Periodic orbits of Linear flows on connected Lie groups
DOI:
https://doi.org/10.22199/issn.0717-6279-5261Keywords:
periodic orbits, linear flows, connected Lie groupsAbstract
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 ?, we show that the existence of periodic orbits of φt is based on the eigenvalues of the derivation ?. 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.
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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
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