Singularities Of Vector Fields Research Articles

Local models for smooth vector fields of the line

We present the local classification of singularities of smooth vector fields on the line, with respect to the equivalence relation of C1–conjugacy. Along the way, we recall the analogous classification, up to C0 and C∞ conjugacy. We also give the transversal unfoldings of the corresponding normal forms and treat the case where the changes of coordinates are tangent to the identity. Thus, a fairly complete description of the 1–d case is achieved.

Planar quartic–quadratic fold–fold singularity of Filippov systems and its bifurcation

In this paper it is exhibited the bifurcation scenario concerning a typical singularity of planar piecewise smooth vector fields in two zones. This singularity is characterized by a quadratic contact of one vector field and a quartic contact of another vector field at the same point of the switching manifold. By means of a three parameter perturbation, we observe the presence of bifurcations like: Hopf Bifurcation, Sliding Hopf Bifurcation, the birth and annihilation of chaotic scenarios, among others. The two and three dimensional bifurcation diagrams are included.

The linear term of the Poincaré map at singularities of planar vector fields

The aim of this work is to give conditions on the parameters of a family of analytic planar vector fields with a fixed Newton diagram and a monodromic singularity in order to guarantee that the coefficient (generalized first Lyapunov constant) of the linear part of the Poincaré map has an explicit formula that depends only on the Newton diagram. As far as we know, all the monodromic classes appearing in the literature satisfy that formula.

Polarization singularity index determination using wedge plate lateral shear interferometry

Polarization singularity is a point in an inhomogeneous polarization distribution at which the azimuth of the polarization ellipse is indeterminate. The azimuth gradient is solenoidal around the polarization singular point. In metrology, shear interferometry enables the visualization of phase gradients in optical beams. Since the polarization azimuth gradient is related to the Stokes phase gradient, we adopt the shear interferometry technique to visualize these phase gradients for the detection of polarization singularity. We use a wedge plate lateral shear interferometer (LSI) in our experiments for the detection of the ellipse and vector field singularities. Being a self-referencing and intensity-based technique, this method is convenient for practical use. Unlike the traditional Stokes polarimetry method, where the polarization distribution at every spatial point is evaluated, this technique can distinguish degenerate states of polarization singularities simply by visual observation of intensity fringes. Index and helicity measurement of these polarization singularities without using polarization measurement methods are novel aspects of this scheme. Both theoretical and experimental results are presented to support our claim.

The Poynting Vector Field Singularities: Effects of Symmetry and Its Violation

AbstractThe phenomenological theory revealing the generic effects of the problem symmetry, its violation, and energy conservation law on the singularities of the Poynting vector field is presented. The bifurcation scenario of formation (annihilation) of the singularities under variations of the problem parameters is elucidated. The results describe the singularities in scattering a linearly polarized plane electromagnetic wave. However, they are valid for any configuration of the incident beam at its scattering by a subwavelength particle. The author shows that topological changes in the pattern of the Poynting vector field occur through a finite number of pitchfork bifurcations. It means that the patterns are topologically stable under variations of the problem parameter(s) that lie between the bifurcation values. The latter ensures that the discussed topological properties of the problem are robust to weak symmetry violation, which is inevitable in any actual experiment. The general consideration is illustrated by a detailed study of singularities in scattering by an infinite right circular germanium cylinder. The results open the possibility of fitting and controlling radiation patterns on subwavelength scales important for various nanotechnologies.

Regularity for parabolic equations with singular non-zero divergence vector fields

We establish two-sided Gaussian bounds on the heat kernel of divergence-form parabolic equation with singular time-inhomogeneous vector field satisfying some minimal assumptions.

Two-Dimensional Ferronematics, Canonical Harmonic Maps and Minimal Connections

We study a variational model for ferronematics in two-dimensional domains, in the “super-dilute” regime. The free energy functional consists of a reduced Landau-de Gennes energy for the nematic order parameter, a Ginzburg–Landau type energy for the spontaneous magnetisation, and a coupling term that favours the co-alignment of the nematic director and the magnetisation. In a suitable asymptotic regime, we prove that the nematic order parameter converges to a canonical harmonic map with non-orientable point defects, while the magnetisation converges to a singular vector field, with line defects that connect the non-orientable point defects in pairs, along a minimal connection.

On real center singularities of complex vector fields on surfaces

One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations [R. Moussu, Une démonstration géométrique d'un théorème de Lyapunov-Poincaré, Astérisque 98–99 (1982), pp. 216–223]. In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with ‘many’ periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two. In this paper we prove versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.

