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In order to reduce errors caused by human factors to identify the linear region, we propose a new method based on the fuzzy C-means clustering for calculating the maximum Lyapunov exponent from small data. The method based on the changing characteristic of divergence index curve is used to identify the linear region. Firstly, the divergence index data are calculated from the small data algorithm for the given chaotic time series. Secondly, the fuzzy C-means clustering method is used for dividing the data into two classes (unsaturated and saturated data), and the unsaturated data are retained. Thirdly, the retained data are divided by the same clustering method into three classes (positive fluctuation data, zero fluctuation data and negative fluctuation data), and the zero fluctuation data are retained. Fourthly, the 3$ criterion is used for excluding gross errors to retain the valid from the selected data. Finally, the regression analysis and statistical test are used to identify the linear region from the valid data. The effectiveness of the proposed method can be demonstrated by the famous chaotic systems of Logistic and Henon. The calculated results are closr to the theoretical values than the subjective method. Experimental results show that the proposed new approach is easier to operate, more efficient and more accurate as compared with the subjective recognition. But this method has its own shortcomings. (1) As the new method is verified by the simulation experiment, there exists no strict mathematical proof. (2) Since the difference algorithm is used in this new method, it will miss some detailed information in some cases. (3) The calculation accuracy still needs to be improved, so this method only serves as a reference to detect the linear region, it can not be applied to high precision engineering field. Considering the deficiencies of the new method, we will make further research to improve the calculation method for maximum Lyapunovexponent, so as to make it solve the real-time problem of the signal detection, and find the accurate location of abrupt climate change in the field of meteorology, to provide accurate satellite launch safety period in the field of space weather and other aspects. In short, studying the largest Lyapunov exponent from chaotic time series has a wide application prospect and practical significance.
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Keywords:
- maximal Lyapunov exponent /
- linear region /
- fuzzy C-means clustering
[1] Yu S M 2011 Chaotic Systems and Chaotic Circuits: Principle, Design and Its Application in Communications (Xian: Xidian University Press) p4 (in Chinese) [禹思敏 2011 混沌系统与混沌电路: 原理、设计及其在通信中的应用 (西安: 西安电子科技大学出版社) 第4页]
[2] Wang W J, Ye M, Chen X W 1994 Advances in Water Science 2 87 (in Chinese) [王文均, 叶敏, 陈显维 1994 水科学进展 2 87]
[3] Zhou S, Feng Y, Wu W Y, Li Y, Liu J 2014 Res. Astron. Astronphys. 14 104
[4] L J H, Zhan Y, Lu J A 2000 Proceedings of the CSEE 20 80 (in Chinese) [吕金虎, 占勇, 陆君安 2000 中国电机工程学报 20 80]
[5] Packard N H, Crutchfield J, Famrmer J 1980 Phys. Rev. Lett. 45 712
[6] Yu S M 2008 Acta Phys. Sin. 57 3374 (in Chinese) [禹思敏 2008 物理学报 57 3374]
[7] Yu S M, Tang W K S, L J H, Chen G R 2008 IEEE T. Circuits - II 55 1168
[8] Shen C W, Yu S M, L J H, Chen G R 2014 IEEE T. Circuits - I 61 854
[9] Shen C W, Yu S M, L J H, Chen G R 2014 IEEE T. Circuits - I 61 2380
[10] Li Q X, Li K J 2007 Publ. Astron. Soc. Jpn. 59 983
[11] Xian M, Zhuang Z W, Xiao S P, Guo G R 1998 Journal of National University of Defense Techonlogy 20 60 (in Chinese) [鲜明, 庄钊文, 肖顺平, 郭桂蓉 1998 国防科技大学学报 20 60]
[12] L J H, Lu J A, Chen S H 2002 Chaotic time series analysis and its application (Wuhan: Wuhan University Press) p73 (in Chinese) [吕金虎, 陆君安, 陈士华 2002 混沌时间序列分析及其应用 (武汉: 武汉大学出版社) 第73页]
[13] Hubertus F, Udwadia F E, Proskurowski W 1997 Physica D 101 1
[14] Shimada I, Nagashima T 1979 Prog. Theor. Phys. 61 1605
[15] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285
[16] Briggs K 1990 Phys. Lett. A 151 27
[17] Rosenstein M T, Collins J J, de Luca C J 1993 Physica D 65 117
[18] Barna G, Tsuda I 1993 Phys. Lett. A 175 421
