Restricted Versions of the Hilbert 16th Problem and Related Topics in the Theory of Analytic Foliations

解析叶理理论中希尔伯特第十六问题的限制版本及相关主题

基本信息

  • 批准号:
    0100404
  • 负责人:
  • 金额:
    --
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2001
  • 资助国家:
    美国
  • 起止时间:
    2001-07-01 至 2004-06-30
  • 项目状态:
    已结题

项目摘要

The Hilbert 16th problem, part 2, is: "What may be said about the number and location of limit cycles of a planar polynomial vector field?" Traditionally this question is interpreted as a problem of finding an upper bound of the number of limit cycles of a polynomial vector field as a function of the degree of the polynomials. Even for the degree two, the upper bound is not yet found, and its existence is not yet proved. It makes sense to consider "restricted versions" of the Hilbert 16th problem. Namely, the set or all polynomial vector fields is replaced by a particular subset of by a similar class. The examples are Lienard and Abel equations (the latter ones are polynomial in the phase variable with coefficients 1-periodic in time). Even for these equations the problem of the number of limit cycles stays open. It is solved by the PI with an extra restriction: Abel equations, and Lienard ones with the polynomial of odd degree, are considered together with an upper bound for the magnitudes of the coefficients; the upper estimate on the number of limit cycles depends on this upper bound of the magnitudes. This is the result from the prior NSF support. In the current project we try to get rid of this latter restriction, and to give an estimate of the number of limit cycles that depends on the degrees of the polynomials in the right hand side only; for Abel equations the coefficients should be trigonometric polynomials of given degree. We hope to use mighty tools of the theory of complex analytic foliations, growth and zeros theorems for holomorphic functions and methods developed under the prior NSF support. Another important problem to be studied is the infinitesimal Hilbert 16th problem. It requires to estimate the number of limit cycles generated by a small perturbation from the ovals of the Hamiltonian polynomial vector field; closed orbits of the latter field form continuous families. This problem is reduced to the estimate of the number of zeros of an Abelian integral, that is, an integral of a polynomial 1-form over the ovals of a real polynomial in the plane; the estimate should be given in terms of the degrees of the polynomial Hamiltonian function and of the integrand. This problem was investigated by the author since 69; later on by Yakovenko, D.Novikov, Horosov, Gavrilov, Petrov, Khovanski, Varchenko and others. Some progress was obtained by Glutsuk and the author for the restricted version of the problem when the Hamiltonian polynomial is taken of a special type and of arbitrary degree. One of the goals of this project is to get an explicit upper estimate that is expected to be an exponential of a polynomial of the degree of the Hamiltonian function, provided that the integrand has a smaller degree.The theory of dynamical systems is the realm of determinism, on one hand, and of chaos, on the other hand. Vector fields in the phase space of dimension higher than two form the realm of chaos. This was understood in 1960's, and henceforth, this realm is the subject of the top interest for mathematicians, computer scientists and physicists. On the other hand, the classical subject of planar differential equations which may be called "realmof order" attracted the interest of researchers during more that one hundred years, beginning with Poincare and Hilbert. Hilbert's 16th problem is the main one in this domain. It persists the efforts of mathematicians during 100 years, and it is clear now that simplified "restricted" versions of the problem should be attacked first. The project suggests some concrete ways of this attack based on new ideas and the progress from the prior NSF support.
希尔伯特第16问题,第2部分,是:“关于平面多项式向量场的极限环的数量和位置可以说什么?“传统上,这个问题被解释为一个问题,找到一个多项式向量场的极限环数量的上限作为多项式次数的函数。即使对于二次,上界也没有找到,它的存在性也没有得到证明。考虑希尔伯特第16问题的“限制版本”是有意义的。也就是说,集合或所有多项式向量场被类似类的特定子集所取代。例子是Lienard方程和Abel方程(后者是相位变量的多项式,系数为1-时间周期)。即使对于这些方程,极限环的数目问题仍然是一个悬而未决的问题。它是解决了PI与一个额外的限制:阿贝尔方程,和Lienard的多项式的奇数次,被认为是与上限的幅度的系数;上限估计的数量极限环取决于这个上限的幅度。这是先前NSF支持的结果。在目前的项目中,我们试图摆脱后一种限制,并给出一个估计的数量极限环,这取决于次数的多项式在右手边只;阿贝尔方程的系数应该是三角多项式的次数。我们希望使用强大的工具,理论的复解析叶理,增长和零定理的全纯函数和方法下开发的先前NSF的支持。另一个需要研究的重要问题是无穷小希尔伯特第16问题。它需要估计由哈密顿多项式向量场的椭圆的小扰动产生的极限环的数量;后者的闭轨道形成连续的家庭。这个问题被简化为阿贝尔积分的零点数的估计,即平面上真实的多项式椭圆上的多项式1-形式的积分;估计应该根据多项式汉密尔顿函数和被积函数的次数给出。提交人从69年起就对这一问题进行了调查;后来Yakovenko、D.Novikov、Horosov、Gavrilov、Petrov、Khovanski、Varchenko和其他人也对这一问题进行了调查。Glutsuk和作者对当Hamilton多项式取特殊类型和任意次数时的限制型问题取得了一些进展。这个项目的目标之一是得到一个显式的上估计,预计是一个指数的多项式的次数的哈密尔顿函数,提供的被积函数有一个较小的degrees.The理论的动力系统是领域的决定论,一方面,混沌,另一方面。相空间中的向量场的维数大于2,形成了混沌的领域。这在20世纪60年代被理解,从此,这个领域是数学家,计算机科学家和物理学家最感兴趣的主题。另一方面,经典的问题,平面微分方程,这可能被称为“realmof order”吸引了研究人员的兴趣,在一百多年来,开始与庞加莱和希尔伯特。希尔伯特第16问题是这一领域的主要问题。它坚持努力的数学家在100年来,现在很明显,简化的“限制”版本的问题应该首先攻击。该项目根据新的想法和先前NSF支持的进展,提出了一些具体的攻击方法。

项目成果

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Yulij Ilyashenko其他文献

Hölder properties of perturbed skew products and Fubini regained
扰动偏斜积和 Fubini 的 Hölder 性质恢复
  • DOI:
    10.1088/0951-7715/25/8/2377
  • 发表时间:
    2010
  • 期刊:
  • 影响因子:
    1.7
  • 作者:
    Yulij Ilyashenko;Andrei Neguț
  • 通讯作者:
    Andrei Neguț
Sternberg Linearization Theorem for Skew Products

Yulij Ilyashenko的其他文献

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{{ truncateString('Yulij Ilyashenko', 18)}}的其他基金

Hilbert 16th Problem and Related Topics in Complex Analysis and Foliations
复分析和叶状结构中的希尔伯特第 16 问题及相关主题
  • 批准号:
    0700973
  • 财政年份:
    2007
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Simplified Versions of Hilbert 16th Problem and Related Topics in Complex Dynamics and Analytic Foliations
希尔伯特第十六问题的简化版本以及复杂动力学和解析叶状结构中的相关主题
  • 批准号:
    0400945
  • 财政年份:
    2004
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Hilbert Type Numbers and Related Topics in Analytic Differential Equations
解析微分方程中的希尔伯特类型数和相关主题
  • 批准号:
    9970372
  • 财政年份:
    1999
  • 资助金额:
    --
  • 项目类别:
    Standard Grant

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