Classification of Complex Polynomial Vector Fields in One Complex Variable

On the centers of the weight-homogeneous polynomial vector fields on the plane

We classify all the centers of a planar weight-homogeneous polynomial vector field of weight degree 1, 2, 3 and 4.

Darboux polynomials and global phase portraits for the D_2 vector field

We study a vector field of R^3 equivariant under the D_2 symmetry group, called "the D_2 field" in the literature. We construct the complete list of Darboux polynomials for it, solving the partial differential equation defining them. We also use these polynomials to comment on its global qualitative behaviour. This is meant to be a first step towards the comparison of vector fields based on the module generated by their Darboux polynomials.

Darboux theory of integrability for real polynomial vector fields on $\sss^n$

This is a survey on the Darboux theory of integrability for polynomial vector fields, first in $\R^n$ and second in the $n$-dimensional sphere $\sss^n$. We also provide new results about the maximum number of parallels and meridians that a polynomial vector field $\X$ on $\sss^n$ can have in function of its degree. These results in some sense extend the known result on the maximum number of hyperplanes that a polynomial vector field $\Y$ in $\R^n$ can have in function of the degree of $\Y$.

Isochronicity and Commutation of Polynomial Vector Fields

We study a connection between the isochronicity of a center of a polynomial vector field and the existence of a polynomial commuting system. We demonstrate an isochronous system of degree 4 which does not commute with any polynomial system. We prove that among the Newton polynomial systems only the Lienard and Abel systems may commute with transversal polynomial fields. We give a full and constructive description of centralizers of the Abel polynomial systems. We give new examples of isochronous systems.

Limiting behavior of trajectories of complex polynomial vector fields

We prove that every trajectory of a polynomial vector field on the complex projective plane accumulates to the singular locus of the vector field. This statement represents a holomorphic version of the Poincare-Bendixson theorem and solves the complex analytic counterpart of Hilbert's 16th problem. The main result can be also reformulated as the nonexistence of "exceptional minimals" of holomorphic foliations on $\pp^2$ and, in particular, implies the nonexistence of real analytic Levi flat hypersurfaces in the complex projective plane. Finally, we describe (in the first approximation) the way…

Combinatorics of line arrangements and dynamics of polynomial vector fields

Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$--derivations. First, we give a dynamical interpretation of $\mathcal{D}(\mathcal{A})$ as the set of polynomial vector fields which posses $\mathcal{A}$ as invariant set. We characterize polynomial vector fields having an infinite number of invariant lines. Then we prove that the minimal degree of polynomial vector fields fixing only a finite set of lines in $\mathcal{D}(\mathcal{A})$ is not determined by the combinatorics of $\mathcal{A}$.

Combinatorics of line arrangements and polynomial vector fields

Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$-derivations view as the set of polynomial vector fields which possess $\mathcal{A}$ as an invariant set. We first characterize polynomial vector fields having an infinite number of invariant lines. Then we prove that the minimal degree of polynomial vector fields fixing only a finite set of lines in $\mathcal{D}(\mathcal{A})$ is not determined by the combinatorics of $\mathcal{A}$.

$n$维球面上多项式向量场的动力学与可积性

Dynamics and integrability of polynomial vector fields on the $n$-dimensional sphere

本文研究球面上多项式向量场的Hamilton性、不变低维球面与超平面界及完全可积性，并给出齐次情形的刻画。

Planar polynomial vector fields having a polynomial first integral can be obtained from linear systems

We consider in this work planar polynomial differential systems having a polynomial first integral. We prove that these systems can be obtained from a linear system through a polynomial change of variables.

On the classification of laminations associated to quadratic polynomials

Given any rational map $f$, there is a lamination by Riemann surfaces associated to $f$. Such laminations were constructed in general by Lyubich and Minsky. In this paper, we classify laminations associated to quadratic polynomials with periodic critical point. In particular, we prove that the topology of such laminations determines the combinatorics of the parameter. We also describe the topology of laminations associated to other types of quadratic polynomials.

