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Synchronization of hyperchaotic systems via improved impulsive control method

Ma Tie-Dong Jiang Wei-Bo Fu Jie Xue Fang-Zheng

Synchronization of hyperchaotic systems via improved impulsive control method

Ma Tie-Dong, Jiang Wei-Bo, Fu Jie, Xue Fang-Zheng
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  • The improved impulsive control method is proposed to realize the complete synchronization of integral and fractional order hyperchaotic systems. Some effective sufficient conditions are produced to realize the asymptotical stability of synchronization error system. In particular, some simple and practical conditions are derived in synchronizing the chaotic systems by choosing constant impulsive distances and control gains. Compared with the existing results, the main results are less conservative by relaxing some unnecessary inequality constraints. Simulation results show the effectiveness and the feasibility of the proposed impulsive controller.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61104080), the Natural Science Foundation of Chongqing, China (Grant No. CSTC, 2010BB2238), Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100191120025), and the China Postdoctoral Science Foundation (Grant Nos. 20100470813, 20100480043).
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    Mandelbrot B B 1983 The Fractal Geometry of Nature (New York: Freeman)

    [2]

    Hartley T T, Lorenzo C F, Qammer H K 1995 IEEE Trans. CAS-I 42 485

    [3]

    Arena P, Caponetto R, Fortuna L, Porto D 1997 Proceedings ECCTD, Budapest 42 p1259

    [4]

    Ahmad W M, Sprott J C 2003 Chaos, Solitons and Fractals 16 339

    [5]

    Yu Y G, Li H X, Wang S, Yu J Z 2009 Chaos, Solitons and Fractals 42 1181

    [6]

    Lu J G, Chen G R 2006 Chaos, Solitons and Fractals. 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]

    Wang X Y, He Y J 2008 Acta Phys. Sin. 57 1485 (in Chinese) [王兴元, 贺毅杰 2008 物理学报 57 1485]

    [10]

    Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821

    [11]

    Yang D S, Zhang H G, Zhao Y, Song C H, Wang Y C 2010 Acta Phys. Sin. 59 1562 (in Chinese) [杨东升, 张化光, 赵琰, 宋崇辉, 王迎春 2010 物理学报 59 1562]

    [12]

    Zhang H G, Zhao Y, Yu W, Yang D S 2008 Chin. Phys. B 17 4056

    [13]

    Zhang H G, Huang W, Wang Z L, Chai T Y 2006 Phys. Lett. A 350 363

    [14]

    Zhang H G, Wang Z L, Liu D R 2004 Int. J. Bifurcat. Chaos 14 3505

    [15]

    Zhao Y, Zhang H G, Zheng C D 2008 Chin. Phys. B 17 529

    [16]

    Sun Q Y, Zhang H G, Zhao Y 2010 Chin. Phys. B 19 070512

    [17]

    Yang D S, Zhang H G, Li A P, Meng Z Y 2007 Acta Phys. Sin. 56 3121 (in Chinese) [杨东升, 张化光, 李爱平, 孟子怡 2007 物理学报 56 3121]

    [18]

    Wang Y C, Zhang H G, Wang X Y, Yang D S 2010 IEEE Trans. Syst. Man Cybern. B 40 1468

    [19]

    Bhalekar S, Daftardar-Gejji V 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 3536

    [20]

    Taghvafard H, Erjaee G H 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 4079

    [21]

    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]

    [22]

    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]

    [23]

    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]

    [24]

    Odibat Z M 2010 Nonlinear Dyn. 60 479

    [25]

    Wu C J, Zhang Y B, Yang N N 2011 Chin. Phys. B 20 060505

    [26]

    Wang X Y, Zhang Y L, Li D, Zhang N 2011 Chin. Phys. B 20 030506

    [27]

    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

    [28]

    Zhang H G, Ma T D, Huang G B, Wang Z L 2010 IEEE Trans. Syst. Man Cybern. B 40 831

    [29]

    Ma T D, Fu J, Sun Y 2010 Chin. Phys. B 19 090502

    [30]

    Zhang H G, Ma T D, Yu W, Fu J 2008 Chin. Phys. B 17 3616

    [31]

    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]

    [32]

    Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3742

    [33]

    Zhang H G, Fu J, Ma T D 2009 Chin. Phys. B 18 969

    [34]

    Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3751

    [35]

    Gao T G, Chen Z Q, Yuan Z Z, Yu D C 2007 Chaos, Solitons and Fractals 33 922

    [36]

    Wu Z M, Xie J Y 2007 Chin. Phys. 16 1901

    [37]

    Fu J, Yu M, Ma T D 2011 Chin. Phys. B 20 120508

    [38]

    Podlubny I 1999 Fractional Differential Equations (New York: Academic)

  • [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 Trans. CAS-I 42 485

    [3]

    Arena P, Caponetto R, Fortuna L, Porto D 1997 Proceedings ECCTD, Budapest 42 p1259

    [4]

    Ahmad W M, Sprott J C 2003 Chaos, Solitons and Fractals 16 339

    [5]

    Yu Y G, Li H X, Wang S, Yu J Z 2009 Chaos, Solitons and Fractals 42 1181

    [6]

    Lu J G, Chen G R 2006 Chaos, Solitons and Fractals. 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]

    Wang X Y, He Y J 2008 Acta Phys. Sin. 57 1485 (in Chinese) [王兴元, 贺毅杰 2008 物理学报 57 1485]

    [10]

    Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821

    [11]

    Yang D S, Zhang H G, Zhao Y, Song C H, Wang Y C 2010 Acta Phys. Sin. 59 1562 (in Chinese) [杨东升, 张化光, 赵琰, 宋崇辉, 王迎春 2010 物理学报 59 1562]

