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In order to understand the complex three-dimensional dynamical system with the unstable nodes, we propose a nonlinear controller. The corresponding controlling system makes the codimension one, two, and three Hopf bifurcations happen. The mathematical deduction demonstrates that the system can be controlled to produce the degenerate Hopf bifurcation at desired location and stability of controllable bifurcation.
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Keywords:
- chaotic system /
- control /
- Hopf bifurcation /
- Lyapunov coefficient
[1] Lorenz E N 1963 Atmos. J. Sci. 20 130
[2] Rössler O E 1976 Phys. Lett. A 57 397
[3] Chua L O, Komuro M, Matsumoto T 1986 IEEE Trans. Circ. Syst. 33 1072
[4] Sprott J C 1994 Phys. Rev. E 50 647
[5] L J H, Chen G R 2002 Int. J. Bifurcat. Chaos 12 659
[6] L J H, Chen G R, Cheng D Z 2004 Int. J. Bifurcat. Chaos 14 1507
[7] Liu L, Su Y C, Liu C X 2007 Acta Phys. Sin. 56 1966 (in Chinese) [刘凌, 苏燕辰, 刘崇新 2007 物理学报 56 1966]
[8] Yang Q G, Chen G R 2008 Int. J. Bifurcat. Chaos 18 1393
[9] Yang Q G, Wei Z C, Chen G R 2010 Int. J. Bifurcat. Chaos 20 1061
[10] Li Z, Han C Z 2002 Chin. Phys. 11 666
[11] Li R H, Xu W, Li S 2007 Chin. Phys. 16 1591
[12] Wei Z C, Yang Q G 2011 Nonlinear Anal-Real. 12 106
[13] Hua C C, Guan X P 2004 Chin. Phys. Lett. 21 1441
[14] Chen S H, Liu J, Feng J W, L J H 2002 Chin. Phys. Lett. 19 1257
[15] Luo X S, Fang J Q 2000 Chin. Phys. 9 333
[16] Zhang X H, Shen K 1999 Chin. Phys. 8 651
[17] Fang J Q, Chen G R 1999 Chin. Phys. 8 526
[18] Wu W G, Gu T X 2000 Acta Phys. Sin. 49 1922 (in Chinese) [伍维根, 古天祥 2000 物理学报 49 1922]
[19] Chen L Q, Liu Y Z 1996 Physics 25 278 (in Chinese) [陈立群, 刘延柱 1996 物理 25 278]
[20] Chen G R, L J H 2003 Dynamic Analysis, Control and Synchronization of Lorenz System (Vol. 1) (Beijing: Science Press) pp1-130 (in Chinese) [陈关荣, 吕金虎 2003 Lorenz系统族的动力学分析、控制与同步 (第一版) (北京: 科学出版社) 第1–130页]
[21] Kuznetsov Y A 1998 Elements of Applied Bifurcation Theory (New York: Springer-Verlag) pp1-20
[22] Messias M, Braga D C, Mello L F 2009 Int. J. Bifurcat. Chaos 19 497
[23] Wei Z C 2010 Appl. Math. Comput. 217 422
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[1] Lorenz E N 1963 Atmos. J. Sci. 20 130
[2] Rössler O E 1976 Phys. Lett. A 57 397
[3] Chua L O, Komuro M, Matsumoto T 1986 IEEE Trans. Circ. Syst. 33 1072
[4] Sprott J C 1994 Phys. Rev. E 50 647
[5] L J H, Chen G R 2002 Int. J. Bifurcat. Chaos 12 659
[6] L J H, Chen G R, Cheng D Z 2004 Int. J. Bifurcat. Chaos 14 1507
[7] Liu L, Su Y C, Liu C X 2007 Acta Phys. Sin. 56 1966 (in Chinese) [刘凌, 苏燕辰, 刘崇新 2007 物理学报 56 1966]
[8] Yang Q G, Chen G R 2008 Int. J. Bifurcat. Chaos 18 1393
[9] Yang Q G, Wei Z C, Chen G R 2010 Int. J. Bifurcat. Chaos 20 1061
[10] Li Z, Han C Z 2002 Chin. Phys. 11 666
[11] Li R H, Xu W, Li S 2007 Chin. Phys. 16 1591
[12] Wei Z C, Yang Q G 2011 Nonlinear Anal-Real. 12 106
[13] Hua C C, Guan X P 2004 Chin. Phys. Lett. 21 1441
[14] Chen S H, Liu J, Feng J W, L J H 2002 Chin. Phys. Lett. 19 1257
[15] Luo X S, Fang J Q 2000 Chin. Phys. 9 333
[16] Zhang X H, Shen K 1999 Chin. Phys. 8 651
[17] Fang J Q, Chen G R 1999 Chin. Phys. 8 526
[18] Wu W G, Gu T X 2000 Acta Phys. Sin. 49 1922 (in Chinese) [伍维根, 古天祥 2000 物理学报 49 1922]
[19] Chen L Q, Liu Y Z 1996 Physics 25 278 (in Chinese) [陈立群, 刘延柱 1996 物理 25 278]
[20] Chen G R, L J H 2003 Dynamic Analysis, Control and Synchronization of Lorenz System (Vol. 1) (Beijing: Science Press) pp1-130 (in Chinese) [陈关荣, 吕金虎 2003 Lorenz系统族的动力学分析、控制与同步 (第一版) (北京: 科学出版社) 第1–130页]
[21] Kuznetsov Y A 1998 Elements of Applied Bifurcation Theory (New York: Springer-Verlag) pp1-20
[22] Messias M, Braga D C, Mello L F 2009 Int. J. Bifurcat. Chaos 19 497
[23] Wei Z C 2010 Appl. Math. Comput. 217 422
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