ABSTRACT
We conceive a new recurrence quantifier for time series based on the concept of information entropy, in which the probabilities are associated with the presence of microstates defined on the recurrence matrix as small binary submatrices. The new methodology to compute the entropy of a time series has advantages compared to the traditional entropies defined in the literature, namely, a good correlation with the maximum Lyapunov exponent of the system and a weak dependence on the vicinity threshold parameter. Furthermore, the new method works adequately even for small segments of data, bringing consistent results for short and long time series. In a case where long time series are available, the new methodology can be employed to obtain high precision results since it does not demand large computational times related to the analysis of the entire time series or recurrence matrices, as is the case of other traditional entropy quantifiers. The method is applied to discrete and continuous systems.
Acknowledgments
The authors acknowledge the support of Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, Brazil (Grant No. 302785/2017-5), Coordenação de Aperfeiçoamento de pessoal de Nível Superior, CAPES, through Project Nos. 88881.119252/2016-01 and BEX: 11264/13-6, and Financiadora de Estudos e Projetos (FINEP).
- 1. E. Bradley and H. Kantz, Chaos 25, 097610 (2015). https://doi.org/10.1063/1.4917289, Google ScholarScitation
- 2. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, 2004), Vol. 7. Google Scholar
- 3. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, 2014). Google Scholar
- 4. J. M. Cushing, R. F. Costantino, B. Dennis, R. Desharnais, and S. M. Henson, Chaos in Ecology: Experimental Nonlinear Dynamics (Elsevier, 2002), Vol. 1. Google Scholar
- 5. D. A. Hsieh, J. Fin. 46, 1839 (1991). https://doi.org/10.1111/j.1540-6261.1991.tb04646.x, Google ScholarCrossref
- 6. H. Herzel, W. Ebeling, and A. O. Schmitt, Phys. Rev. E 50, 5061 (1994). https://doi.org/10.1103/PhysRevE.50.5061, Google ScholarCrossref
- 7. E. M. Izhikevich, Dynamical Systems in Neuroscience (MIT Press, 2007). Google Scholar
- 8. T. L. Prado, S. R. Lopes, C. A. S. Batista, J. Kurths, and R. L. Viana, Phys. Rev. E 90, 032818 (2014). https://doi.org/10.1103/PhysRevE.90.032818, Google ScholarCrossref
- 9. H. Poincaré, Acta Math. 13, A3 (1890). https://doi.org/10.1007/BF02392506, Google ScholarCrossref
- 10. J.-P. Eckmann, S. O. Kamphorst, and D. Ruelle, Europhys. Lett. 4, 973 (1987). https://doi.org/10.1209/0295-5075/4/9/004, Google ScholarCrossref
- 11. S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, 1999). Google Scholar
- 12. N. Marwan, M. C. Romano, M. Thiel, and J. Kurths, Phys. Rep. 438, 237 (2007). https://doi.org/10.1016/j.physrep.2006.11.001, Google ScholarCrossref
- 13. J. P. Zbilut and C. L. Webber, Phys. Lett. A 171, 199 (1992). https://doi.org/10.1016/0375-9601(92)90426-M, Google ScholarCrossref
- 14. C. L. Webber and J. P. Zbilut, J. Appl. Physiol. 76, 965 (1994). https://doi.org/10.1152/jappl.1994.76.2.965, Google ScholarCrossref
- 15. C. Letellier, Phys. Rev. Lett. 96, 254102 (2006). https://doi.org/10.1103/PhysRevLett.96.254102, Google ScholarCrossref
- 16. D. Eroglu, T. K. D. Peron, N. Marwan, F. A. Rodrigues, L. d. F. Costa, M. Sebek, I. Z. Kiss, and J. Kurths, Phys. Rev. E 90, 042919 (2014). https://doi.org/10.1103/PhysRevE.90.042919, Google ScholarCrossref
- 17. C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002). https://doi.org/10.1103/PhysRevLett.88.174102, Google ScholarCrossref
- 18. N. Marwan, J. F. Donges, Y. Zou, R. V. Donner, and J. Kurths, Phys. Lett. A 373, 4246 (2009). https://doi.org/10.1016/j.physleta.2009.09.042, Google ScholarCrossref
- 19. D. B. Vasconcelos, S. R. Lopes, R. L. Viana, and J. Kurths, Phys. Rev. E 73, 056207 (2006). https://doi.org/10.1103/PhysRevE.73.056207, Google ScholarCrossref
- 20. T. L. Prado, P. P. Galúzio, S. Lopes, and R. L. Viana, Chaos 24, 013106 (2014). https://doi.org/10.1063/1.4861895, Google ScholarScitation
- 21. K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos (Springer, 1996). Google ScholarCrossref
- 22. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, 2nd ed. (Springer-Verlag, 1992), Vol. 1. Google ScholarCrossref
- 23. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Physica D 16, 285 (1985). https://doi.org/10.1016/0167-2789(85)90011-9, Google ScholarCrossref
- 24. L. Lacasa, B. Luque, F. Ballesteros, J. Luque, and J. C. Nuno, Proc. Natl. Acad. Sci. U.S.A. 105, 4972 (2008). https://doi.org/10.1073/pnas.0709247105, Google ScholarCrossref
Please Note: The number of views represents the full text views from December 2016 to date. Article views prior to December 2016 are not included.
