Homogeneous Vector Fields Research Articles

Non Completely Solvable Systems of Complex First Order PDE's

We revisit the lack of local solvability for homogeneous vector fields with smooth complex valued coefficients, in the spirit of Nirenberg's three dimensional example. First we provide a short expository proof, in the case of CR dimension one, with arbitrary CR codimension. Next we pass to Lorenzian structures with any CR codimension \geq 1 and CR dimension \geq 2 . Several different approaches are presented. Finally we discuss the connection with the absence of the Poincare lemma and the failure of local CR embeddability, and present a global example.

Randomized Spectral and Fourier-Wavelet Methods for multidimensional Gaussian random vector fields

We develop some new variations of the Randomized Spectral Method (RSM) and Fourier-Wavelet Method (FWM) for the simulation of homogeneous Gaussian random vector fields based on plane wave decompositions. We then compare the various random field simulation methods through the examination of their efficiency and the quality of the Eulerian and Lagrangian statistical characteristics of a three-dimensional isotropic incompressible random field as simulated by ensemble or by spatial averaging. We find that the RSM, which is easier to implement, can simulate random fields with accurate second order Eulerian and Lagrangian correlations more efficiently than the FWM when the statistics are collected through ensemble averages. When the correlation structure is inferred from spatial averages of a single realization of the random field (through appeal to an ergodic theorem), the FWM is more efficient in simulating an accurate Eulerian correlation function, and both the FWM and RSM are of comparable efficiency in simulating second order Lagrangian statistics accurately.

Isotropy theorem for cosmological Yang-Mills theories

We consider homogeneous non-abelian vector fields with general potential terms in an expanding universe. We find a mechanical analogy with a system of N interacting particles (with N the dimension of the gauge group) moving in three dimensions under the action of a central potential. In the case of bounded and rapid evolution compared to the rate of expansion, we show by making use of a generalization of the virial theorem that for arbitrary potential and polarization pattern, the average energy-momentum tensor is always diagonal and isotropic despite the intrinsic anisotropic evolution of the vector field. We consider also the case in which a gauge-fixing term is introduced in the action and show that the average equation of state does not depend on such a term. Finally, we extend the results to arbitrary background geometries and show that the average energy-momentum tensor of a rapidly evolving Yang-Mills fields is always isotropic and has the perfect fluid form for any locally inertial observer.

Isotropy theorem for cosmological vector fields

We consider homogeneous abelian vector fields in an expanding universe. We find a mechanical analogy in which the system behaves as a particle moving in three dimensions under the action of a central potential. In the case of bounded and rapid evolution compared to the rate of expansion, we show by making use of the virial theorem that for arbitrary potential and polarization pattern, the average energy-momentum tensor is always diagonal and isotropic despite the intrinsic anisotropic evolution of the vector field. For simple power-law potentials of the form V=\lambda (A^\mu A_\mu)^n, the average equation of state is found to be w=(n-1)/(n+1). This implies that vector coherent oscillations could act as natural dark matter or dark energy candidates. Finally, we show that under very general conditions, the average energy-momentum tensor of a rapidly evolving bounded vector field in any background geometry is always isotropic and has the perfect fluid form for any locally inertial observer.

On The Number of Characteristic Lines for Homogeneous Vector Fields

We give the exact number of characteristic lines for homogeneous vector fields, when the number is finite. This in particular answers an open question posed by Llibre and Pilyugina (J Dynam Diff Equ 21:487–499, 2009).

On the polynomial vector fields on

Let X be a polynomial vector field of degree n on M, M = ℝm. The dynamics and the algebraic-geometric properties of the vector fields X have been studied intensively, mainly for the case when M = ℝm, and especially when n = 2. Several papers have been dedicated to the study of the homogeneous polynomial vector field of degree n on $\mathbb{S}^2$, mainly for the case where n = 2 and M = $\mathbb{S}^2$. But there are very few results on the non-homogeneous polynomial vector fields of degree n on $\mathbb{S}^2$. This paper attempts to rectify this slightly.

