Dimensional Vector Fields Research Articles

Detecting multiplicity for systems of second-order equations: an alternative approach

In this paper we are concerned with a system of second-order differential equations of the form $x''+A(t,x)x=0$, $t\in [0,\pi]$, $x\in {{\bf R}}^N$, where $A(t,x)$ is a symmetric $N\times N$ matrix. We concentrate on an asymptotically linear situation and we prove the existence of multiple solutions to the Dirichlet problem associated to the system. Multiplicity is obtained by a comparison between the number of moments of verticality of the matrices $A_0(t)$ and $A_\infty(t)$, which are the uniform limits of $A(t,x)$ for $|x|\to 0$ and $|x|\to +\infty$, respectively. For the proof, which is based on a generalized shooting approach, we provide a theorem on the existence of zeros of a class of $N$-dimensional vector fields.

CODIMENSION 3 BIFURCATIONS OF HOMOCLINIC ORBITS WITH ORBIT FLIPS AND INCLINATION FLIPS

The homoclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near the homoclinic orbit. This homoclinic orbit is non-principal in the meanings that its positive semi-orbit takes orbit flip and its unstable foliation takes inclination flip. The existence, nonexistence, uniqueness and coexistence of the 1-homoclinic orbit and the 1-periodic orbit are studied. The existence of the two-fold periodic orbit and three-fold periodic orbit are also obtained.

Localization of non-Abelian gauge fields on domain walls at weak coupling: D-brane prototypes

Building on our previous results, we study D-brane/string prototypes in weakly coupled (3+1)-dimensional supersymmetric field theory engineered to support (2+1)-dimensional domain walls, "non-Abelian" strings and various junctions. Our main but not exclusive task is the study of localization of non-Abelian gauge fields on the walls. The model we work with is N=2 QCD, with the gauge group SU(2)x$U(1) and N_f=4 flavors of fundamental hypermultiplets (referred to as quarks), perturbed by the Fayet-Iliopoulos term of the U(1) factor. In the limit of large but almost equal quark mass terms a set of vacua exists in which this theory is at weak coupling. We focus on these vacua (called the quark vacua). We study elementary BPS domain walls interpolating between selected quark vacua, as well as their bound state, a composite wall. The composite wall is demonstrated to localize a non-Abelian gauge field on its world sheet. Next, we turn to the analysis of recently proposed "non-Abelian" strings (flux tubes) which carry orientational moduli corresponding to rotations of the "color-magnetic" flux direction inside a global O(3). We find a 1/4-BPS solution for the string ending on the composite domain wall. The end point of this string is shown to play the role of a non-Abelian (dual) charge in the effective world volume theory of non-Abelian (2+1)-dimensional vector fields confined to the wall.

Complete algebraic vector fields on affine surfaces, Part I

In this work we study the singularities at infinity of algebraic vector fields in dimension 2.These singularities will be classified under a mild assumption. The general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in Part II. Our main results are local and therefore can be carried over more general surfaces. Whereas we deal with ℂ -complete vector fields, the results also apply to ℝ-complete ones thanks to a theorem of Forstneric [7].

Evolutionary programming-based univector field navigation method for past mobile robots

Most of navigation techniques with obstacle avoidance do not consider the robot orientation at the target position. These techniques deal with the robot position only and are independent of its orientation and velocity. To solve these problems this paper proposes a novel univector field method for fast mobile robot navigation which introduces a normalized two dimensional vector field. The method provides fast moving robots with the desired posture at the target position and obstacle avoidance. To obtain the sub-optimal vector field, a function approximator is used and trained by evolutionary programming. Two kinds of vector fields are trained, one for the final posture acquisition and the other for obstacle avoidance. Computer simulations and real experiments are carried out for a fast moving mobile robot to demonstrate the effectiveness of the proposed scheme.

