Dynamics and Feedback Control for a New Hyperchaotic System
In this paper, stability and Hopf bifurcation analysis of
a novel hyperchaotic system are investigated. Four feedback control
strategies, the linear feedback control method, enhancing feedback
control method, speed feedback control method and delayed feedback
control method, are used to control the hyperchaotic attractor to
unstable equilibrium. Moreover numerical simulations are given to
verify the theoretical results.
[1] E.N. Lorenz, "Deterministic non-periodic flows," J. Atmospheric. Sci.,
vol. 20, pp. 130-141, 1963.
[2] G. Chen, and T. Ueta, "Yet another chaotic attractor," Int. J. Bifur. Chaos,
vol. 9, pp. 1465-1466, 1999.
[3] G. Tigan, "Analysis of a 3D chaotic system," Chaos, Solitons and
Fractals, vol. 36, pp. 1315-1319, 2008.
[4] X. Wang, and M. Wang. "A hyperchaos generated from Lorenz system,"
Physica A, vol. 387, pp. 3751-3758, 2008.
[5] Song Zheng, Gaogao Dong, and Qinsheng Bi, "A new hyperchaotic
system and its synchronization," Applied Mathematics and Computation,
vol. 215, pp. 3192-3200, 2010.
[6] Jaume Llibre, and Xiang Zhang, "On the Hopf-zero bifurcation of the
Michelson system," Nonlinear Analysis: RWA, vol. 12, pp. 1650-1653,
2011.
[7] Song Zheng, Xiujing Han, and Qinsheng Bi, "Bifurcations and fast-slow
behaviors in a hyperchaotic dynamical system," Commun. Nonlinear Sci.
Numer. Simulat., vol. 16, pp. 1998-2005, 2011.
[8] Feng Li, and Yinlai Jin, "Hopf bifurcation analysis and numerical
simulation in a 4D-hyperchaotic system," Nonlinear Dynamics, vol. 67,
pp. 2857-2864, 2012.
[9] Xing He, Yonglu Shu, Chuandong Li, and Huan Jin, "Nonlinear analysis
of a novel three-scroll chaotic system,", J. Appl. Math. Comput., doi:
10.1007/s12190-011-0523-y.
[10] Haojie Yu, Guoliang Cai, and Yuxiu Li, "Dynamic analysis and control
of a new hyperchaotic finance system," Nonlinear Dynamics, vol. 67, pp.
2171-2182, 2012.
[11] Wenguang Yu, "Stabilization of three-dimensional chaotic systems via
single state feedback controller," Physics Letters A, vol. 374, pp. 1488-
1492, 2009.
[12] Chaohai Tao, Chunde Yang, Yan Luo, Hongxia Xiong, and Feng Hu,
"Speed feedback control of chaotic system," Chaos, Solitons and Fractals,
vol.23, pp. 259-263, 2005.
[13] Chunde Yang, Chaohai Tao, and Ping Wang, "Comparison of feedback
control methods for a hyperchaotic Lorenz system," Physics Letters A,
vol. 374, pp. 729-732, 2010.
[14] Wuneng Zhou, Lin Pan, Zhong Li, and Wolfgang A. Halang, "Non-
linear feedback control of a novel chaotic system," International Journal
of Control, Automation, and Systems, vol. 7, pp. 939-944, 2009.
[15] Lin Pan, Wuneng Zhou, and Jian-an Fang, "Dynamics analysis of a new
simple chaotic attractor," International Journal of Control, Automation,
and Systems, vol. 8, pp. 468-472, 2010.
[16] Yue Wu, Xiaobing Zhou, Jia Chen, and Bei Hui, "Chaos synchronization
of a new 3D chaotic system," Chaos, Solitons and Fractals, vol. 42, pp.
1812-1819, 2009.
[17] Wenguang Yu, "Synchronization of three dimensional chaotic systems
via a single state feedback," Commun. Nonlinear Sci. Numer. Simulat.,
vol. 16, pp. 2880-2886, 2011.
[18] Woo-Sik Son, and Young-Jai Park, "Delayed feedback on the dynamical
model of a financial system," Chaos, Solitons and Fractals, vol. 44, pp.
208-217, 2011.
[19] Qin Gao, and Junhai Ma, "Chaos and Hopf bifurcation of a finance
sysem," Nonlinear Dynamics, vol. 58, pp. 209-216, 2009.
[20] Min Xiao, and Jinde Cao, "Bifurcation analysis and chaos control for
L¨u system with delayed feedback," International Journal of Bifurcation
and Chaos, vol. 17, No. 12, pp.4309-4322, 2007.
[21] Shigui Ruan, and Junjie Wei, "On the zeros of transcendental functions
with applications to stability of delay differential equations with two
delays," Dyna. Cont. Disc. Impu. Syst. Series A: Math. Anal., vol. 10,
pp. 863-874, 2003.
[22] B.D. Hassard, N.D. Kazarinoff, and Y.H. Wan, Theory and Applications
of Hopf bifurcation, Cambridge University Press, Cambridge, 1981.
[1] E.N. Lorenz, "Deterministic non-periodic flows," J. Atmospheric. Sci.,
vol. 20, pp. 130-141, 1963.
