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The European Physical Journal Special Topics

, Volume 191, Issue 1, pp 53–70 | Cite as

Beyond the odd number limitation of time-delayed feedback control of periodic orbits

  • B. Fiedler
  • V. Flunkert
  • P. Hövel
  • E. Schöll
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Abstract.

We discuss the stabilization of odd-number orbits by time-delayed feedback control. In particular, we review the stabilization of odd-number orbits born in a subcritical Hopf bifurcation or a saddle-node bifurcation of periodic orbits. These examples refute the often invoked odd-number theorem.

Keywords

Periodic Orbit Hopf Bifurcation European Physical Journal Special Topic Unstable Periodic Orbit Floquet Multiplier 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. Lett. 64, 1196 (1990)zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Handbook of Chaos Control, edited by E. Schöll, H.G. Schuster (Wiley-VCH, Weinheim, 2008), second completely revised and enlarged editionGoogle Scholar
  3. 3.
    S. Boccaletti, C. Grebogi, Y.C. Lai, H. Mancini, D. Maza, Phys. Rep. 329, 103 (2000)CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    D.J. Gauthier, Am. J. Phys. 71, 750 (2003)CrossRefADSGoogle Scholar
  5. 5.
    K. Pyragas, Phys. Lett. A 170, 421 (1992)CrossRefADSGoogle Scholar
  6. 6.
    K. Pyragas, A. Tamaševičius, Phys. Lett. A 180, 99 (1993)CrossRefADSGoogle Scholar
  7. 7.
    S. Bielawski, D. Derozier, P. Glorieux, Phys. Rev. E 49, R971 (1994)CrossRefADSGoogle Scholar
  8. 8.
    T. Pierre, G. Bonhomme, A. Atipo, Phys. Rev. Lett. 76, 2290 (1996)CrossRefADSGoogle Scholar
  9. 9.
    K. Hall, D.J. Christini, M. Tremblay, J.J. Collins, L. Glass, J. Billette, Phys. Rev. Lett. 78, 4518 (1997)CrossRefADSGoogle Scholar
  10. 10.
    D.W. Sukow, M.E. Bleich, D.J. Gauthier, J.E.S. Socolar, Chaos 7, 560 (1997)CrossRefADSGoogle Scholar
  11. 11.
    O. Lüthje, S. Wolff, G. Pfister, Phys. Rev. Lett. 86, 1745 (2001)CrossRefADSGoogle Scholar
  12. 12.
    P. Parmananda, R. Madrigal, M. Rivera, L. Nyikos, I.Z. Kiss, V. Gáspár, Phys. Rev. E 59, 5266 (1999)CrossRefADSGoogle Scholar
  13. 13.
    J.M. Krodkiewski, J.S. Faragher, J. Sound Vib. 234, 591 (2000)CrossRefADSGoogle Scholar
  14. 14.
    T. Fukuyama, H. Shirahama, Y. Kawai, Phys. Plasmas 9, 4525 (2002)CrossRefADSGoogle Scholar
  15. 15.
    C. von Loewenich, H. Benner, W. Just, Phys. Rev. Lett. 93, 174101 (2004)CrossRefADSGoogle Scholar
  16. 16.
    M.G. Rosenblum, A.S. Pikovsky, Phys. Rev. Lett. 92, 114102 (2004)CrossRefADSGoogle Scholar
  17. 17.
    O.V. Popovych, C. Hauptmann, P.A. Tass, Phys. Rev. Lett. 94, 164102 (2005)CrossRefADSGoogle Scholar
  18. 18.
    S. Schikora, P. Hövel, H.J. Wünsche, E. Schöll, F. Henneberger, Phys. Rev. Lett. 97, 213902 (2006)CrossRefADSGoogle Scholar
  19. 19.
    W. Just, T. Bernard, M. Ostheimer, E. Reibold, H. Benner, Phys. Rev. Lett. 78, 203 (1997)CrossRefADSGoogle Scholar
  20. 20.
    H. Nakajima, Phys. Lett. A 232, 207 (1997)zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    H. Nakajima, Y. Ueda, Physica D 111, 143 (1998)zbMATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    I. Harrington, J.E.S. Socolar, Phys. Rev. E 64, 056206 (2001)CrossRefADSGoogle Scholar
