Class Of Vector Fields Research Articles

On a class of vector fields with discontinuities of divide-by-zero type and its applications to geodesics in singular metrics

We study phase portraits at singular points of vector fields of the special type where all components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. Also we assume some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three applications in differential geometry: singularities of geodesic flows in various singular metrics on two-dimensional manifolds.

The global dynamics of a class of vector fields in ℝ3

In this paper, we find a bridge connecting a class of vector fields in ℝ3 with the planar vector fields and give a criterion of the existence of closed orbits, heteroclinic orbits and homoclinic orbits of a class of vector fields in ℝ3. All the possible nonwandering sets of this class of vector fields fall into three classes: (i) singularities; (ii) closed orbits; (iii) graphs of unions of singularities and the trajectories connecting them. The necessary and sufficient conditions for the boundedness of the vector fields are also obtained.

On the substantiation of a projection method for the stationary Fokker-Planck equation

A projection-type condition is discussed that is sufficient for the stationary Fokker-Planck equation Δu - div(uf) = 0 to be solvable within a class of probability density functions. Based on existence theorems and estimates of positive solutions obtained by the author, a fairly large class of vector fields f satisfying this condition is proposed.

Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups

AbstractLet 𝔾 be a step-two nilpotent group of H-type with Lie algebra 𝔊 = V ⊕ t. We define a class of vector fields X = {Xj} on 𝔾 depending on a real parameter k ≥ 1, and we consider the corresponding p-Laplacian operator Lp,ku = divX(|∇Xu|p−2∇Xu). For k = 1 the vector fields X = {Xj} are the left invariant vector fields corresponding to an orthonormal basis of V; for 𝔾 being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator Lp,k and as an application, we get a Hardy type inequality associated with X.

On one class of Lipschitz vector fields in ℝ3

Under consideration are some continuous metric functions induced by one class of Lipschitz vector fields in ℝ3. These functions are showed to be quasimetrics within the domain of definition of the vector fields. We prove some analogs of the Rashevsky-Chow Theorem and the Ball-Box Theorem under some restriction on the class of vector fields. The methods of proofs do not use the existence of the nilpotent tangent cone.

Newton vector fields on the plane and on the torus

In this article, we show that complex vector fields on the punctured Riemann sphere are Newton vector fields and as a consequence all meromorphic vector fields and all elliptic vector fields are Newton vector fields. Moreover, for a large class of vector fields, which includes all elliptic vector fields, the proof is constructive in the sense that one can explicitly construct the function characterizing the Newton vector field.

Renormalization of vector fields and Diophantine invariant tori

AbstractWe extend the renormalization group techniques that were developed originally for Hamiltonian flows to more general vector fields on 𝕋d×ℝℓ. Each Diophantine vector ω∈ℝd determines an analytic manifold 𝒲 of infinitely renormalizable vector fields, and each vector field on 𝒲 is shown to have an elliptic invariant d-torus with frequencies ω1,ω2,…,ωd. Analogous manifolds for particular classes of vector fields (Hamiltonian, divergence-free, symmetric, reversible) are obtained simply by restricting 𝒲 to the corresponding subspace. We also discuss non-degeneracy conditions, and the resulting reduction in the number of parameters needed in parametrized families to guarantee the existence of invariant tori.

C k Solvability near the Characteristic set for a Class of Vector Fields of Infinite Type

C k Solvability near the Characteristic set for a Class of Vector Fields of Infinite Type

Involutive orbits of non-Noether symmetry groups

We consider set of functions on Poisson manifold related by continues one-parameter group of transformations. Class of vector fields that produce involutive families of functions is investigated and relationship between these vector fields and non-Noether symmetries of Hamiltonian dynamical systems is outlined. Theory is illustrated with sample models: modified Boussinesq system and Broer-Kaup system.

On the geomagnetic directional problem: a uniqueness result

We consider the following nonlinear boundary value problem in the exterior space V ^ = { x ∈ R 3 : | x | > 1 } of the unit sphere S : Given a vector field D : S → R 3 we ask for all harmonic vector fields B : V ^ → R 3 which decay at least as fast as a dipole field at infinity and are parallel to D on S , i.e. there is f : S → R such that B = f D . This problem is related to the problem of reconstructing the geomagnetic field outside the earth from directional data measured on the earth's surface. We prove in this note the unique solvability of the above problem for a certain class of vectorfields, viz. all those vector fields D n k , n ∈ N , 0 < | k | < n , which are obtained by restricting a single nonaxisymmetric multipole field on S .

Traces and fine properties of a $BD$ class of vector fields and applications

Dans cet. article, nous etudions les proprietes fines et les proprietes de trace pour une classe de champs de vecteurs de la. forme C = wB, ou w est: une fonction scalaire bornee et B est localement bornee et avec deformation bornee. Dans l'hypothese que la divergence au sens de la distribution de C sur une hypersurface, avec le comportement ponctuel de w. Nous etudions aussi le comportement de ces traces par rapport a la transformation wB → h(w)B, avec h ∈ C 1 , et nous demontrons une formule explicite pour les traces. Comme consequence de ces resultats nous demontrons que la theorie de DiPerna-Lions peut s'etendre a.ux champs de vecteurs speciaux avec une deformation bornee. Lorsque B est localement BV nous obtenons aussi les estimations sur la taille des ensembles de « discontinuite » approximative de w.