[2] G. Chen, and T. Ueta, "Yet another chaotic attractor," Int. J. Bifur. Chaos,
vol. 9, pp. 1465-1466, 1999.
[3] G. Tigan, "Analysis of a 3D chaotic system," Chaos, Solitons and
Fractals, vol. 36, pp. 1315-1319, 2008.
[4] X. Wang, and M. Wang. "A hyperchaos generated from Lorenz system,"
Physica A, vol. 387, pp. 3751-3758, 2008.
[5] Song Zheng, Gaogao Dong, and Qinsheng Bi, "A new hyperchaotic
system and its synchronization," Applied Mathematics and Computation,
vol. 215, pp. 3192-3200, 2010.
[6] Jaume Llibre, and Xiang Zhang, "On the Hopf-zero bifurcation of the
Michelson system," Nonlinear Analysis: RWA, vol. 12, pp. 1650-1653,
2011.
[7] Song Zheng, Xiujing Han, and Qinsheng Bi, "Bifurcations and fast-slow
behaviors in a hyperchaotic dynamical system," Commun. Nonlinear Sci.
Numer. Simulat., vol. 16, pp. 1998-2005, 2011.
[8] Feng Li, and Yinlai Jin, "Hopf bifurcation analysis and numerical
simulation in a 4D-hyperchaotic system," Nonlinear Dynamics, vol. 67,
pp. 2857-2864, 2012.
[9] Xing He, Yonglu Shu, Chuandong Li, and Huan Jin, "Nonlinear analysis
of a novel three-scroll chaotic system,", J. Appl. Math. Comput., doi:
10.1007/s12190-011-0523-y.
[10] Haojie Yu, Guoliang Cai, and Yuxiu Li, "Dynamic analysis and control
of a new hyperchaotic finance system," Nonlinear Dynamics, vol. 67, pp.
2171-2182, 2012.
[11] Wenguang Yu, "Stabilization of three-dimensional chaotic systems via
single state feedback controller," Physics Letters A, vol. 374, pp. 1488-
1492, 2009.
[12] Chaohai Tao, Chunde Yang, Yan Luo, Hongxia Xiong, and Feng Hu,
"Speed feedback control of chaotic system," Chaos, Solitons and Fractals,
vol.23, pp. 259-263, 2005.
[13] Chunde Yang, Chaohai Tao, and Ping Wang, "Comparison of feedback
control methods for a hyperchaotic Lorenz system," Physics Letters A,
vol. 374, pp. 729-732, 2010.
[14] Wuneng Zhou, Lin Pan, Zhong Li, and Wolfgang A. Halang, "Non-
linear feedback control of a novel chaotic system," International Journal
of Control, Automation, and Systems, vol. 7, pp. 939-944, 2009.
[15] Lin Pan, Wuneng Zhou, and Jian-an Fang, "Dynamics analysis of a new
simple chaotic attractor," International Journal of Control, Automation,
and Systems, vol. 8, pp. 468-472, 2010.
[16] Yue Wu, Xiaobing Zhou, Jia Chen, and Bei Hui, "Chaos synchronization
of a new 3D chaotic system," Chaos, Solitons and Fractals, vol. 42, pp.
1812-1819, 2009.
[17] Wenguang Yu, "Synchronization of three dimensional chaotic systems
via a single state feedback," Commun. Nonlinear Sci. Numer. Simulat.,
vol. 16, pp. 2880-2886, 2011.
[18] Woo-Sik Son, and Young-Jai Park, "Delayed feedback on the dynamical
model of a financial system," Chaos, Solitons and Fractals, vol. 44, pp.
208-217, 2011.
[19] Qin Gao, and Junhai Ma, "Chaos and Hopf bifurcation of a finance
sysem," Nonlinear Dynamics, vol. 58, pp. 209-216, 2009.
[20] Min Xiao, and Jinde Cao, "Bifurcation analysis and chaos control for
L¨u system with delayed feedback," International Journal of Bifurcation
and Chaos, vol. 17, No. 12, pp.4309-4322, 2007.
[21] Shigui Ruan, and Junjie Wei, "On the zeros of transcendental functions
with applications to stability of delay differential equations with two
delays," Dyna. Cont. Disc. Impu. Syst. Series A: Math. Anal., vol. 10,
pp. 863-874, 2003.
[22] B.D. Hassard, N.D. Kazarinoff, and Y.H. Wan, Theory and Applications
of Hopf bifurcation, Cambridge University Press, Cambridge, 1981.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:50906", author = "Kejun Zhuang and Hailong Zhu", title = "Dynamics and Feedback Control for a New Hyperchaotic System", abstract = "In this paper, stability and Hopf bifurcation analysis of
a novel hyperchaotic system are investigated. Four feedback control
strategies, the linear feedback control method, enhancing feedback
control method, speed feedback control method and delayed feedback
control method, are used to control the hyperchaotic attractor to
unstable equilibrium. Moreover numerical simulations are given to
verify the theoretical results.", keywords = "Feedback control, Hopf bifurcation, hyperchaotic system,
stability.", volume = "6", number = "8", pages = "842-4", }
