Periodic orbits of Linear flows on connected Lie groups

• Stelmastchuk, Simão Nicolau ^[1]
  1. [1] Universidade Federal do Paraná

     Universidade Federal do Paraná

     Brasil

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 42, Nº. 3, 2023, págs. 663-680
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-5261
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • V. Ayala and J. Tirao, “Linear control systems on Lie groups and controllability”, Proceedings of Symposia in Pure Mathematics, vol. 64, pp....
  • R. Biggs and C. Remsing, “Control systems on three-dimensional Lie groups: equivalence and controllability”, Journal of Dynamical and Control...
  • F. Cardetti and D. Mittenhuber, “Local controllability for linear control systems on Lie groups”, Journal of Dynamical and Control Systems,...
  • 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,...
  • C. Kawan, O. G. Rocio and A.J. Santana, “On topological conjugacy of left invariant flows on semisimple and affine Lie groups”, Proyecciones...
  • 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...