Global centres in a class of quintic polynomial differential systems

Author(s)

da Cruz, Leonardo P. C. ; Llibre, Jaume

• Document type: Journal article
• Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS; v. N/A, p. 16-pg., 2024-04-11.

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$. (AU)

• FAPESP's process: 21/14987-8 - Bifurcation of limit cycles in smooth piecewise systems and an application in Medicine
• Grantee: Leonardo Pereira Costa da Cruz
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• FAPESP's process: 22/14484-9 - Applications in biology of piecewise differential equations
• Grantee: Leonardo Pereira Costa da Cruz
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor