Coalescing Complex Planar Stationary Points

In this work, inspired by the technique of the complete transversal, used for the classification of plane branches, developed by Hefez, A. and Hernandes, M., as well as Bruce, J.W., Kirk, N.P. and du Plesis, A.A., study the singularities of applications, we establish a classification of vector fields through their normal forms. In the case of vector fields with non zero linear part in $(\mathbb{C}^{2}, 0) $ and nilpotent fields in $(\mathbb {C}^{n}, 0), n\geq 2$ we recover the classical normal forms for those fields, and we provide a formal normal form different from Takens in dimension 2. Likewise, we obtain the normal form for the vector fields in $(\mathbb{C},0)$ of any multiplicity.

We consider a planar vector field $X$ near a saddle type $p:-q$ resonant singular point. Assuming that it has a normal form with a Gevrey-$d$ expansion (like $d=p+q$ which is in particular the case when starting from an analytic vector field) we show that $X$ can be linearized working with a change of coordinates that is of Gevrey order $d$ in certain $\log$-like variables, called compensators or also tags, multiplied by the first integral $u=x^qy^p$ of the linear part. Next we consider the unfolding of such a resonance, and provide (weaker) Gevrey-type linearization using compensators.

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.

Topological Equivalence of Planar Vector Fields and Their Generalised Principal Part

It is well known that isolated singularities of two dimensional analytic vector fields can be desingularized: after a finite number of blowing up operations we obtain a vector field that exhibits only elementary singularities. In the present paper we introduce a similar method to simplify the periodic limit sets of analytic families of vector fields. Although the method is applied here only to reduce to families in which the zero set has codimension at least two, we conjecture that it can be used in general. This is related to the famouss Hibert's problem about planar vector fields.

In this article, we show that complex vector fields on the punctured Riemann sphere are Newton vector fields and as a consequence all meromorphic vector fields and all elliptic vector fields are Newton vector fields. Moreover, for a large class of vector fields, which includes all elliptic vector fields, the proof is constructive in the sense that one can explicitly construct the function characterizing the Newton vector field.

We study self-gravitating bound states of a complex vector field, known as Proca stars, with a new type of quartic-order self-interaction, which does not exist in the case of either a complex scalar field or a real vector field. Depending on the sign of the coupling constant, this quartic self-interaction can yield a distinct feature of Proca stars from the previously investigated self-interaction of the vector field. We find that self-gravitating solutions can be so compact that the photon sphere could form. However, we also show that the self-interaction gives rise to a ghost instability for the stars whose compactness is close that for the formation of a photon sphere, which might invalidate the formation of the photon sphere.

In this work we consider the problem of classifying all configurations of invariant lines of total multiplicity eight (including the line at infinity) of real planar cubic differential systems. The classification was initiated in Bujac and Vulpe (J Math Anal Appl 423:1025–1080, 2015) where the cubic systems which possess 4 distinct infinite singularities are considered and 17 such distinct configurations were detected. That work was continued in Bujac and Vulpe (Qual Theory Dyn Syst 14(1):109–137, 2015) where the classification was done for the family of cubic systems with three distinct infinite singularities and where the existence of 5 distinct configurations was proved. The cubic systems with 2 infinite singularities are considered in Bujac (Bul Acad Stiinte Repub Mold Mat 1:48–86, 2015) and Bujac and Vulpe (Classification of a subfamily of cubic differential systems with invariant straight lines of total multiplicity eight. Universitat Autonoma de Barselona, 35, 2015) where 25 configurations were constructed. This article deals with the last remaining subfamily of cubic systems. We prove here a classification theorem (Main Theorem) of real planar cubic vector fields which possess exactly one infinite singular point and eight invariant straight lines, including the line at infinity and including their multiplicities. We show that these systems could possess only 4 such configurations. We underline that the classifications of all the above mentioned families of cubic systems are taken modulo the action of the group of real affine transformations and time rescaling and are given in terms of invariant polynomials. The algebraic invariants and comitants allow one to verify for any given real cubic system whether or not it has invariant lines of total multiplicity eight, and to specify its configuration of lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems, given in any normal form.

