A Transfer Principle in the Real Plane from Nonsingular Algebraic Curves to Polynomial Vector Fields

For every nonsingular algebraic curve C of degree m in the real plane a polynomial vector field of degree 2m–1 is constructed, which has exactly the ovals of C as attracting limit cycles. Therefore, every progress on the algebraic part of Hilbert's 16th problem automatically yields progress on its dynamical part.     

Counterexample to a Conjecture on the Algebraic Limit Cycles of Polynomial Vector Fields

In Geometriae Dedicata 79 (2000), 101–108, Rudolf Winkel conjectured: for a given algebraic curve f=0 of degree m 4 there is in general no polynomial vector field of degree less than 2m -1 leaving invariant f=0 and having exactly the ovals of f=0 as limit cycles. Here we show that this conjecture is not true. Mathematics Subject Classifications (2000). Primary 14P25, 34C05, 34A34.     

Limit Cycles of a Planar Vector Field

This paper presents a solution to the problem to find isolated closed trajectories of two-dimensional dynamic systems. In contrast to the method of Bendixsons ring regions, the new method is constructive. It allows the determination of the location of closed trajectories and therefore gives an upper bound for their number. The method is based on the idea to use inherent geometrical and physical extremal properties of these trajectories to transform the problem into an optimization task (isoperimetric problem of variational calculus) that can be solved by numerical algorithms, e.g., by hillclimbing.     

Rational and Homological Equivalence of Real Algebraic Cycles

In this paper we study algebraic cycles on smooth quasiprojective real algebraic varieties. We are interested under what conditions an algebraic cycle homologous to zero is rationally equivalent to a cycle supporting no real points.     

弱化希尔伯特第16问题及其研究现状

V.I.Arnold多次提出如下问题:对于给定的自然数n≥2,所有n次多项式1-形式,沿一切可能的m≥3次闭代数曲线族的阿贝尔积分的孤立零点的最大个数Z(m,n)=?由Poincare-Pontryagin定理可知,当阿贝尔积分不恒为零时,A(n)=Z(n+1,n)给出n次Hamilton系统在n次多项式扰动下从原有周期环域分支出极限环的最大个数,因此Arnold把这个问题称为弱化的希尔伯特第16问题.30多年来,对此问题的研究取得了一定进展,也遇到了很大困难.本文拟对这个问题和相关研究工作做一个粗浅的介绍.     

Complex Zeros of Algebraic Polynomial with Non-Zero Mean Random Coefficients

In this paper we obtain a formula for the average density of the distribution of complex zeros of an algebraic polynomial with random coefficients. The coefficients are assumed independent identical normally distributed random variables with mean and variance ². The value of the average density for the case of =0 and ²=1 was obtained previously. Some limits of the distribution of the complex zeros are provided using the presented formula.     

A cubic system with thirteen limit cycles

We construct a planar cubic system and demonstrate that it has at least 13 limit cycles. The construction is essentially based on counting the number of zeros of some Abelian integrals.     

具有两个二次代数曲线解的三次系统的代数极限环

本文证明了具有椭圆和抛物线解的三次系统可以存在代数极限环，纠正了文[4]的主要结果.     

Abel-like differential equations with no periodic solutions

We present various criteria for the non-existence of positive periodic solutions of generalized Abel differential equations with periodic coefficients that can change sign. As an application, we obtain some families of planar vector fields without limit cycles.     

Perturbation from an asymmetric cubic Hamiltonian

This paper concerns with the number of limit cycles from an asymmetric Hamiltonian of degree three under cubic perturbation. Eleven limit cycles are found and three different distributions are given by using the methods of bifurcation theory and qualitative analysis, two of which are new.     

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x²+y²+ax⁴+bx²y²+cy⁴=h, h∈Σ, ac(4ac−b²)≠0, Σ=(0,h₁) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.     

一类哈密顿系统的Abel积分之比单调性的判据

研究了一类哈密顿系统的两个Abel积分比值的单调性的条件,指出这个单调性条件可由文中给出的两个判定函数直接确定.     

The Expected Number of Local Maxima of a Random Algebraic Polynomial

In this work we obtain an asymptotic estimate for the expected number of maxima of the random algebraic polynomial , where a _j (j=0, 1,...,n–1) are independent, normally distributed random variables with mean and variance one. It is shown that for nonzero , the expected number of maxima is asymptotic to log n, when n is large.     

Algebraic Polynomials with Non-identical Random Coefficients

The asymptotic estimate for the expected number of real zeros of a random algebraic polynomial is known. The identical random coefficients a_j(ω) are normally distributed defined on a probability space , ω ∈Ω. The estimate for the expected number of zeros of the derivative of the above polynomial with respect to x is also known, which gives the expected number of maxima and minima of Q_n(x, ω). In this paper we provide the asymptotic value for the expected number of zeros of the integration of Q_n(x,ω) with respect to x. We give the geometric interpretation of our results and discuss the difficulties which arise when we consider a similar problem for the case of .     

Algebraic polynomials with non-identical random coefficients

There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial The coefficients are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all sufficiently large, the above expected value is shown to be . Also, it is known that if the have non-identical variance , then the expected number of real zeros increases to . It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than . In this work for two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain . In fact, so far the case of is the only case that can significantly increase the expected number of real zeros.

     

Real Quotient Singularities and Nonsingular Real Algebraic Curves in the Boundary of the Moduli Space

The quotient of a real analytic manifold by a properly discontinuous group action is, in general, only a semianalytic variety. We study the boundary of such a quotient, i.e., the set of points at which the quotient is not analytic. We apply the results to the moduli space M_g/ of nonsingular real algebraic curves of genus g (g2). This moduli space has a natural structure of a semianalytic variety. We determine the dimension of the boundary of any connected component of M_g/. It turns out that every connected component has a nonempty boundary. In particular, no connected component of M_g/ is real analytic. We conclude that M_g/ is not a real analytic variety.     

The number and distributions of limit cycles for a class of cubic near-Hamiltonian systems

This paper concerns with the number of limit cycles for a cubic Hamiltonian system under cubic perturbation. The fact that there exist 9-11 limit cycles is proved. The different distributions of limit cycles are given by using methods of bifurcation theory and qualitative analysis, among which two distributions of eleven limit cycles are new.     

n次代数多项式有m个不同根的充要条件

本文利用方阵的迹及顺序主子式 ,给出了 n次代数多项式有 m( m n)个不同根的充要条件 .     

一类Hamilton系统的Abel积分比值单调性的判别条件

用直接计算的方法对一类Hamilton系统的两个Abel积分比值的单调性进行讨论,指出该单词性条件可由两个判定函数直接确定．