On the computation of Darboux first integrals of a class of planar polynomial vector fields

We study the class of planar polynomial vector fields admitting Darboux first integrals of the type $\prod_{i=1}^r f_i^{α_i}$, where the $α_i$'s are positive real numbers and the $f_i$'s are polynomials defining curves with only one place at infinity. We show that these vector fields have an extended reduction procedure and give an algorithm which, from a part of the extended reduction of the vector field, computes a Darboux first integral for generic exponents.

Invariant algebraic surfaces of polynomial vector fields in dimension three

We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria…

Complete polynomial vector fields on $\C^2$, Part I

In this work, under a mild assumption, we give the classification of the complete polynomial vector fields in two variables up to algebraic automorphisms of $\C^2$. The general problem is also reduced to the study of the combinatorics of certain resolutions of singularities. Whereas we deal with $\C$-complete vector fields, our results also apply to $\R$-complete ones thanks to a theorem of Forstneric \cite{forst}.

Polynomial entropy for the circle homeomorphisms and for $C^1$ nonvanishing vector fields on $\T^2$

We prove that the polynomial entropy of an orientation preserving homeomorphism of the circle equals 1 when the homeomorphism is not conjugate to a rotation and that it is 0 otherwise. In a second part we prove that the polynomial entropy of a flow on the two dimensional torus associated with a $C^1$ nonvanishing vector field is less that $1$. We moreover prove that when the flow possesses periodic orbits its polynomial entropy equals 1 unless it is conjugate to a rotation (in this last case, the polynomial entropy is zero).

Phase portraits for quadratic homogeneous polynomial vector fields on S^2

Let X be a homogeneous polynomial vector field of degree 2 on 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 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 16^{th} 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 homo…

平面多项式向量场中非双曲极限环的结构不稳定性研究

On the structural instability of non-hyperbolic limit cycles on planar polynomial vector fields

该研究聚焦平面多项式向量场非双曲极限环的结构稳定性问题，证明在惠特尼拓扑下奇数次非双曲极限环也存在结构不稳定性，补全了多项式情形下的相关结论。

Realization problems for limit cycles of planar polynomial vector fields

We show that for any finite configuration of closed curves $Γ\subset \mathbb{R}^2$, one can construct an explicit planar polynomial vector field that realizes $Γ$, up to homeomorphism, as the set of its limit cycles with prescribed periods, multiplicities and stabilities. The only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities. The constructed vector fields are Darboux integrable and admit a polynomial inverse integrating factor.

n维椭球面上实多项式向量场的Darboux可积性理论

Darboux theory of integrability for real polynomial vector fields on the $n-$dimensional ellipsoid

该研究将n维球面的Darboux可积性理论推广到n维椭球面，给出了椭球面上多项式向量场不变平行面与子午面最大数量的新结果，拓展了欧氏空间中多项式向量场不变超平面数量上界的相关结论。

Topological enumeration of complex polynomial vector fields

The enumeration of combinatorial classes of the complex polynomial vector fields in C presented in [Dia13] is extended here to a closed form enumeration of combinatorial classes for degree d polynomial vector fields up to rotations of 2(d-1)st roots of unity. The main tool in the proof of this result is based on a general method of enumeration developed by V.A.Liskovets [Lis98].

Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles

We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups $G$ such that $G$ is generated by a compact family $Γ$, the planar postcritical set of $G$ is bounded, and $G$ is (semi-) hyperbolic. In one of the classes, we have that for almost every sequence $γ\in Γ^{\Bbb{N}}$, the Julia set $J_γ$ of $γ$ is a Jordan curve but not a quasicircle, t…

Poincaré compactification for non-polynomial vector fields

In this work a theorical framework to apply the Poincaré compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of…

Complex polynomial vector fields with many algebraic orbits

We state some generalizations of a theorem due to G. Darboux, which originally states that a polynomial vector field in the complex plane exhibits a rational first integral and has all its orbits algebraic provided that it exhibits infinitely many algebraic orbits. In this paper, we give an interpretation of this result in terms of the classical Reeb stability theorems, for compact leaves of (non-singular) smooth foliations. Then we give versions of Darboux's theorem, assuring, for a (non-singular) holomorphic foliation of any codimension, the existence of an open set of compact leaves provide…

