Real Analytic Vector Fields Research Articles

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.

Algebraic and Computational Formulas for the Index of Real Analytic Vector Fields

The signature formula of Eisenbud–Levine and Khimshiashvili for computing the Poincare–Hopf index of a real analytic vector field at an algebraically isolated singularity is well known. We present in this paper an algebraic formula which allows to compute the index in the non–algebraically isolated case when the complex zeros associated to the complexified vector field have codimension one. We also analyse some instances in the codimension 2 case and describe a computer implementation that permits the calculation of the index in both the algebraically and non–algebraically isolated cases.

On the finite cyclicity of open period annuli

Let $\Pi$ be an open, relatively compact period annulus of real analytic vector field $X_0$ on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from $\Pi$ under a given multi-parameter analytic deformation $X_\lambda$ of $X_0$ is finite, provided that $X_0$ is either Hamiltonian, or generic Darbouxian vector field.

A Rudin–Carleson theorem for planar vector fields

This paper generalizes the Rudin–Carleson theorem for homogeneous solutions of locally solvable real analytic vector fields.

A conjugated version of the countable dense homogeneity

We are interested in finding a homeomorphism h of a space X with h −1 ○ Φ ○ h ( A ) = B for a given bijection Φ of X and every pair of countable dense subsets A and B of X. For a separable Banach space X, such a homeomorphism h always exists provided the fixed-point set of Φ has the empty interior. Moreover, h can be chosen to be real-analytic. As a consequence, there exists a real analytic flow that sends A onto B after time t = 1 . Actually, for X = R n , any bounded real-analytic vector field can be approximated by a real-analytic vector field whose induced flow sends A onto B after time t = 1 . Topological and C p smooth counterparts of these results are also obtained.

Dynamical systems and nilpotent sub-Riemannian geometry

We consider dynamical systems determined by distributions of real analytic vector fields, the equations of motion, and the associated gauge transformation are presented in detail. A gradation of the associated Lie algebras leads to the consideration of polynomial vector fields. The minimization of kinetic energy action in the class of horizontal curves associated with the distribution is formulated as a sub-Riemannian geodesic problem. Normal geodesics are fully described for the so-called Gaveau–Brockett distribution. The exponential mapping, the unitary spheres, and the wave fronts are calculated for particular cases.

Globally analytic hypoelliptic vector fields on compact surfaces

We present a characterization of the global analytic hypoellipticity of a complex, non-singular, real analytic vector field defined on a compact, connected, orientable, two-dimensional, real analytic manifold. In particular, we show that such vector fields exist only on the torus.

Multiplicity of limit cycles and analytic m-solutions for planar differential systems

This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Γ of an analytic vector field it always exists a real analytic function f 0 ( x , y ) , defined in a neighborhood of Γ, and such that Γ is contained in its zero level set. In this work we introduce the notion of f 0 ( x , y ) being an m- solution, which is a merely analytic concept. Our main result is that a limit cycle Γ is of multiplicity m if and only if f 0 ( x , y ) is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles.

Quasi-analytic solutions of analytic ordinary differential equations and o-minimal structures

It is well known that the non-spiraling leaves of real analytic foliations of codimension 1 all belong to the same o-minimal structure. Naturally, the question arises if the same statement is true for non-oscillating trajectories of real analytic vector fields. We show, under certain assumptions, that such a trajectory generates an o-minimal and model complete structure together with the analytic functions. The proof uses the asymptotic theory of irregular singular ordinary differential equations in order to establish a quasi-analyticity result from which the main theorem follows. As applications, we present an infinite family of o-minimal structures such that any two of them do not admit a common extension, and we construct a non-oscillating trajectory of a real analytic vector field in dimension 5 that is not definable in any o-minimal extension of the reals

Resolution of singularities of real-analytic vector fields in dimension three

Let χ be an analytic vector field defined in a real-analytic manifold of dimension three. We prove that all the singularities of χ can be made elementary by a finite number of blowing-ups in the ambient space.

