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

TL;DR

This paper investigates the existence and bifurcation of limit cycles in linear vector fields on product manifolds of spheres and Euclidean spaces, revealing differences from Euclidean cases and posing open problems.

Contribution

It analyzes limit cycles of linear vector fields on $( imes ext{S}^2)^m imes ext{R}^n$ manifolds using averaging theory, highlighting new phenomena not present in Euclidean spaces.

Findings

Limit cycles can bifurcate from periodic orbits on these manifolds.

The study identifies conditions under which limit cycles appear.

An open problem on the maximum number of limit cycles is proposed.

Abstract

It is well known that linear vector fields defined in $\mathbb{R}^n$ can not 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 $(\mathbb{S}^2)^m \times \mathbb{R}^n$ when such vector fields are perturbed inside the class of all linear vector fields. The study is done using the averaging theory. We also present an open problem concerning the maximum number of limit cycles of linear vector fields on $(\mathbb{S}^2)^m \times \mathbb{R}^n$.