Stable piecewise polynomial vector fields

Consider in R^2 the semi-planes N={y>0} and S={y<0}$ having as common boundary the straight line D={y=0}$. In N and S are defined polynomial vector fields X and Y, respectively, leading to a discontinuous piecewise polynomial vector field Z=(X,Y). This work pursues the stability and the transition analysis of solutions of Z between N and S, started by Filippov (1988) and Kozlova (1984) and reformulated by Sotomayor-Teixeira (1995) in terms of the regularization method. This method consists in analyzing a one parameter family of continuous vector fields Z_ε$, defined by averaging X and Y. This …

一个具有有理系数且无局部多项式李雅普诺夫函数的全局渐近稳定的多项式向量场

A Globally Asymptotically Stable Polynomial Vector Field with Rational Coefficients and no Local Polynomial Lyapunov Function

本文提出一个二维七次多项式向量场，具有有理系数，全局渐近稳定但无局部解析李雅普诺夫函数。

三维多项式向量场的希尔伯特第16问题版本：孤立不变环面的计数

A version of Hilbert's 16th Problem for 3D polynomial vector fields: Counting isolated invariant tori

本文将希尔伯特第16问题扩展至三维多项式向量场，基于平均方法提出从平面极限环构造三维法向双曲不变环面的机制，并证明孤立不变环面数量上界$N(m)$的增长速度至少为$m^3/128$。

Discrete and continuous dynamics of real $3$-dimensional nilpotent polynomial vector fields

A large class of real $3$-dimensional nilpotent polynomial vector fields of arbitrary degree is considered. The aim of this work is to present general properties of the discrete and continuous dynamical systems induced by these vector fields. In the discrete case, it is proved that each dynamical system has a unique fixed point and no $2$-cycles. Moreover, either the fixed point is a global attractor or there exists a $3$-cycle which is not a repeller. In the continuous setting, it is proved that each dynamical system is polynomially integrable. In addition, for a subclass of the considered ve…

Algebraic and qualitative aspects of quadratic vector fields related with classical orthogonal polynomials

This paper is a sequel of the reference \cite[§4.2, p.p. 1782--1783]{almp}, in where some families of quadratic polynomial vector fields related with orthogonal polynomials were studied. We extend such results that contain some details related with differential Galois Theory as well the inclusion of Darboux theory of integrability and qualitative theory of dynamical systems.

一个没有多项式李雅普诺夫函数的全局渐近稳定平面齐次多项式向量场

A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field With No Polynomial Lyapunov Function

研究反驳了全局渐近稳定齐次多项式向量场必有齐次多项式李雅普诺夫函数的猜想，以具整数系数的平面齐次三次多项式向量场为反例，其无特定性质的齐次多项式李雅普诺夫函数，但有明确二次齐次李雅普诺夫函数，还给出机器验证形式化。

A Characterization of Multiplicity-Preserving Global Bifurcations of Complex Polynomial Vector Fields

For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree d, it is proved that any bifurcation which preserves the multiplicity of equilibrium points can be realized as a composition of a finite number of simpler bifurcations, and these bifurcations are characterized.

On Parameter Space of Complex Polynomial Vector Fields in the Complex Plane

The space of degree d single-variable monic and centered complex polynomial vector fields can be decomposed into loci in which the vector fields have the same topological structure. We analyze the geometric structure of these loci and describe some bifurcations, in particular, it is proved that new homoclinic separatrices can form under small perturbation. By an example, we show that this decomposition of parameter space by combinatorial data is not a cell decomposition. The appendix to this article, joint work with Tan Lei, shows that landing separatrices are stable under small perturbation o…

The bifurcation diagram of cubic polynomial vector fields on $\mathbb{C}\mathbb{P}^1$

In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot z=z^3+ ε_1z+ε_0$ for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of $ε_1,ε_0\in\mathbb{C}$. The bifurcation diagram is in $\mathbb{R}^4$, but its conic structure allows describing it for parameter values in $\mathbb{S}^3$. There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points.…

平面多项式向量场族的达布型中心条件

Darboux-type center conditions for families of planar polynomial vector fields

研究平面多项式向量场的中心焦点问题，基于李雅普诺夫函数法，给出两类微分系统中心焦点条件新结果，找到由李雅普诺夫常数生成的鲍廷理想的一个包含关系，结果适用于任意次数\(n\geq2\)。

