singular varieties have been considered by various authors from several different viewpoints. All these indices coincide with the classical local index of Poincar\'e-Hopf when the ambient…

Gradient vector fields are fundamental objects from both theoretical and practical perspectives, since various phenomena can be modeled within this framework. The ``moduli space'' of such vector fields provides the foundation for describing…

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological…

In this paper, we give a full description of all possible singular points that occur in generic 2-parameter families of vector fields on compact 2-manifolds. This is a part of a large project aimed to a complete study of global bifurcations…

We discuss the notions of indices of vector fields and 1-forms and their generalizations to singular varieties and varieties with actions of finite groups, as well as indices of collections of vector fields and 1-forms.

We give a generalization of an algebraic formula of Gomez-Mont for the index of a vector field with isolated zero in (C^n,0) and tangent to an isolated hypersurface singularity. We only assume that the vector field has an isolated zero on…

We study holomorphic vector fields on isolated hypersurface singularities and derive global obstructions to the existence of holomorphic vector fields on compact singular varieties. For a hypersurface germ $(V,0)$ with an isolated…

We introduce several sufficient conditions to guarantee the existence of the Milnor vector field for new classes of singularities of map germs. This special vector field is related with the equivalence problem of the Milnor fibrations for…

We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is…

The existence of a nowhere zero real vector field implies a well-known restriction on a compact manifold. But all manifolds admit nowhere zero complex vector fields. The relation between these observations is clarified.

The aim of this paper is to investigate the point spectra of vector fields. We will define the point spectrum of a vector field and study some of its basic properties. In particular, we will prove that point spectra are well-behaved under…

In an unpublished note [H1] we have described a method to obtain a formula for the index of an analytic vector field with (complex) isolated zero on a real analytic hypersurface with (complex) isolated singularity. This formula, like the…

We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also…

The paper studies the complex 1-dimensional polynomial vector fields with real coefficients under topological orbital equivalence preserving the separatrices of the pole at infinity. The number of generic strata is determined, and a…

We prove a sufficient condition for the existence of explicit first integrals for vector fields which admit an integrating factor. This theorem recovers and extends previous results in the literature on the integrability of vector fields…