Polynomial normal forms with exponentially small remainder for analytic vector fields

A key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms, invented by Poincaré, which gives simple forms to which a vector field can be reduced close to the equilibrium. In the class of formal vector valued vector fields the problem can be easily solved, whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs. Nevertheless the study of the dynamics in a neighborhood of the origin can very often be carried out via a normalization up to finite order. This paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in R^m. More precisely we prove that for any vector field in R^m admitting the origin as a fixed point with a semi-simple linearization, the order of the normal form can be optimized so that the remainder is exponentially small. We also give several examples of non-semi-simple linearization for which this result is still true.     

Complete polynomial vector fields on the complex plane

We give a full classification, up to polynomial automorphisms, of complete polynomial vector fields in two complex variables.     

On the domain invariance of countably condensing vector fields

Using homotopy theory, we give the domain invariance theorem for countably condensing vector fields, where the notion of countably condensing maps is due to Väth. A starting point of this investigation is that there is a symmetric characteristic set for a countably condensing map.     

Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse

We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into the form x=x2＋y2-1＋y（ax＋by＋c）,y=x（ax＋by＋c）,and the ellipse becomes x2＋y2=1.We prove that（i） this quadratic system is analytic integrable if and only if a=0;（ii） if x2＋y2=1 is a periodic orbit,then this quadratic system is Liouvillian integrable if and only if x2＋y2=1 is not a limit cycle;and（iii） if x2＋y2=1 is not a periodic orbit,then this quadratic system is Liouvilian integrable if and only if a=0.     

Global analytic regularity for sums of squares of vector fields

We consider a class of operators in the form of a sum of squares of vector fields with real analytic coefficients on the torus and we show that the zero order term may influence their global analytic hypoellipticity. Also we extend a result of Cordaro-Himonas.

     

On the index of finite determinacy of vector fields with resonances

In this paper we study normal forms for a class of germs of 1-resonant vector fields on R^n with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of finite determinacy from above as well as the essentially simplified polynomial normal forms for such vector fields. In the case that a vector field has a zero eigenvalue, the result leads to an interesting corollary, a linear dependence of the derivatives of the hyperbolic variables on the central variable.     

The linearization of an autonomous system in the neighborhood of a “node”-type singular point

In the article we study the connection between the smoothness class of the diffeomorphism, linearizing an autonomous system of class C in the neighborhood of a node-type singular point, and the form of the resonance relations between the roots of the matrix of the linear part. We establish sufficient conditions for the existence of the linearizing diffeomorphism from a fixed smoothness class.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 833–842, December, 1973.In conclusion we remark that the foundation on which the present paper was based was Kondrat'ev's report at a scientific-research seminar at the Moscow State University.     

Lie symmetries for the orbital linearization of smooth planar vector fields around singular points

This work is a generalization of the method proposed in [I.A. García, S. Maza, Linearization of analytic isochronous centers from a given commutator, J. Math. Anal. Appl. 339 (1) (2008) 740-745] of linearization of analytic isochronous centers from a given commutator. In this paper we propose a constructive procedure to get the change of variables that orbitally linearizes a smooth planar vector field on C² around an elementary singular point (i.e., a singular point with associated eigenvalues λ,μ∈C satisfying λ≠0) or a nilpotent singular point from a given infinitesimal generator of a Lie symmetry.     

Harmonicity of vector fields on the oscillator groups with neutral signature

In this paper, we mainly investigate curvature properties and harmonicity of invariant vector fields on the four-dimensional Oscillator groups endowed with three left-invariant pseudo-Riemannian metrics of signature (2,2). We determine all harmonic vector fields, vector fields which define harmonic maps and the vector fields which are critical points for the energy functional restricted to vector fields of the same length.     

Conformal vector fields on tangent bundle of a Riemannian manifold

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometers using some Riemannian and pseudo-Riemannian lift metrics on TM. Here we consider the Riemannian or pseudo-Riemannian lift metric G on TM which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on (TM,G).     

Asymptotic stability at infinity for differentiable vector fields of the plane

Let be a differentiable (but not necessarily C¹) vector field, where σ>0 and . Denote by R(z) the real part of z∈C. If for some ?>0 and for all , no eigenvalue of DpX belongs to , then: (a) for all , there is a unique positive semi-trajectory of X starting at p; (b) it is associated to X, a well-defined number I(X) of the extended real line [−∞,∞) (called the index of X at infinity) such that for some constant vector v∈R² the following is satisfied: if I(X) is less than zero (respectively greater or equal to zero), then the point at infinity ∞ of the Riemann sphere R²∪{∞} is a repellor (respectively an attractor) of the vector field X+v.     

Orbital topological classification of analytic differential equations in a neighborhood of a degenerate elementary singular point on the two-dimensional complex plane

We consider the germ of an analytic differential equation in (C², 0) defined by a vector field with a singularity at zero. A singular point of the equation is called degenerate elementary if one eigenvalue of the linear part of the field is equal to zero and the other is different from zero. The orbital topological classification of differential equations in a neighborhood of a degenerate elementary singular point is obtained for all cases except the so-called Liouville case. This case defines a subset of equations which does not separate the corresponding function space.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 13, pp. 137–165, 1988.     

Holomorphic vector bundles on a neighborhood of non-Stein compact subsets of Stein spaces

LetK be a compact subset of a complex spaceX. Here we give conditions onX andK assuring the existence of a fundamental systemU of open neighborhoods, ofK such that for everyU∈U there is a holomorphic vector bundleE onU which is not holomorphically trivial.

---

Sunto SiaX uno spazio complesso eK∩X un compatto. In questo lavoro diamo condizioni suX eK che garantiscono l'esistenza di un sistema fondamentale di intorni apertiU diK inX tali che per ogniU∈U esiste un fibrato olomorfo non-triviale suU.

     

A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields

We introduce a hierarchy of integrable partial differential equations in 2+1 dimensions arising from the commutation of one-parameter families of vector fields, and we construct the formal solution of the associated Cauchy problems using the inverse scattering method for one-parameter families of vector fields. Because the space of eigenfunctions is a ring, the inverse problem can be formulated in three distinct ways. In particular, one formulation corresponds to a linear integral equation for a Jost eigenfunction, and another formulation is a scalar nonlinear Riemann problem for suitable analytic eigenfunctions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 147–156, July, 2007.     

Zaremba's problem for elliptic equations in the neighborhood of a singular point or in the infinity

In the present paper the behavior of solutions of the mixed Zaremba's problem in the neighborhood of a boundary point and at infinity is studied. In part I of this paper[4] the concept of Wiener's generalized solution of Zaremba's problem was introduced and the so called Growth Lemma for the class of domains, satisfying isoperimetric condition, was proven. In part II regularity criterion for joining points of Neumann's and Dirichlet's boundary conditions is formulated. Generalized solution in unlimited domains as a limit of Zaremba's problem's solutions in a sequence of limited domains is introduced and a regularity condition allowed to obtain an analogue of Phragmen-Lindeloeff theorem for the solutions of Zaremba's problem. Main results of the present paper are formulated in terms of divergence of Wiener's type series.