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Global centres in a class of quintic polynomial differential systems

Part of: Qualitative theory

Published online by Cambridge University Press:  11 April 2024

Leonardo P. C. da Cruz Open the ORCID record for Leonardo P. C. da Cruz [Opens in a new window]  and
Jaume Llibre Open the ORCID record for Jaume Llibre [Opens in a new window]
Leonardo P. C. da Cruz
Affiliation:
Departamento de Matemática, Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, Avenida Trabalhador São Carlense, 400, 13566-590, São Carlos, SP, Brazil (leonardocruz@icmc.usp.br)
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain (Jaume.Llibre@uab.cat)
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Abstract

A centre of a differential system in the plane $ {\mathbb {R}}^2$ is an equilibrium point $p$ having a neighbourhood $U$ such that $U\setminus \{p\}$ is filled with periodic orbits. A centre $p$ is global when $ {\mathbb {R}}^2\setminus \{p\}$ is filled with periodic orbits. In general, it is a difficult problem to distinguish the centres from the foci for a given class of differential systems, and also it is difficult to distinguish the global centres inside the centres. The goal of this paper is to classify the centres and the global centres of the following class of quintic polynomial differential systems

\begin{align*} \dot{x}= y,\quad \dot{y}={-}x+a_{05}\,y^5+a_{14}\,x\,y^4+a_{23}\,x^2\,y^3+a_{32}\,x^3\,y^2+a_{41}\,x^4\,y+a_{50}\,x^5, \end{align*}
in the plane $ {\mathbb {R}}^2$.

Keywords

centreglobal centrepolynomial differential systemsLyapunov quantitiesblow upquintic polynomial

MSC classification

Primary: 34C05: Location of integral curves, singular points, limit cycles

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 156 , Issue 1 , February 2026 , pp. 39 - 54
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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