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
• 负责人: Yulij Ilyashenko
• 金额: --
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400945&HistoricalAwards=false
• 关键词: Simplified; Versions; Hilbert; 16th; Problem

The project deals with numerous topics in the theory of planar differential equations related to Hilbert's 16th problem and continues two previous projects. It considers both real and complex equations. There are two major achievements in the work over the previous project that will be developed in the current one. First, rather unexpectedly, the Kupka-Smale (KS) property was proved for polynomial automorphisms both of real and complex planes. The proof is based on "persistence theorems" and "Petrovski-Landis (PL) strategy". Generally speaking, persistence theorems claim the possibility of global extension of some geometric properties of polynomial dynamical systems over the whole parameter space. PL strategy makes use of persistence theorems to prove or disprove such geometric properties. New persistence theorems for heteroclinic points of polynomial automorphisms of a complex space are expected. Genericity of Kupka-Smale property for such automorphisms is suggested as a consequence. The main tool would be the PL strategy. Another achievement is an upper estimate by Glutsyuk and the PI of the number of zeros of Abelian integrals both in real and complex domains. This estimate is the best amidst other estimates of this kind due to Yakovenko and his students. On the other hand, it provides an approach to the complete solution of the restricted version of the Infinitesimal Hilbert 16th Problem. Together with the PL strategy, this gives an approach to the Infinitesimal Hilbert 16th Problem itself: give an upper bound of the number of real zeros of an integral of a polynomial one-form over the ovals of another polynomial in the plane. The project suggests numerous problems on the persistence properties for polynomial dynamical systems, simultaneous uniformization and topological properties of polynomial foliations. Study of the relations between these branches of the theory is an important part of the project. Moreover, new simultaneous uniformization theorems, together with new generic properties of polynomial and analytic foliation of the complex space are expected.Theory of dynamical systems is split into two parts: multidimensional systems (realm of chaos); two-dimensional systems (realm of order). Hilbert 16th problem is a central one in the theory of two-dimensional systems. The problem itself persists the efforts of mathematicians during more than a hundred years. Centennial history of investigations related to Hilbert 16th problem is reviewed in a survey article by the PI published in the Bulletin of the AMS in 2002. The survey contains, in particular, many results of the previous NSF projects, as well as problems that are subject to the current project. Note that two-dimensional dynamical systems provide models for various problems in physics, engineering and biology (predator-prey models). Understanding of real two-dimensional dynamics is therefore a subject of general scientific interest. On the other hand, study of complex extensions of real dynamical systems provides important new information about real systems and is interesting in itself.

该项目涉及与希尔伯特第16问题相关的平面微分方程理论中的许多主题，并继续以前的两个项目。它考虑了真实的和复杂的方程。在前一个项目的工作中，有两个主要成果将在本项目中得到发展。首先，相当意外地，Kupka-Smale（KS）性质证明了多项式自同构的真实的和复杂的平面。证明是基于“持久性定理”和“彼得罗夫斯基-兰迪斯（PL）策略”。一般来说，持续性定理要求多项式动力系统的某些几何性质在整个参数空间上的全局扩张的可能性。PL策略利用持久性定理来证明或反驳这些几何性质。期望得到复空间中多项式自同构的异宿点的新的持久性定理。Kupka-Smale性质的通用性，这样的自同构的建议作为一个结果。主要工具是PL策略。另一个成就是上估计Glutsyuk和PI的阿贝尔积分的零点的数量都在真实的和复杂的域。这一估计是雅科文科和他的学生做出的其他此类估计中最好的。另一方面，它提供了一种方法来完全解决的限制版本的无穷小希尔伯特第16问题。 与PL策略一起，这给出了无穷小希尔伯特第16问题本身的一种方法：给出一个多项式1-形式在平面上另一个多项式的椭圆上的积分的真实的零点个数的上界。该项目提出了许多关于多项式动力系统的持久性，多项式叶理的同时一致化和拓扑性质的问题。研究这些理论分支之间的关系是该项目的重要组成部分。此外，新的同时一致化定理，以及多项式和复空间的解析叶理的新的一般性质也是人们所期望的。动力系统理论分为两部分：多维系统（混沌领域）和二维系统（有序领域）。Hilbert第16问题是二维系统理论中的一个中心问题。这个问题本身坚持了一百多年来数学家的努力。 在2002年发表在AMS公报上的一篇调查文章中，PI回顾了与希尔伯特第16问题相关的百年研究历史。特别是，该调查包含了以前NSF项目的许多结果，以及当前项目的问题。请注意，二维动力系统为物理学、工程学和生物学中的各种问题提供了模型（捕食者-被捕食者模型）。因此，对真实的二维动力学的理解是一个具有普遍科学意义的课题。另一方面，对真实的动力系统的复扩张的研究提供了关于真实的系统的重要的新信息，并且其本身是有趣的。