    [12]

    Zhang H G, Zhao Y, Yu W, Yang D S 2008 Chin. Phys. B 17 4056

    [13]

    Zhang H G, Huang W, Wang Z L, Chai T Y 2006 Phys. Lett. A 350 363

    [14]

    Zhang H G, Wang Z L, Liu D R 2004 Int. J. Bifurcat. Chaos 14 3505

    [15]

    Zhao Y, Zhang H G, Zheng C D 2008 Chin. Phys. B 17 529

    [16]

    Sun Q Y, Zhang H G, Zhao Y 2010 Chin. Phys. B 19 070512

    [17]

    Yang D S, Zhang H G, Li A P, Meng Z Y 2007 Acta Phys. Sin. 56 3121 (in Chinese) [杨东升, 张化光, 李爱平, 孟子怡 2007 物理学报 56 3121]

    [18]

    Wang Y C, Zhang H G, Wang X Y, Yang D S 2010 IEEE Trans. Syst. Man Cybern. B 40 1468

    [19]

    Bhalekar S, Daftardar-Gejji V 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 3536

    [20]

    Taghvafard H, Erjaee G H 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 4079

    [21]

    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]

    [22]

    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]

    [23]

    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]

    [24]

    Odibat Z M 2010 Nonlinear Dyn. 60 479

    [25]

    Wu C J, Zhang Y B, Yang N N 2011 Chin. Phys. B 20 060505

    [26]

    Wang X Y, Zhang Y L, Li D, Zhang N 2011 Chin. Phys. B 20 030506

    [27]

    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

    [28]

    Zhang H G, Ma T D, Huang G B, Wang Z L 2010 IEEE Trans. Syst. Man Cybern. B 40 831

    [29]

    Ma T D, Fu J, Sun Y 2010 Chin. Phys. B 19 090502

    [30]

    Zhang H G, Ma T D, Yu W, Fu J 2008 Chin. Phys. B 17 3616

    [31]

    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]

    [32]

    Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3742

    [33]

    Zhang H G, Fu J, Ma T D 2009 Chin. Phys. B 18 969

    [34]

    Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3751

    [35]

    Gao T G, Chen Z Q, Yuan Z Z, Yu D C 2007 Chaos, Solitons and Fractals 33 922

    [36]

    Wu Z M, Xie J Y 2007 Chin. Phys. 16 1901

    [37]

    Fu J, Yu M, Ma T D 2011 Chin. Phys. B 20 120508

    [38]

    Podlubny I 1999 Fractional Differential Equations (New York: Academic)

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    [3] Hu Jian-Bing, Han Yan, Zhao Ling-Dong. Synchronizing fractional chaotic systems based on Lyapunov equation. Acta Physica Sinica, 2008, 57(12): 7522-7526. doi: 10.7498/aps.57.7522
    [4] Zhang Xiao-Dan, Cui Li-Juan. The bound for a class of four-dimensional hyperchaotic system and its synchronization. Acta Physica Sinica, 2011, 60(11): 110511. doi: 10.7498/aps.60.110511
    [5] Wang Shi-Long, Li Dong, Zhang Xiao-Hong, Yang Dan. Fuzzy impulsive control of chaos in permanent magnet synchronous motors with parameter uncertainties. Acta Physica Sinica, 2009, 58(5): 2939-2948. doi: 10.7498/aps.58.2939
    [6] Zhang Ruo-Xun, Cao He-Fei. Adaptive synchronization of fractional-order chaotic system via sliding mode control. Acta Physica Sinica, 2011, 60(5): 050510. doi: 10.7498/aps.60.050510
    [7] Huang Li-Lian, Qi Xue. The synchronization of fractional order chaotic systems with different orders based on adaptive sliding mode control. Acta Physica Sinica, 2013, 62(8): 080507. doi: 10.7498/aps.62.080507
    [8] Chen Ye, Li Sheng-Gang, Liu Heng. Synchronization of fractional-order chaotic systems based on adaptive fuzzy control. Acta Physica Sinica, 2016, 65(17): 170501. doi: 10.7498/aps.65.170501
    [9] Ma Tie-Dong, Zhang Hua-Guang, Wang Zhi-Liang. Impulsive synchronization for unified chaotic systems with channel time-delay and parameter uncertainty. Acta Physica Sinica, 2007, 56(7): 3796-3802. doi: 10.7498/aps.56.3796
    [10] Liu Ding, Yan Xiao-Mei. Projective synchronization of fractional-order chaotic systems based on sliding mode control. Acta Physica Sinica, 2009, 58(6): 3747-3752. doi: 10.7498/aps.58.3747
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  • Received Date:  21 September 2011
  • Accepted Date:  28 May 2012
  • Published Online:  20 May 2012

Synchronization of hyperchaotic systems via improved impulsive control method

  • 1. College of Automation, Chongqing University, Chongqing 400044, China;
  • 2. Key Laboratory of Optoelectronic Technology and System, Ministry of Education, College of Optoelectronic Engineering, Chongqing University, Chongqing 400044, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 61104080), the Natural Science Foundation of Chongqing, China (Grant No. CSTC, 2010BB2238), Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100191120025), and the China Postdoctoral Science Foundation (Grant Nos. 20100470813, 20100480043).

Abstract: The improved impulsive control method is proposed to realize the complete synchronization of integral and fractional order hyperchaotic systems. Some effective sufficient conditions are produced to realize the asymptotical stability of synchronization error system. In particular, some simple and practical conditions are derived in synchronizing the chaotic systems by choosing constant impulsive distances and control gains. Compared with the existing results, the main results are less conservative by relaxing some unnecessary inequality constraints. Simulation results show the effectiveness and the feasibility of the proposed impulsive controller.

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