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Impact of damping amplitude on chaos detection reliability of the improved 0–1 test for oversampled and noisy observations

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Abstract

The novel chaos detection method (i.e., 0–1 test for chaos) determines the median \(K_m(c)\) of asymptotic growth rates K(c) to identify whether the measured time series is chaotic or not and has been widely used in several applications. Motivated by the fact that the validity of improved 0–1 test for chaos has been confirmed for noisy and oversampled observations from various dynamics, the effect of damping term amplitude on it is further discussed for various dynamical behavior (for instance, sine-circle and Peter de Jong dynamical systems) in this paper. For the magnitude order of the limit value over \(10^3\), the diagnostic indicator \(K_m(c)\) has more accurate values corresponding to a quasi-periodic route to chaos with chaotic behavior. These numerical results also clearly present that the value \(K_m(c)\) is sensitive to the amplitude and the speed of decline of \(K_m(c)\) changes with the amplitude can reflect the degree of chaos. The insights gained from the present study accelerate the development of advanced noise-robust methods for detecting the presence (or absence) chaos from noisy empirical measurements.

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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

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

    Article  Google Scholar 

  2. Lai, Y., Ye, N.: Recent developments in chaotic time series analysis. Int. J. Bifurc. Chaos 13, 1383–1422 (2003)

    Article  MathSciNet  Google Scholar 

  3. Ye, B., Chen, J., Ju, C., Li, H., Wang, X.: Distinguishing chaotic time series from noise: a random matrix approach. Commun. Nonlinear Sci. Numer. Simul. 44, 284–291 (2016)

    Article  MathSciNet  Google Scholar 

  4. Politi, A.: Quantifying the dynamical complexity of chaotic time series. Phys. Rev. Lett. 118, 144101 (2017)

    Article  Google Scholar 

  5. Ding, S.-L., Song, E.-Z., Yang, L.-P., Litak, G., Wang, Y.-Y., Yao, C., Ma, X.-Z.: Analysis of chaos in the combustion process of premixed natural gas engine. Appl. Therm. Eng. 121, 768–778 (2017)

    Article  Google Scholar 

  6. Xiao, Q., Wang, S., Zhang, Z., Xu, J.: Analysis of sunspot time series (1749–2014) by means of 0–1 test for chaos detection. In: International Conference on Computational Intelligence and Security, pp. 215–218

  7. Gottwald, G.A., Melbourne, I.: A new test for chaos in deterministic systems. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 460, pp. 603–611. The Royal Society

  8. Gottwald, G.A., Melbourne, I.: Testing for chaos in deterministic systems with noise. Physica D 212, 100–110 (2005)

    Article  MathSciNet  Google Scholar 

  9. Gottwald, G.A., Melbourne, I.: The 0–1 Test for Chaos: A Review. Springer, Berlin (2016)

    MATH  Google Scholar 

  10. Fouda, J.S.A.E., 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)

    Article  MathSciNet  Google Scholar 

  11. Gottwald, G.A., Melbourne, I.: On the implementation of the 0–1 test for chaos. SIAM J. Appl. Dyn. Syst. 8, 129–145 (2009)

    Article  MathSciNet  Google Scholar 

  12. Bernardini, D., Litak, G.: An overview of 0–1 test for chaos. J. Braz. Soc. Mech. Sci. Eng. 38, 1433–1450 (2016)

    Article  Google Scholar 

  13. Martinsen-Burrell, N., Julien, K., Petersen, M.R., Weiss, J.B.: Merger and alignment in a reduced model for three-dimensional quasigeostrophic ellipsoidal vortices. Phys. Fluids 18, 057101 (2006)

    Article  MathSciNet  Google Scholar 

  14. Falconer, I., Gottwald, G.A., Melbourne, I., Wormnes, K.: Application of the 0–1 test for chaos to experimental data. SIAM J. Appl. Dyn. Syst. 6, 395–402 (2007)

    Article  MathSciNet  Google Scholar 

  15. Litak, G., Syta, A., Budhraja, M., Saha, L.M.: Detection of the chaotic behaviour of a bouncing ball by the 0–1 test. Chaos Solitons Fractals 42, 1511–1517 (2009)

    Article  Google Scholar 

  16. Webel, K.: Chaos in German stock returns—new evidence from the 0–1 test. Econ. Lett. 115, 487–489 (2012)

    Article  Google Scholar 

  17. Krese, B., Govekar, E.: Analysis of traffic dynamics on a ring road-based transportation network by means of 0–1 test for chaos and Lyapunov spectrum. Transp. Res. Part C Emerg. Technol. 36, 27–34 (2013)