Further Reduction of Normal Forms for Vector Fields

We study Lie's method in the way of Ushiki for further reduction of normal forms for vector fields with singularity at the origin. We give further reduction of normal forms in two typical cases for vector fields in dimension 2: one with a rotation as its linear part and the other with a nilpotent linear part.

Dynamics of localized structures in vectorial waves

Dynamical properties of topological defects in a two dimensional complex vector field are considered. These objects naturally arise in the study of polarized transverse light waves. Dynamics is modeled by a vector complex Ginzburg-Landau equation with parameter values appropriate for linearly polarized laser emission. Creation and annihilation processes, and self-organization of defects in lattice structures, are described. We find "glassy" configurations dominated by vectorial defects and a melting process associated with topological-charge unbinding.

A weighted inverse matrix approach to searching for the electric field sources

In this paper, we propose an approach to solving for an inverse problem. We try to evaluate a unique solution of an inverse problem by means of the weighted inverse matrix. We have applied our weighted inverse matrix method to searching for the electric field source and demonstrating three dimensional electric field distributions. As a result, we have succeeded in evaluating the unique electric field source and in visualizing the three dimensional electric vector field distributions from the locally measured electric fields.

Strange Attractors Across the Boundary¶of Hyperbolic Systems

We study a bifurcation of Axiom A (hyperbolic) vector fields in dimension three leading to robust strange attractors with singularities. The Axiom A vector fields involved in the bifurcation exhibit a basic set equivalent to the suspension of a three symbol subshift. The attractors arising from this kind of bifurcation are not equivalent to the geometric Lorenz attractors.

Correlation functions of the energy–momentum tensor on spaces of constant curvature

An analysis of one- and two-point functions of the energy–momentum tensor on homogeneous spaces of constant curvature is undertaken. The possibility of proving a c-theorem in this framework is discussed, in particular in relation to the coefficients c, a, which appear in the energy–momentum tensor trace on general curved backgrounds in four dimensions. Ward identities relating the correlation functions are derived and explicit expressions are obtained for free scalar, spinor field theories in general dimensions and also free vector fields in dimension four. A natural geometric formalism which is independent of any choice of coordinates is used and the role of conformal symmetries on such constant curvature spaces is analysed. The results are shown to be constrained by the operator product expansion. For negative curvature the spectral representation, involving unitary positive energy representations of O( d−1,2), for two-point functions of vector currents is derived in detail and extended to the energy–momentum tensor by analogy. It is demonstrated that, at non-coincident points, the two-point functions are not related to a in any direct fashion and there is no straightforward demonstration obtainable in this framework of irreversibility under renormalisation group flow of any function of the couplings for four-dimensional field theories which reduces to a at fixed points.

Homoclinic saddle-node bifurcations in singularly perturbed systems

In this paper we study the creation of homoclinic orbits by saddle-node bifurcations. Inspired on similar phenomena appearing in the analysis of so-called “localized structures” in modulation or amplitude equations, we consider a family of nearly integrable, singularly perturbed three dimensional vector fields with two bifurcation parameters a and b. The O(e) perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, Ws(Γ) and Wu(Γ), the stable and unstable manifolds of a slow manifold Γ. Homoclinic orbits to Γ correspond to intersections Ws(Γ)∩Wu(Γ); Ws(Γ)∩Wu(Γ)=∅ for a a*. The bifurcation at a=a* is followed by a sequence of nearby, O(e2(loge)2) close, homoclinic saddle-node bifurcations at which pairs of N-pulse homoclinic orbits are created (these orbits make N circuits through the fast field). The second parameter b distinguishes between two significantly different cases: in the cooperating (respectively counteracting) case the averaged effect of the fast field is in the same (respectively opposite) direction as the slow flow on Γ. The structure of Ws(Γ)∩Wu(Γ) becomes highly complicated in the counteracting case: we show the existence of many new types of sometimes exponentially close homoclinic saddle-node bifurcations. The analysis in this paper is mainly of a geometrical nature.

