On real center singularities of complex vector fields on surfaces

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Dynamical Systems-An International Journal Pub Date : 2023-10-25 DOI:10.1080/14689367.2023.2270931

V. León, B. Scárdua

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引用次数: 0

Abstract

AbstractOne of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations [R. Moussu, Une démonstration géométrique d'un théorème de Lyapunov-Poincaré, Astérisque 98–99 (1982), pp. 216–223]. In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with ‘many’ periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two. In this paper we prove versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.Keywords: Foliationcenter singularityfirst integralintegrable form AcknowledgmentThe first author is grateful to Edital n°77/2022/PRPPG-PAAP-UNILA for partially supporting this research work.Disclosure statementNo potential conflict of interest was reported by the author(s).

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曲面上复向量场的实中心奇点

【摘要】经典李亚普诺夫-庞加莱中心定理的一个版本指出，一个非简并实解析中心型平面向量场奇点允许一个解析第一积分。在对这一结果的进一步证明中，R. Moussu在这一结果与全纯叶的奇点理论之间建立了重要的联系。穆苏，李亚普诺夫-波因卡罗，《通讯通讯与通讯通讯》，《通讯通讯与通讯通讯》(1982)，第216-223页。本文考虑了两个主要框架的推广:(i)在奇点附近具有“许多”周期轨道的平面实解析向量场和(ii)在二维空间中具有合适奇点的全纯叶理的胚芽。本文证明了庞加莱-李亚普诺夫中心定理的几个版本，包括全纯向量场的证明。我们还给出了一些应用程序，暗示在这个框架中还有更多的东西需要探索。第一作者感谢Edital n°77/2022/PRPPG-PAAP-UNILA对本研究工作的部分支持。披露声明作者未报告潜在的利益冲突。

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来源期刊

Dynamical Systems-An International Journal 物理-力学

CiteScore

0.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Dynamical Systems: An International Journal is a world-leading journal acting as a forum for communication across all branches of modern dynamical systems, and especially as a platform to facilitate interaction between theory and applications. This journal publishes high quality research articles in the theory and applications of dynamical systems, especially (but not exclusively) nonlinear systems. Advances in the following topics are addressed by the journal: •Differential equations •Bifurcation theory •Hamiltonian and Lagrangian dynamics •Hyperbolic dynamics •Ergodic theory •Topological and smooth dynamics •Random dynamical systems •Applications in technology, engineering and natural and life sciences