Poincaré Compactification for Non-polynomial Vector Fields

We consider certain vector fields in the plane which possess a centre. The main result is that for Hamiltonian polynomial systems which are of even degree, which possess homogeneous nonlinearities, and which have a centre located at the origin, the period function is a strictly increasing function of the energy, throughout its interval of definition. It is also shown that for nonlinear homogeneous Hamiltonian polynomial vector fields of arbitrary degree which possess a centre, the period function is a strictly decreasing function of the energy. With appropriate modifications, this result is extended to arbitrary homogeneous vector fields which possess a centre, irrespective of their being Hamiltonian or polynomial; the period function is then strictly monotonic, except when the degree of homogeneity is one, when the systems are isochronous.

Hilbert's 16th problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree , has been one of the most important driving forces for new developments in the qualitative theory of vector fields. Increasing the dimension, one cannot expect the existence of a finite upper bound for the number of limit cycles of, for instance, 3D polynomial vector fields of a given degree . Here, as an extension of such a problem in the 3D space, we investigate the number of isolated invariant tori in 3D polynomial vector fields. In this context, given a natural number , we denote by the upper bound for the number of isolated invariant tori of 3D polynomial vector fields of degree . Based on a recently developed averaging method for detecting invariant tori, our first main result provides a mechanism for constructing 3D differential vector fields with a number of normally hyperbolic invariant tori from a given planar differential vector field with hyperbolic limit cycles. The strength of our mechanism in studying the number lies in the fact that the constructed 3D differential vector field is polynomial provided that the given planar differential vector field is polynomial. Accordingly, our second main result establishes a lower bound for in terms of lower bounds for the number of hyperbolic limit cycles of planar polynomial vector fields of degree . Based on this last result, we apply a methodology due to Christopher and Lloyd to show that grows as fast as . Finally, the above‐mentioned problem is also formulated for higher dimensional polynomial vector fields.

Simulation of homogeneous nonGaussian stochastic vector fields

We discuss the application of ambit fields to the construction of stochastic vector fields in two dimensions that are divergence-free and statistically homogeneous and isotropic but are not invariant under the parity operation. These vector fields are derived from a stochastic stream function defined as a weighted integral with respect to a Lévy basis. By construction, the stream function is homogeneous and isotropic and the corresponding vector field is, in addition, divergence-free. From a decomposition of the kernel in the Lévy-based integral, necessary conditions for the violation of parity symmetry are inferred. In particular, we focus on such fields that allow for skewness of projected increments, which is one of the cornerstones of the Kraichnan–Leith–Bachelor theory of two-dimensional turbulence.

We consider a class of nonvariational linear operators formed by homogeneous left invariant Hormander's vector fields with respect to a structure of Carnot group. The bounded coefficients of the operators belong to "vanishing logarithmic mean oscillation" class with respect to the distance induced by the vector fields (in particular they can be discontinuous). We prove local estimates in "local BMO" spaces intersected with the Lebesgue spaces. Even in the uniformly elliptic case our estimates improve the known results.

Projections of Polynomial Vector Fields and the Poincaré Sphere

LexX be a homogeneous polynomial vector field of degreen≥3 on S2 having finitely many invariant circles. Then, for such a vector fieldX we find upper bounds for the number of invariant circles, invariant great circles, invariant circles intersecting at a same point and parallel circles with the same director vector. We give examples of homogeneous polynomial vector fields of degree 3 on S2 having finitely many invariant circles which are not great circles, which are limit cycles, but are not great circles and invariant great circles that are limit cycles. Moreover, for the casen=3 we determine the maximum number of parallel invariant circles with the same director vector.

Several recent works have combined discrete and continuous motion planning methods for robot navigation and control. The basic idea of some of these works is to plan a path, by determining a sequence of neighboring discrete regions of the configuration space, and to assign a vector field that drives the robots through these regions. This paper addresses the problem of efficiently computing vector fields over a sequence of consecutive triangles. Differently from previous numerical approaches, which were not able to compute fully continuous fields in triangulated spaces, this paper presents an algorithm that is able to compute guaranteed continuous vector fields over a sequence of adjacent triangles.

We study the solvability and removable singularities of the equation divF=µ , with measure data µ , in the class of continuous or Lp vector fields F , where 1=p=8 . In particular, we show that, for a signed measure µ , the equation divF=µ has a solution F?L8(Rn) if and only if |µ(U)|=CHn-1(?U) for any open set U with smooth boundary. For non-negative measures µ , we obtain explicit characterizations of the solvability of divF=µ in terms of potential energies of µ for p?8 , and in terms of densities of µ for continuous vector fields. These existence results allow us to characterize the removable singularities of the corresponding equation divF=µ with signed measures µ .

In this paper we present an n-dimensional canonical piecewise-linear electrical circuit. It contains 2n two-terminal elements: n linear dynamic elements (capacitors and inductors), n - 1 linear resistors and one nonlinear (piecewise-linear) resistor. This circuit can realize any prescribed eigenvalue pattern, except for a set of measure zero, associated with (i) any n-dimensional two-region continuous piecewise-linear vector fields and (ii) any n-dimensional three-region symmetric (with respect to the origin) piecewise-linear continuous vector fields. We also proved a theorem that specifies the conditions for a vector field, realized with our canonical circuit, to have two or three equilibrium points.

We give a possible extension of the definition of quaternionic power
\nseries, partial derivatives and vector fields in the case of two (and then several)
\nnon commutative (quaternionic) variables. In this setting we also investigate
\nthe problem of describing zero functions which are not null functions in the for-
\nmal sense. A connection between an analytic condition and a graph theoretic
\nproperty of a subgraph of a Hamming graph is shown, namely the condition
\nthat polynomial vector field has formal divergence 0 is equivalent to connect-
\nedness of subgraphs of Hamming graphs H(d, 2). We prove that monomials in
\nvariables z and w are always linearly independent as functions only in bidegrees
\n(p, 0), (p, 1), (0, q), (1, q) and (2, 2).

Algorithms and methods in differential algebra

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two nonequivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space where p is larger than the space dimension, under the assumption that the so-called “forward-backward integral curves” associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable. Additionally, for such vector fields it is possible to prove almost everywhere uniqueness of (standard) integral curves, which also implies uniqueness of positive measure solutions to the continuity equation with absolutely continuous initial datum.

The well-known Cauchy integral decomposition in the plane is a representation of a continuous function on a curve as a sum of two functions analytic inside and outside the curve. These analytic components inherit smoothness of the original function almost independently of the smoothness of the curve itself. Let G be a domain in homeomorphic to a ball with C 1-smooth boundary S. Let F be a continuous vector field on S. When it is possible to represent it as a sumF = F + + F −, where vector fields F ± are harmonic in G and respectively, and smooth of prescribed order s up to the boundary? The following two conditions on F turn out to be necessary and sufficient: 1. The circulation of F along any closed path on S vanishes. 2. For any ball B with radius r centered on S there exists a polynomial harmonic vector field H of degree m = [s] such that on B∩S. This is a multidimensional analog of the Cauchy integral decomposition. The result does not presuppose any extra smoothness of S.

The problem of vector field approximation emerges in the wide range of fields such as motion control, computer vision and so on. The paper discusses an approximation method for reconstructing an entire continuous vector field from a sparse set of sample data by neural networks. In order to improve approximation accuracy and efficiency, we incorporate the inherent property of vector fields into the learning problem of neural networks and derive a new learning algorithm. It is shown through numerical experiments that the proposed method makes it possible to reconstruct vector fields accurately and efficiently.