Periodic orbits of Linear flows on connected Lie groups
Keywords:periodic orbits, linear flows, connected Lie groups
Our main goal is to study the periodic orbits of linear flows on a real, connected Lie group. Since each linear flow φt has a derivation associated 𝒟, we show that the existence of periodic orbits of φt is based on the eigenvalues of the derivation 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 a connected, simply connected, solvable Lie groups of dimension 2 or 3.
V. Ayala and J. Tirao, “Linear control systems on Lie groups and controllability”, Proceedings of Symposia in Pure Mathematics, vol. 64, pp. 47-64, 1999.
R. Biggs and C. Remsing, “Control systems on three-dimensional Lie groups: equivalence and controllability”, Journal of Dynamical and Control Systems, vol. 20, 2014. https://doi.org/10.1007/s10883-014-9212-0
F. Cardetti and D. Mittenhuber, “Local controllability for linear control systems on Lie groups”, Journal of Dynamical and Control Systems, vol. 11, no. 3, 353-373, 2005. https://doi.org/10.1007/s10883-005-6584-1
C. Chicone, Ordinary differential equations with applications. New York: Springer, 2006.
F. Colonius and W. Kliemman, Dynamical System and Linear Algebra. Providence: AMS, 2014.
Ph. Jouan, “Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces”, Journal of Dynamics and Control Systems, vol. 17, pp. 591-616, 2011.
C. Kawan, O. G. Rocio and A.J. Santana, “On topological conjugacy of left invariant flows on semisimple and affine Lie groups”, Proyecciones (Antofagasta), vol. 30, no. 2, pp. 175-188, 2011. https://doi.org/10.4067/s0716-09172011000200004
C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos. London: CRC Press, 1999.
A.L Onishchik and E.B. Vinberg, Lie Groups and Lie Algebras III- Structure of Lie Groups and Lie Algebras. Berlin: Springer, 1990.
L.B. San Martin, Grupos de Lie. Editora da Unicamp, Campinas, 2017.
S. N. Stelmastchuk, “Linear flows on compact, semisimple Lie groups: stability and periodic orbits”, Electronic Journal of Qualitative Theory of Differential Equations, no. 84, pp. 1-12, 2021. https://doi.org/10.14232/ejqtde.2021.1.84
How to Cite
Copyright (c) 2023 Simão Nicolau Stelmastchuk
This work is licensed under a Creative Commons Attribution 4.0 International License.
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.