具有五次齐次非线性的可逆全局中心

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Dynamical Systems-An International Journal Pub Date : 2023-07-10 DOI:10.1080/14689367.2023.2228737

J. Llibre, C. Valls

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引用次数: 2

摘要

微分系统在平面上的中心是一个具有邻域U的奇异点，该邻域充满了周期轨道。全球中心是一个充满周期性轨道的中心。确定一个给定的微分系统是否有一个中心通常是一个难题，但更难知道它是否有全局中心。在本文中，我们讨论了一类形式为（1）的多项式微分系统，其中P和Q是n次齐次多项式。已知这些系统只有当n是奇数时才能具有全局中心，并且在n的情况下具有全局中心 = 1和n = 3是已知的。在这里，我们处理案例n = 5，我们对一个四参数系统族的全局中心进行了分类（1）。特别地，我们说明了如何仅使用垂直爆炸来研究线性部分为零的奇异点的局部相图。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Reversible global centres with quintic homogeneous nonlinearities

A centre of a differential system in the plane is a singular point having a neighbourhood U such that is filled of periodic orbits. A global centre is a centre such that is filled of periodic orbits. To determine if a given differential system has a centre is in general a difficult problem, but it is even harder to know if it has a global centre. In the present paper we deal with the class of polynomial differential systems of the form (1) where P and Q are homogeneous polynomials of degree n. It is known that these systems can have global centres only if n is odd and the global centres in the cases n = 1 and n = 3 are known. Here we work with the case n = 5 and we classify the global centres of a four parameter family of systems (1). In particular we illustrate how to study the local phase portraits of the singular points whose linear part is identically zero using only vertical blow ups.

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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