Effect of the Energy Conservation Law, Space Dimension, and Problem Symmetry on the Poynting Vector Field Singularities

Effect of the Energy Conservation Law, Space Dimension, and Problem Symmetry on the Poynting Vector Field Singularities

Bursting Dynamics in a Singular Vector Field with Codimension Three Triple Zero Bifurcation

As a kind of dynamical system with a particular nonlinear structure, a multi-time scale nonlinear system is one of the essential directions of the current development of nonlinear dynamics theory. Multi-time scale nonlinear systems in practical applications are often complex forms of coupling of high-dimensional and high codimension characteristics, leading to various complex bursting oscillation behaviors and bifurcation characteristics in the system. For exploring the complex bursting dynamics caused by high codimension bifurcation, this paper considers the normal form of the vector field with triple zero bifurcation. Two kinds of codimension-2 bifurcation that may lead to complex bursting oscillations are discussed in the two-parameter plane. Based on the fast–slow analysis method, by introducing the slow variable W=Asin(ωt), the evolution process of the motion trajectory of the system changing with W was investigated, and the dynamical mechanism of several types of bursting oscillations was revealed. Finally, by varying the frequency of the slow variable, a class of chaotic bursting phenomena caused by the period-doubling cascade is deduced. Developing related work has played a positive role in deeply understanding the nature of various complex bursting phenomena and strengthening the application of basic disciplines such as mechanics and mathematics in engineering practice.

Restoration of Well-Posedness of Infinite-Dimensional Singular ODE’s via Noise

In this paper we aim at generalizing the results of A. K. Zvonkin (That removes the drift, 22, 129, 41) and A. Y. Veretennikov (Theory Probab. Appl., 24, 354, 39) on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

Perturbation of V-point polarization singular vector beams

The structured beams of light endowed with vector field singularities can be expressed as a superposition of spin and orbital angular momentum states. This article investigates the effect of perturbation of vector field singularities by superposing it with a homogeneously polarized beam and discusses its possible applications in studying the chiroptical effects. This perturbation leads to the splitting of vector field singularities (V-points) into stable lower-index ellipse field singularities (C-points). The strength of perturbation decides the separation between the two C-points in the resultant beam. It is found that the unfolding of V-points into lower-index C-points is always accompanied by a measurable centroid shift in the total and component intensity distribution of the resultant beam. Moreover, the centroid shift is more pronounced in the component intensity, whose state of polarization matches the state of polarization of the perturbing beam. The strength of perturbation can be measured by analyzing the radial shift in the centroid of the beam. It is observed that in the process of perturbation, the polarization singularity index is always conserved while the net helicity in the resultant beam is decided by the helicity of the perturbing beam. The strength of the perturbing beam can be modified by introducing a chiral sample in the beam path which may be useful to investigate the chiroptical effects.

On admissible singular drifts of symmetric α‐stable process

AbstractWe consider the problem of existence of a (unique) weak solution to the SDE describing symmetric α‐stable process with a locally unbounded drift , , . In this paper, b belongs to the class of weakly form‐bounded vector fields, the class providing the L2 theory of the non‐local operator behind the SDE, i.e. . It contains as proper sub‐classes other classes of singular vector fields studied in the literature in connection with this operator, such as the Kato class, the weak class and the Campanato–Morrey class (in general, such b makes invalid the standard heat kernel estimates in terms of the heat kernel of the fractional Laplacian). We show that the operator with weakly form‐bounded b admits a realization as (minus) Feller generator, and that the probability measures determined by the Feller semigroup (uniquely in appropriate sense) admit description as weak solutions to the corresponding SDE. The proof is based on detailed regularity theory of in , .

Self-healing of vector field singularities

We report here the self-healing ability of vector field singularities. While few studies discuss intensity healing in scalar beams, there is no report on polarization healing and healing of vector field singularities. Healing of polarization distributions from obstructions encountered by the beam is the high point of observation reported here as it entails healing of relative phases and amplitudes of the polarization component fields, in addition to intensity healing. This feature of robustness may play a significant role in applied research. The Poynting vector flow in the transverse plane of vector vortex beams is the key to the healing process. Since our investigation here is with vector field singularities, we also monitored the distribution of the Stokes parameter S3 to confirm the healing process.

Nature of the Poynting Vector Field Singularities in Resonant Light Scattering by Nanoparticles.

Singularities of the Poynting vector field subwavelength patterns in resonant light scattering by nanoparticles are discussed and classified. There are two generic types of the singularities, namely, (i) the singularities related to the vanishing of the magnetic (and/or electric) field at the singular points and (ii) the singularities related to the formation of standing waves in proximity to the singular points. The connection of these types of singularities to the topology of the singular points, space dimension (3D vs. 2D), and energy conservation law are revealed. In particular, it is shown that in 2D cases in non-dissipative media, the energy conservation reduces the possible types of generic singular points to saddles and centers only. In 3D cases, a universal expression connecting different components of the Poynting vector and valid for any generic singularities is derived and numerically checked for various types of singular points.

