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A new simple hyperbolic-type three-dimensional autonomous chaotic system is proposed. It is of interest that the chaotic system has only five terms which mainly rely on a nonlinear quadratic hyperbolic sine term and a quadratic cross-product term. Compared with other three-dimensional chaotic systems, the new system has not only less terms, but also a wider range of chaos when the parameter varies. Basic dynamical properties of the system are studied by numerical and theoretical analysis. Moreover the projective synchronization of the five-term hyperbolic-type chaotic system with fully uncertain parameters is also investigated in this paper. Based on Lyapunov stability theory and Barbalat's lemma, a new adaptive controller with parameter update law is designed to projectivly synchronize two chaotic systems asymptotically and globally, including two identical exponential-type chaotic systems and two non-identical chaotic systems. Numerical simulations show the effectiveness and the feasibility of the developed methods.
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Keywords:
- three-dimensional autonomous chaotic system /
- five-term hyperbolic-type chaotic system /
- projective synchronization /
- adaptive controller
[1] Han J J, Fu W J 2010 Chin. Phys. B 19 010205
[2] Di G H, Xu Y, Xu W, Gu R C 2011 Acta Phys. Sin. 60 020504 (in Chinese)[狄根虎, 许勇, 徐伟, 顾仁财 2011 物理学报 60 020504]
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[6] R? ssler O E 1976 Phys. Lett. A 57 397
[7] Chen G, Ueta T 1999 Int. J. Bifur. Chaos 9 1465
[8] Lü J, Chen G 2002 Int. J. Bifur. Chaos 12 659
[9] Lü J, Chen G, Cheng D, Celikovsky S 2002 Int. J. Bifur. Chaos 12 2917
[10] Celikovsky S, Chen G 2002 Int. J. Bifur. Chaos 12 1789
[11] Liu C, Liu T, Liu L, Liu K 2004 Chaos, Solitons and Fractals 22 1031
[12] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[13] Pan L, Zhou W, Zhou L, Sun K 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 2628
[14] Liu Y Z, Jiang C S, Lin C S 2008 Acta Phys. Sin. 57 709 (in Chinese)[刘扬正, 姜长生, 林长圣 2008 物理学报 57 709]
[15] Wang F Q, Liu C X 2006 Acta Phys. Sin. 55 5055 (in Chinese)[王发强, 刘崇新 2006 物理学报 55 5055]
[16] Xu F, Yu P 2010 J. Math. Anal. Appl. 362 252
[17] Chen Z S, Sun K H, Zhang T S 2005 Acta Phys. Sin. 54 2580 (in Chinese)[陈志盛, 孙克辉, 张泰山 2005 物理学报 54 2580]
[18] Liu Y Z, Jiang C S, Lin C S, Jiang Y M 2007 Chin. Phys. 16 660
[19] Wu X J, Wang X Y 2006 Acta Phys. Sin. 55 6261 (in Chinese)[武相军, 王兴元 2006 物理学报 55 6261]
[20] Cai G L, Huang J J 2006 Acta Phys. Sin. 55 3997 (in Chinese)[蔡国梁,黄娟娟 2006 物理学报 55 3997]
[21] Wang F Q, Liu C X 2006 Chin. Phys. 15 1971
[22] Guo H J, Yin Y W, Wang H M 2008 Chin. Phys. B 17 1652
[23] Li X C, Xu W, Xiao Y Z 2008 Acta Phys. Sin. 57 4721 (in Chinese)[李秀春, 徐伟, 肖玉柱 2008 物理学报 57 4721]
[24] Hua C C, Guan X P 2004 Chin. Phys. 13 1391
[25] Chen X R, Liu C X, Li Y X 2008 Acta Phys. Sin. 57 1453 (in Chinese)[陈向荣, 刘崇新, 李永勋 2008 物理学报 57 1453]
[26] Njah A N 2010 Nonlinear Dyn. 61 1
[27] Lü L, Zhang Q L, Guo Z A 2008 Chin. Phys. B 17 0498
[28] Zheng H Q, Jing Y W 2011 Chin. Phys. B 20 060504
