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In this paper, a novel impulsive control method based on comparison system is proposed to realize complete synchronization of a class of fractional order chaotic systems. By constructing the suitable response system, the original fractional order error system can be converted into the integral order one. Based on the theory of Lyapunov stability and impulsive differential equations, some effective sufficient conditions are derived to guarantee the asymptotical stability of synchronization error system. In particular, some simpler and more convenient conditions are derived by taking the same impulsive distances and control gains. Compared with the existing results, the main results in this paper are more practical and rigorous. Simulation results for fractional order Chen system show the effectiveness and the feasibility of the proposed impulsive control method.
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Keywords:
- fractional order chaotic systems /
- chaos synchronization /
- impulsive control /
- comparison system
[1] Mandelbrot B B 1983 The Fractal Geometry of Nature (New York: Freeman)
[2] Hartley T T, Lorenzo C F, Qammer H K 1995 IEEE Transactions CAS-I 42 485
[3] Arena P, Caponetto R, Fortuna L, Porto D 1997 In: Proceedings ECCTD, Budapest 42 1259
[4] Ahmad W M, Sprott J C 2003 Chaos Solitons and Fract. 16 339
[5] Li C P, Peng G J 2004 Chaos Solitons and Fract. 22 443
[6] Lu J G, Chen G R 2006 Chaos Solitons and Fract. 27 685
[7] Lu J G 2006 Phys. Lett. A 354 305
[8] Li C G, Chen G R 2004 Physica A 341 55
[9] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[10] Bhalekar S, Daftardar-Gejji V 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 3536
[11] Taghvafard H, Erjaee G H 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 4079
[12] Cao H F, Zhang R X 2011 Acta Phys. Sin. 60 050510 (in Chinese) [曹鹤飞, 张若洵 2011 物理学报 60 050510]
[13] Sun N, Zhang H G, Wang Z L 2011 Acta Phys. Sin. 60 050511 (in Chinese) [孙宁, 张化光, 王智良 2011 物理学报 60 050511]
[14] Zhao L D, Hu J B, Liu X H 2010 Acta Phys. Sin. 59 2305 (in Chinese) [赵灵冬, 胡建兵, 刘旭辉 2010 物理学报 59 2305]
[15] Zhang R X, Yang S P 2010 Chin. Phys. B 19 020510
[16] Wu C J, Zhang Y B, Yang N N 2011 Chin. Phys. B 20 060505
[17] Wang X Y, Zhang Y L, Li D, Zhang N 2011 Chin. Phys. B 20 030506
[18] Sheu L J, Tam L M, Lao S K, Kang Y, Lin K T, Chen J H, Chen H K 2009 Int. J. Nonlinear Sci. Numer. Simulat. 10 33
[19] Zhang H G, Ma T D, Huang G B, Wang Z L 2010 IEEE Trans. Syst. Man Cybern. B 40 831
[20] Ma T D, Fu J, Sun Y 2010 Chin. Phys. B 19 090502
[21] Ma T D, Zhang H G, Wang Z L 2007 Acta Phys. Sin. 56 3796 (in Chinese) [马铁东, 张化光, 王智良 2007 物理学报 56 3796]
[22] Zhang H G, Ma T D, Yu W, Fu J 2008 Chin. Phys. B 17 3616
[23] Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3742
[24] Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3751
[25] Yang T 1999 IEEE Trans. Autom. Contr. 44 1081
[26] Yang T 2001 Impulsive Control Theory (Berlin: Spinger-Verlag)
[27] Podlubny I 1999 Fractional Differential Equations (New York: Academic)
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[1] Mandelbrot B B 1983 The Fractal Geometry of Nature (New York: Freeman)
[2] Hartley T T, Lorenzo C F, Qammer H K 1995 IEEE Transactions CAS-I 42 485
[3] Arena P, Caponetto R, Fortuna L, Porto D 1997 In: Proceedings ECCTD, Budapest 42 1259
[4] Ahmad W M, Sprott J C 2003 Chaos Solitons and Fract. 16 339
[5] Li C P, Peng G J 2004 Chaos Solitons and Fract. 22 443
[6] Lu J G, Chen G R 2006 Chaos Solitons and Fract. 27 685
[7] Lu J G 2006 Phys. Lett. A 354 305
[8] Li C G, Chen G R 2004 Physica A 341 55
[9] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[10] Bhalekar S, Daftardar-Gejji V 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 3536
[11] Taghvafard H, Erjaee G H 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 4079
[12] Cao H F, Zhang R X 2011 Acta Phys. Sin. 60 050510 (in Chinese) [曹鹤飞, 张若洵 2011 物理学报 60 050510]
[13] Sun N, Zhang H G, Wang Z L 2011 Acta Phys. Sin. 60 050511 (in Chinese) [孙宁, 张化光, 王智良 2011 物理学报 60 050511]
[14] Zhao L D, Hu J B, Liu X H 2010 Acta Phys. Sin. 59 2305 (in Chinese) [赵灵冬, 胡建兵, 刘旭辉 2010 物理学报 59 2305]
[15] Zhang R X, Yang S P 2010 Chin. Phys. B 19 020510
[16] Wu C J, Zhang Y B, Yang N N 2011 Chin. Phys. B 20 060505
[17] Wang X Y, Zhang Y L, Li D, Zhang N 2011 Chin. Phys. B 20 030506
[18] Sheu L J, Tam L M, Lao S K, Kang Y, Lin K T, Chen J H, Chen H K 2009 Int. J. Nonlinear Sci. Numer. Simulat. 10 33
[19] Zhang H G, Ma T D, Huang G B, Wang Z L 2010 IEEE Trans. Syst. Man Cybern. B 40 831
[20] Ma T D, Fu J, Sun Y 2010 Chin. Phys. B 19 090502
[21] Ma T D, Zhang H G, Wang Z L 2007 Acta Phys. Sin. 56 3796 (in Chinese) [马铁东, 张化光, 王智良 2007 物理学报 56 3796]
[22] Zhang H G, Ma T D, Yu W, Fu J 2008 Chin. Phys. B 17 3616
[23] Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3742
[24] Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3751
[25] Yang T 1999 IEEE Trans. Autom. Contr. 44 1081
[26] Yang T 2001 Impulsive Control Theory (Berlin: Spinger-Verlag)
[27] Podlubny I 1999 Fractional Differential Equations (New York: Academic)
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