Analytic Vector Fields Research Articles

On the number of limit cycles which appear by perturbation of two-saddle cycles of planar vector fields

We prove that the number of limit cycles which bifurcate from a two-saddle loop of an analytic planar vector field X 0 under an arbitrary finite-parameter analytic deformation X λ , λ ∈ (ℝN, 0), is uniformly bounded with respect to λ.

Orbital linearization around equilibria of vector fields in [formula omitted] and Lie symmetries

In this paper we study some relationships between analytic vector fields X in Rn which are analytically orbitally linearizable in a neighborhood of an equilibrium point and the structure of the set N(X) of its locally analytic normalizers (analytic infinitesimal generators of Lie point orbitally symmetries in that neighborhood). Characterization and necessary and sufficient conditions for the orbital linearization of X via the existence of special elements in N(X) are given.

Classification of complex simply connected homogeneous spaces of dimension not greater than 2

The paper is devoted to the classification of finite-dimensional complex Lie algebras of analytic vector fields on the complex plane and the corresponding actions of Lie groups on complex two-dimensional manifolds. These Lie algebras were specified by Sophus Lie. He specified vector fields which form bases of the Lie algebras. However the structure of the Lie algebras was not clarified, and isomorphic Lie algebras among listed were not established. Thus, the classification was far from complete, and the situation has not been essentially changed until now. This paper is devoted to the completion of the above mentioned classification. We consider the part of this classification which concerns transitive actions of Lie groups.

On a Hsu-Unified Structure Manifold with a Quarter-Symmetric Non-Metric Connection

In the present paper, we have defined a Hsu-unified structure manifold and a Hsu-Kahler manifold and studied some properties of the quarter-symmetric non-metric connection. Certain interesting results on such manifolds have been obtained. We have also studied the properties of the contravariant almost analytic vector field on these manifolds equipped with the quarter-symmetric non-metric connection.

Toward a Model Theory for Transseries

The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, o-minimal structures, to name a few. We give an overview of the algebraic and model-theoretic aspects of this differential field, and report on our efforts to understand its first-order theory.

Analytic invariants for the 1:-1 resonance

Associated to analytic Hamiltonian vector fields in $\mathbb{C}^4$ having an equilibrium point satisfying a non semisimple $1:-1$ resonance, we construct two universal constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.

Asymptotics of the monodromy transformation in certain classes of monodromy germs

We calculate the second term of the asymptotics of the monodromy transformation of a monodromic singular point of an analytic vector field on a plane whose Newton diagram consists of one or two edges. In the cases under consideration the principal term of the monodromy transformation coincides with the identity function. The case of two edges is characterized by the fact that, as a result of blowing up the singularity by the Newton diagram, a singular point emerges that is a degenerate saddle. The results obtained make it possible to state a sufficient condition for the existence of a focus and to construct the stability boundary in the classes of vector fields under consideration.

ON THE CENTER CONDITIONS FOR ANALYTIC MONODROMIC DEGENERATE SINGULARITIES

In this paper, we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields and can be applied independently so that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allow to study monodromic degenerate singularities with characteristic directions.

Shape and period of limit cycles bifurcating from a class of Hamiltonian period annulus

In this work we are concerned with the problem of shape and period of isolated periodic solutions of perturbed analytic radial Hamiltonian vector fields in the plane. Françoise developed a method to obtain the first non vanishing Poincaré–Pontryagin–Melnikov function. We generalize this technique and we apply it to know, up to any order, the shape of the limit cycles bifurcating from the period annulus of the class of radial Hamiltonians. We write any solution, in polar coordinates, as a power series expansion in terms of the small parameter. This expansion is also used to give the period of the bifurcated periodic solutions. We present the concrete expression of the solutions up to third order of perturbation of Hamiltonians of the form H=H(r). Necessary and sufficient conditions that show if a solution is simple or double are also presented.

Renormalization of Planar Analytic Saddles

Let \({X = \lambda_{1} x_{1}{\frac{\partial}{\partial {x_1}}} + \lambda_2 x_2 \frac{\partial}{\partial {x_2}} + O(|x|^2)}\) be an analytic vector field near x = 0. We suppose that the linear part of this vector field has real eigenvalues λ1, λ2 and that the ratio \({\eta = -\frac{\lambda_1}{\lambda_2}}\) is a positive irrational number. In a previous paper of the first author and P. De Maesschalck, it was shown that any analytic saddle can be conjugated analytically to a form ‘as close as desired’ to the formal normal form. In this paper we will iterate and renormalize these conjugacies. The iteration of this procedure will be strongly connected to the diophantine properties of η and we will establish the convergence of this process. A consequence of this convergence will be the two dimensional version of the by now classical linearization theorem of Bruno.