In this article we prove a classification theorem (Main Theorem) of real planar cubic vector fields which possess eight invariant straight lines, including the line at infinity and including their multiplicities and in addition they possess three distinct infinite singularities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of affine invariant polynomials. The invariant polynomials allow one to verify for any given real cubic system whether or not it has invariant straight lines of total multiplicity eight, and to specify its configuration of straight lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems in this class, given in any normal form.

Hypernormal form theory: foundations and algorithms

Unique normal forms for lambda calculus with surjective pairing

At this time the Cheshire Cat vanished quite slowly, beginning with the end of the tail, ending with a grin, which remained some time after the rest of it had gone, [Ca].0. Introduction.Morse theory for distance functions was initiated by Grove-Shiohama [GS] and Gromov's [Gl] paper where basic notions of the theory were formulated; even this very initial level of the theory leads to important geometric applications.Grove and Shiohama [GS] established a generalized sphere theorem by constructing a vector field on a Riemannian manifold, with the property that the distance function had no stationary points along the integral curves at non-singular points of the vector field.They showed that this vector field had exactly two singular points and hence the manifold was homeomorphic to a sphere.Later Gromov [Gl] was able to bound the sum of the Betti numbers of a positively curved Riemannian manifold.He controlled the location of critical points of a Riemannian distance function on a positively curved manifold using Toponogov's theorem and then was able to bound the number of critical points of this function and hence the homology of the manifold, using a spectral sequence argument.Morse theory for Riemannian distance functions was discussed in [Gr], [L] and other papers, which explained its importance for geometric applications rather than developed the theory itself.Even a suitable concept of the index of a critical point has not been developed.As a result, the most powerful tools of the classical Morse theory such as Morse inequalities and the correspondence between critical points and the handle decomposition of the manifold cannot be used.The relationship with the classical Morse theory has not been investigated either; in particular, the connection between notions of the critical points in both theories is not obvious.In a different direction, the structure of Alexandrov spaces with curvatures bounded below was investigated using distance functions in several papers starting with [BGP] and continuing with [PI], [P2].A key result obtained is a canonical stratification of such Alexandrov spaces into topological manifolds and again the technique is a type of Morse theory, using the distance function.As Alexandrov spaces do not have as much structure as Riemannian manifolds, our theory gives more detailed information on the nature of critical points and index.M. Gromov pointed out in [Gl] that the Morse theory for Riemannian distance functions can be developed by analogy with the classical Morse theory.The aim of this paper is to construct a Morse theory for functions which are minima of finite families of smooth functions and clarify the connection with Riemannian distance functions for non-positively curved manifolds.We develop the theory in the classical style, including the notion of the index for critical points, and clarify relations with the Grove-Shiohama-Gromov approach.

On moduli of conjugation for some n-dimensional vector fields

The universal unfolding of a normal form can be employed to reveal the general behaviors of a specific local bifurcation, while the computation of the normal form for high codimensional bifurcation still remains unsolved. This paper focuses on a vector field with codimension-3 triple Hopf bifurcation. Besides 1:1 internal resonance for two frequencies in semi-simple form, two cases are considered, corresponding to internal resonance and noninternal resonance between the first two frequencies and the third frequency, respectively. Based on a combination of center manifold and normal theory, all the coefficients in the normal form and the nonlinear transformation are derived explicitly in terms of the coefficients of the original vector field. Upon the recursive procedure established, a user friendly computer program can be easily developed using a symbolic computation language Maple to compute the coefficients up to an arbitrary order for a specific vector field with triple Hopf bifurcation. Furthermore, universal unfolding of the normal form is obtained, which can be used to display the topological structure in the neighborhood of bifurcation point. It is pointed out that different choices of the remaining terms in the nonlinear transformation may lead to different expressions of the normal form and the unfolding, which are qualitatively equivalent to each other.

In this talk we give a report on a paper where we consider a model sum of squares of planar complex vector fields, being close to Kohn's operator but with a point singularity. The characteristic variety of the operator is the same symplectic real analytic manifold as Kohn's. We show that this operator is hypoelliptic and Gevrey hypoelliptic provided certain conditions are satisfied. We show that in the Gevrey spaces below a certain index the operator is not hypoelliptic. Moreover there are cases in which the operator is not even hypoelliptic. This fact leads to some general negative statement on the hypoellipticity properties of sums of squares of complex vector fields, even when the complex H\ormander condition is satisfied.