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In this paper, a class of four-dimensional continuous autonomous system is studied. It has only one balance under certain conditions but it shows a complicated dynamic behavior. The equilibrium points and lyapunov exponents of the system are analyzed. The bound of this system is estimated and the expression of the bound is presented. In addition, the complete synchronization is also discussed by designing a linear controller. Finally, the corresponding numerical simulations are performed.
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Keywords:
- hyperchaotic system /
- Lyapunov exponent /
- bound of chaotic system /
- synchronization
[1] Lorenz E N 1963 J. Atmos. Sci. 20 130
[2] Lü J H, Chen G R 2002 Int. J. Bifurc. Chaos. 12 659
[3] Zhang X D, Wang Z, Zhao P D 2008 Chin. Phys. Lett. 25 397
[4] Zhao P D, Zhang X D 2008 Acta Phys. Sin. 57 2791 (in Chinese) [赵品栋、张晓丹 2008 57 2791 ]
[5] Cenys A, Tamaservicius A, Baziliauskas A, Krivickas R, Lindberg E 2003 Chaos, Soliton. Fract. 17 349
[6] Zhang X D, Liu X, Zhao P D 2009 Acta Phys. Sin. 58 4415 (in Chinese) [张晓丹、刘 翔、赵品栋 2009 58 4415]
[7] Cafagna D, Grassi G 2003 Int. J. Bifurc. Chaos. 13 2889
[8] Rossler O E 1979 Phys. Lett. A 71 155
[9] Wang J, Chen Z, Yuan Z 2006 Chin. Phys. 15 1216
[10] Hua C C, Guan X P 2004 Chin. Phys. 13 1391
[11] Leonov G, Bunin A, Koksch N 1987 Z. Angrew. Math. Mech. 67 649
[12] Zhou T, Tang Y, Chen G R 2003 Int. J. Bifurc. Chaos. 13 2561
[13] Li D M, Lu J A,Wu X Q, Chen G R 2005 Chaos, Soliton. Fract. 23 529
[14] Li D M, Lu J A, Wu X Q, Chen G R 2006 J. Math. Appl. 323 844
[15] Li D M, Wu X Q, Lu J A 2007 Chaos Soliton. Fract. 156 121
[16] Wang P, Li D M, Hu Q L 2010 Commun. Nonlinear Sci. Number Simulat. 15 2514
[17] Zheng Y, Zhang X D 2010 Chin. Phys. B 19 010505
[18] Chen G R, Lv J H 2003 Dynamical analysis, control and synchronization of a class of Lorenz systems (Beijing: Science Press) p9(in Chinese) [陈关荣、吕金虎 2003 Lorenz系统族的动力学分析、控制与同步(北京:科学出版社)第9页]
[19] Min L Q, Zhang X D, Chen G R 2005 Int. J. Bifurc. Chaos. 15 119
[20] Chen Z Q, Yang Y, Qi G Y, Yuan Z Z 2007 Phys. Lett. A 360 696
[21] Qi G Y, Chen G R, Du S, Chen Z Q, Yuan Z 2005 Physic. A 352 295
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[1] Lorenz E N 1963 J. Atmos. Sci. 20 130
[2] Lü J H, Chen G R 2002 Int. J. Bifurc. Chaos. 12 659
[3] Zhang X D, Wang Z, Zhao P D 2008 Chin. Phys. Lett. 25 397
[4] Zhao P D, Zhang X D 2008 Acta Phys. Sin. 57 2791 (in Chinese) [赵品栋、张晓丹 2008 57 2791 ]
[5] Cenys A, Tamaservicius A, Baziliauskas A, Krivickas R, Lindberg E 2003 Chaos, Soliton. Fract. 17 349
[6] Zhang X D, Liu X, Zhao P D 2009 Acta Phys. Sin. 58 4415 (in Chinese) [张晓丹、刘 翔、赵品栋 2009 58 4415]
[7] Cafagna D, Grassi G 2003 Int. J. Bifurc. Chaos. 13 2889
[8] Rossler O E 1979 Phys. Lett. A 71 155
[9] Wang J, Chen Z, Yuan Z 2006 Chin. Phys. 15 1216
[10] Hua C C, Guan X P 2004 Chin. Phys. 13 1391
[11] Leonov G, Bunin A, Koksch N 1987 Z. Angrew. Math. Mech. 67 649
[12] Zhou T, Tang Y, Chen G R 2003 Int. J. Bifurc. Chaos. 13 2561
[13] Li D M, Lu J A,Wu X Q, Chen G R 2005 Chaos, Soliton. Fract. 23 529
[14] Li D M, Lu J A, Wu X Q, Chen G R 2006 J. Math. Appl. 323 844
[15] Li D M, Wu X Q, Lu J A 2007 Chaos Soliton. Fract. 156 121
[16] Wang P, Li D M, Hu Q L 2010 Commun. Nonlinear Sci. Number Simulat. 15 2514
[17] Zheng Y, Zhang X D 2010 Chin. Phys. B 19 010505
[18] Chen G R, Lv J H 2003 Dynamical analysis, control and synchronization of a class of Lorenz systems (Beijing: Science Press) p9(in Chinese) [陈关荣、吕金虎 2003 Lorenz系统族的动力学分析、控制与同步(北京:科学出版社)第9页]
[19] Min L Q, Zhang X D, Chen G R 2005 Int. J. Bifurc. Chaos. 15 119
[20] Chen Z Q, Yang Y, Qi G Y, Yuan Z Z 2007 Phys. Lett. A 360 696
[21] Qi G Y, Chen G R, Du S, Chen Z Q, Yuan Z 2005 Physic. A 352 295
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