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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Rosenblum, M., Pikovsky, A., and Kurths, J., ‘Phase synchronization of chaotic oscillators’, Physical Review Letters 76, 1996, 1804–1807.
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.
Pikovsky, A., Rosenblum, M., and Kurths, J.Synchronization, Cambridge Nonlinear Science Series 12, 2001.
Boccaletti, S., Kurths, J., Osipov, G. V., Valladares, D., and Zhou, C., ‘The synchronization of chaotic systems’, Physics Reports 366, 2002, 1–101.
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.
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.
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.
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.
Blasius, B., Huppert, A., and Stone, L., ‘Complex dynamics and phase synchronization in spatially extended ecological systems’, Nature 399, 1999, 354–359.
Schäfer, C., Rosenblum, M. G., Kurths, J., and Abel, H.-H., ‘Heartbeat synchronized with ventilation’, Nature 392, 1998, 239–240.
DeShazer, D. J., Breban, R., Ott, E., and Roy, R., ‘Detecting phase synchronization in a chaotic laser array’, Physical Review Letters 87, 2001, 044101.
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, 194101
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.
Fisher, G. Plane algebraic curves, American Mathematical Soceity, Providence, RI, 2001.
Sparrow, C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, Berlin, 1982.
Madan, R. N. Chua circuit: A paradigm for chaos, World Scientific, Singapore, 1993.
Lauterborn, W., Kurz, T., and Wiesenfeldt, M. Coherent Optics. Fundamentals and Applications. Springer-Verlag, Berlin, Heidelberg, New York, 1993.
Kiss, I. Z., Lv, Q., and Hudson, J. L., ‘Synchronization of non-phase-coherent chaotic electrochemical oscillations’, Physical Review E 71, 2005, 035201(R).
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.
Poincaré, H., ‘Sur le problme des trios corps et les equations de la dynamique’, Acta Mathmatica 13, 1890, 1–27.
Eckmann, J. P., Kamphorst, S. O., and Ruelle, D., ‘Recurrence plots of dynamical systems’, Europhysics Letters 4, 1987, 973–977.
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.
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.
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.
Park, E.-H., Zaks, M., and Kurths, J., ‘Phase synchronization in the forced Lorenz system’, Physical Review E 60, 1999, 6627–6638.
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.
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.
Romano, M. C., Thiel, M., Kurths, J., and von Bloh, W., ‘Multivariate recurrence plots’, Physics Letters A 330, 2004, 214–223.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kurths, J., Romano, M.C., Thiel, M. et al. Synchronization Analysis of Coupled Noncoherent Oscillators. Nonlinear Dyn 44, 135–149 (2006). https://doi.org/10.1007/s11071-006-1957-x
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11071-006-1957-x