Singularities Of Vector Fields Research Articles

Robust Polylines Tracing for N-Symmetry Direction Field on Triangulated Surfaces

We are proposing an algorithm for tracing polylines that are oriented by a direction field defined on a triangle mesh. The challenge is to ensure that two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is impossible to enforce with existing algorithms. The core of our contribution is to determine how polylines cross each triangle. Our solution is inspired by EdgeMaps where each triangle boundary is decomposed into inflow and outflow intervals such that each inflow interval is mapped onto an outflow interval. To cross a triangle, we find the inflow interval that contains the entry point, and link it to the corresponding outflow interval, with the same barycentric coordinate. To ensure that polylines cannot merge or cross, we introduce a new direction field representation, we resolve the inflow/outflow interval pairing with a guaranteed combinatorial algorithm, and propagate the barycentric positions with arbitrary precision number representation. Using these techniques, two streamlines crossing the same triangle cannot merge or cross, but only locally overlap when all streamline extremities are located on the same edge. Cross-free and merge-free polylines can be traced on the mesh by iteratively crossing triangles. Vector field singularities and polyline/vertex crossing are characterized and consistently handled.

Some remarks on inverse Jacobi multipliers around Hopf singularities

We study several properties of inverse Jacobi multipliers V around Hopf singularities of analytic vector fields X in Rn which are relevant to the study of the local bifurcation of periodic orbits. When n=3 and the singularity is a saddle-focus we show that: (i) any two locally smooth and non-flat linearly independent inverse Jacobi multipliers have the same Taylor expansion; (ii) any smooth and non-flat V has associated exactly one smooth center manifold Wc of X such that Wc⊂V−1(0). We also study whether the properties of the vanishing set V−1(0) proved in the 3-dimensional case remain valid when n≥4.

On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations

A two-fold singularity is a point on a discontinuity surface of a piecewise-smooth vector field at which the vector field is tangent to the surface on both sides. Due to the double tangency, forward evolution from a two-fold is typically ambiguous. This is an especially serious issue for two-folds that are reached by the forward orbits of a non-zero measure set of initial points. However, arbitrarily small perturbations of the vector field can make forward evolution well-defined, and from an applied perspective, such perturbations may represent additional model features that enhance the realism of a piecewise-smooth mathematical model. Three physically motivated forms of perturbation: hysteresis, time-delay, and noise, are analysed individually. The purpose of this paper is to characterise the perturbed dynamics in the limit that the size of the perturbation tends to zero. This concept is applied to a two-fold in two dimensions. In each case the limit leads to a novel probabilistic notion of forward evolution from the two-fold.

Index of singularities of real vector fields on singular hypersurfaces

Gomez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincare-Hopf index to complex vector fields tangent to singular hypersurfaces. The GSV-index extends to the real case. This is a survey paper on the joint research with Gomez-Mont and Giraldo about calculating the GSV-index $\Ind_{V_\pm,0}(X)$ of a real vector field $X$ tangent to a singular hypersurface $V=f^{-1}(0)$. The index $\Ind_{V_{\pm,0}}(X)$ is calculated as a combination of several terms. Each term is given as a signature of some bilinear form on a local algebra associated to $f$ and $X$. Main ingredients in the proof are Gomez-Mont's formula for calculating the GSV-index on \emph{ singular complex} hypersurfaces and the formula of Eisenbud, Levine and Khimshiashvili for calculating the Poincare-Hopf index of a singularity of a \emph{real} vector field in $\R^{n+1}$

Singularities of complex vector fields having many closed orbits

Singularities of complex vector fields having many closed orbits

Birth of limit cycles bifurcating from a nonsmooth center

This paper is concerned with a codimension analysis of a two-fold singularity of piecewise smooth planar vector fields, when it behaves itself like a center of smooth vector fields (also called nondegenerate Σ-center). We prove that any nondegenerate Σ-center is Σ-equivalent to a particular normal form Z0. Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from Z0 that have k hyperbolic limit cycles bifurcating from the nondegenerate Σ-center of Z0 (the same holds for k=∞). Moreover, we also exhibit families of piecewise smooth vector fields of codimension k emerging from Z0. As a consequence we prove that Z0 has infinite codimension.

An inexact proximal point algorithm for maximal monotone vector fields on Hadamard manifolds

In this paper, an inexact proximal point algorithm concerned with the singularity of maximal monotone vector fields is introduced and studied on Hadamard manifolds, in which a relative error tolerance with squared summable error factors is considered. It is proved that the sequence generated by the proposed method is convergent to a solution of the problem. Moreover, an application to the optimization problem on Hadamard manifolds is given. The main results presented in this paper generalize and improve some corresponding known results given in the literature.

Flows near compact invariant sets. Part I

In this paper it is proved that near a compact, invariant, proper subset of a continuous flow on a compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. This result shows that assuming the connectedness of the phase space implies the existence of a considerably deeper classification of topological flow behaviour, in the vicinity of compact invariant sets, than that described in the classical theorems of Ura-Kimura and Bhatia. The proposed classification brings to light, in a systematic way, the possibility of occurrence of orbits of infinite height arbitrarily near the compact invariant in question, and this under relatively simple conditions. Singularities of smooth vector fields displaying this strange phenomenon occur in every dimension greater than 2 (in this paper, a smooth flow on the 3-dimensional sphere exhibiting such an equilibrium is constructed). Near periodic orbits, the same phenomenon is observable already in dimension 4 (and on every manifold of dimension greater than 4). As a corollary to the main result, an elegant characterization of the topological Hausdorff structure of the set of all compact minimal sets of the flow is obtained (Theorem 2). Keywords: topological behaviour of C0 flows, compact invariant sets, compact minimal sets, topological Hausdorff structure, non-hyperbolic singularities and periodic orbits, orbits of infinite height.