Normalizations with Exponentially Small Remainders for Nonautonomous Analytic Periodic Vector Fields

In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependancy, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E0 and E1. Under light diophantine conditions on the eigenvalues of the linear part, we prove that there is a polynomial change of coordinates in E1 allowing to eliminate up to a finite polynomial order all terms depending only on the coordinate u0 of E0 in the E1 component of the vector field. We moreover show that, optimizing the choice of the degree of the polynomial change of coordinates, we get an exponentially small remainder. In the second part, we prove a normal form theorem with exponentially small remainder. Similar theorems have been proved before in the autonomous case : this paper generalizes those results to the nonautonomous periodic case.

Local behavior of planar analytic vector fields via integrability

We present an algorithm to study the local behavior of singular points of planar analytic vector fields having a first integral which is a quotient of analytic functions. The algorithm is based on the blow-up method. It emphasizes the curves passing through the singular points and avoids the computation of the desingularized systems. Vector fields having a rational first integral are a particular case.

Characterization of a monodromic singular point of a planar vector field

The Newton diagram and the lowest-degree quasi-homogeneous terms of an analytic planar vector field allow us to determine whether an isolated singular point of the vector field is monodromic or has a characteristic trajectory.

Global solvability of real analytic complex vector fields in two variables

This paper deals with the global solvability of a complex vector field with real analytic coefficients in two real variables. The vector field is assumed to satisfy the Nirenberg–Treves condition ( P ) for local solvability. Normal forms for the vector field near the one-dimensional orbits are obtained and a generalization of the Riemann–Hilbert problem is considered.

Real hypersurfaces of a complex projective space

In this paper, we classify real hypersurfaces in the complex projective space \(C P^{\frac{n+1}{2}}\) whose structure vector field is a φ-analytic vector field (a notion similar to analytic vector fields on complex manifolds). We also define Jacobi-type vector fields on a Riemannian manifold and classify real hypersurfaces whose structure vector field is a Jacobi-type vector field.

Convergence of the Adomian decomposition method for initial-value problems

We prove convergence of the Adomian decomposition method for an abstract initial-value problem using the method of majorants from the Cauchy-Kowalevskaya theorem for differential equations with analytic vector fields. Convergence rates of the Adomian method are investigated in the context of the nonlinear Schrödinger equation. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 749–766, 2011

Singularities of spacelike constant mean curvature surfaces in Lorentz–Minkowski space

AbstractWe study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space L3. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve f0(x), and a real analytic null vector field v(x) parallel to the tangent field of f0, find a conformally parameterized (generalized) CMC H surface in L3 which contains this curve as a singular set and such that the partial derivatives fx and fy are given by df0/dx and v along the curve. Within the class of generalized surfaces considered, the solution is unique and we give a formula for the generalized Weierstrass data for this surface. This gives a framework for studying the singularities of non-maximal CMC surfaces in L3. We use this to find the Björling data – and holomorphic potentials – which characterize cuspidal edge, swallowtail and cuspidal cross cap singularities.

On the number of limit cycles which appear by perturbation of Hamiltonian two-saddle cycles of planar vector fields

We find an upper bound to the maximal number of limit cycles, which bifurcate from a hamiltonian two-saddle loop of an analytic vector field, under an analytic deformation.

Conformal vector fields on Kaehler manifolds

It is known that a conformal vector field on a compact Kaehler manifold is a Killing vector field. In this paper, we are interested in finding conditions under which a conformal vector field on a non-compact Kaehler manifold is Killing. First we prove that a harmonic analytic conformal vector field on a 2n-dimensional Kaehler manifold (n ≠ 2) of constant nonzero scalar curvature is Killing. It is also shown that on a 2n-dimensional Kaehler Einstein manifold (n > 1) an analytic conformal vector field is either Killing or else the Kaehler manifold is Ricci flat. In particular, it follows that on non-flat Kaehler Einstein manifolds of dimension greater than two, analytic conformal vector fields are Killing.

A Note on the Convergence of Parametrised Non-Resonant Invariant Manifolds

Truncated Taylor series representations of invariant manifolds are abundant in numerical computations. We present an aposteriori method to compute the convergence radii and error estimates of analytic parametrisations of non-resonant local invariant manifolds of a saddle of an analytic vector field, from such a truncated series. This enables us to obtain local enclosures, as well as existence results, for the invariant manifolds.