Skip to main content
SpringerLink
Log in

Efficient detection of the quasi-periodic route to chaos in discrete maps by the three-state test

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The three-state test (3ST) is a method based on ordinal pattern analysis for detecting chaos and determining the period in time series. For some well-known chaotic dynamical systems, we showed that the test behaves similar to Lyapunov exponents. However, the 3ST is detecting quasi-periodic motions both as regular and non-regular. In this paper, we propose to use the sensitivity of its chaos indicator \(\lambda \) on time delay for clear discernment between quasi-periodic and chaotic dynamics. Simulation results obtained using the logistic map and the sine-circle map attest that the sensitivity of \(\lambda \) on time delay is sufficient for the detection of the periodic and quasi-periodic route to chaos. A comparison with the permutation entropy confirms the effectiveness of the 3ST for the analysis of discrete time series data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Huikuri, H.V., Perkiomaki, J.S., Maestri, R., Pinna, G.D.: The analysis of observed chaotic data in physical systems. Philos. Trans. A 367, 1223–1238 (2009)

    Article  Google Scholar 

  2. Pincus, S.M., Viscarello, R.R.: The analysis of observed chaotic data in physical systems. Obstet. Gynecol. 79, 249 (1992)

    Google Scholar 

  3. Voss, A., Schulz, S., Schroeder, R., Baumert, M., Caminal, P.: The analysis of observed chaotic data in physical systems. Philos. Trans. R. Soc. A 367, 277 (2009)

    Article  MATH  Google Scholar 

  4. Ching, E.S.C., Tsang, Y.K.: The analysis of observed chaotic data in physical systems. Phys. Rev. E 76, 041910 (2007)

    Article  Google Scholar 

  5. Penaand, M.A., Echeverria, J.C., Garcia, M.T., González- Camarena, R.: The analysis of observed chaotic data in physical systems. Med. Biol. Eng. Comput. 47, 709 (2009)

    Article  Google Scholar 

  6. Lin, D.C., Sharif, A.: The analysis of observed chaotic data in physical systems. Chaos 20, 023121 (2010)

    Article  Google Scholar 

  7. Acharya, R.U., Lim, C.M., Joseph, P.: The analysis of observed chaotic data in physical systems. ITBM-RBM 23, 333 (2002)

    Article  Google Scholar 

  8. Bogaert, C., Beckers, F., Ramaekers, D., Aubert, A.E.: The analysis of observed chaotic data in physical systems. Auton. Neurosci. 90, 142 (2001)

    Article  Google Scholar 

  9. Carvajal, R., Wessel, N., Vallverdú, M., Caminal, P., Voss, A.: The analysis of observed chaotic data in physical systems. Comput. Methods Program Biomed. 78, 133 (2005)

    Article  Google Scholar 

  10. Casaleggio, A., Cerutti, S., Signorini, M.G.: The analysis of observed chaotic data in physical systems. Meth. Inform. Med. 36, 274 (1997)

    Google Scholar 

  11. Guzzetti, S., Signorini, M.G., Cogliati, C., Mezzetti, S.A., Porta, S.C., Malliani, A.: The analysis of observed chaotic data in physical systems. Cardiovasc. Res. 31, 441 (1996)

    Article  Google Scholar 

  12. Hagerman, I., Berglund, M., Lorin, M., Nowak, J., Sylvén, C.: The analysis of observed chaotic data in physical systems. Cardiovasc. Res. 31, 410 (1996)

    Article  Google Scholar 

  13. Hu, J., Gao, J.B., Tung, W.W.: The analysis of observed chaotic data in physical systems. Chaos 19, 028506 (2009)

    Article  MathSciNet  Google Scholar 

  14. Brudno, A.A.: The analysis of observed chaotic data in physical systems. Trans. Mosc. Math. Soc. 2, 127 (1983)

    Google Scholar 

  15. Galatolo, S.: The analysis of observed chaotic data in physical systems. Discrete Contin. Dyn. Syst. 7, 477 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pincus, S.M., Goldberger, A.L.: The analysis of observed chaotic data in physical systems. Am. J. Physiol. Heart Circ. Physiol. 266, 1643 (1994)

    Google Scholar 

  17. Pincus, S.M.: The analysis of observed chaotic data in physical systems. Proc. Natl. Acad. Sci. 88, 2297 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hao, B.: The analysis of observed chaotic data in physical systems. Phys. D 51, 161 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88, 174102 (2002)

    Article  Google Scholar 

  20. Bian, C., Qin, C., Ma, Q.D.Y., Shen, Q.: Modified permutation-entropy analysis of heartbeat dynamics. Phys. Rev. E 85, 021,906 (2012)

    Article  Google Scholar 

  21. Amigó, J.M.: Permutation Complexity in Dynamical Systems. Springer, New York (2010)

  22. Fadlallah, B., Príncipe, J., Chen, B., Keil, A.: Weighted-permutation entropy: an improved complexity measure for time series. Phys. Rev. E 87, 022911 (2013)

    Article  Google Scholar 

  23. Bruzzo, A., Gesierich, B., Santi, M., Tassinari, C., Bir-baumer, N., Rubboli, G.: Permutation entropy to detect vigilance changes and preictal states from scalp eeg in epileptic patients: a preliminary study. Neurol. Sci. 29, 39 (2008)

    Article  Google Scholar 

  24. Li, X., Cui, S., Voss, L.: Using permutation entropy to measure the electroencephalographic effects of sevoflurane. Anesthesiology 109, 448456 (2008)

    Article  Google Scholar 

  25. Li, X., Ouyang, G., Richards, D.: Predictability analysis of absence seizures with permutation entropy. Epilepsy Res. 77, 7074 (2007)

    Article  Google Scholar 

  26. Cao, Y., Tung, W., Gao, J., Protopopescu, V., Hively, L.: Detecting dynamical changes in time series using the permutation entropy. Phys. Rev. E 70, 046217 (2004)

    Article  MathSciNet  Google Scholar 

  27. Li, Z., Ouyang, G., Li, D., Li, X.: Characterization of the causality between spike trains with permutation conditional mutual information. Phys. Rev. E 84, 021929 (2011)

    Article  Google Scholar 

  28. Fouda, J.S.A.E., Effa, J.Y., Kom, M., Ali, M.: The three-state test for chaos detection in discrete maps. Appl. Soft Comput. 13, 47314737 (2013)

    Google Scholar 

  29. Collet, P., Eckmann, J.-P.: Iterated Maps on the Interval as Dynamical Systems. Birkhäeuser, Boston (1980)

    MATH  Google Scholar 

  30. Afsar, O., Bagei, G.B., Tirnakli, U.: Renormalized entropy for one dimensional discrete map: Periodic and quasi-periodic route to chaos and their robustness. Eur. Phys. J. B 86, 307–320 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported by a DAAD scholarship.

Author information

Authors and Affiliations

  1. Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon

    J. S. Armand Eyebe Fouda

  2. Institute of Mathematics, University of Kassel, Heinrich-Plett Str. 40, 34132 , Kassel, Germany

    Wolfram Koepf

Authors
  1. J. S. Armand Eyebe Fouda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfram Koepf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. S. Armand Eyebe Fouda.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eyebe Fouda, J.S.A., Koepf, W. Efficient detection of the quasi-periodic route to chaos in discrete maps by the three-state test. Nonlinear Dyn 78, 1477–1487 (2014). https://doi.org/10.1007/s11071-014-1529-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1529-4

Keywords

Search

Navigation