[29] Zhang Q, L¨u J, Chen S 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 3067
[30] Li X F, Leung A C S, Han X P, Liu X J, Chu Y D 2011 Nonlinear Dyn. 63 263
[31] Taghvafard H, Erjaee G H 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 4079
[32] Hu M F, Xu Z Y 2007 Chin. Phys. 16 3231
[33] Cai N, Jing Y, Zhang S 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 1613
[34] Ghosh D, Bhattacharya S 2010 Nonlinear Dyn. 61 11
[35] Dai H, Jia L X, Hui M, Si G Q 2011 Chin. Phys. B 20 040507
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[1] Han J J, Fu W J 2010 Chin. Phys. B 19 010205
[2] Di G H, Xu Y, Xu W, Gu R C 2011 Acta Phys. Sin. 60 020504 (in Chinese)[狄根虎, 许勇, 徐伟, 顾仁财 2011 物理学报 60 020504]
[3] Liu Y M, Zhang Y H, Yang J Q 2009 J. Circuits Syst. 14 116 (in Chinese)[刘玉民, 张雨虹, 杨金泉 2009 电路与系统学报 14 116]
[4] Huang C H, Lin C H, Kuo C L 2011 IEEE Trans. Power Delivery 26 944
[5] Lorenz E N 1963 J. Atmos. Sci. 20 130
[6] R? ssler O E 1976 Phys. Lett. A 57 397
[7] Chen G, Ueta T 1999 Int. J. Bifur. Chaos 9 1465
[8] Lü J, Chen G 2002 Int. J. Bifur. Chaos 12 659
[9] Lü J, Chen G, Cheng D, Celikovsky S 2002 Int. J. Bifur. Chaos 12 2917
[10] Celikovsky S, Chen G 2002 Int. J. Bifur. Chaos 12 1789
[11] Liu C, Liu T, Liu L, Liu K 2004 Chaos, Solitons and Fractals 22 1031
[12] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[13] Pan L, Zhou W, Zhou L, Sun K 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 2628
[14] Liu Y Z, Jiang C S, Lin C S 2008 Acta Phys. Sin. 57 709 (in Chinese)[刘扬正, 姜长生, 林长圣 2008 物理学报 57 709]
[15] Wang F Q, Liu C X 2006 Acta Phys. Sin. 55 5055 (in Chinese)[王发强, 刘崇新 2006 物理学报 55 5055]
[16] Xu F, Yu P 2010 J. Math. Anal. Appl. 362 252
[17] Chen Z S, Sun K H, Zhang T S 2005 Acta Phys. Sin. 54 2580 (in Chinese)[陈志盛, 孙克辉, 张泰山 2005 物理学报 54 2580]
[18] Liu Y Z, Jiang C S, Lin C S, Jiang Y M 2007 Chin. Phys. 16 660
[19] Wu X J, Wang X Y 2006 Acta Phys. Sin. 55 6261 (in Chinese)[武相军, 王兴元 2006 物理学报 55 6261]
[20] Cai G L, Huang J J 2006 Acta Phys. Sin. 55 3997 (in Chinese)[蔡国梁,黄娟娟 2006 物理学报 55 3997]
[21] Wang F Q, Liu C X 2006 Chin. Phys. 15 1971
[22] Guo H J, Yin Y W, Wang H M 2008 Chin. Phys. B 17 1652
[23] Li X C, Xu W, Xiao Y Z 2008 Acta Phys. Sin. 57 4721 (in Chinese)[李秀春, 徐伟, 肖玉柱 2008 物理学报 57 4721]
[24] Hua C C, Guan X P 2004 Chin. Phys. 13 1391
[25] Chen X R, Liu C X, Li Y X 2008 Acta Phys. Sin. 57 1453 (in Chinese)[陈向荣, 刘崇新, 李永勋 2008 物理学报 57 1453]
[26] Njah A N 2010 Nonlinear Dyn. 61 1
[27] Lü L, Zhang Q L, Guo Z A 2008 Chin. Phys. B 17 0498
[28] Zheng H Q, Jing Y W 2011 Chin. Phys. B 20 060504
[29] Zhang Q, L¨u J, Chen S 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 3067
[30] Li X F, Leung A C S, Han X P, Liu X J, Chu Y D 2011 Nonlinear Dyn. 63 263
[31] Taghvafard H, Erjaee G H 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 4079
[32] Hu M F, Xu Z Y 2007 Chin. Phys. 16 3231
[33] Cai N, Jing Y, Zhang S 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 1613
[34] Ghosh D, Bhattacharya S 2010 Nonlinear Dyn. 61 11
[35] Dai H, Jia L X, Hui M, Si G Q 2011 Chin. Phys. B 20 040507
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