Generalised power series solutions of sub-analytic differential equations

We show that if a solution y ( x ) of a sub-analytic differential equation admits an asymptotic expansion ∑ i = 1 ∞ c i x μ i , μ i ∈ R + , then the exponents μ i belong to a finitely generated semi-group of R + . We deduce a similar result for the components of non-oscillating trajectories of real analytic vector fields in dimension n. To cite this article: M. Matusinski, J.-P. Rolin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Local analytic models for families of hyperbolic vector fields

We study local analytic models for hyperbolic singularities of analytic families of real analytic vector fields. We emphasize on those cases where the normal form theorems of Poincare and Siegel do not apply, for example for a family of saddles. We also consider the preservation of possible symmetries.

Topological equivalence for chains of saddle-connections

In this paper we give a complete topological classification for multiple saddle-connections of a real analytic vector field along an axis Γ in an ambient space of dimension three, under the assumption that Γ is the intersection of two invariant surfaces D 1 and D 2.

Champs de vecteurs analytiques et champs de gradients

A theorem of Łojasiewicz asserts that any relatively compact solution of a real analytic gradient vector field has finite length. We show here a generalization of this result for relatively compact solutions of an analytic vector field X with a smooth invariant hypersurface, transversally hyperbolic for X, where the restriction of the field is a gradient. This solves some instances of R. Thom's Gradient Conjecture. Furthermore, if the dimension of the ambient space is three, these solutions do not oscillate (in the sense that they cut an analytic set only finitely many times); this can also be applied to some gradient vector fields.

Poincaré's Reversibility Condition

We consider a real planar analytic vector field, X, such that the origin, O, is a centre for the linearization of X. Poincaré's condition of reversibility with respect to a line passing through O is then a sufficient condition for O to be a centre for the vector field X. We provide necessary and sufficient conditions, involving the vanishing of certain polynomials in the coefficients in the expansion of X, for reversibility. We also show that if the linearization, L(x), of the divergence of X is non-trivial, then the only possible reversibility line is given by L(x)=0; in such cases, this provides the basis for a simple test of reversibility. We examine the consequences of our various tests for quadratic and cubic vector fields; all non-Hamiltonian cases are discussed. When L(x)≡0 in cubic systems, it is possible for the reversibility line (if it exists) to be unique, but it is also possible for there to be two such lines. These possibilities are characterized algebraically, and a prescription is provided for determining the reversibility line(s) in each case.

Invariant Curves and Topological Invariants for Real Plane Analytic Vector Fields

Let Z be a germ of a singular real analytic vector field at O∈R2. We give conditions on the multiplicity and the Milnor number of Z which imply that the foliation defined by Z has a characteristic orbit or an analytic invariant curve with the hypothesis that Z is a real generalized curve. Then it is proved that for a non-dicritical real generalized curve, the multiplicity mod2 is invariant under bilipshitz homeomorphisms preserving foliations.

Non-analytic hypoellipticity in the presence of symplecticity

Here we construct non-analytic solutions to a class of hypoelliptic operators with symplectic characteristic set and in the form of a sum of squares of real analytic vector fields.

The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant

Given a real analytic vector field tangent to a hypersurface V with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra B Ann B (h) associated with the singularity of the vector field on V. We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of number and for even dimensional hypersurfaces it gives a method to compute the Poincaré-Hopf index of the vector field restricted to the hypersurface.

Global properties of a class of planar vector fields of infinite type

This paper studies some global and semi global properties of infinite type, planar, C-valued real analytic vector fields that are invariant under the rotation group. Results are proved on the integrability, kernel, range and classification of such operators.

Observability of real analytic vector fields on compact manifolds

A smooth vector field on a smooth compact manifold is said to be observable if there exists a smooth R 1- valued output function that distinguishes initial states on any time interval. In this paper we prove that a real analytic vector field on a compact manifold is observable if and only if its singularities are isolated. The result remains true for R p- valued functions. In this case one has to assume that the set of singular points is ( p − 1)-dimensional.