Skip to main content
SpringerLink
Log in

Chaos, attraction, and control for semigroup actions

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

This manuscript studies sensitivity and chaos for semigroup actions on completely regular spaces. The main results explain how the notions of attraction and control play a fundamental role in the investigation of Auslander–Yorke and Li–Yorke chaos. A general type of non-chaotic semigroup action is exhibited and a criterium for chaos in control systems is presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Alves, R.W.M., Rocha, V.H.L., Souza, J.A.: A characterization of completely regular spaces. Int. J. Math. 26, 1550032 (2015)

    Article  MathSciNet  Google Scholar 

  2. Alves, R.W.M., Souza, J.A.: Cantor-Kuratowski theorem in uniformizable spaces. Topol. Appl. 252, 158–168 (2019)

    Article  Google Scholar 

  3. Braga Barros, C.J., Reis, R.A.: On the number of control sets on compact homogeneous spaces. Port. Math. 60, 359–371 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Braga Barros, C.J., Rocha, V.H.L., Souza, J.A.: Lyapunov stability for semigroup actions. Semigroup Forum 88, 227–249 (2014)

    Article  MathSciNet  Google Scholar 

  5. Braga Barros, C.J., Rocha, V.H.L., Souza, J.A.: Lyapunov stability and attraction under equivariant maps. Can. J. Math. 67, 1247–1269 (2015)

    Article  MathSciNet  Google Scholar 

  6. Braga Barros, C.J.,  San Martin, L.A.B.: On the number of control sets on projective spaces. Syst. Control Lett. 29, 21–26 (1996)

    Article  MathSciNet  Google Scholar 

  7. Ceccherini-Silberstein, T., Coornaert, M.: Sensitivity and Devaney’s chaos in uniform spaces. J. Dyn. Control Syst. 19, 568–579 (2013)

    Article  MathSciNet  Google Scholar 

  8. Colonius, F., Kliemann, W.: Some aspects of control systems as dynamical systems. J. Dyn. Differ. Equ. 5, 469–494 (1993)

    Article  MathSciNet  Google Scholar 

  9. Colonius, F., Kliemann, W.: The Dynamics of Control. Birkhäuser, Boston (2000)

    Book  Google Scholar 

  10. Devaney, R.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Redwood (1989)

    MATH  Google Scholar 

  11. Huang, W., Ye, X.: Devaney’s chaos or 2-scattering implies Li–Yorke chaos. Topol. Appl. 117, 259–272 (2002)

    Article  MathSciNet  Google Scholar 

  12. Kliemann, W.: Recurrence and invariant measures for degenerate diffusions. Ann. Probab. 15, 690–707 (1987)

    Article  MathSciNet  Google Scholar 

  13. Kontorovich, E., Megrelishvili, M.: A note on sensitivity of semigroup actions. Semigroup Forum 76, 133–141 (2008)

    Article  MathSciNet  Google Scholar 

  14. Li, T., Yorke, J.: Period 3 implies chaos. Am. Math. Mon. 82, 985–992 (1975)

    Article  MathSciNet  Google Scholar 

  15. Polo, F.: Sensitive dependence on initial conditions and chaotic group actions. Proc. Am. Math. Soc. 138, 2815–2826 (2010)

    Article  MathSciNet  Google Scholar 

  16. Rybak, O.V.: Li–Yorke sensitivity for semigroup actions. Ukr. Math. J. 65, 752–759 (2013)

    Article  MathSciNet  Google Scholar 

  17. San Martin, L.A.B.: Invariant control sets on flag manifolds. Math. Control Signals Syst. 6, 41–61 (1993)

    Article  MathSciNet  Google Scholar 

  18. San Martin, L.A.B., Tonelli, P.A.: Semigroup actions on homogeneous spaces. Semigroup Forum 50, 59–88 (1995)

    Article  MathSciNet  Google Scholar 

  19. Schneider, F.M., Kerkhoff, S., Behrisch, M., Siegmund, S.: Chaotic actions of topological semigroups. Semigroup Forum 87, 590–598 (2013)

    Article  MathSciNet  Google Scholar 

  20. Souza, J.A.: Lebesgue covering lemma on nonmetric spaces. Int. J. Math. 24, 1350018 (2013)

    Article  MathSciNet  Google Scholar 

  21. Souza, J.A.: Recurrence theorem for semigroup actions. Semigroup Forum 83, 351–370 (2011)

    Article  MathSciNet  Google Scholar 

  22. Souza, J.A., Tozatti, H.V.M., Rocha, V.H.L.: On stability and controllability for semigroup actions. Topol. Methods Nonlinear Anal. 48, 1–29 (2016)

    Article  MathSciNet  Google Scholar 

  23. Wang, H., Long, X., Fu, H.: Sensitivity and chaos of semigroup actions. Semigroup Forum 84, 81–89 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Estadual de Maringá, Maringá, PR, Brazil

    Josiney A. Souza

  2. Departamento de Matemática, Universidade Tecnológica Federal do Paraná, Campo Mourão, PR, Brazil

    Hélio V. M. Tozatti

Authors
  1. Josiney A. Souza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hélio V. M. Tozatti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Josiney A. Souza.

Additional information

Communicated by Jimmie Lawson.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research supported by Fundação Araucária 476/14 and CNPq 476024/2012-9 Universal 14/2012.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Souza, J.A., Tozatti, H.V.M. Chaos, attraction, and control for semigroup actions. Semigroup Forum 101, 202–225 (2020). https://doi.org/10.1007/s00233-020-10093-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-020-10093-7

Keywords

Search

Navigation