Vector Field Problem Research Articles

Surfaces with Central Configuration and Dulac’s Problem for a Three Dimensional Isolated Hopf Singularity

AbstractLet $$\xi $$ ξ be a real analytic vector field with an elementary isolated singularity at $$0\in \mathbb {R}^3$$ 0 ∈ R 3 and eigenvalues $$\pm bi,c$$ ± b i , c with $$b,c\in \mathbb {R}$$ b , c ∈ R and $$b\ne 0$$ b ≠ 0 . We prove that all cycles of $$\xi $$ ξ in a sufficiently small neighborhood of 0, if they exist, are contained in the union of finitely many subanalytic invariant surfaces, each one entirely composed of a continuum of cycles. In particular, we solve Dulac’s problem for such vector fields, i.e., finiteness of limit cycles.

Monodromy and Dulac's problem for piecewise analytical planar vector fields

Consider an analytical function f:V⊂R2→R having 0 as its regular value, a switching manifold Σ=f−1(0) and a piecewise analytical vector field X=(X+,X−), i.e. X± are analytical vector fields defined on Σ±={p∈V:±f(p)>0}. We characterize when the vector field X has a monodromic singular point in Σ, called Σ-monodromic singular point. Moreover, under certain conditions, we show that a Σ-monodromic singular point of X has a neighborhood free of limit cycles.

Perception-Aware Image-Based Visual Servoing of Aggressive Quadrotor UAVs

The maintenance of visual features within the sensor field of view poses a significant challenge for underactuated aerial vehicles such as quadrotors, especially during aggressive maneuvers. However, the existing image-based visual servo (IBVS) control methods rely on strict target visibility assumptions or impose excessive constraints on the quadrotor's agility to meet this requirement. Furthermore, the effectiveness of the visibility constraint defined in prior works remains unverified in aggressive flight tests. To address these issues, we present a robust IBVS scheme for quadrotors to perform aggressive maneuvers while ensuring target visibility. Based on the nonlinear model-predictive control framework, we propose a novel underactuation compensation scheme to eliminate the need for a virtual camera frame, which enables us to formulate the true target visibility constraint. We then introduce a quaternion-based representation of spherical visual features to handle the nonsmooth vector field problem on the 2-sphere and derive its corresponding image kinematics. We validate our method through three challenging visual servo tasks where agile maneuvers are desired: fast landing, aggressive long-distance flight, and dynamic object tracking. Extensive simulation and experiment show that our method consistently achieves a target-visible rate of 100% in all the image frames, even under a maximum pitch of 21.04 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{\circ }$</tex-math></inline-formula> . The results validate the effectiveness of our visibility constraint under large robot rotations and underscore its importance in enabling robust and aggressive flights.

Generalized magnetostatic target field method for shielded magnetic field coils in a separable coordinate system.

A theoretical method is described to analytically calculate a pair of surface current densities, which produce a desired static magnetic field in one region of the space and zero magnetic field in another. The analysis is based on the known relationship between a surface current density and a stream function, the equivalence of stream functions and surface magnetic dipole density, and the scalar potential representation of the associated magnetic field in free space. From these relations, we formulate the magnetostatic problem, which is often treated as a vector field problem, as a scalar field problem in which a two-dimensional scalar field (stream function) is related to a three-dimensional one (magnetic scalar potential) via the differentiation of the electrostatic Green's function 1/|r-rs|. It is shown that, in a coordinate system in which a separated form of the Green's function exists (separable coordinate system), there exists a simple relationship between a harmonic component of a stream function and a harmonic component of the magnetic scalar potential. The method is applied to calculate idealized surface current patterns for actively shielded, linear gradient field coils in the Cartesian, cylindrical, and spherical coordinates.

The well-posedness of the Cauchy problem for self-interacting vector fields

We point out that the initial-value (Cauchy) problem for self-interacting vector fields presents the same well-posedness issues as for first-order derivative self-interacting scalar fields (often referred to as k-essence). For the latter, suitable strategies have been employed in the last few years to successfully evolve the Cauchy problem at the level of the infrared theory, without the need for an explicit ultraviolet completion. We argue that the very same techniques can also be applied to self-interacting vector fields, avoiding a number of issues and “pathologies” recently found in the literature.

Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

Some positive answers to the problem of endowing a dynamical system with a Hamiltonian formulation are presented within the class of Poisson structures in a geometric framework. We address this problem on orientable manifolds and by using decomposable Poisson structures. In the first case, the existence of a Hamiltonian formulation is ensured under the vanishing of some topological obstructions, improving a result of Gao. In the second case, we apply a variant of the Hojman construction to solve the problem for vector fields admitting a transversally invariant metric and, in particular, for infinitesimal generators of proper actions. Finally, we also consider the hamiltonization problem for Lie group actions and give solutions in the particular case in which the acting Lie group is a low-dimensional torus.

A layer potential approach to functional and clinical brain imaging

In this work, we consider the inverse source recovery problem from sEEG, EEG and MEG point-wise data. We regard this as an inverse source recovery problem for L2 vector-fields normally oriented and supported on the grey/white matter interface, which together with the brain, skull and scalp form a non-homogeneous layered conductor. We assume that the quasistatic approximation of Maxwell’s equation holds for the electro-magnetic fields considered. The electric data is measured point-wise inside and outside the conductor while the magnetic data is measured only point-wise outside the conductor. These ill-posed problems are solved via Tikhonov regularization on triangulations of the interfaces and a piecewise linear model for the current on the triangles. Both in the continuous and discrete formulation the electric potential is expressed as a linear combination of double layer potentials while the magnetic flux density in the continuous case is a vector-surface integral whose discrete formulation features single layer potentials. A main feature of our approach is that these contributions can be computed exactly. Due to the consideration of the regularity conditions of the electric potential in the inverse source recovery problem, the Cauchy transmission problem for the electric potential is inadvertently solved as well. In the problem, we propagate only the electric potential while the normal derivatives at the interfaces of discontinuity of the electric conductivities are computed directly from the resulting solution. This reduces the computational complexity of the problem. There is a direct connection between the magnetic flux density and the electrical potential in conductors such as the one we explore, hence a coupling of the sEEG, EEG and MEG data for solving the respective inverse source recovery problems simultaneously is direct. We treat these problems in a unified approach that uses only single and/or double layer potentials. We provide numerical examples using realistic meshes of the head with synthetic data.

Center problem for generic degenerate vector fields

We generalize the method of construction of an integrating factor for Abel differential equations, developed in Briskin et al. (1998), for any generic monodromic singularity. Here generic means that the vector field has not characteristic directions in the quasi-homogeneous leading term in certain coordinates. We apply this method to some degenerate differential systems.

M[formula omitted]-spectral estimation: A relative entropy approach

This paper deals with M2-signals, namely multivariate (or vector-valued) signals defined over a multidimensional domain. In particular, we propose an optimization technique to solve the covariance extension problem for stationary random vector fields. The multidimensional Itakura–Saito distance is employed as an optimization criterion to select the solution among the spectra satisfying a finite number of moment constraints. In order to avoid technicalities that may happen on the boundary of the feasible set, we deal with the discrete version of the problem where the multidimensional integrals are approximated by Riemann sums. The spectrum solution is also discrete, which occurs naturally when the underlying random field is periodic. We show that a solution to the discrete problem exists, is unique and depends smoothly on the problem data. Therefore, we have a well-posed problem whose solution can be tuned in a smooth manner. Finally, we have applied our theory to the target parameter estimation problem in an integrated system of automotive modules. Simulation results show that our spectral estimator has promising performance.

A modification of the method of fundamental solutions for solving 2D problems with concave and complicated domains

To solve various problems in complicated and concave domains by the method of fundamental solutions (MFS), it is required to consider a large number of source and collocation points that increases the computational time of the analysis. This paper suggests a modification to the MFS, which can make it more efficient and reliable for solving applied problems in complicated domains. In the proposed method, each pseudo source is converted to two or more number of sub-sources with equal intensities. Therefore, the total number of sources is increased while the number of unknowns is not increased. By this approach we will be able to reduce the distance between the main and the pseudo boundaries and therefore modeling the problems with complicated boundaries can be performed effectively. The proposed method is investigated for a scalar field problem, i.e. the Laplace equation, and a vector field problem, i.e. elastostatic problem. It is shown that by using the proposed method, in addition to reducing the calculation time, the condition number of the coefficient matrix also significantly decreases. By solving several numerical example problems, it is observed that the presented method can obtain accurate solutions by using a small number of collocation points.