Visualization of Vector Field by Virtual Reality

A visualization software program is developed in order to analyze three dimensional vector fields by means of today’s advanced virtual reality technology. This program enables simulation researchers to interactively visualize stream lines, tracer particle motions, isosurfaces, etc. with stereo view. The program accepts any kindof vector field s on structured mesh as an input data. A virtual reality hardware system, on which the program operates, and details of the program are described.

The energy of Hopf vector fields

The 3-dimensional Hopf vector field is shown to be a stable harmonic section of the unit tangent bundle. In contrast, higher dimensional Hopf vector fields are unstable harmonic sections; indeed, there is a natural variation through smooth unit vector fields which is locally energy-decreasing, and whose asymptotic limit is a singular vector field of finite energy. This energy is explicitly calculated, and conjectured to be the infimum of the energy functional over all smooth unit vector fields.

Successional stability of vector fields in dimension three

A topological invariant, the community transition graph, is introduced for dissipative vector fields that preserve the skeleton of the positive orthant. A vector field is defined to be successionally stable if it lies in an open set of vector fields with the same community transition graph. In dimension three, it is shown that vector fields for which the origin is a connected component of the chain recurrent set can be approximated in the C 1 C^1 Whitney topology by a successionally stable vector field.

Symmetric singularities of reversible vector fields in dimension three

For a large class of reversible vector fields on the three-dimensional space we present all the topological types and their respective normal forms of the symmetric singularities of codimension 0, 1.

Instability of equilibria in dimension three

In this paper we show that if v is an analytic vector field on ℝ 3 having an isolated singular point at 0, then there exists a trajectory of v which converges to 0 in the past or in the future. The proof is based on certain results concerning desingularizaton of vector fields in dimension three and on index-type arguments à la Poincaré-Hopf.

A new method for 3-D vector field visualization utilizing streamlines and volume rendering techniques

To grasp the physical behavior of a three dimensional vector field phenomenon, not only it's magnitude, but also its direction, loci and the streamlines of the vector field should be visualized simultaneously. In this paper, a new method for vector field visualization which satisfies the above mentioned criteria is presented. The method employs in parallel the volume rendering and the streamline visualization methods, providing highly sophisticated techniques for three dimensional vector field visualization. Two successful applications of the proposed visualization method for three dimensional eddy-current distribution inside an aluminum plate are also presented.

Batalin–Fradkin Approach to (1 + 1)-Dimensional Massive Vector Fields and the Gluonic Effective Action for Hard Thermal Loops

The Lagrangian of (1 + 1)-dimensional massive vector fields is studied. Since this system has second class constraints, the method of Batalin–Fradkin, which introduces new fields to convert second class constraints into first class ones, is applied in an extended manner. Instead of the usual treatment, which uses the Stueckelberg field as a new field, we can use a pseudoscalar field. We will show there are at least two ways to introduce a pseudoscalar. At the quantum level, one way leads to the system that is equivalent to the original system, and the other way gives an inequivalent system. The relation of these two ways is clarified. As an application of the latter way, we consider QCD at finite temperature and the gluonic effective action for hard thermal loops is constructed.

Regularization of discontinuous vector fields in dimension three

In this paper vector fields around the origin in dimension three which are approximations of discontinuous ones are studied. In a former work of Sotomayor and Teixeira [6] it is shown, via regularization, that Filippov's conditions are the natural ones to extend the orbit solutions through the discontinuity set for vector fields in dimension two. In this paper we show that this is also the case for discontinuous vector fields in dimension three. Moreover, we analyse the qualitative dynamics of the local flow in a neighborhood of the codimension zero regular and singular points of the discontinuity surface.

Bifurcation and stability of families of hyperbolic vector fields in dimension three

In this article we study the global stability of one-parameter families of hyperbolic vector fields with simple bifurcations in three-dimensional manifolds at least in all known cases (see introduction).