Skip to main content
SpringerLink
Log in

Equivariant Nonautonomous Normal Forms

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

For a nonautonomous dynamics with discrete time, we show that if the dynamics is equivariant (respectively, reversible), then any normal form as well as the coordinate change bringing the dynamics to this normal form have equivariance (respectively, reversibility) properties. The resonances of the linear part of the dynamics are expressed in terms of the nonuniform spectrum, that in its turn is defined in terms of the notion of a tempered exponential dichotomy.

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

Similar content being viewed by others

References

  1. Barreira, L., Pesin, Ya.: Nonuniform Hyperbolicity, Encyclopedia of Mathematics and its Applications 115. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  2. Barreira, L., Valls, C.: Reversibility and equivariance in center manifolds of nonautonomous dynamics. Discrete Contin. Dyn. Syst. 18, 677–699 (2007)

    Article  MathSciNet  Google Scholar 

  3. Barreira, L., Valls, C.: Spectrum of a nonautonomous dynamics for growth rates. Publ. Math. Debrecen 91, 43–62 (2017)

    Article  MathSciNet  Google Scholar 

  4. Barreira, L., Valls, C.: Normal forms via nonuniform hyperbolicity. J. Differ. Equ. 266, 2175–2213 (2019)

    Article  MathSciNet  Google Scholar 

  5. Lamb, J., Roberts, J.: Time-reversal symmetry in dynamical systems: a survey, in Time-Reversal Symmetry in Dynamical Systems (Coventry, 1996). Phys. D 112, 1–39 (1998)

    Article  MathSciNet  Google Scholar 

  6. Mielke, A.: Hamiltonian and Lagrangian Flows on Center Manifolds. Lecture Notes in Mathematics, vol. 1489. Springer, Berlin (1991)

    Book  Google Scholar 

  7. Roberts, J., Quispel, R.: Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems. Phys. Rep. 216, 63–177 (1992)

  8. Sacker, R., Sell, G.: A spectral theory for linear differential systems. J. Differ. Equ. 27, 320–358 (1978)

    Article  MathSciNet  Google Scholar 

  9. Sevryuk, M.: Reversible Systems, Lecture Notes in Mathematics 1211, Spinger, (1986)

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001, Lisboa, Portugal

    Luis Barreira & Claudia Valls

Authors
  1. Luis Barreira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Claudia Valls
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luis Barreira.

Additional information

Publisher's Note

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

Supported by FCT/Portugal through the project UID/MAT/04459/2019. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barreira, L., Valls, C. Equivariant Nonautonomous Normal Forms. Qual. Theory Dyn. Syst. 20, 71 (2021). https://doi.org/10.1007/s12346-021-00513-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-021-00513-6

Keywords

Mathematics Subject Classification

Search

Navigation