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

Ye Fei; Chen Zeng-Qiang; Jia Hong-Yan

doi:10.7498/aps.60.010203

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.

Keywords:

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

References

[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

Cited By

[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

[1]	Gu Zhao-Long, Li Jian-Xin. Topological order and fractionalized excitations in quantum many-body systems. Acta Physica Sinica, 2024, 73(7): 070301. doi: 10.7498/aps.73.20240222
[2]	Lin Ji-Yan, Chen Cheng, Guo Lin-Wei, Li Yao, Lin Shu-Yu, Sun Jiao-Xia, Xu Jie. Research on acoustic control of coupled vibration system of transducers using acoustic surface and topological defect structures. Acta Physica Sinica, 2024, 73(22): 224301. doi: 10.7498/aps.73.20241199
[3]	Cai De-Huan, Qu Su-Ping. Dynamic topological phenomena in periodically driven Raman lattice. Acta Physica Sinica, 2024, 73(14): 140301. doi: 10.7498/aps.73.20240535
[4]	Wang Zi-Yao, Chen Fu-Jia, Xi Xiang, Gao Zhen, Yang Yi-Hao. Non-reciprocal topological photonics. Acta Physica Sinica, 2024, 73(6): 064201. doi: 10.7498/aps.73.20231850
[5]	Jiang Jing, Wang Xiao-Yun, Kong Peng, Zhao He-Ping, He Zhao-Jian, Deng Ke. Dislocation defect states in acoustic quadrupole topological insulators. Acta Physica Sinica, 2024, 73(15): 154302. doi: 10.7498/aps.73.20240640
[6]	Wang Huan, He Chun-Juan, Xu Sheng, Wang Yi-Yan, Zeng Xiang-Yu, Lin Jun-Fa, Wang Xiao-Yan, Gong Jing, Ma Xiao-Ping, Han Kun, Wang Yi-Ting, Xia Tian-Long. Single crystal growth of topological semimetals and magnetic topological materials. Acta Physica Sinica, 2023, 72(3): 038103. doi: 10.7498/aps.72.20221574
[7]	Gu Kai-Yuan, Luo Tian-Chuang, Ge Jun, Wang Jian. Superconductivity in topological materials. Acta Physica Sinica, 2020, 69(2): 020301. doi: 10.7498/aps.69.20191627
[8]	Luo Kai-Fa, Yu Rui. Topological states in electric circuit. Acta Physica Sinica, 2019, 68(22): 220305. doi: 10.7498/aps.68.20191398
[9]	Zhang Wei-Feng, Li Chun-Yan, Chen Xian-Feng, Huang Chang-Ming, Ye Fang-Wei. Topological zero-energy modes in time-reversal-symmetry-broken systems. Acta Physica Sinica, 2017, 66(22): 220201. doi: 10.7498/aps.66.220201
[10]	Luo Ming-Wei, Luo Xiao-Hua, Li Hua-Qing. A family of four-dimensional multi-wing chaotic system and its circuit implementation. Acta Physica Sinica, 2013, 62(2): 020512. doi: 10.7498/aps.62.020512
[11]	Zhou Xuan, Yang Fan, Zhang Feng-Ming, Zhou Wei-Ping, Zou Wei. Control method for complex network topological connection optimization. Acta Physica Sinica, 2013, 62(15): 150201. doi: 10.7498/aps.62.150201
[12]	Yu Yue, Zhang Chun, Han Xiu-Jing, Jiang Hai-Bo, Bi Qin-Sheng. Oscillations and non-smooth bifurcation analysis of Chen system with periodic switches. Acta Physica Sinica, 2013, 62(2): 020508. doi: 10.7498/aps.62.020508
[13]	Li Qing-Du, Tang Song. Algorithm for finding horseshoes in three-dimensional hyperchaotic maps and its application. Acta Physica Sinica, 2013, 62(2): 020510. doi: 10.7498/aps.62.020510
[14]	Hao Jian-Hong, Sun Na-Yan. The characteristics of the chaotic parameters for a loss type of modified coupled dynamic system. Acta Physica Sinica, 2012, 61(15): 150504. doi: 10.7498/aps.61.150504
[15]	Yu Fei, Wang Chun-Hua, Yin Jin-Wen, Xu Hao. A 4-D chaos with fully qualified four-wing type. Acta Physica Sinica, 2012, 61(2): 020506. doi: 10.7498/aps.61.020506
[16]	Wang Yong, Guan Miao, Zhang Lei. A three-phase four-leg topology of the new matrix converter. Acta Physica Sinica, 2010, 59(2): 867-876. doi: 10.7498/aps.59.867
[17]	Hu Guo-Si. A family of hyperchaotic systems with four-wing attractors. Acta Physica Sinica, 2009, 58(6): 3734-3741. doi: 10.7498/aps.58.3734
[18]	CHEN RUI-XIONG. DETERMINATION OF THE TOPOLOGICAL ENTROPY OF ONE-DIMENSIONAL MAPS BY THE INVERSE ORBITS ANALYSIS. Acta Physica Sinica, 1989, 38(9): 1501-1505. doi: 10.7498/aps.38.1501
[19]	WENG JIA-QIANG, KONG LING-JIANG, CHEN GUANG-ZHI. A PROOF ON THE EQUIVALENCE OF TOPOLOGICAL ENTROPY AT SUBREGION WINDOW OF RLm. Acta Physica Sinica, 1987, 36(12): 1583-1589. doi: 10.7498/aps.36.1583
[20]	Chen Rui-xiong, Chen Shi-gang. THE TOPOLOGICAL ENTROPY OF ONE-DIMENSIONAL UNIMODAL MAPS. Acta Physica Sinica, 1986, 35(10): 1338-1346. doi: 10.7498/aps.35.1338

Catalog

Vol. 60, Issue 1 - 2011-01-05

Metrics

Abstract views: 8091
PDF Downloads: 975
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板