一个三维四翼自治混沌系统的拓扑马蹄分析

贾红艳; 陈增强; 叶菲

doi:10.7498/aps.60.010203

摘要

基于拓扑马蹄映射理论,验证了一个三维四翼自治的混沌系统的拓扑马蹄的存在.由于该混沌系统是连续系统,首先选取了一个Poincaré截面,并在该截面下定义了该混沌系统的一个一次回归Poincaré映射.通过利用计算机辅助证明方法,得出了该映射与一个2移位映射拓扑半共扼,说明该三维四翼自治系统的拓扑熵大于或等于ln2,进而证明了该系统的混沌行为.

关键词:

Based on topological horseshoe map theory, the paper analyses the existence of topological horseshoe in a 3-D four-wing chaotic system. As the chaotic system is continuous, the paper first choses a Poincaré section, then under which defines a first return Poincaré map. A conclusion that the Poincaré map is semi-conjugate to 2-shift map can be obtained by utilizing computer-assisted verification, showing that the topological entropy of the 3-D four-wing system is larger than or equal to ln2, which further verifies the chaotic characteristic of the system.

Keywords:

作者及机构信息

(1)南开大学自动化系,天津 300071; (2)天津大学自动化系,天津 300072; (3)天津科技大学自动化系,天津 300222;南开大学自动化系,天津 300071

基金项目: 国家自然科学基金(批准号:60774088和10772135),高等学校博士学科点专项科研基金(批准号:20090031110029 ),天津市市应用基础及前沿技术研究计划(批准号:08JCZDJC21900), 天津市高等学校科技发展基金(批准号:20088026),国家教育部留学回国人员科研启动基金资助的课题.

Authors and contacts

(1)Department of Automation, Nankai University, Tianjin 300071, China; (2)Department of Automation, Tianjin University of Science and Technology, Tianjin 300222, China;Department of Automation, Nankai University, Tianjin 300071, China; (3)Department of Automation, Tianjin University, Tianjin 300072, China

参考文献

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[2]	Chen G R, Ueta T 1999 Int. J. Bifur. Chaos. 9 1465
[3]	Lü J H, Chen G R, Cheng D Z, Celikovsky S 2002 Int. J. Bifur. Chaos. 12 2917
[4]	Lü J H, Chen G R 2002 Int. J. Bifur. Chaos 12 659
[5]	Liu C X, Liu T, Liu L, Liu K 2004 Chaos, Solitons & Fractals 22 1031
[6]	Qi G Y, Chen G R, Du S Z, Chen Z Q, Yuan Z Z 2005 Physica A 352 295
[7]	Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) [刘崇新 2007 56 6865]
[8]	Wang F Q, Liu C X 2006 Acta Phys. Sin. 55 3922(in Chinese) [王发强、刘崇新 2006 55 3922]
[9]	Chen X R, Liu C X, Wang F Q, Liu Y X 2008 Acta Phys. Sin. 57 1416(in Chinese) [陈向荣、刘崇新、王发强、李永勋 2008 57 1416]
[10]	Wang F Z, Qi G Y, Chen Z Q, Yuan Z Z 2007 Acta Phys. Sin. 56 3137(in Chinese)[王繁珍、齐国元、陈增强、袁著祉 2007 56 3137]
[11]	Qi G Y, Chen G R, van Wyk M A, van Wyk B J, Zhang Y H 2008 Chaos Solitons & Fractals 38 705
[12]	Chen Z Q, Yang Y, Yuan Z Z 2008 Chaos, Solitons & Fractals 38 1187
[13]	Li Y X, Wallace K. S. Tang, Chen G R 2005 Int. J. Circ. Theor. Appl. 33 235
[14]	Wang J Z, Chen Z Q, Chen G R, Yuan Z Z 2008 Int. J. Bifur. Chaos. 18 3309
[15]	Jia H Y, Chen Z Q, Yuan Z Z 2009 Acta Phys. Sin.58 4469[贾红艳、陈增强、袁著祉 2009 58 4469]
[16]	Yu S M, Lü J H, Chen G R. 2007 Physics letter A 364 244
[17]	Cang S J, Chen Z Q, Yuan Z Z 2008 Acta Phys. Sin. 57 1493 (in Chinese) [仓诗建、陈增强、袁著祉 2008 57 1493]
[18]	Udaltsov VS, Goedgebuer J P, Larger L, Cuenot J B, Rhodes W T 2003 Optics and Spectroscopy. 95 114
[19]	Hsieh J Y, Hwang C C, Wang A P, Li W J 1999 International Journal of Control. 72 882
[20]	Song Y Z 2007 Chin. Phys. 16 1918
[21]	Song Y Z Zhao G Z, Qi D L 2006 Chin. Phys. 15 2266
[22]	Wiggins S 1990 Introduction to Applied Nonlinear Dynamical Systems and Chaos (New York: Springer-Verlag) p421
[23]	Kennedy J, Kocak S, Yorke J A 2001 Amer. Math.Mon. 208 411
[24]	Kennedy J, Yorke J A 2001 Trans. Amer. Math.Soc. 353 2513
[25]	Yang X S, Tang Y 2004 Chaos, Solitons & Fractals 19 841
[26]	Yang X S 2004 Chaos, Solitons & Fractals 20 1149
[27]	Yang X S, Yu Y G, Zhang S C 2003 Chaos, Solitons & Fractals 18 223
[28]	Yang X S, Li Q D 2005 Int. J. Bifur. Chaos. 15 1823
[29]	Huang Y, Yang X S 2005 Chaos, Solitons & Fractals 26 79
[30]	Yang X S, Li Q D 2004 Int. J. Bifur. Chaos. 14 1847
[31]	Wu W J, Chen Z Q, Yuan Z Z 2009 Solitons & Fractals 41 2756
[32]	Wu W J, Chen Z Q, Yuan Z Z 2008 The 9th International Conference for Young Computer Scientists. Zhang Jia Jie, Hunan, China, November 18—21, 2008 p3033
[33]	Wu W J, Chen Z Q, Chen G R 2009 International Workshop on Chaos-Fractals Theories and Applications. Shenyang, Liaoning, China, November 6—8, 2009 p277
[34]	Chen Z Q, Yang Y, Qi G Y, Yuan Z Z 2007 Phys. Lett. A 360 696
[35]	Wang J Z, Chen Z Q, Yuan Z Z 2006 Chin. Phys. 15 1216