ON THE CENTER CONDITIONS FOR ANALYTIC MONODROMIC DEGENERATE SINGULARITIES

In this paper, we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields and can be applied independently so that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allow to study monodromic degenerate singularities with characteristic directions.

Singularities of the divergence of continuous vector fields and uniform Hausdorff estimates

We prove that every closed set which is not sigma-finite with respect to the Hausdorff measure H^{N-1} carries singularities of continuous vector fields in the Euclidean space R^N for the divergence operator. We also show that finite measures which do not charge sets of sigma-finite Hausdorff measure H^{N-1} can be written as an L^1 perturbation of the divergence of a continuous vector field. The main tool is a property of approximation of measures in terms of the Hausdorff content.

An axiomatic version of Zariski’s patching theorem

We state six axioms concerning any regularity property P in a given birational equivalence class of algebraic threefolds. Axiom 5 states the existence of a Local Uniformization in the sense of valuations for P. If Axioms 1 to 5 are satisfied by P, then the function field has a projective model which is everywhere regular with respect to P. Adding Axiom 6 ensures the existence of a P-Resolution of Singularities for any projective model. Applications concern Resolution of Singularities of vector fields and a weak version of Hironaka’s Strong Factorization Conjecture for birational morphisms of nonsingular projective threefolds, both of them in characteristic zero. The last section contains open problems about axiomatizing regularity properties which have P-Resolution of Singularities.

SINGULARITIES OF DIVERGENCE-FREE VECTOR FIELDS WITH VALUES INTO S1OR S2: APPLICATIONS TO MICROMAGNETICS

In this survey, we present several results on the regularizing effect, rigidity and approximation of 2D unit-length divergence-free vector fields. We develop the concept of entropy (coming from scalar conservation laws) in order to analyze singularities of such vector fields. In particular, based on entropies, we characterize lower semicontinuous line-energies in 2D and we study by Γ-convergence method the associated regularizing models (like the 2D Aviles–Giga and the 3D Bloch wall models). We also present some applications to the analysis of pattern formation in micromagnetics. In particular, we describe domain walls in the thin ferromagnetic films (e.g. symmetric Néel walls, asymmetric Néel walls, asymmetric Bloch walls) together with interior and boundary vortices.

Asymptotic Property of Divergent Formal Solutions in Linearization of Singular Vector Fields

We study asymptotic properties of divergent formal solutions appearing in the linearization problem of a singular vector field without a Diophantine condition or existence of additional first integrals. We give an asymptotic meaning to divergent formal solutions constructed from a singular perturbative solution (cf. [6]).

Sectorial solution to semilinear singular partial differential equation

We shall show the solvability of semilinear Fuchsian partial differential systems in a multi-sectorial domain. Our equation contains a linearizing equation of a singular vector field to its linear part when so-called small denominators occur. We will show the existence of an analytic solution in a multi-sectorial domain without assuming any Diophantine condition.

The index of singularities of vector fields and finite jets

We describe when the index of a singularity of a smooth vector field is determined by a finite jet at the singularity. We also give some criteria to determine some terms from the formal series expansion of the vector field at the singularity which determine the index.

A Legendre Transform on an Exotic S3

Abstract We consider an exotic contact form α on S3 and we establish explicitly the existence of a non singular vector field v in ker(α) such that the non-singular one-differential form β(・) := dα(v, ・) is a contact form on S3 with the same orientation as α. In particular this means that a Legendre transform can be completed.

Fractional Laplacian with singular drift

For $\alpha \in (1,2)$ we consider the equation $\partial _t u = \varDelta ^{\alpha /2} u + b \cdot \nabla u$, where $b$ is a time-independent, divergence-free singular vector field of the Morrey class $M_1^{1-\alpha }$. We show that if the Morrey norm $

Asymptotic Property of Divergent Formal Solutions in Linearization of Singular Vector Fields

Asymptotic Property of Divergent Formal Solutions in Linearization of Singular Vector Fields

Una Trasformata di Legendre su un modello non standard

We consider an exotic contact form on the three-dimensional sphere and we establish explicitly the existence of a non singular vector field in its kernel such that the dual form is still a contact form with the same orientation than the given one. In particular this means that a Legendre transform can be completed.

Symmetries of center singularities of plane vector fields

Let $ {D^2} \subset {\mathbb{R}^2} $ be a closed unit 2-disk centered at the origin $ O \in {\mathbb{R}^2} $ and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus, topologically O is a “center” singularity. Let $ \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty } \right) $ be the function associating with each z ≠ O its period with respect to F. In general, such a function cannot be even continuously defined at O. Let also $ {\mathcal{D}^{+} }(F) $ be the group of diffeomorphisms of D 2 that preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D 2 if and only if the 1-jet of F at O is a “rotation,” i.e., $ {j^1}F(O) = - y\frac{\partial }{{\partial x}} + x\frac{\partial }{{\partial y}} $ . Then $ {\mathcal{D}^{+} }(F) $ is homotopy equivalent to a circle.