  23. 23.
    K. Pyragas, V. Pyragas, H. Benner, Phys. Rev. E 70, 056222 (2004)CrossRefADSGoogle Scholar
  24. 24.
    V. Pyragas, K. Pyragas, Phys. Rev. E 73, 036215 (2006)CrossRefMathSciNetADSGoogle Scholar
  25. 25.
    J.E.S. Socolar, D.W. Sukow, D.J. Gauthier, Phys. Rev. E 50, 3245 (1994)CrossRefADSGoogle Scholar
  26. 26.
    H.G. Schuster, M.B. Stemmler, Phys. Rev. E 56, 6410 (1997)CrossRefADSGoogle Scholar
  27. 27.
    H. Nakajima, Y. Ueda, Phys. Rev. E 58, 1757 (1998)CrossRefADSGoogle Scholar
  28. 28.
    K. Pyragas, Phys. Rev. Lett. 86, 2265 (2001)CrossRefADSGoogle Scholar
  29. 29.
    B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, E. Schöll, Phys. Rev. Lett. 98, 114101 (2007)CrossRefADSGoogle Scholar
  30. 30.
    B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, E. Schöll, in Handbook of Chaos Control, edited by E. Schöll, H.G. Schuster (Wiley-VCH, Weinheim, 2008), p. 73, second completely revised and enlarged editionGoogle Scholar
  31. 31.
    W. Just, B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, E. Schöll, Phys. Rev. E 76, 026210 (2007)CrossRefMathSciNetADSGoogle Scholar
  32. 32.
    B. Fiedler, S. Yanchuk, V. Flunkert, P. Hövel, H.J. Wünsche, E. Schöll, Phys. Rev. E 77, 066207 (2008)CrossRefMathSciNetADSGoogle Scholar
  33. 33.
    O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.O. Walther, Delay Equations (Springer-Verlag, New York, 1995)Google Scholar
  34. 34.
    Y.A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, New York, 1995)Google Scholar
  35. 35.
    C.M. Postlethwaite, M. Silber, Phys. Rev. E 76, 056214 (2007)CrossRefMathSciNetADSGoogle Scholar
  36. 36.
    B. Fiedler, Time-delayed feedback control: qualitative promise and quantitative constraints, in Proc. 6th EUROMECH Nonlinear Dynamics Conference (ENOC-2008), edited by A. Fradkov, B. Andrievsky (2008) http://lib.physcon.ru/?item=1568Google Scholar
  37. 37.
    M.G. Rosenblum, A.S. Pikovsky, Phys. Rev. E 70, 041904 (2004)CrossRefMathSciNetADSGoogle Scholar
  38. 38.
    N. Baba, A. Amann, E. Schöll, W. Just, Phys. Rev. Lett. 89, 074101 (2002)CrossRefADSGoogle Scholar
  39. 39.
    J. Sieber, Nonlinearity 19, 2489 (2006)zbMATHCrossRefMathSciNetADSGoogle Scholar
  40. 40.
    M. Kehrt, P. Hövel, V. Flunkert, M.A. Dahlem, P. Rodin, E. Schöll, Eur. Phys. J. B 68, 557 (2009)zbMATHCrossRefADSGoogle Scholar
  41. 41.
    B. Fiedler, V. Flunkert, P. Hövel, E. Schöll, Phil. Trans. R. Soc. A 368, 319 (2010)zbMATHCrossRefADSGoogle Scholar
  42. 42.
    G. Brown, C.M. Postlethwaite, M. Silber, Physica D (2010), doi: 10.1016/j.physd.2010. 12.011Google Scholar

Copyright information

© EDP Sciences and Springer 2011

Authors and Affiliations

  • B. Fiedler
    • 1
  • V. Flunkert
    • 2
  • P. Hövel
    • 2
  • E. Schöll
    • 2
  1. 1.Institut für Mathematik I, FU BerlinBerlinGermany
  2. 2.Institut für Theoretische Physik, Technische Universität BerlinBerlinGermany

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