一般可逆复多项式向量场

Generic reversible complex polynomial vector fields

本文研究了一般复一维多项式向量场在保持无穷远点极点分离轨的同胚轨道等价下，确定了一般分层数目，给出了由组合和解析部分构成的模的完整参数化，描述了 3 次和 4 次的分岔图，并证明了任意一般模的实现定理。

Abelian integrals and limit cycles for a class of cubic polynomial vector fields of Lotka-Volterra type with a rational first integral of degree 2

In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree $n$. The analysis is carried out by estimating the number of zeros of the corresponding Abelian integrals. Moreover, using \emph{Chebyshev criterion}, we show that the sharp upper bound for the number of zeros of the Abelian integrals defined on each period annulus is 3 for $n=3$. The simultaneous bifurcation and distribution of limit cycles for the system with …

Discriminantly separable polynomials and quad-equations

We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable each. Our classification is based on the study of structures of zeros of a polynomial component $P$ of a discriminant. This classification is related to the classification of pencils of conics in a delicate way. We establish a relationship between our classification and the classification of integrable quad-equations which has been suggested recently by Adler,…

平面实多项式向量场的中心和不变直线及其配置

Centers and invariant straight lines of planar real polynomial vector fields and its configurations

本文研究了平面实多项式哈密顿向量场的中心和不变直线数量及其配置规律，证明了当中心数达到上限时的配置规则，并探讨了中心数与极限环存在的关系。

关于拟齐次多项式向量场的周期环域的循环性

On the cyclicity of the period annulus of quasi-homogeneous polynomial vector fields

研究在单参数多项式扰动下，具有中心的拟齐次多项式向量场从周期环域分岔出的极限环个数，通过改进的Francoise算法和组合技术建立了k阶Melnikov函数孤立零点数的上界公式。

向量场在凸多面体域上的多项式插值

Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain

本文提出了一种在凸多面体域上基于最小二乘法的多项式向量场重建方法，能够精确满足无穿透边界条件。

The Complete Classification of Rational Preperiodic Points of Quadratic Polynomials over Q: A Refined Conjecture

We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over $\bold Q$, assuming the conjecture that it is impossible to have rational points of period $4$ or higher. In particular, we show under this assumption that the number of preperiodic points is at most~$9$. Elliptic curves of small conductor and the genus~$2$ modular curves $X_1(13)$, $X_1(16)$, and $X_1(18)$ all arise as curves classifying quadratic polynomials with various combinations of preperiodic points. To complete the classification, we compute the rational points on a non-mod…

多项式向量场的可积性与一个对偶问题

Integrability of polynomial vector fields and a dual problem

本文利用参数空间对偶性研究多项式向量场的可积性，建立微分算子、可积性簇与李群不变量及局部解析首次积分之间的联系，并以二次向量场为例加以说明。

A topological classification of plane polynomial systems having a globally attracting singular point

In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible set" consisting of finitely many vectors with components in the set $\{n/3: n=0,1,2,\ldots\}$), so that two such systems are equivalent if and only if (after appropriately fixing an orientation in $\mathbb{R}^2$ and a heteroclinic separatrix) they have the same feasible set. In fact, this classification is achieved in the more general setting of continuous flo…

Upper bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles"

The paper deals with planar polynomial vector fields. We aim to estimate the number of orbital topological equivalence classes for the fields of degree n. An evident obstacle for this is the second part of Hilbert's 16th problem. To circumvent this obstacle we introduce the notion of equivalence modulo limit cycles. This paper is the continuation of the author's paper in [Mosc. Math. J. 1 (2001), no. 4] where the lower bound of the form 2^{cn^2} has been obtained. Here we obtain the upper bound of the same form. We also associate an equipped planar graph to every planar polynomial vector field…

Commuting planar polynomial vector fields for conservative Newton systems

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. Let $f \in K[x]$, where $K$ is a field of characteristic zero, and $d$ the derivation that corresponds to the differential equation $\ddot x = f(x)$ in a standard way. Let also $H$ be the Hamiltonian polynomial…