Critical periods of perturbations of reversible rigidly isochronous centers

In this paper we discuss bifurcation of critical periods in an m-th degree time-reversible system, which is a perturbation of an n-th degree homogeneous vector field with a rigidly isochronous center at the origin. We present period-bifurcation functions as integrals of analytic functions which depend on perturbation coefficients and reduce the problem of critical periods to finding zeros of a judging function. This procedure gives not only the number of critical periods bifurcating from the period annulus but also the location of these critical periods. Applying our procedure to the case n = m = 2 we determine the maximum number of critical periods and their location; to the case n = m = 3 we investigate the bifurcation of critical periods up to the first order in ε and obtain the expression of the second period-bifurcation function when the first one vanishes.

Poincaré–Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields

We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincaré–Bendixson theorem describing recurrence properties and ω-limit sets of geodesics for a meromorphic connection on P 1 ( C ) . We then show how to associate to a homogeneous vector field Q in C n a rank 1 singular holomorphic foliation F of P n − 1 ( C ) and a (partial) meromorphic connection ∇ o along F so that integral curves of Q are described by the geodesic flow of ∇ o along the leaves of F , which are Riemann surfaces. The combination of these results yields powerful tools for a detailed study of the dynamics of homogeneous vector fields. For instance, in dimension two we obtain a description of recurrence properties of integral curves of Q, and of the behavior of the geodesic flow in a neighborhood of a singularity, classifying the possible singularities both from a formal point of view and (for generic singularities) from a holomorphic point of view. We also get examples of unexpected new phenomena, we put in a coherent context scattered results previously known, and we obtain (as far as we know for the first time) a complete description of the dynamics in a full neighborhood of the origin for a substantial class of holomorphic maps tangent to the identity. Finally, as an example of application of our methods we study in detail the dynamics of quadratic homogeneous vector fields in C 2 .

Magnetic field correlation tensor in a universe with a multiply connected space section

The concept of statistical homogeneity and isotropy for vector fields for cosmological models with multiply connected space sections is analyzed. Considering a flat 3D torus as an example, it is shown that the correlation tensor of a statistically homogeneous and isotropic (locally) solenoidal vector field in this case depends on a countable set of functions corresponding to various classes of geodesics connecting the points in which the tensor is calculated. In contrast, such a tensor in a simply connected Universe depends on just one function.

Averaging techniques without requiring a fast time-varying differential equation

An averaging result is presented for local uniform asymptotic stability of nonlinear differential equations without requiring a fast time-varying vectorfield. The nonlinearity plays a crucial role: close to the origin, the trajectories vary slowly compared to the time dependence of the vectorfield. The result generalises averaging results which prove stability properties for systems having a homogeneous vectorfield with positive order. The result is illustrated with several examples.

Phase portraits for quadratic homogeneous polynomial vector fields on % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJX % wAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-jj8tnaaCaaa % leqabaGaaGOmaaaaaaa!4488! $$ \mathbb{S}^2 $$

Let X be a homogeneous polynomial vector field of degree 2 on \( \mathbb{S}^2 \). We show that if X has at least a non-hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on \( \mathbb{S}^2 \) is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16th Hilbert’s problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on \( \mathbb{S}^2 \) of degree n.

Number of Invariant Straight Lines for Homogeneous Polynomial Vector Fields of Arbitrary Degree and Dimension

We study the number of invariant straight lines through the origin of the homogeneous polynomial differential systems of degree m in \({{\mathbb R}^d}\) or \({{\mathbb C}^d}\) , when this number is finite. This notion extends in the natural way the classical notion of eigenvectors of homogeneous linear differential systems to homogeneous polynomial differential systems. This number provides un upper bound for the number of infinite singular points of the polynomial differential systems of degree m in \({{\mathbb R}^d}\) . This upper bound is reached if all the invariant straight lines through the origin are real.