C ∞ -Normal Forms of Local Vector Fields

The problem of a local classification of vector fields is considered. A construction of formal and C∞-normal forms is described.

Stochastic Jacobi fields and vector fields induced by varying area on path spaces

We study two classes of vector fields on the path space over a closed manifold with a Wiener Riemannian measure. By adopting the viewpoint of Yang-Mills field theory, we study a vector field defined by varying a metric connection. We prove that the vector field obtained in this way satisfies a Jacobi field equation which is different from that of classical one by taking in account that a Brownian motion is invariant under the orthogonal group action, so that it is a geometric vector field on the space of continuous paths, and induces a quasi-invariant solution flow on the path space. The second object of this paper is vector fields obtained by varying area. Here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense, but also its Levy area. We prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the Cameron-Martin space in the Malliavin calculus sense, and at the same time changing its Levy area in an appropriate way. This leads to a new derivation of the integration by parts formula on the path space.

A Class of Vector Fields on Path Spaces

In this paper we show that the vector fieldX∇, hon a based path spaceWo(M) over a Riemannian manifoldMdefined by parallel translating a curvehin the initial tangent spaceToMvia an affine connection ∇ induces a solution flow which preserves the Wiener measure on the based path spaceWo(M), provided the affine connection ∇ is adjoint skew-symmetric. In the case when ∇ is a metric connection, then ∇ is adjoint skew-symmetric if and only if ∇ is torsion skew-symmetric.

ON THE GENERALITY OF THE UNFOLDED CHUA'S CIRCUIT

In this paper, we study the generality of Chua's oscillator by deriving a class of vector fields that Chua's oscillator is equivalent to. For the class of vector fields with a scalar nonlinearity, we prove that under certain conditions, two such vector fields are topologically conjugate if the Jacobian matrices at each point have the same eigenvalues and the equilibrium points are matched up. We show how these conditions are related to the complete state observability of a corresponding linear system. These results are used to show that the n-dimensional Chua's oscillator is topologically conjugate to almost every vector field in this class. We comment on the special case when the vector field is piecewise-linear and in particular when the vector field is 2-segment piecewise-linear. These results are illustrated by transforming several systems studied in the literature into equivalent Chua's oscillators. We also extend some of these results to the case of several scalar nonlinearities. As a corollary we prove that almost all piecewise-linear vector fields with parallel boundary planes are topologically conjugate if the boundary planes and equilibrium points are the same and the eigenvalues in corresponding regions are the same. We also give a dual result of topological conjugacy.

A class of vector fields on manifolds containing second order ODEs

A class of vector fields on manifolds containing second order ODEs

On the two-dimensional initial boundary value problem for the Navier-Stokes equations with discontinuous boundary data

We consider the initial boundary value problem for the Navier-Stokes equations with boundary conditions\(\vec v|\partial \Omega = \vec a\). We assume that\(\vec a\) may have jump discontinuities at finitely many points ξ1;. . .,ξm of the boundary ϖΩ of a bounded domain Ω ⊂ ℝ2. We prove that this problem has a unique generalized solution in a finite time interval or for small initial and boundary data. The solution is found in a class of vector fields with infinite energy integral. The case of a moving boundary is also considered. Bibliography: 11 titles.

FINDING PARAMETERS FOR CHUA’S OSCILLATOR WHICH IS TOPOLOGICALLY CONJUGATE TO SYSTEMS WITH A SCALAR NONLINEARITY

Chua’s oscillator is topologically conjugate to a large class of vector fields with a scalar non-linearity. In this letter, we give an algorithm which, given a vector field in this class, finds the parameters for Chua’s oscillator for which Chua’s oscillator is topologically conjugate to it. We illustrate this by transforming Sparrow’s system and the chaotic Colpitts oscillator into equivalent Chua’s oscillators.

On weak relative degree and the McGehee transformation

Points where jumps in relative degree occur are presented as singularities in the vector field of the internal dynamics of systems that are candidates for input-output linearization via nonlinear feedback. For a certain class of vector fields, it is shown that the point at which the relative degree is weak can be transformed to a saddle equilibrium point at the origin. >

Knowledge-Based Nonlinear Boundary Integral Models of Compressible Viscous Flows Over Arbitrary Bodies: Taking CFD Back to Basics

The paper starts with a discussion of a knowledge-based CFD methodology. A new incompressible formulation known as SAVER is first introduced, employing a novel relaxation approach. This is then generalized through a modification of the boundary conditions to the GENESIS methodology, for analysis or design in compressible, rotational flow. A discussion is presented of how the basic causal nature of integral methods offers new insights into certain flow phenomena, such as shocks and separations, and facilitates aerodynamic sensitivity analysis. The paper presents a new class of vector fields approximating the Euler equations for transonic flows, and shows how GENESIS can be used to construct first an exact solution of these approximate fields, then a numerical solution of the residual error fields. The explicit representation of a shock discontinuity on the body boundary exploits its causal link with conditions at the sonic point to suppress, non-dissipatively, the mathematically-valid but physically-impossible formation of an expansion shock.