Linear flows on compact, semisimple Lie groups: stability, periodic orbits, and Poincar\'e-Bendixon's Theorem

TL;DR

This paper investigates the stability and periodic orbits of linear flows on compact, semisimple Lie groups, providing classifications and a version of Poincaré-Bendixson's Theorem, with applications to specific groups like SO(3), SU(2), and SO(4).

Contribution

It introduces a classification of periodic orbits and a version of Poincaré-Bendixson's Theorem for linear flows on compact, semisimple Lie groups, extending understanding of their dynamics.

Findings

Classified periodic orbits on SO(3), SU(2), and SO(4)

Established stability criteria for linear flows on these groups

Presented a version of Poincaré-Bendixson's Theorem for the setting

Abstract

Our first purpose is to study the stability of linear flows on real, connected, compact, semisimple Lie groups. After, we study and classify periodic orbits of linear and invariant flows. In particular, we obtain a version of Poincar\'e-Bendixon's Theorem. As an application, we present periodic orbits of linear or invariant flows on $SO(3)$ or $SU(2)$, and we classify periodic orbits of a linear or invariant system on $SO(4)$.