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Journal of Dynamics and Differential Equations

, Volume 31, Issue 3, pp 1279–1299 | Cite as

A Sternberg Theorem for Nonautonomous Differential Equations

  • L. V. Cuong
  • T. S. Doan
  • S. SiegmundEmail author
Article
  • 169 Downloads

Abstract

We show that a hyperbolic nonautonomous differential equation can be smoothly linearized if the associated Sacker–Sell spectrum satisfies a non-resonance condition. This result extends the classical Sternberg theorem to nonautonomous differential equations.

Notes

Acknowledgements

The work of the first and the second author is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03–2017.01.

References

  1. 1.
    Aulbach, B., Wanner, T.: Integral manifolds for Carathéodory type differential equations in Banach spaces. In: Aulbach, B., Colonius, F. (eds.) Six Lectures on Dynamical Systems, pp. 45–119. World Scientific, Singapore (1996)CrossRefGoogle Scholar
  2. 2.
    Aulbach, B., Wanner, T.: Topological simplification of nonautonomous difference equations. J. Differ. Equ. Appl. 12, 283–296 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonckaert, P., Dumortier, F.: On a linearization theorem of Sternberg for germs of diffeomorphisms. Math. Z. 185(1), 115–135 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonckaert, P.: On the continuous dependence of the smooth change of coordinates in parametrized normal form theorems. J. Differ. Equ. 106, 107–120 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonckaert, P., De Maesschalck, P., Doan, T.S., Siegmund, S.: Partial linearization for planar nonautonomous differential equations. J. Differ. Equ. 258, 1618–1652 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bruhat, F.: Travaux de Sternberg. (French) [Works of Sternberg]. Séminaire Bourbaki 6(217), 179–196 (1995)MathSciNetGoogle Scholar
  7. 7.
    Coppel, W.A.: Dichotomies in Stability Theory. Springer Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1978)CrossRefzbMATHGoogle Scholar
  8. 8.
    Dumortier, F., Rodrigues, P.R., Roussarie, R.: Germs of Diffeomorphisms in the Plane. Springer Lecture Notes in Mathematics, vol. 902. Springer, Berlin (1981)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hartman, P.: Ordinary Differential Equations. Birkhäuser, Boston (1982)zbMATHGoogle Scholar
  10. 10.
    Neirynck, K.: Local equivalence and conjugacy of families of vector fields and diffeomorphisms. Dissertation, UHasselt Diepenbeek (2005)Google Scholar
  11. 11.
    Palmer, K.: A generalization of Hartman’s linearization theorem. J. Math. Anal. Appl. 41, 753–758 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sacker, R.J., Sell, G.R.: A spectral theory for linear differential systems. J. Differ. Equ. 27(3), 320–358 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Siegmund, S.: Spektraltheorie, glatte Faserungen und Normalformen für Differentialgleichungen vom Carathéodory-Typ. Dissertation, University of Augsburg (1999)Google Scholar
  14. 14.
    Siegmund, S.: Normal forms for nonautonomous differential equations. J. Differ. Equ. 178, 541–573 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Siegmund, S.: Dichotomy spectrum for nonautonomous differential equations. J. Dyn. Differ. Equ. 14(1), 243–258 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Siegmund, S.: Reducibility of nonautonomous linear differential equations. J. Lond. Math. Soc. 65(2), 397–410 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Siegmund, S.: Normal forms for nonautonomous difference equations. Comput. Math. Appl. 45, 1059–1073 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sternberg, S.: Local contractions and a theorem of Poincaré. Am. J. Math. 79, 809–824 (1957)CrossRefzbMATHGoogle Scholar
  19. 19.
    Sternberg, S.: On the structure of local homeomorphisms of Euclidian \(n\)-space, I. Am. J. Math. 80, 623–631 (1958)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Information TechnologyNational University of Civil EngineeringHanoiVietnam
  2. 2.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  3. 3.Center for Dynamics, Department of MathematicsTechnische Universität DresdenDresdenGermany

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