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Detection and identification of chaotic signal is very important in the chaotic time series analysis. It is not easy to distinguish chaotic time series from stochastic processes since they share some similar natures. The detection methods to capture and utilize the structure of state-space dynamics can be very effective. In practice, it is very hard to obtain full information about the structure, and accurate phase-space reconstruction from scalar time series data is also a real challenge. However, the chaotic signals also show fundamental dynamical structure in the incomplete two-dimensional phase-space for the reason that they are generated by the deterministic chaotic systems or maps. Based on the fact that the distribution of chaotic signals is quite different from that of the noise signals in the incomplete two-dimensional phase-space, a novel detection method, which depends on the component permutation of the incomplete two-dimensional phase-space, is proposed. The incomplete two-dimensional phase-space is first obtained through the time series. Then, the first component is sorted in the ascending order, and the second component is permutated accordingly. The permutated component shows more structure characteristics for chaotic signals because of the relation between these two components. But this phenomenon does not appear in the noise because these components are independent of each other. And then, the permutated component is segmented into several groups properly. Finally, the sample mean and sample variance of different groups are calculated to obtain the sequence of sample mean (SSM) and the sequence of sample variance (SSV). Meanwhile, by calculating the variance of the SSM and the mean of the SSV, the test statistic is obtained. Furthermore, it is proved that this test statistic follows the F distribution under the null hypothesis of Gaussian noise. The proposed method is first adopted for detecting the several chaotic signals under different data lengths in Gaussian noise conditions. The simulation results show that the proposed method can detect chaotic signals effectively under low signal-to-noise ratio and it also has a good robustness against noise compared with the permutation entropy test. The time consumptions of the proposed method under different data lengths are evaluated and also compared with the results of permutation entropy test, showing that the proposed method can detect chaotic signals quickly, and the time complexity is much lower than that of the permutation entropy test. The theoretical analysis and simulation results demonstrate that the proposed method not only outperforms the permutation entropy test with lower complexity, but also has a better robustness against noise.
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Keywords:
- component permutation /
- chaos detection /
- incomplete phase-space
[1] Kaplan D T, Glass L 1992 Phys. Rev. Lett. 68 427
[2] Wayland R, Bromley D, Pickett D, Passamante A 1993 Phys. Rev. Lett. 70 580
[3] Salvino L W, Cawley R 1994 Phys. Rev. Lett. 73 1091
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[17] Matilla-García M, Marín M R 2011 Geogr. Anal. 43 228
[18] Riedl M, Mller A, Wessel N 2013 Eur. Phys. J.- Spec. Top. 222 249
[19] Ouyang G, Dang C, Richards D A, Li X 2010 Clin. Neurophysiol. 121 694
