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Nonlinear Dynamics

, Volume 44, Issue 1–4, pp 135–149 | Cite as

Synchronization Analysis of Coupled Noncoherent Oscillators

  • Jürgen KurthsEmail author
  • M. Carmen Romano
  • Marco Thiel
  • Grigory V. Osipov
  • Mikhail V. Ivanchenko
  • István Z. Kiss
  • John L. Hudson
Article

Abstract

We present two different approaches to detect and quantify phase synchronization in the case of coupled non-phase coherent oscillators. The first one is based on the general idea of curvature of an arbitrary curve. The second one is based on recurrences of the trajectory in phase space. We illustrate both methods in the paradigmatic example of the Rössler system in the funnel regime. We show that the second method is applicable even in the case of noisy data. Furthermore, we extend the second approach to the application of chains of coupled systems, which allows us to detect easily clusters of synchronized oscillators. In order to illustrate the applicability of this approach, we show the results of the algorithm applied to experimental data from a population of 64 electrochemical oscillators.

Key Words

chaotic oscillators data analysis synchronization 

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References

  1. 1.
    Rosenblum, M., Pikovsky, A., and Kurths, J., ‘Phase synchronization of chaotic oscillators’, Physical Review Letters 76, 1996, 1804–1807.CrossRefGoogle Scholar
  2. 2.
    Pikovsky, A., Rosenblum, M., Osipov, G., and Kurths, J., ‘Nonlinear Phenomena, Phase synchronization of chaotic oscillators by external driving’, Physica D 104, 1997, 219–238.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Pikovsky, A., Rosenblum, M., and Kurths, J.Synchronization, Cambridge Nonlinear Science Series 12, 2001.Google Scholar
  4. 4.
    Boccaletti, S., Kurths, J., Osipov, G. V., Valladares, D., and Zhou, C., ‘The synchronization of chaotic systems’, Physics Reports 366, 2002, 1–101.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Elson, R. C., Selverston, A. I., Huerta, R., Rulkov, N. F., Rabinovich, M. I., and Abarbanel, H. D. I., ‘Synchronous behavior of two coupled biological neurons’, Physical Review Letters 81, 1998, 5692–5695.CrossRefGoogle Scholar
  6. 6.
    Tass, P., Rosenblum, M. G., Weule, J., Kurths, J., Pikovsky, A., Volkmann, J., Schnitzler, A., and Freund, H.-J., ‘Detection of n:m phase locking from noisy data: Application to magnetoencephalography’, Physical Review Letters 81, 1998, 3291–3294.CrossRefGoogle Scholar
  7. 7.
    Ticos, C. M., Rosa, E., Jr., Pardo, W. B., Walkenstein, J. A., and Monti, M., ‘Experimental real-time phase synchronization of a paced chaotic plasma discharge’, Physical Review Letters 85, 2000, 2929–2932.CrossRefGoogle Scholar
  8. 8.
    Makarenko, V. and Llinas, R., ‘Experimentally determined chaotic phase synchronization in a neuronal system’, in Proceedings of the National Academy of Sciences of the United States of America, 95, 1998, 15747–15752.CrossRefGoogle Scholar
  9. 9.
    Blasius, B., Huppert, A., and Stone, L., ‘Complex dynamics and phase synchronization in spatially extended ecological systems’, Nature 399, 1999, 354–359.CrossRefGoogle Scholar
  10. 10.
    Schäfer, C., Rosenblum, M. G., Kurths, J., and Abel, H.-H., ‘Heartbeat synchronized with ventilation’, Nature 392, 1998, 239–240.CrossRefGoogle Scholar
  11. 11.
    DeShazer, D. J., Breban, R., Ott, E., and Roy, R., ‘Detecting phase synchronization in a chaotic laser array’, Physical Review Letters 87, 2001, 044101.CrossRefGoogle Scholar
  12. 12.
    Boccaletti, S., Allaria, E., Meucci, R., and Arecchi, F.T., ‘Experimental characterization of the transition to phase synchronization of chaotic CO2 laser systems’, Physical Review Letters 89, 2002, 194101CrossRefGoogle Scholar
  13. 13.
