Periodic orbits of Linear and invariant flows on connected Lie groups

Our main is to study periodic orbits of linear and invariant flows on a real, connected Lie group. Since each linear flow ${\phi }_t$ has a derivation associated $\mathcal{D}$, we show that the existence of periodic orbits of ${\phi }_t$ is based on the eigenvalues of the derivation $\mathcal{D}$. From this, we study periodic orbits of a linear flow on noncompact, semisimple Lie groups, and we work with periodic orbits of a linear flow on connected, simply connected, solvable Lie groups of dimension 2 or 3.