  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 $SO(4)$.