  18. Dawesand, J.H.P., Freeland, M.C.: The ‘0–1 test for chaos’ and strange nonchaotic attractors. Preprint (2008)

  19. Gopal, R., Venkatesan, A., Lakshmanan, M.: Applicability of 0–1 test for strange nonchaotic attractors. Chaos Interdiscip. J. Nonlinear Sci. 23, 023123 (2013)

    Article  MathSciNet  Google Scholar 

  20. Savi, M.A., Pereira-Pinto, F.H.I., Viola, F.M., Paula, A.S.D., Bernardini, D., Litak, G., Rega, G.: Using 0–1 test to diagnose chaos on shape memory alloy dynamical systems. Chaos Solitons Fractals 103, 307–324 (2017)

    Article  MathSciNet  Google Scholar 

  21. Wontchui, T.T., Effa, J.Y., Fouda, H.P.E., Fouda, J.S.A.E.: Dynamical behavior of Peter-De-Jong map using the modified 0–1 and 3ST tests for chaos. Ann. Rev. Chaos Theory Bifurc. Dyn. Syst. 7, 1–21 (2017)

    Google Scholar 

  22. Fouda, J.S.A.E., Bodo, B., Sabat, S.L., Effa, J.Y.: A modified 0–1 test for chaos detection in oversampled time series observations. Int. J. Bifurc. Chaos 24, 1450063 (2014)

    Article  MathSciNet  Google Scholar 

  23. 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, 4731–4737 (2013)

    Article  Google Scholar 

  24. Budhraja, M., Kumar, N., Saha, L.: The 0–1 test applied to Peter-De-Jong map. Int. J. Eng. Innov. Technol. (IJEIT) 2, 253–257 (2012)

    Google Scholar 

  25. Armand Eyebe Fouda, J., Bodo, B., Sabat, S.L., Effa, J.Y.: A modified 0–1 test for chaos detection in oversampled time series observations. Int. J. Bifurc. Chaos 24, 1450063 (2014)

    Article  MathSciNet  Google Scholar 

  26. Afsar, O., Bagci, G.B., Tirnakli, U.: Renormalized entropy for one dimensional discrete maps: periodic and quasi-periodic route to chaos and their robustness. Eur. Phys. J. B 86, 307 (2013)

    Article  MathSciNet  Google Scholar 

  27. Hamdi, B., Hassen, S.: A new hypersensitive hyperchaotic system with no equilibria. Int. J. Bifurc. Chaos 27, 1750064 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors wish to acknowledge the Editors and anonymous reviewers for their constructive comments and suggestions, which helped us to improve the manuscript. This work was financially supported by the Natural Science Foundation of Yunnan Province, China (No. 202101AU070031), the Scientific Research Fund Project of Yunnan Education Department, China (No. 2021J0063), and Young Elite Scientist Sponsorship Program by CAST, China (No. YESS20210106). The authors are grateful for reference material from Dr. G. A. Gottwald (University of Sydney, Australia). Dr. Q. Xiao is grateful to Dr. C. Zhang who was a visiting researcher to Arizona State University (USA) for her comments and suggestions. Dr. J. Chen wishes to thank Dr. B. Feng (University of Texas Rio Grande Valley, USA) for directing his attention to chaos theory.

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Authors and Affiliations

  1. State Key Laboratory of Complex Nonferrous Metal Resources Clean Utilization, Kunming University of Science and Technology, Kunming, 650093, People’s Republic of China

    Qingtai Xiao, Yanan Liao, Wei Xu & Hua Wang

  2. Faculty of Metallurgical and Energy Engineering, Kunming University of Science and Technology, Kunming, 650093, People’s Republic of China

    Qingtai Xiao, Yanan Liao, Wei Xu & Hua Wang

  3. School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX, 78541, USA

    Qingtai Xiao & Junchao Chen

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  1. Qingtai Xiao
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  2. Yanan Liao
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  3. Wei Xu
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  4. Junchao Chen
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  5. Hua Wang
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Correspondence to Hua Wang.

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When this paper was submitted, Dr. Q. Xiao and Dr. J. Chen were visiting scholars at School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg TX 78541, USA.

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Xiao, Q., Liao, Y., Xu, W. et al. Impact of damping amplitude on chaos detection reliability of the improved 0–1 test for oversampled and noisy observations. Nonlinear Dyn 108, 4385–4398 (2022). https://doi.org/10.1007/s11071-022-07416-4

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  • DOI: https://doi.org/10.1007/s11071-022-07416-4

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