Global geometry of families of polynomial vector fields

多项式向量场族的全局几何

• 批准号: RGPIN-2020-05145
• 负责人: Schlomiuk, Dana
• 金额: $ 1.31万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759381
• 关键词: Global; geometry; families; polynomial; vector

Together with my collaborators Artés, Llibre and Vulpe we obtained, after 8 years of work, the geometric classification of the global configurations of singularities of the class QS of planar quadratic vector fields, yielding 1765 such configurations. Our book (680 pp.) is due to appear in the Birkhäuser series of Springer this year. We introduced the geometric equivalence relation for polynomial vector fields. This relation is deeper than the topological one, taking into account also algebraic and geometric features of singularities such as the order of weak singularities, multiplicity of singularities, level of isochronicity, etc. We later obtained the topological classification of the global configurations of singularities of QS yielding 208  configurations, due to appear in in the first issue of 2020 of Qual.Theory of Dyn. Syst. This opened the road for obtaining the topological classification, modulo limit cycles, of all QS. Indeed, each global topological configuration of singularities constitutes a skeleton over which the topological phase portraits modulo limit cycles could be constructed, based on a combinatorial study of possible connections. This may take a several years. The general theoretical framework we introduced for polynomial vector fields in our book, opened new roads for studying families of polynomial systems. In particular we intend to study the family QW1 of systems in QS possessing a weak focus of order 1. Weak foci are very important in the production of limit cycles and hence for Hilbert's 16th problem (H16). This class is 4-dimensional, modulo the action of the affine group and time rescaling and so far no 4-dimensional subclass of QS was studied. (The 2 and 3-dimensional families QW3 and QW2 of families in QS with 3rd and 2nd weak foci were studied by Llibre, Artés and myself.) The applicant also proposes to study several families of cubic systems mentioned in the proposal.  Another direction of my research is on Darboux, algebraic and Liouvillian integrability. Darboux gave only sufficient conditions for Darboux integrability.  Poincaré's problem on recognizing when a system is algebraically integrable (i.e. having a rational first integral), is still open even for systems in QS. Together with my Ph.D. student Ana Maria Travaglini and my collaborator Regilene Oliveira, we plan to study from the viewpoint of integrability, the family QSH of systems in QS having an invariant hyperbola. This family displays a rich variety of geometric structures. It is a good testing ground for seeing how the geometric and dynamic properties of the systems blend in producing the different kinds of integrability. This work lies at the interface of algebraic-geometry and dynamical systems. Finally the applicant together with Llibre intend to give a sound geometrical background (including a needed geometrical equivalence relation) for studying the family of graphics in QS and then apply it to the existential part of H16.

我们与我的合作者Artés、Llibre和Vulpe一起，经过8年的工作，获得了平面二次向量场QS类奇点的全局配置的几何分类，产生了1765个这样的配置。我们的书（680页）将出现在今年施普林格的Birkhäuser系列中。引入了多项式向量场的几何等价关系。这种关系是深于拓扑之一，同时考虑到代数和几何特征的奇点，如秩序的弱奇点，多重性的奇点，水平的等时性等，我们后来得到的拓扑分类的全球配置的奇点的QS产生208配置，由于出现在第一期的2020年的Qual。动力学理论。这开辟了道路，获得拓扑分类，模极限环，所有QS。事实上，每一个全球性的拓扑配置的奇点构成了一个骨架的拓扑相图模极限环可以构建的基础上，可能的连接的组合研究。这可能需要几年时间。我们在书中介绍了多项式向量场的一般理论框架，为研究多项式系统族开辟了新建道路。特别是，我们打算研究家庭QW1系统在QS拥有一个弱焦点的顺序1。弱焦点在极限环的产生中非常重要，因此对于希尔伯特第16问题（H16）也是如此。这个类是4维的，模仿射群和时间重新标度的作用，到目前为止还没有研究过QS的4维子类。(The Llibre，Artés和我研究了具有第3和第2弱焦点的QS中的2维和3维族QW3和QW2。申请人还建议研究几个家庭的三次系统中提到的建议。我的研究的另一个方向是达布，代数和刘维可积性。达布只给出了达布可积的充分条件。庞加莱关于系统何时代数可积（即有一个有理第一积分）的问题，即使对于QS中的系统仍然是开放的。加上我的博士学位。学生Ana Maria Travaglini和我的合作者Regilene Oliveira，我们计划从可积性的角度研究QS中具有不变双曲线的系统族QSH。该族显示出丰富多样的几何结构。这是一个很好的试验场，可以用来观察系统的几何和动力学性质如何混合产生不同类型的可积性。这项工作是在代数几何和动力系统的接口。最后，申请人与Llibre一起打算给出用于研究QS中的图形族的合理几何背景（包括所需的几何等价关系），然后将其应用于H16的存在部分。