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Realization of synchronization between hyperchaotic systems by using a scheme of intermittent linear coupling

Ma Jun Wu Xin-Yi Qin Hui-Xin

Realization of synchronization between hyperchaotic systems by using a scheme of intermittent linear coupling

Ma Jun, Wu Xin-Yi, Qin Hui-Xin
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  • Based on the Lyapunov stability theory, it is confirmed that complete synchronization can be realized under intermittent linear coupling. The linear controller is selected as ‘stop’ or ‘on control’ by using a switch function; while the first switch function is realized by using a rectangular wave with the same amplitude, and the controller turns on/off in the peiod Ta, Tb alternately. The second switch function is adjusted by a square wave with the same amplitude, and the interval period is marked as T0. At first, a class of exponential Lyapunov function is designed to discuss the reliability and possibility of complete synchronization induced by indirectional linear coupling when the controller is adjusted by two types of switch function. The averaged power consumption of controller within a transient period is defined to measure the cost and efficiency of this scheme. In numerical studies, for the case of first switch function (rectangular wave), the distribution of the largest conditional Lyapunov function for the response system is calculated in the two-parameter space for interval period Ta vs. Tb, the synchronization area vs. nonsynchronization area, the distribution of averaged power consumption in the parameter space Ta vs. Tb. It is also confirmed that complete synchronization can be reached at appropriate Ta, Tb, and coupling intensity. In the case of the second switch function, the distribution of the largest conditional Lyapunov function for the response system is calculated in the two-parameter space for coupling intensity k vs. interval period T0, and the series of error function and averaged power consumption. It is found that complete synchronization can be realized at appropriate coupling intensity and interval period T0. It is also found that the averaged power consumption of controller within a transient period can reach a smallest value at an appropriate coupling intensity. Numerical results are consistent with the theoretical analysis.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11265008).
    [1]

    Boccaletti S, Grebogi C, Lai Y C 2000 Phys. Rep. 329 103

    [2]

    Perc M, Marhl M 2003 Biophys Chem. 104 509

    [3]

    Li Q D, Yang X S 2003 Electron Lett. 39 1306

    [4]

    Kodba S, Perc M, Marhl M 2005 Eur. J. Phys. 26 205

    [5]

    Krese B, Perc M, Govekar E 2010 Chaos 20 013129

    [6]

    Alsing P M, Gavrielides A, Kovanis V 1997 Phys. Rev. E 56 6302

    [7]

    VanWiggeren G D, Roy R 1998 Science 279 1198

    [8]

    Xia W, Cao J D 2008 Chaos 18 023128

    [9]

    Wu D, Li J J 2010 Chin. Phys. B 19 120505

    [10]

    Wang X Y, Zhang N, Ren X L 2011 Chin. Phys. B 20 020507

    [11]

    Boccaletti S, Kurths J, Osipov G 2002 Phys. Rep. 366 1

    [12]

    DeShazer D J, Breban R, Ott E 2004 Int. J. Bifurcat Chaos 14 3205

    [13]

    Lu J G, Xi Y G, Wang X F 2004 Int. J. Bifurcat Chaos 14 1431

    [14]

    Kim M Y, Sramek C, Uchida A 2006 Phys. Rev. E 74 016211

    [15]

    Lu J G, Hill D J 2008 IEEE Trans Circ. Syst. II 55 586

    [16]

    Cao J D, Ho W C, Yang Y 2009 Phys. Lett. A 373 3128

    [17]

    Lu J, Cao J D, Ho W C 2008 IEEE Trans Circ. Syst. I 55 1347

    [18]

    Yu W, Cao J D 2007 Physica A 375 467

    [19]

    Guan J B 2010 Chin. Phys. Lett. 27 020502

    [20]

    Feng Y F, Zhang Q L 2010 Chin. Phys. B 19 120504

    [21]

    Li S Y, Ge Z M 2011 Nonlinear Dynam 64 77

    [22]

    Wang Z L, Shi X R 2011 Commun. Nonlinear Sci. Numer Simulat 16 46

    [23]

    Wang C N, Ma J, Jin W Y 2012 Dynam Syst. 27253

    [24]

    Wang T B, Qin T F, Chen G Z 2001 Acta Phys. Sin. 50 1851 (in Chinese) [王铁邦, 覃团发, 陈光旨 2001 物理学报 50 1851]

    [25]

    Jiang P Q, Luo X S, Wang B H 2002 Acta Phys. Sin. 51 1937 (in Chinese) [蒋品群, 罗晓曙, 汪秉宏 2002 物理学报 51 1937]

    [26]

    Ma J, Liao G H, Mo X H 2005 Acta Phys. Sin. 54 5585 (in Chinese) [马军, 廖高华, 莫晓华 2005 物理学报 54 5585]

    [27]

    Sarasola C, Torrealdea F J, d’Anjou A 2002 Math Comput Simulat 58 309

    [28]