    Kiss, I. Z. and Hudson, J. L., ‘Phase synchronization and suppression of chaos through intermittency in forcing of an electrochemical oscillator’, Physical Review E 64, 2001, 046215.CrossRefGoogle Scholar
  14. 14.
    Fisher, G. Plane algebraic curves, American Mathematical Soceity, Providence, RI, 2001.Google Scholar
  15. 15.
    Sparrow, C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, Berlin, 1982.Google Scholar
  16. 16.
    Madan, R. N. Chua circuit: A paradigm for chaos, World Scientific, Singapore, 1993.Google Scholar
  17. 17.
    Lauterborn, W., Kurz, T., and Wiesenfeldt, M. Coherent Optics. Fundamentals and Applications. Springer-Verlag, Berlin, Heidelberg, New York, 1993.Google Scholar
  18. 18.
    Kiss, I. Z., Lv, Q., and Hudson, J. L., ‘Synchronization of non-phase-coherent chaotic electrochemical oscillations’, Physical Review E 71, 2005, 035201(R).CrossRefGoogle Scholar
  19. 19.
    Chen, J. Y., Wong, K. W., Zheng, H. Y., and Shuai, J. W., ‘Intermittent phase synchronization of coupled spatiotemporal chaotic systems’, Physical Review E 64, 2001, 016212.CrossRefGoogle Scholar
  20. 20.
    Poincaré, H., ‘Sur le problme des trios corps et les equations de la dynamique’, Acta Mathmatica 13, 1890, 1–27.zbMATHGoogle Scholar
  21. 21.
    Eckmann, J. P., Kamphorst, S. O., and Ruelle, D., ‘Recurrence plots of dynamical systems’, Europhysics Letters 4, 1987, 973–977.Google Scholar
  22. 22.
    Thiel, M., Romano, M. C., and Kurths, J., ‘How much information is contained in a recurrence plot?’, Physics Letters A 330(5), 2004, 343–349.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Thiel, M., Romano, M. C., Read, P., and Kurths, J., ‘Estimation of dynamical invariants without embedding by recurrence plots’, Chaos 14(2), 2004, 234–243.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Marwan, N., Trauth, M. H., Vuille, M., Kurths, J., ‘Comparing modern and Pleistocene ENSO-like influences in NW Argentina using nonlinear time series analysis methods’, Climate Dynamics 21(3–4), 2003, 317–326.CrossRefGoogle Scholar
  25. 25.
    Park, E.-H., Zaks, M., and Kurths, J., ‘Phase synchronization in the forced Lorenz system’, Physical Review E 60, 1999, 6627–6638.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Osipov, G. V., Hu, B., Zhou, C., Ivanchenko, M. V., and Kurths, J., ‘Three types of transitions to phase synchronization in coupled chaotic oscillators’, Physical Review Letters 91, 2003, 024101.CrossRefGoogle Scholar
  27. 27.
    Thiel, M., Romano, M. C., Kurths, J., Meucci, R., Allaria, E., and Arecchi, F. T., ‘Nonlinear Dynamics, Influence of observational noise on the recurrence quantification analysis’, Physica D 171(3), 2002, 138–152.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Romano, M. C., Thiel, M., Kurths, J., and von Bloh, W., ‘Multivariate recurrence plots’, Physics Letters A 330, 2004, 214–223.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Osipov, G. V., Pikovsky, A., Rosenblum, M., and Kurths, J., ‘Phase synchronization effects in a lattice of nonidentical Rssler oscillators’, Physical Review E 55, 1997, 2353–2361.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kiss, I. Z., Zhai, Y., and Hudson, J. L., ‘Collective dynamics of chaotic chemical oscillators and the law of large numbers’, Physical Review Letters 88, 2002, 238301.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Jürgen Kurths
    • 1
    Email author
  • M. Carmen Romano
    • 1
  • Marco Thiel
    • 1
  • Grigory V. Osipov
    • 2
  • Mikhail V. Ivanchenko
    • 2
  • István Z. Kiss
    • 3
  • John L. Hudson
    • 3
  1. 1.Institute of PhysicsUniversität PotsdamPotsdamGermany
  2. 2.Department of RadiophysicsNizhny Novgorod UniversityNizhny NovgorodRussia
  3. 3.Department of Chemical EngineeringUniversity of VirginiaCharlottesvilleU.S.A.

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