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Topological horseshoe analysis for a three-dimensional four-wing autonomous chaotic system

Chen Zeng-Qiang Jia Hong-Yan Ye Fei

Topological horseshoe analysis for a three-dimensional four-wing autonomous chaotic system

Chen Zeng-Qiang, Jia Hong-Yan, Ye Fei
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  • 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.
    • Funds:
    [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 物理学报 56 6865]

    [8]

    Wang F Q, Liu C X 2006 Acta Phys. Sin. 55 3922(in Chinese) [王发强、刘崇新 2006 物理学报 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 物理学报 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 物理学报 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 物理学报 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 物理学报 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

  • [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 物理学报 56 6865]

    [8]

    Wang F Q, Liu C X 2006 Acta Phys. Sin. 55 3922(in Chinese) [王发强、刘崇新 2006 物理学报 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 物理学报 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 物理学报 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 物理学报 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 物理学报 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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  • Received Date:  06 February 2010
  • Accepted Date:  27 April 2010
  • Published Online:  15 January 2011

Topological horseshoe analysis for a three-dimensional four-wing autonomous chaotic system

  • 1. (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

Abstract: 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.

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