Limit cycles of linear vector fields on $(\mathbb{S}^2)^m \times \mathbb{R}^n$

Clara Cufí-Cabré and Jaume Llibre

Vol. 324 (2023), No. 2, 249–263

DOI: 10.2140/pjm.2023.324.249

Abstract

It is well known that linear vector fields defined in ${ℝ}^n$ cannot have limit cycles, but this is not the case for linear vector fields defined in other manifolds. We study the existence of limit cycles bifurcating from a continuum of periodic orbits of linear vector fields on manifolds of the form ${({\mathbb{𝕊}}^2)}^m\times {ℝ}^n$ when such vector fields are perturbed inside the class of all linear vector fields. The study is done using averaging theory. We also present an open problem about the maximum number of limit cycles of linear vector fields on ${({\mathbb{𝕊}}^2)}^m\times {ℝ}^n$.

Keywords

limit cycle, periodic orbit, isochronous center, averaging method

Mathematical Subject Classification

Primary: 34A30, 34C25, 34C29

Milestones

Received: 13 July 2022

Revised: 13 January 2023

Accepted: 19 June 2023

Published: 26 July 2023