    Li F, Jin W Y, Ma J 2012 Acta Phys. Sin. 61 240501 (in Chinese) [李凡, 靳伍银, 马军 2012 物理学报 61 240501]

    [29]

    Tamaševičius A, Namajūnas A, Čenys A 1996 Electron Lett. 32 957

    [30]

    Yalçin M E 2007 Chaos, Solitons & Fractals 34 1659

    [31]

    Li N, Li J F 2011 Acta Phys. Sin. 60 110512 (in Chinese) [李农, 李建芬 2011 物理学报 60 110512]

    [32]

    L J H, Chen G R, Yu X G, Leung H 2004 IEEE Trans Circ. Sys. I 51 2476

    [33]

    Yu S M, Lin Q H, Qiu S S 2003 Acta Phys. Sin. 52 25 (in Chinese) [禹思敏, 林清华, 丘水生 2003 物理学报 52 25]

    [34]

    Yu S M 2005 Acta Phys. Sin. 54 1500 (in Chinese) [禹思敏 2005 物理学报 54 1500]

    [35]

    Wang F Q, Liu C X, Lu J J 2006 Acta Phys. Sin. 55 3289 (in Chinese) [王发强, 刘崇新, 逯俊杰 2006 物理学报 55 3289]

    [36]

    L J H, Chen G R 2006 Int. J. Bifurcat. Chaos 16 775

    [37]

    Wang F Q, Liu C X 2007 Acta Phys. Sin. 56 1983 (in Chinese) [王发强, 刘崇新 2007 物理学报 56 1983 ]

    [38]

    Chen L, Peng H J, Wang D S 2008 Acta Phys. Sin. 57 3337 (in Chinese) [谌 龙, 彭海军, 王德石 2008 物理学报 57 3337 ]

    [39]

    Hu G S 2009 Acta Phys. Sin. 58 3734 (in Chinese) [胡国四 2009 物理学报 58 3734 ]

    [40]

    Bao B C, Liu Z, Xu J P, Zhu L 2010 Acta Phys. Sin. 59 1540 (in Chinese) [包伯成, 刘中, 许建平, 朱雷 2010 物理学报 59 1540]

    [41]

    Chen S B, Ceng Y C, Xu M L, Chen J S 2011 Acta Phys. Sin. 60 020507 (in Chinese) [陈仕必, 曾以成, 徐茂林, 陈家胜 2011 物理学报 60 020507]

    [42]

    Lin Y, Wang C Y, Xu H 2012 Acta Phys. Sin. 61 240503 (in Chinese) [林愿, 王春华, 徐浩 2012 物理学报 61 240503]

  • [1]

    Boccaletti S, Grebogi C, Lai Y C 2000 Phys. Rep. 329 103

    [2]

    Perc M, Marhl M 2003 Biophys Chem. 104 509

    [3]

    Li Q D, Yang X S 2003 Electron Lett. 39 1306

    [4]

    Kodba S, Perc M, Marhl M 2005 Eur. J. Phys. 26 205

    [5]

    Krese B, Perc M, Govekar E 2010 Chaos 20 013129

    [6]

    Alsing P M, Gavrielides A, Kovanis V 1997 Phys. Rev. E 56 6302

    [7]

    VanWiggeren G D, Roy R 1998 Science 279 1198

    [8]

    Xia W, Cao J D 2008 Chaos 18 023128

    [9]

    Wu D, Li J J 2010 Chin. Phys. B 19 120505

    [10]

    Wang X Y, Zhang N, Ren X L 2011 Chin. Phys. B 20 020507

    [11]

    Boccaletti S, Kurths J, Osipov G 2002 Phys. Rep. 366 1

    [12]

    DeShazer D J, Breban R, Ott E 2004 Int. J. Bifurcat Chaos 14 3205

    [13]

    Lu J G, Xi Y G, Wang X F 2004 Int. J. Bifurcat Chaos 14 1431

    [14]

    Kim M Y, Sramek C, Uchida A 2006 Phys. Rev. E 74 016211

    [15]

    Lu J G, Hill D J 2008 IEEE Trans Circ. Syst. II 55 586

    [16]

    Cao J D, Ho W C, Yang Y 2009 Phys. Lett. A 373 3128

    [17]

    Lu J, Cao J D, Ho W C 2008 IEEE Trans Circ. Syst. I 55 1347

    [18]

    Yu W, Cao J D 2007 Physica A 375 467

    [19]

    Guan J B 2010 Chin. Phys. Lett. 27 020502

    [20]

    Feng Y F, Zhang Q L 2010 Chin. Phys. B 19 120504

    [21]

    Li S Y, Ge Z M 2011 Nonlinear Dynam 64 77

    [22]

    Wang Z L, Shi X R 2011 Commun. Nonlinear Sci. Numer Simulat 16 46

    [23]

    Wang C N, Ma J, Jin W Y 2012 Dynam Syst. 27253

    [24]

    Wang T B, Qin T F, Chen G Z 2001 Acta Phys. Sin. 50 1851 (in Chinese) [王铁邦, 覃团发, 陈光旨 2001 物理学报 50 1851]

    [25]

    Jiang P Q, Luo X S, Wang B H 2002 Acta Phys. Sin. 51 1937 (in Chinese) [蒋品群, 罗晓曙, 汪秉宏 2002 物理学报 51 1937]

    [26]

    Ma J, Liao G H, Mo X H 2005 Acta Phys. Sin. 54 5585 (in Chinese) [马军, 廖高华, 莫晓华 2005 物理学报 54 5585]

    [27]

    Sarasola C, Torrealdea F J, d’Anjou A 2002 Math Comput Simulat 58 309

    [28]

    Li F, Jin W Y, Ma J 2012 Acta Phys. Sin. 61 240501 (in Chinese) [李凡, 靳伍银, 马军 2012 物理学报 61 240501]

    [29]

    Tamaševičius A, Namajūnas A, Čenys A 1996 Electron Lett. 32 957

    [30]

    Yalçin M E 2007 Chaos, Solitons & Fractals 34 1659

    [31]

    Li N, Li J F 2011 Acta Phys. Sin. 60 110512 (in Chinese) [李农, 李建芬 2011 物理学报 60 110512]

    [32]

    L J H, Chen G R, Yu X G, Leung H 2004 IEEE Trans Circ. Sys. I 51 2476

    [33]

    Yu S M, Lin Q H, Qiu S S 2003 Acta Phys. Sin. 52 25 (in Chinese) [禹思敏, 林清华, 丘水生 2003 物理学报 52 25]

    [34]

    Yu S M 2005 Acta Phys. Sin. 54 1500 (in Chinese) [禹思敏 2005 物理学报 54 1500]

    [35]

    Wang F Q, Liu C X, Lu J J 2006 Acta Phys. Sin. 55 3289 (in Chinese) [王发强, 刘崇新, 逯俊杰 2006 物理学报 55 3289]

    [36]

    L J H, Chen G R 2006 Int. J. Bifurcat. Chaos 16 775

    [37]

    Wang F Q, Liu C X 2007 Acta Phys. Sin. 56 1983 (in Chinese) [王发强, 刘崇新 2007 物理学报 56 1983 ]

    [38]

    Chen L, Peng H J, Wang D S 2008 Acta Phys. Sin. 57 3337 (in Chinese) [谌 龙, 彭海军, 王德石 2008 物理学报 57 3337 ]

    [39]

    Hu G S 2009 Acta Phys. Sin. 58 3734 (in Chinese) [胡国四 2009 物理学报 58 3734 ]

    [40]

    Bao B C, Liu Z, Xu J P, Zhu L 2010 Acta Phys. Sin. 59 1540 (in Chinese) [包伯成, 刘中, 许建平, 朱雷 2010 物理学报 59 1540]

    [41]

    Chen S B, Ceng Y C, Xu M L, Chen J S 2011 Acta Phys. Sin. 60 020507 (in Chinese) [陈仕必, 曾以成, 徐茂林, 陈家胜 2011 物理学报 60 020507]

    [42]

    Lin Y, Wang C Y, Xu H 2012 Acta Phys. Sin. 61 240503 (in Chinese) [林愿, 王春华, 徐浩 2012 物理学报 61 240503]

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  • Received Date:  18 March 2013
  • Accepted Date:  15 May 2013
  • Published Online:  05 September 2013

Realization of synchronization between hyperchaotic systems by using a scheme of intermittent linear coupling

  • 1. Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 11265008).

Abstract: Based on the Lyapunov stability theory, it is confirmed that complete synchronization can be realized under intermittent linear coupling. The linear controller is selected as ‘stop’ or ‘on control’ by using a switch function; while the first switch function is realized by using a rectangular wave with the same amplitude, and the controller turns on/off in the peiod Ta, Tb alternately. The second switch function is adjusted by a square wave with the same amplitude, and the interval period is marked as T0. At first, a class of exponential Lyapunov function is designed to discuss the reliability and possibility of complete synchronization induced by indirectional linear coupling when the controller is adjusted by two types of switch function. The averaged power consumption of controller within a transient period is defined to measure the cost and efficiency of this scheme. In numerical studies, for the case of first switch function (rectangular wave), the distribution of the largest conditional Lyapunov function for the response system is calculated in the two-parameter space for interval period Ta vs. Tb, the synchronization area vs. nonsynchronization area, the distribution of averaged power consumption in the parameter space Ta vs. Tb. It is also confirmed that complete synchronization can be reached at appropriate Ta, Tb, and coupling intensity. In the case of the second switch function, the distribution of the largest conditional Lyapunov function for the response system is calculated in the two-parameter space for coupling intensity k vs. interval period T0, and the series of error function and averaged power consumption. It is found that complete synchronization can be realized at appropriate coupling intensity and interval period T0. It is also found that the averaged power consumption of controller within a transient period can reach a smallest value at an appropriate coupling intensity. Numerical results are consistent with the theoretical analysis.

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