施引文献

[1]	Lorenz E N 1963 J. Atmos. Sci. 20 130
[2]	Chen G R, Ueta T 1999 Int. J. Bifur. Chaos. 9 1465
[3]	Lü J H, Chen G R, Cheng D Z, Celikovsky S 2002 Int. J. Bifur. Chaos. 12 2917
[4]	Lü J H, Chen G R 2002 Int. J. Bifur. Chaos 12 659
[5]	Liu C X, Liu T, Liu L, Liu K 2004 Chaos, Solitons & Fractals 22 1031
[6]	Qi G Y, Chen G R, Du S Z, Chen Z Q, Yuan Z Z 2005 Physica A 352 295
[7]	Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) [刘崇新 2007 56 6865]
[8]	Wang F Q, Liu C X 2006 Acta Phys. Sin. 55 3922(in Chinese) [王发强、刘崇新 2006 55 3922]
[9]	Chen X R, Liu C X, Wang F Q, Liu Y X 2008 Acta Phys. Sin. 57 1416(in Chinese) [陈向荣、刘崇新、王发强、李永勋 2008 57 1416]
[10]	Wang F Z, Qi G Y, Chen Z Q, Yuan Z Z 2007 Acta Phys. Sin. 56 3137(in Chinese)[王繁珍、齐国元、陈增强、袁著祉 2007 56 3137]
[11]	Qi G Y, Chen G R, van Wyk M A, van Wyk B J, Zhang Y H 2008 Chaos Solitons & Fractals 38 705
[12]	Chen Z Q, Yang Y, Yuan Z Z 2008 Chaos, Solitons & Fractals 38 1187
[13]	Li Y X, Wallace K. S. Tang, Chen G R 2005 Int. J. Circ. Theor. Appl. 33 235
[14]	Wang J Z, Chen Z Q, Chen G R, Yuan Z Z 2008 Int. J. Bifur. Chaos. 18 3309
[15]	Jia H Y, Chen Z Q, Yuan Z Z 2009 Acta Phys. Sin.58 4469[贾红艳、陈增强、袁著祉 2009 58 4469]
[16]	Yu S M, Lü J H, Chen G R. 2007 Physics letter A 364 244
[17]	Cang S J, Chen Z Q, Yuan Z Z 2008 Acta Phys. Sin. 57 1493 (in Chinese) [仓诗建、陈增强、袁著祉 2008 57 1493]
[18]	Udaltsov VS, Goedgebuer J P, Larger L, Cuenot J B, Rhodes W T 2003 Optics and Spectroscopy. 95 114
[19]	Hsieh J Y, Hwang C C, Wang A P, Li W J 1999 International Journal of Control. 72 882
[20]	Song Y Z 2007 Chin. Phys. 16 1918
[21]	Song Y Z Zhao G Z, Qi D L 2006 Chin. Phys. 15 2266
[22]	Wiggins S 1990 Introduction to Applied Nonlinear Dynamical Systems and Chaos (New York: Springer-Verlag) p421
[23]	Kennedy J, Kocak S, Yorke J A 2001 Amer. Math.Mon. 208 411
[24]	Kennedy J, Yorke J A 2001 Trans. Amer. Math.Soc. 353 2513
[25]	Yang X S, Tang Y 2004 Chaos, Solitons & Fractals 19 841
[26]	Yang X S 2004 Chaos, Solitons & Fractals 20 1149
[27]	Yang X S, Yu Y G, Zhang S C 2003 Chaos, Solitons & Fractals 18 223
[28]	Yang X S, Li Q D 2005 Int. J. Bifur. Chaos. 15 1823
[29]	Huang Y, Yang X S 2005 Chaos, Solitons & Fractals 26 79
[30]	Yang X S, Li Q D 2004 Int. J. Bifur. Chaos. 14 1847
[31]	Wu W J, Chen Z Q, Yuan Z Z 2009 Solitons & Fractals 41 2756
[32]	Wu W J, Chen Z Q, Yuan Z Z 2008 The 9th International Conference for Young Computer Scientists. Zhang Jia Jie, Hunan, China, November 18—21, 2008 p3033
[33]	Wu W J, Chen Z Q, Chen G R 2009 International Workshop on Chaos-Fractals Theories and Applications. Shenyang, Liaoning, China, November 6—8, 2009 p277
[34]	Chen Z Q, Yang Y, Qi G Y, Yuan Z Z 2007 Phys. Lett. A 360 696
[35]	Wang J Z, Chen Z Q, Yuan Z Z 2006 Chin. Phys. 15 1216

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