[20] Nicolaou N, Georgiou J 2012 Expert Syst. Appl. 39 202
[21] Zunino L, Zanin M, Tabak B M, Pérez D G, Rosso O A 2009 Physica A 388 2854
[22] Ruiz M C, Guillamón A, Gabaldón A 2012 Entropy-Switz 14 74
[23] Li J, Yan J, Liu X, Ouyang G 2014 Entropy-Switz 16 3049
[24] Toomey J P, Kane D M 2014 Opt. Express 22 1713
[25] Weck P J, Schaffner D A, Brown M R 2015 Phys. Rev. E 91 023101
[26] Chen X, Jin N D, Zhao A, Gao Z K, Zhai L S, Sun B 2015 Physica A 417 230
[27] Wang F P, Wang Z J, Guo J B 2002 Acta Phys. Sin. 51 474 (in Chinese) [汪芙平, 王赞基, 郭静波 2002 51 474]
[28] Zheng H Z, Hu J F, Liu L D, He Z S 2011 Acta Phys. Sin. 60 110507 (in Chinese) [郑皓洲, 胡进峰, 刘立东, 何子述 2011 60 110507]
[29] Packard N H, Crutchfield J P, Farmer J D, Show R S 1980 Phys. Rev. Lett. 45 712
[30] Takens F 1981 Dynamical Systems and Turbulence (Berlin: Springer Verlag) p366
[31] Zhang J, Luo X, Small M 2006 Phys. Rev. E 73 016216
[32] Garland J, Bradley E, Meiss J D 2015 arXiv: 1506.01128[math.DS]
[33] Bradley E, Kantz H 2015 Chaos 25 097610
[34] Garland J, Bradley E 2015 arXiv:1503.01678[nlin.CD]
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[1] Kaplan D T, Glass L 1992 Phys. Rev. Lett. 68 427
[2] Wayland R, Bromley D, Pickett D, Passamante A 1993 Phys. Rev. Lett. 70 580
[3] Salvino L W, Cawley R 1994 Phys. Rev. Lett. 73 1091
[4] Ortega G J, Louis E 1998 Phys. Rev. Lett. 81 4345
[5] Jeong J, Gore J C, Peterson B S 2002 IEEE T. Bio-med. Eng. 49 1374
[6] Barahona M, Poon C S 1996 Nature 381 215
[7] Poon C S, Barahona M 2001 P. Natl. Acad. Sci.USA 98 7107
[8] Lei M, Meng G 2008 Chaos Solitons Fract. 36 512
[9] Bandt C, Pompe B 2002 Phys. Rev. Lett. 88 174102
[10] Liu X F, Wang Y 2009 Chin. Phys. B 18 2690
[11] Bian C, Qin C, Ma Q D Y, Shen Q 2012 Phys. Rev. E 85 021906
[12] Amigó J M, Kocarev L, Szczepanski J 2006 Phys. Lett. A 355 27
[13] Amigó J M, Zambrano S, Sanjuán M A F 2007 Europhys. Lett. 79 5001
[14] Rosso O A, Fuentes M A 2007 Phys. Rev. Lett. 99 154102
[15] Matilla-García M, Marín M R 2008 J. Econometris 144 139
[16] López F, Matilla-García M, Mur J, Marín M R 2010 Reg. Sci. Urban Econ. 40 106
[17] Matilla-García M, Marín M R 2011 Geogr. Anal. 43 228
[18] Riedl M, Mller A, Wessel N 2013 Eur. Phys. J.- Spec. Top. 222 249
[19] Ouyang G, Dang C, Richards D A, Li X 2010 Clin. Neurophysiol. 121 694
[20] Nicolaou N, Georgiou J 2012 Expert Syst. Appl. 39 202
[21] Zunino L, Zanin M, Tabak B M, Pérez D G, Rosso O A 2009 Physica A 388 2854
[22] Ruiz M C, Guillamón A, Gabaldón A 2012 Entropy-Switz 14 74
[23] Li J, Yan J, Liu X, Ouyang G 2014 Entropy-Switz 16 3049
[24] Toomey J P, Kane D M 2014 Opt. Express 22 1713
[25] Weck P J, Schaffner D A, Brown M R 2015 Phys. Rev. E 91 023101
[26] Chen X, Jin N D, Zhao A, Gao Z K, Zhai L S, Sun B 2015 Physica A 417 230
[27] Wang F P, Wang Z J, Guo J B 2002 Acta Phys. Sin. 51 474 (in Chinese) [汪芙平, 王赞基, 郭静波 2002 51 474]
[28] Zheng H Z, Hu J F, Liu L D, He Z S 2011 Acta Phys. Sin. 60 110507 (in Chinese) [郑皓洲, 胡进峰, 刘立东, 何子述 2011 60 110507]
[29] Packard N H, Crutchfield J P, Farmer J D, Show R S 1980 Phys. Rev. Lett. 45 712
[30] Takens F 1981 Dynamical Systems and Turbulence (Berlin: Springer Verlag) p366
[31] Zhang J, Luo X, Small M 2006 Phys. Rev. E 73 016216
[32] Garland J, Bradley E, Meiss J D 2015 arXiv: 1506.01128[math.DS]
[33] Bradley E, Kantz H 2015 Chaos 25 097610
[34] Garland J, Bradley E 2015 arXiv:1503.01678[nlin.CD]
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