Tailoring polarization singularity lattices by phase engineering of three-beam interference

Structured optical lattices obtained from interference of three linearly polarized plane waves arranged in a cone geometry produce interlaced polarization singularity structures. These lattices are known to be populated with both C-point and V-point singularities. All the lattice fields abide by the principle of index conservation. In this article, we focused on tailoring these polarization singularity lattices by perturbing the interfering beams with a relative encoded phase. We have shown that a spatial lateral-shifting of the ellipse- and vector field singularities by an order of lattice parameter can be achieved through an appropriate phase engineered design. Similar phase engineering in six and eight beam interference method of lattice generation produces different results. In six and eight beam interference, phase engineering enables realization of C-point only lattices and provides handedness control for the ellipse field singularities, whereas in three beam interference case, there is only a translation shift in the lattice structure.

Heat kernel bounds for parabolic equations with singular (form-bounded) vector fields

We consider Kolmogorov operator $$-\nabla \cdot a \cdot \nabla + b \cdot \nabla $$ with measurable uniformly elliptic matrix a and prove Gaussian lower and upper bounds on its heat kernel under minimal assumptions on the vector field b and its divergence $$\mathrm{div\,}b$$ . More precisely, we prove: (1) Gaussian lower bound, provided that $$\mathrm{div\,}b \ge 0$$ , and b is in the class of form-bounded vector fields (containing e.g. the class $$L^d$$ , the weak $$L^d$$ class, as well as some vector fields that are not even in $$L_{\mathrm{loc}}^{2+\varepsilon }$$ , $$\varepsilon >0$$ ); in these assumptions, the Gaussian upper bound is in general invalid; (2) Gaussian upper bound, provided that b is form-bounded, and the positive part of $$\mathrm{div\,}b$$ is in the Kato class; in these assumptions, the Gaussian lower bound is in general invalid; (3) Gaussian upper and lower bounds, provided that b is form-bounded, $$\mathrm{div\,}b$$ is in the Kato class; (4) A priori Gaussian upper and lower bounds, provided that b is in a large class containing the class of form-bounded vector fields, $$\mathrm{div\,}b$$ is in the Kato class.

Dynamics of singular complex analytic vector fields with essential singularities II

The singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb C}_z$ belonging to the family ${\mathscr E}(r,d)=\left\{ X(z)=\frac{1}{P(z)} e^{E(z)}\frac{\partial }{\partial z}\ \Big\vert \ P, E\in\mathbb{C}[z]\right\}$, where $P$ is monic, $deg(P)=r$, $deg(E)=d$, $r+d\geq 1$, have a finite number of poles on the complex plane and an isolated essential singularity at infinity (for $d\geq 1$). Our aim is to describe geometrically $X$, particularly the singularity at infinity. We use the natural one to one correspondence between $X$, a global singular analytic distinguished parameter $\Psi_X(z)=\int^z P(\zeta) e^{-E(\zeta)}d\zeta$, and the Riemann surface ${\mathcal R}_X$ of this distinguished parameter. We introduce $(r,d)$-configuration trees which are weighted directed rooted trees. An $(r,d)$-configuration tree completely encodes the Riemann surface ${\mathcal R}_X$ and the singular flat metric associated on ${\mathcal R}_X$. The $(r,d)$-configuration trees provide "parameters" for the complex manifold ${\mathscr E}(r,d)$, which give explicit geometrical and dynamical information; a valuable tool for the analytic description of $X\in{\mathscr E}(r,d)$. Furthermore, given $X$, the phase portrait of the associated real vector field $Re(X)$ on the Riemann sphere is decomposed into $Re(X)$-invariant components: half planes and finite height strips. The germ of $X$ at infinity is described as a combinatorial word (consisting of hyperbolic, elliptic, parabolic and entire angular sectors having the point at infinity of $\widehat{\mathbb C}_z$ as center). The structural stability, under perturbation in ${\mathscr E}(r,d)$, of the phase portrait of $Re(X)$ is characterized by using the $(r,d)$-configuration trees. We provide explicit conditions, in terms of $r$ and $d$, as to when the number of topologically equivalent phase portraits of $Re(X)$ is unbounded.

Local generic behavior of a planar Filippov system with non-smooth switching curve

This paper discusses the generic local classification of typical singularities of 2D piecewise smooth vector fields when the switching set is an non smooth curve. The main goal is to obtain classification results concerning structural stability and generic codimension one local bifurcations.

Conical Singular Points and Vector Fields

Conical Singular Points and Vector Fields