Introducing totally nonparallel immersions

An immersion of a smooth n-dimensional manifold M→Rq is called totally nonparallel if, for every distinct x,y∈M, the tangent spaces at f(x) and f(y) contain no parallel lines; equivalently, they span a 2n-dimensional space. Given a manifold M, we seek the minimum dimension TN(M) such that there exists a totally nonparallel immersion M→RTN(M). In analogy with the totally skew embeddings studied by Ghomi and Tabachnikov, we find that totally nonparallel immersions are related to the generalized vector field problem, the immersion and embedding problems for real projective spaces, and nonsingular symmetric bilinear maps.Our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration spaceFk(M) of k-tuples of distinct points of M; for example, k-regular embeddings, k-skew embeddings, k-neighborly embeddings, and several others. Typically, a map satisfying one of these configuration space conditions induces some Sk-equivariant map on the configuration space Fk(M) (or on a bundle thereof) and obstructions can be computed in the form of Stiefel-Whitney classes. However, the existence problem for such conditions is relatively unstudied.Our main result is a Whitney-type theorem: every smooth n-manifold M admits a totally nonparallel immersion into R4n−1, one dimension less than given by genericity. We begin by studying the local problem, which requires a thorough understanding of the space of nonsingular symmetric bilinear maps, after which the main theorem is established using the removal-of-singularities h-principle technique due to Gromov and Eliashberg. When combined with a recent non-immersion theorem of Davis, we obtain the exact value TN(RPn)=4n−1 when n is a power of 2. This is the first optimal-dimension result for any closed manifold M besides S1, for any of the recently-studied configuration space conditions.

Complex isochronous centers and linearization transformations for cubic Z2-equivariant planar systems

In this paper, we study complex isochronous center problem for cubic complex planar vector fields, which are assumed to be Z2-equivariant with two symmetric centers. Such integrable systems can be classified as 11 cases. A complete classification is given on the complex isochronous centers and proven to have a total of 54 cases. All the algebraic conditions for the 54 cases are derived and, moreover, all the corresponding linearization transformations are obtained. This problem for the Z2-equivariant with two symmetric centers has been completely solved.

The biharmonic homotopy problem for unit vector fields on 2-tori

The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form different sets. Working with surfaces, we first obtain general characterizations of biharmonic unit vector fields and biharmonic unit sections under conformal change of the metric. In the case of a 2-dimensional torus, this leads to a proof that biharmonic unit sections are always harmonic and a general existence theorem, in each homotopy class, for biharmonic unit vector fields.

Weak solutions for the Oseen system in 2D and when the given velocity is not sufficiently regular

The aim of this work is twofold: proving the existence of solution (u,π)∈H1(Ω)×L2(Ω) in bounded domains of R2 and the whole plane for the Oseen problem (O) for solenoidal vector fields v in L2(Ω), and analyzing the same problem in bounded domains of Rn for n=2,3 when h=0, g=0 and the solenoidal field v belongs to Ls(Ω) for s<n.

On the Dulac's problem for piecewise analytic vector fields

This paper presents a version of Dulac's Problem for piecewise analytic vector fields that states conditions for the number of limit cycles around certain minimal sets. A suitable model-theoretic structure is introduced under which a qualitative investigation of the problem is settled.

Hyperbolic Evolution Equations, Lorentzian Holonomy, and Riemannian Generalised Killing Spinors

We prove that the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds is well posed. The proof is based on the derivation and analysis of suitable hyperbolic evolution equations given in terms of the Ricci tensor and other geometric objects. Moreover, we classify Riemannian manifolds satisfying the constraint conditions for this Cauchy problem. It is then possible to characterise certain holonomy reductions of globally hyperbolic manifolds with parallel null vector in terms of flow equations for Riemannian special holonomy metrics. For exceptional holonomy groups these flow equations have been investigated in the literature before in other contexts. As an application, the results provide a classification of Riemannian manifolds admitting imaginary generalised Killing spinors. We will also give new local normal forms for Lorentzian metrics with parallel null spinor in any dimension.

