Advertisement

The European Physical Journal Special Topics

, Volume 191, Issue 1, pp 15–27 | Cite as

Dynamics between order and chaos revisited

  • E.J. NgamgaEmail author
  • D.V. SenthilkumarEmail author
  • J. KurthsEmail author
Regular article

Abstract.

We show that dynamics between order and chaos, namely strange nonchaotic dynamics can be efficiently studied by means of recurrence properties. Different transitions to this dynamics in coupled Rössler oscillators are revealed by some measures of complexity based on the recurrence time, which is the time needed for a system to recur to a former visited neighborhood. Furthermore, regions of the parameter space where the system is in non-phase, imperfect-phase or phase synchronization are depicted by means of recurrence based indices such as the generalized autocorrelation function and the correlation of probability of recurrence.

Keywords

European Physical Journal Special Topic Chaotic Attractor Phase Synchronization Large Lyapunov Exponent Generalize Synchronization 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. Grebogi, E. Ott, S. Pelikan, J.A. Yorke, Physica D 13, 261 (1984)zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    D. Ruelle, F. Takens, Comm. Math. Phys. 20, 167 (1971)zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    M. Ding, J.A.S. Kelso, Int. J. Bifur. Chaos Appl. Sci. Eng. 4, 553 (1994)zbMATHCrossRefGoogle Scholar
  4. 4.
    A.J. Mandell, K.A. Selz, J. Stat. Phys. 70, 355 (1993)zbMATHCrossRefADSGoogle Scholar
  5. 5.
    A. Bondeson, E. Ott, T.M. Antonsen, Jr., Phys. Rev. Lett. 55, 2103 (1985)CrossRefMathSciNetADSGoogle Scholar
  6. 6.
    C.-S. Zhou, T.-L. Chen, Eur. Phys. Lett. 38, 261 (1997)CrossRefMathSciNetADSGoogle Scholar
  7. 7.
    W.L. Ditto, M.L. Spano, H.T. Savage, S.N. Rauseo, J. Heagy, E. Ott, Phys. Rev. Lett. 65, 533 (1990)CrossRefADSGoogle Scholar
  8. 8.
    W.X. Ding, H. Deutsch, A. Dinklage, C. Wilke, Phys. Rev. E 55, 3769 (1997)CrossRefADSGoogle Scholar
  9. 9.
    T. Zhou, F. Moss, A. Bulsara, Phys. Rev. A 45, 5394 (1992)CrossRefADSGoogle Scholar
  10. 10.
    K.-P. Zeyer, A.F. Miinster, F.W. Schneider, J. Phys. Chem. 99, 13173 (1995)CrossRefGoogle Scholar
  11. 11.
    B.P. Bezruchko, S.P. Kuznetsov, Y.P. Seleznev, Phys. Rev. E 62, 7828 (2000)CrossRefADSGoogle Scholar
  12. 12.
    G. Ruiz, P. Parmananda, Phys. Lett. A 367, 478 (2007)CrossRefADSGoogle Scholar
  13. 13.
    E.J. Ngamga, A. Buscarino, M. Frasca, L. Fortuna, A. Prasad, J. Kurths, Chaos 18, 013128 (2008)CrossRefADSGoogle Scholar
  14. 14.
    S. Negi, R. Ramaswamy, Pramana J. Phys. 56, 47 (2001)CrossRefADSGoogle Scholar
  15. 15.
    X. Wang, Y-C. Lai, C.H. Lai, Phys. Rev. E 74, 016203 (2006)CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    D.V. Senthilkumar, K. Srinivasan, K. Thamilmaran, M. Lakshmanan, Phys. Rev. E 78, 066211 (2008)CrossRefADSGoogle Scholar
  17. 17.
    U. Feudel, S. Kuznetsov, A. Pikovsky, World Scientific Series on Nonlinear Science, Series A, Vol. 56 (2006)Google Scholar
  18. 18.
    U. Feudel, J. Kurths, A.S. Pikovsky, Physica D 88, 176 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    E.J. Ngamga, A. Nandi, R. Ramaswamy, M.C. Romano, M. Thiel, J. Kurths, Phys. Rev. E 75, 036222 (2007)CrossRefADSGoogle Scholar
  20. 20.
    J.-P. Eckmann, S.O. Kamphorst, D. Ruelle, Eur. Phys. Lett. 4, 973 (1987)CrossRefADSGoogle Scholar
  21. 21.
    N. Marwan, PhD Thesis, University of Potsdam, 2003Google Scholar
  22. 22.
    J.P. Zbilut, C.L. Webber Jr., Phys. Lett. A 171, 199 (1992)CrossRefADSGoogle Scholar
  23. 23.
    C.L. Webber Jr., J.P. Zbilut, J. Appl. Physiol. 76, 965 (1994)Google Scholar
  24. 24.
    N. Marwan, N. Wessel, U. Meyerfeldt, A. Schirdewan, J. Kurths, Phys. Rev. E 66, 026702 (2002)CrossRefADSGoogle Scholar
  25. 25.
    M. Agrawal, A. Prasad, R. Ramaswamy, Phys. Rev. E 81, 026202 (2010)CrossRefADSGoogle Scholar
  26. 26.
    M.C. Romano, M. Thiel, J. Kurths, I.Z. Kiss, J.L. Hudson, Europhys. Lett. 71, 466 (2005)CrossRefADSGoogle Scholar
  27. 27.
    N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Phys. Rep. 438, 237 (2007)CrossRefMathSciNetADSGoogle Scholar
  28. 28.
    A. Prasad, V. Mehra, R. Ramaswamy, Phys. Rev. Lett. 79, 4127 (1997)CrossRefADSGoogle Scholar
  29. 29.
    S.P. Kuznetsov, Phys. Rev. E 65, 066209 (2002)CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    D.V. Senthilkumar, M. Lakshmanan, J. Kurths, Phys. Rev. E 74, 035205(R) (2006)CrossRefADSGoogle Scholar
  31. 31.
    D.V. Senthilkumar, M. Lakshmanan, J. Kurths, Chaos 18, 023118 (2008)CrossRefADSGoogle Scholar
  32. 32.
    D.V. Senthilkumar, M. Lakshmanan, J. Kurths, Eur. Phys. J. Special Topics 164, 35 (2008)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences and Springer 2011

Authors and Affiliations

  1. 1.Potsdam Institute for Climate Impact ResearchPotsdamGermany
  2. 2.Centre for Dynamics of Complex Systems, University of PotsdamPotsdamGermany
  3. 3.Institute of Physics, Humboldt University BerlinBerlinGermany

Personalised recommendations