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In this paper, a novel method, to synchronize two identical or different chaotic/hyperchaotic systems with variable coefficients, is proposed based on the strictly positive real transfer function matrix. By adding a synchronization controller to the response system, the nonlinear parts of the error system derived from the synchronized systems are identified as the inputs of the error system, and the error state variables are identified as the outputs of the error system. Then the transfer function matrix of the error system can be strictly positive real. As a result, the error system can be asymptotically stable at the origin, i.e., the two chaotic/hyperchaotic systems can reach stable synchronization. Moreover, the designed synchronization controllers are linear, clear in parameter selections and robust to the changes of the coefficients of the error system. The specific design processes of the synchronization controllers and the corresponding results are presented in the paper. Also, the numerical simulation results are given to verify the feasibility and effectiveness of this method.
[1] Lorenz E N 1963 Atmos. J. Sci. 20 130
[2] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[3] Pyragas K 1993 Phys. Lett. A 181 203
[4] Wu C J, Zhang Y B, Yang N N 2011 Chin. Phys. B 20 060505
[5] Kocarev L, Parlitz U 1995 Phys. Rev. Lett. 74 5028
[6] Roy R, Thornburg K S J 1994 Phys. Rev. Lett. 72 2009
[7] Huberman B A, Lumer E 1990 IEEE Trans. CAS 37 547
[8] Liu F C, Li J Y, Zang X F 2011 Acta Phys. Sin. 60 030504 (in Chinese) [刘福才, 李俊义, 臧秀凤 2011 60 030504]
[9] Yang T, Chua L O 1997 IEEE Trans. CAS 44 976
[10] Khalil H K (translated by Zhu Y S) 2005 Nonlinear Systems (3rd Ed.) (Beijing: Publishing House of Electronics Industry) pp173-176 (in Chinese) [哈里尔著 (朱义胜译) 2005 非线性系统 (第三版) (北京: 电子工业出版社) 第173–176页]
[11] Xia C Y 1998 Adaptive control of AC and DC drive systems (Beijing: China Machine Press) p21 (in Chinese) [夏超英 1998 交直流传动系统的自适应控制 (北京: 机械工业出版社) 第21页]
[12] Liao X X, Chen G R 2003 Control Theory & Applications 20 253
[13] Liao X X, Fu Y L, Xie S L 2005 Sci. China Ser. F 48 304
[14] Tanaka T, Langbort C 2011 IEEE T Automat. Contr. 56 2218
[15] Liu B, Tang W S 2006 Modern Control Theory (3rd Ed.) (Beijing: China Machine Press) pp161-165 (in Chinese) [刘豹, 唐万生 2006 现代控制理论 (第3版) (北京: 机械工业出版社) 第161–165页]
[16] Liu B Z, Peng J H 2007 Nonlinear Dynamics (Beijing: Higher Education Press) p147 (in Chinese) [刘秉正, 彭建华 2007 非线性动力学 (北京: 高等教育出版社) 第147页]
[17] L J H, Chen G R, Cheng D Z, Celikovsky S 2002 Int. J. Bifur. Chaos 12 2917
[18] Chen G R, L J H 2003 Dynamic Analysis, Control and Synchronization of Lorenz System Families (Beijing: Science Press) pp131-149 (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131–149页]
[19] Chen G R, Ueta T 1999 Int. J. Bifur. Chaos 9 1465
[20] L J H, Chen G R 2002 Int. J. Bifur. Chaos 12 659
[21] Zhang G S, Niu H 2012 Acta Phys. Sin. 61 110503 (in Chinese) [张国山, 牛弘 2012 61 110503]
[22] Wang X Y, Zhao G B 2010 Int. J. Mod. Phys. B 24 4619
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[1] Lorenz E N 1963 Atmos. J. Sci. 20 130
[2] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[3] Pyragas K 1993 Phys. Lett. A 181 203
[4] Wu C J, Zhang Y B, Yang N N 2011 Chin. Phys. B 20 060505
[5] Kocarev L, Parlitz U 1995 Phys. Rev. Lett. 74 5028
[6] Roy R, Thornburg K S J 1994 Phys. Rev. Lett. 72 2009
[7] Huberman B A, Lumer E 1990 IEEE Trans. CAS 37 547
[8] Liu F C, Li J Y, Zang X F 2011 Acta Phys. Sin. 60 030504 (in Chinese) [刘福才, 李俊义, 臧秀凤 2011 60 030504]
[9] Yang T, Chua L O 1997 IEEE Trans. CAS 44 976
[10] Khalil H K (translated by Zhu Y S) 2005 Nonlinear Systems (3rd Ed.) (Beijing: Publishing House of Electronics Industry) pp173-176 (in Chinese) [哈里尔著 (朱义胜译) 2005 非线性系统 (第三版) (北京: 电子工业出版社) 第173–176页]
[11] Xia C Y 1998 Adaptive control of AC and DC drive systems (Beijing: China Machine Press) p21 (in Chinese) [夏超英 1998 交直流传动系统的自适应控制 (北京: 机械工业出版社) 第21页]
[12] Liao X X, Chen G R 2003 Control Theory & Applications 20 253
[13] Liao X X, Fu Y L, Xie S L 2005 Sci. China Ser. F 48 304
[14] Tanaka T, Langbort C 2011 IEEE T Automat. Contr. 56 2218
[15] Liu B, Tang W S 2006 Modern Control Theory (3rd Ed.) (Beijing: China Machine Press) pp161-165 (in Chinese) [刘豹, 唐万生 2006 现代控制理论 (第3版) (北京: 机械工业出版社) 第161–165页]
[16] Liu B Z, Peng J H 2007 Nonlinear Dynamics (Beijing: Higher Education Press) p147 (in Chinese) [刘秉正, 彭建华 2007 非线性动力学 (北京: 高等教育出版社) 第147页]
[17] L J H, Chen G R, Cheng D Z, Celikovsky S 2002 Int. J. Bifur. Chaos 12 2917
[18] Chen G R, L J H 2003 Dynamic Analysis, Control and Synchronization of Lorenz System Families (Beijing: Science Press) pp131-149 (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131–149页]
[19] Chen G R, Ueta T 1999 Int. J. Bifur. Chaos 9 1465
[20] L J H, Chen G R 2002 Int. J. Bifur. Chaos 12 659
[21] Zhang G S, Niu H 2012 Acta Phys. Sin. 61 110503 (in Chinese) [张国山, 牛弘 2012 61 110503]
[22] Wang X Y, Zhao G B 2010 Int. J. Mod. Phys. B 24 4619
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