Néel walls with prescribed winding number and how a nonlocal term can change the energy landscape

We study a nonlocal Allen–Cahn type problem for vector fields of unit length, arising from a model for domain walls (called Néel walls) in ferromagnetism. We show that the nonlocal term gives rise to new features in the energy landscape; in particular, we prove existence of energy minimisers with prescribed winding number that would be prohibited in a local model.

Algebraic geometry of the center-focus problem for Abel differential equations

The Abel differential equation $y^{\prime }=p(x)y^{3}+q(x)y^{2}$ with polynomial coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem of giving conditions on $(p,q,a,b)$ implying a center for the Abel equation is analogous to the classical Poincaré center-focus problem for plane vector fields. Center conditions are provided by an infinite system of ‘center equations’. During the last two decades, important new information on these equations has been obtained via a detailed analysis of two related structures: composition algebra and moment equations (first-order approximation of the center ones). Recently, one of the basic open questions in this direction—the ‘polynomial moments problem’—has been completely settled in Pakovich and Muzychuk [Solution of the polynomial moment problem. Proc. Lond. Math. Soc. (3)99(3) (2009), 633–657] and Pakovich [Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem. Compositio Math.149 (2013), 705–728]. In this paper, we present a progress in the following two main directions: first, we translate the results of Pakovich and Muzychuk [Solution of the polynomial moment problem. Proc. Lond. Math. Soc. (3)99(3) (2009), 633–657] and Pakovich [Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem. Compositio Math.149 (2013), 705–728] into the language of algebraic geometry of the center equations. Applying these new tools, we show that the center conditions can be described in terms of composition algebra, up to a ‘small’ correction. In particular, we significantly extend the results of Briskin, Roytvarf and Yomdin [Center conditions at infinity for Abel differential equations. Ann. of Math. (2)172(1) (2010), 437–483]. Second, applying these tools in combination with explicit computations, we start in this paper the study of the ‘second Melnikov coefficients’ (second-order approximation of the center equations), showing that in many cases vanishing of the moments and of these coefficients is sufficient in order to completely characterize centers.

Parametric Center-Focus Problem for Abel Equation

The Abel differential equation $$y'=p(x)y^3 + q(x) y^2$$ with meromorphic coefficients $$p,q$$ is said to have a center on $$[a,b]$$ if all its solutions, with the initial value $$y(a)$$ small enough, satisfy the condition $$y(a)=y(b)$$ . The problem of giving conditions on $$(p,q,a,b)$$ implying a center for the Abel equation is analogous to the classical Poincaré Center-Focus problem for plane vector fields. Following Briskin et al. (Ergodic Theory Dyn Syst 19(5):1201–1220, 1999; Isr J Math 118:61–82, 2000); Cima et al. (Qual Theory Dyn Syst 11(1):19–37, 2012; J Math Anal Appl 398(2):477–486, 2013) we say that Abel equation has a “parametric center” if for each $$\epsilon \in \mathbb C$$ the equation $$y'=p(x)y^3 + \epsilon q(x) y^2$$ has a center. In the present paper we use recent results of Briskin et al. (Algebraic Geometry of the Center-Focus problem for Abel differential equations, arXiv:1211.1296 , 2012); Pakovich (Comp Math 149:705–728, 2013) to show show that for a polynomial Abel equation parametric center implies strong “composition” restriction on $$p$$ and $$q$$ . In particular, we show that for $$\deg p,q \le 10$$ parametric center is equivalent to the so-called “Composition Condition” (CC) (Alwash and Lloyd in Proc R Soc Edinburgh 105A:129–152, 1987; Briskin et al. Ergodic Theory Dyn Syst 19(5):1201–1220, 1999) on $$p,q$$ . Second, we study trigonometric Abel equation, and provide a series of examples, generalizing a recent remarkable example given in Cima et al. (Qual Theory Dyn Syst 11(1):19–37, 2012), where certain moments of $$p,q$$ vanish while (CC) is violated.

On parallelizability and span of the Dold manifolds

The Dold manifold P (m,n) is obtained from the product Sm × CPn of the m-dimensional sphere and n-dimensional complex projective space by identifying (x, [z1, . . . , zn+1]) with (−x, [z1, . . . , zn+1]), where z denotes the complex conjugate of z. We answer the parallelizability question for the Dold manifolds P (m,n) and, by completing an earlier (2008) result due to Peter Novotný, we solve the vector field problem for the manifolds P (m, 1).