[19] Jiang H F, Ma H J, Wei X Y, Wen W G 2006 Journal of the China Railway Society 28 63 (in Chinese) [蒋海峰, 马瑞军, 魏学业, 温伟刚 2006 铁道学报 28 63]
[20] Xu X K 2008 Ph.D.Dissertation (Dalian: Dalian Maritime University) (in Chinese) [许小可 2008 博士论文 (大连: 大连海事大学)]
[21] Kantz H 1994 Phys. Lett. A 185 77
[22] Liang Y, Meng Q, Lu J R 2006 Technical Acoustics 25 463 (in Chinese) [梁勇, 孟桥, 陆佶人 2006 声学技术 25 463]
[23] Yao T L 2013 Ph.D.Dissertation (Shanghai: East China University of Science and Technology) (in Chinese) [姚天亮 2013 博士论文 (上海: 华东理工大学)]
[24] Yang S Q, Zhang X H, Zhao C A 2000 Acta Phys. Sin. 49 636 (in Chinese) [杨绍清, 章新华, 赵长安 2000 物理学报 49 636]
[25] Lu S, Wang H Y 2006 Acta Phys. Sin. 55 572 (in Chinese) [卢山, 王海燕 2006 物理学报 55 572]
[26] Lu Z B 2008 Ph. D. Dissertation (Wuhan: Chinese People's Liberation Army Navy Project University) (in Chinese) [陆振波 2008 博士论文 (武汉: 中国人民解放军海军工程大学)]
[27] Yang Y F, Wu M J, Gao Z, Wu Y F, Ren X M 2012 Journal of Vibration Measurement Diagnosis 32 371 (in Chinese) [杨永锋, 仵敏娟, 高喆, 吴亚锋, 任兴民 2012 振动、测试与诊断 32 371]
[28] Jain A K 2010 Pattern Recogn. Lett. 31 651
[29] May R M 1976 Nature 261 459
[30] Hnon M 1976 Commun. Math. Phys. 50 69
[31] Zhang Y P, Sun W H, Liu C A 2010 Chin. Phys. B 19 050512
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[1] Yu S M 2011 Chaotic Systems and Chaotic Circuits: Principle, Design and Its Application in Communications (Xian: Xidian University Press) p4 (in Chinese) [禹思敏 2011 混沌系统与混沌电路: 原理、设计及其在通信中的应用 (西安: 西安电子科技大学出版社) 第4页]
[2] Wang W J, Ye M, Chen X W 1994 Advances in Water Science 2 87 (in Chinese) [王文均, 叶敏, 陈显维 1994 水科学进展 2 87]
[3] Zhou S, Feng Y, Wu W Y, Li Y, Liu J 2014 Res. Astron. Astronphys. 14 104
[4] L J H, Zhan Y, Lu J A 2000 Proceedings of the CSEE 20 80 (in Chinese) [吕金虎, 占勇, 陆君安 2000 中国电机工程学报 20 80]
[5] Packard N H, Crutchfield J, Famrmer J 1980 Phys. Rev. Lett. 45 712
[6] Yu S M 2008 Acta Phys. Sin. 57 3374 (in Chinese) [禹思敏 2008 物理学报 57 3374]
[7] Yu S M, Tang W K S, L J H, Chen G R 2008 IEEE T. Circuits - II 55 1168
[8] Shen C W, Yu S M, L J H, Chen G R 2014 IEEE T. Circuits - I 61 854
[9] Shen C W, Yu S M, L J H, Chen G R 2014 IEEE T. Circuits - I 61 2380
[10] Li Q X, Li K J 2007 Publ. Astron. Soc. Jpn. 59 983
[11] Xian M, Zhuang Z W, Xiao S P, Guo G R 1998 Journal of National University of Defense Techonlogy 20 60 (in Chinese) [鲜明, 庄钊文, 肖顺平, 郭桂蓉 1998 国防科技大学学报 20 60]
[12] L J H, Lu J A, Chen S H 2002 Chaotic time series analysis and its application (Wuhan: Wuhan University Press) p73 (in Chinese) [吕金虎, 陆君安, 陈士华 2002 混沌时间序列分析及其应用 (武汉: 武汉大学出版社) 第73页]
[13] Hubertus F, Udwadia F E, Proskurowski W 1997 Physica D 101 1
[14] Shimada I, Nagashima T 1979 Prog. Theor. Phys. 61 1605
[15] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285
[16] Briggs K 1990 Phys. Lett. A 151 27
[17] Rosenstein M T, Collins J J, de Luca C J 1993 Physica D 65 117
[18] Barna G, Tsuda I 1993 Phys. Lett. A 175 421
[19] Jiang H F, Ma H J, Wei X Y, Wen W G 2006 Journal of the China Railway Society 28 63 (in Chinese) [蒋海峰, 马瑞军, 魏学业, 温伟刚 2006 铁道学报 28 63]
[20] Xu X K 2008 Ph.D.Dissertation (Dalian: Dalian Maritime University) (in Chinese) [许小可 2008 博士论文 (大连: 大连海事大学)]
[21] Kantz H 1994 Phys. Lett. A 185 77
[22] Liang Y, Meng Q, Lu J R 2006 Technical Acoustics 25 463 (in Chinese) [梁勇, 孟桥, 陆佶人 2006 声学技术 25 463]
[23] Yao T L 2013 Ph.D.Dissertation (Shanghai: East China University of Science and Technology) (in Chinese) [姚天亮 2013 博士论文 (上海: 华东理工大学)]
[24] Yang S Q, Zhang X H, Zhao C A 2000 Acta Phys. Sin. 49 636 (in Chinese) [杨绍清, 章新华, 赵长安 2000 物理学报 49 636]
[25] Lu S, Wang H Y 2006 Acta Phys. Sin. 55 572 (in Chinese) [卢山, 王海燕 2006 物理学报 55 572]
[26] Lu Z B 2008 Ph. D. Dissertation (Wuhan: Chinese People's Liberation Army Navy Project University) (in Chinese) [陆振波 2008 博士论文 (武汉: 中国人民解放军海军工程大学)]
[27] Yang Y F, Wu M J, Gao Z, Wu Y F, Ren X M 2012 Journal of Vibration Measurement Diagnosis 32 371 (in Chinese) [杨永锋, 仵敏娟, 高喆, 吴亚锋, 任兴民 2012 振动、测试与诊断 32 371]
[28] Jain A K 2010 Pattern Recogn. Lett. 31 651
[29] May R M 1976 Nature 261 459
[30] Hnon M 1976 Commun. Math. Phys. 50 69
[31] Zhang Y P, Sun W H, Liu C A 2010 Chin. Phys. B 19 050512
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