Billiard in a triangle and quadratic homogeneous foliations on [formula omitted

This Note is partly an announcement. We describe the geodesic foliation of the translation surface associated to a triangular billiard in terms of real and holomorphic foliations defined by quadratic homogeneous vector fields on C 2 . Our technique allows, in particular, to state the existence of real foliations by curves on C 2 arising from homogeneous quadratic vector fields and having a dense set of periodic orbits. To cite this article: F. Valdez, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Nonstationary random acoustic and electromagnetic fields as wave diffusion processes**A summary of selected results in this paper appeared in [1

We investigate the effects of relatively rapid variations of the boundaries of an overmoded cavity on the stochastic properties of its interior acoustic or electromagnetic field. For quasi-static variations, this field can be represented as an ideal incoherent and statistically homogeneous isotropic random scalar or vector field, respectively. A physical model is constructed showing that the field dynamics can be characterized as a generalized diffusion process. The Langevin–It and Fokker–Planck equations are derived and their associated statistics and distributions for the complex analytic field, its magnitude and energy density are computed. The energy diffusion parameter is found to be proportional to the square of the ratio of the standard deviation of the source field to the characteristic time constant of the dynamic process, but is independent of the initial energy density, to first order. The energy drift vanishes in the asymptotic limit. The time–energy probability distribution is in general not separable, as a result of nonstationarity. A general solution of the Fokker–Planck equation is obtained in integral form, together with explicit closed-form solutions for several asymptotic cases. The findings extend known results on statistics and distributions of quasi-stationary ideal random fields (pure diffusions), which are retrieved as special cases.

Correction to Generic polynomial vector fields are not integrable

In the paper Generic polynomial vector fields are not integrable [1], we study some generic aspects of polynomial vector fields or polynomial derivations with respect to their integration. Using direct sums of derivations together with our previous results we showed that, for all n ≥ 3 and s ≥ 2, the absence of polynomial first integrals, or even of Darboux polynomials, is generic for homogeneous polynomial vector fields of degree s in n variables. To achieve this task, we need an example of such vector fields of degree s ≥ 2 for any prime number n ≥ 3 of variables and also for n = 4. The purpose of this note is to correct a gap in our paper for n = 4 by completing the corresponding proof.

HOMOGENEOUS EIGENVALUE ANALYSIS FOR COMPLEX HOMOGENEOUS SYSTEMS

In this paper, we analyze the stability of complex homogeneous systems by homogeneous eigenvalues. First, we redefine the complex homogeneity. Second, we obtain some properties of complex homogeneous functions, complex homogeneous vector fields, and complex homogeneous systems. Finally, we analyze the stability of complex homogeneous systems by homogeneous eigenvalues.

On the number of limit cycles of some systems on the cylinder

In this article we give two criteria for bounding the number of non-contractible limit cycles of a family of differential systems on the cylinder. This family includes Abel equations as well as the polar expression of several types of planar polynomial systems given by the sum of three homogeneous vector fields.

Invariant circles for homogeneous polynomial vector fields on the 2-dimensional 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.

Semicompleteness of homogeneous quadratic vector fields

On étudie les champs de vecteurs holomorphes quadratiques et homogènes de C n qui sont semicomplets : ceux dont les solutions sont uniformes dans leurs domaines maximaux de définition. À un champ générique on associe de façon rationnelle quelques nombres complexes qui s’avèrent entiers dans le cas semicomplet. Ceci montre que, dans l’espace des classes d’équivalence linéaire de champs de vecteurs, les semicomplets sont contenus dans une sorte de réseau. On prouve que les feuilletages de CP n-1 induits par des champs quadratiques semicomplets sont linéarisables au voisinage de leurs points singuliers et on donne quelques familles nouvelles d’exemples dans C 3 . Finalement, on classifie les champs semicomplets de C 3 qui sont isochores et à singularité isolée.

A note on the completeness of homogeneous quadratic vector fields on the plane

In this paper we give a full description of the complete homogeneous quadratic vector fields defined on the plane.