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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
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• #### 摘要

基于李亚普诺夫稳定性理论, 严格证明了一类超混沌系统在间歇线性单向耦合下可以实现完全同步. 线性控制器通过一个开关函数来调节来实现‘停’和‘控’. 第一类开关函数由一个等幅度矩形波来控制, 控制器的打开和关闭选取不同的间隔周期(Ta, Tb); 第二类开关函数由一个等幅度方波来控制, 方波间隔周期记为T0; 首先通过构造指数类型的李亚普诺夫论证了两类开关函数调制下两个超混沌 系统在单向线性耦合下实现同步的可行性问题. 为了定量分析控制效果, 定义了一定周期内控制器的平均能耗. 在数值计算中, 对第一类矩形波函数情形则计算了二参数空间(Ta, Tb)下响应系统的最大李亚普诺夫指数分布, 同步区域/非同步区域分布, 控制器平均能耗分布, 确认在恰当的间隔周期(Ta, Tb)和耦合强度下,两个超混沌系统可以达到完全同步. 对第二类方波函数情形则计算了耦合强度和方波间隔周期T0而参 数区域内响应系统最大条件李亚普诺夫指数分布, 给定耦合强度下选择不同间隔周期下误差函数演化和平均能耗, 研究结果表明: 在恰当的耦合强度和间隔周期T0下两个超混沌系统可以达到完全同步. 同时发现, 在恰当的耦合强度下控制器的平均能耗最小, 数值计算结果验证了理论分析的可靠性.

#### 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.

#### 作者及机构信息

###### 1. 兰州理工大学物理系, 兰州 730050
• 基金项目: 国家自然科学基金(批准号: 11265008)资助的课题.

#### Authors and contacts

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

#### 参考文献

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#### 施引文献

•  [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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##### 出版历程
• 收稿日期:  2013-03-18
• 修回日期:  2013-05-15
• 刊出日期:  2013-09-05

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