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June  2017, 37(6): 3411-3421. doi: 10.3934/dcds.2017144

Topological conjugacy of linear systems on Lie groups

Adriano Da Silva 1, , Alexandre J. Santana 2, and  Simão N. Stelmastchuk 3,

1. 

Departamento de Matemática, Universidade de Campinas, Campinas, Brazil
2. 

Departamento de Matemática, Universidade Estadual de Maringá, Maringá, Brazil
3. 

Universidade Federal do Paraná, Jandáia do Sul, Brazil

Received  November 2016 Revised  January 2017 Published  February 2017

Fund Project: The first author was supported by CAPES grant n° 4792/2016-PRO and partially supported by FAPESP grant n° 2013/19756-8. The second author was parcially supported by Fundação Araucária grant n° 20134003. This work was partially supported by CNPq/Universal grant n° 476024/2012-9.

In this paper we study a classification of linear systems on Lie groups with respect to the conjugacy of the corresponding flows. We also describe stability according to Lyapunov exponents.

Keywords: Topological conjugacy, linear vector fields, flows, Lie groups, Lyapunov exponents.
Mathematics Subject Classification: 22E20, 54H20, 34D20, 37B99.
Citation: Adriano Da Silva, Alexandre J. Santana, Simão N. Stelmastchuk. Topological conjugacy of linear systems on Lie groups. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3411-3421. doi: 10.3934/dcds.2017144
References:
[1]

V. Ayala, F. Colonius and W. Kliemann, On topological equivalence of linear flows with applications to bilinear control systems, J. Dyn. Control Syst., 13 (2007), 337-362.  doi: 10.1007/s10883-007-9021-9.

[2]

F. Colonius and W. Kliemann, Dynamical Systems and Linear Algebra, American Mathematical Society, 2014.

[3]

F. Colonius and A.J. Santana, Topological conjugacy for affine-linear flows and control systems, Commun. Pure Appl. Anal., 10 (2011), 847-857.  doi: 10.3934/cpaa.2011.10.847.

[4]

A. Da Silva, Controllability of linear systems on solvable Lie groups, SIAM J. Control Optim., 54 (2016), 372-390.  doi: 10.1137/140998342.

[5]

C. Kawan, O.G. Rocio and A.J. Santana, On topological conjugacy of left invariant flows on semisimple and affine Lie groups, Proyecciones, 30 (2011), 175-188. 

[6]

N. H. Kuiper and J. W. Robbin, Topological classification of linear endomorphisms, Invent. Math., 19 (1973), 83-106.  doi: 10.1007/BF01418922.

[7]

H. Poincaré, Sur Les Courbes Definies Par Les Equations Differentielles, In Oeuvres de H. Poincaré I, Gauthier-Villars, Paris, 1928.

[8]

J. W. Robbin, Topological conjugacy and structural stability for discrete dynamical systems, Bull. Amer. Math. Soc., 78 (1972), 923-952.  doi: 10.1090/S0002-9904-1972-13058-1.

[9]

C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, 2nd Edition, CRC Press, London, 1999.

show all references

References:
[1]

V. Ayala, F. Colonius and W. Kliemann, On topological equivalence of linear flows with applications to bilinear control systems, J. Dyn. Control Syst., 13 (2007), 337-362.  doi: 10.1007/s10883-007-9021-9.

[2]

F. Colonius and W. Kliemann, Dynamical Systems and Linear Algebra, American Mathematical Society, 2014.

[3]

F. Colonius and A.J. Santana, Topological conjugacy for affine-linear flows and control systems, Commun. Pure Appl. Anal., 10 (2011), 847-857.  doi: 10.3934/cpaa.2011.10.847.

[4]

A. Da Silva, Controllability of linear systems on solvable Lie groups, SIAM J. Control Optim., 54 (2016), 372-390.  doi: 10.1137/140998342.

[5]

C. Kawan, O.G. Rocio and A.J. Santana, On topological conjugacy of left invariant flows on semisimple and affine Lie groups, Proyecciones, 30 (2011), 175-188. 

[6]

N. H. Kuiper and J. W. Robbin, Topological classification of linear endomorphisms, Invent. Math., 19 (1973), 83-106.  doi: 10.1007/BF01418922.

[7]

H. Poincaré, Sur Les Courbes Definies Par Les Equations Differentielles, In Oeuvres de H. Poincaré I, Gauthier-Villars, Paris, 1928.

[8]

J. W. Robbin, Topological conjugacy and structural stability for discrete dynamical systems, Bull. Amer. Math. Soc., 78 (1972), 923-952.  doi: 10.1090/S0002-9904-1972-13058-1.

[9]

C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, 2nd Edition, CRC Press, London, 1999.

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