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Spurious Structures in Recurrence Plots Induced by Embedding


In this paper we show that delay embedding produces spurious structures in a recurrence plot (RP) that are not present in the real attractor. We analyze typical sets of simulated data, such as white noise and data from the chaotic Rössler system to show the relevance of this effect.

In the second part of the paper we show that the second order Rényi entropy and the correlation dimension are dynamical invariants that can be estimated from Recurrence Plots with arbitrary embedding dimension and delay.

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  1. 1.

    Eckmann, J.-P., Kamphorst, S. O., and Ruelle, D., ‘Recurrence plots of dynamical systems’, Europhysics Letters 4, 1987, 973–977.

    Google Scholar 

  2. 2.

    Marwan, N. and Kurths, J., ‘Nonlinear analysis of bivariate data with cross recurrence plots’, Physics Letters A 302(5–6), 2002, 299–307.

    MathSciNet  Article  Google Scholar 

  3. 3.

    Kurths, J., Schwarz, U., Sonett, C. P., and Parlitz, U., ‘Testing nonlinearity in radiocarbon data’, Nonlinear Processes in Geophysics 1, 1994, 72–75.

    Google Scholar 

  4. 4.

    Thiel, M., Romano, M. C., Kurths, J., Meucci, R., Allaria, E., and Arecchi, T., ‘Influence of observational noise on the recurrence quantification analysis’, Physica D: Nonlinear Phenomena 171(3), 2002, 138–152.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Thiel, M., Romano, M. C., and Kurths, J., ‘Analytical description of recurrence plots of white noise and chaotic processes’, Izvestija VUZov—Applied Nonlinear Dynamics 11(3), 2003, 20–30.

    Google Scholar 

  6. 6.

    Webber, C. L. Jr. and Zbilut, J. P., ‘Dynamical assessment of physiological systems and states using recurrence plot strategies’, Journal of Applied Physiology 76, 1994, 965–973.

    Google Scholar 

  7. 7.

    Takens, F., ‘Detecting strange attractors in turbulence’, Dynamical Systems and Turbulence Lecture Notes in Mathematics, Vol. 898, Springer, Berlin, 1981, pp. 366–381.

  8. 8.

    Kennel, M. B., Brown, R., and Abarbanel, H. D. I., ‘Determining embedding dimension for phase-space reconstruction using a geometrical construction’, Physical Review A 45(6), 1992, 3403–3411.

    Article  Google Scholar 

  9. 9.

    Grassberger, P., Schreiber, T., and Schaffrath, C., ‘Non-linear time sequence analysis’, International Journal of Bifurcation and Chaos 1(3), 1991, 521–547.

    MathSciNet  Article  Google Scholar 

  10. 10.

    Zbilut, J. P. and Webber, C. L. Jr., ‘Embeddings and delays as derived from quantification of recurrence plots’, Physics Letters A 171, 1992, 199–203.

    Article  Google Scholar 

  11. 11.

    Ott, E., Chaos in Dynamical Systems, Cambridge University Press, 1993.

  12. 12.

    Thiel, M., Romano, M. C., and Kurths, J., ‘Estimation of dynamical invariants without embedding by recurrence plots’, Chaos 14(2), 2004, 234–243.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Rényi, A., Probability Theory, North-Holland, 1970 (appendix).

  14. 14.

    Grassberger, P., ‘Generalized dimensions of strange attractors’, Physics Letters 97A, 1983, 227–230.

    MathSciNet  Google Scholar 

  15. 15.

    Grassberger, P. and Procaccia, I., ‘Dimensions and entropies of strange attractors from a fluctuating dynamics approach’, Physica D: Nonlinear Phenomena 13, 1984, 34–54.

    MathSciNet  Article  Google Scholar 

  16. 16.

    Grassberger, P. and Procaccia, I., ‘Estimation of the Kolmogorov entropy from a chaotic signal’, Physical Review A 28, 1983, 2591–2593.

    Article  Google Scholar 

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Correspondence to Marco Thiel.

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Thiel, M., Romano, M.C. & Kurths, J. Spurious Structures in Recurrence Plots Induced by Embedding. Nonlinear Dyn 44, 299–305 (2006).

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Key Words

  • data analysis
  • recurrence plots