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To solve the problem of indeterminate synchronization time in different chaotic systems, this paper presents a time-controllable synchronization scheme. A general synchronization controller and parameter update laws are proposed to stabilize the error system, thus the drive and response systems could be synchronized up to a given scaling matrix at a pre-specified exponential convergence rate. The synchronization time formula is strictly deduced, which suggests that the speed of synchronization is determined by several parameters, such as exponential rate, initial system value and other parameters brought in by the controller. By adjusting these parameters, the performance of the synchronization can be effectively improved. In numerical simulation, two nonidentical 3D autonomous chaotic systems are chosen to verify this method. The error system can be rapidly stabilized, and unknown parameters are also identi?ed correctly. Firally, two groups of time-controllable parameters are given to verify the theory, wherein synchronization of both cases can be obtained quickly and each result of the synchronization is consistent with the theoretical calculation. The synchronization scheme is characterized by high safety and efficiency, and has its potential value in secure communication.
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Keywords:
- chaos /
- adaptive control /
- projective synchronization /
- time-controllable
[1] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[2] Zhang R X, Tian G, Li P, Yang S P 2008 Acta. Phys. Sin. 57 2073 (in Chinese) [张若洵, 田钢, 栗苹, 杨世平 2008 物理学报 57 2073]
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[12] [13] Pourmahmood M, Khanmohammadi S, Alizadeh G 2011 Commun Nonlinear Sci Numer. Simulat 16 2853
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[17] [18] Jing X D, Lv L 2008 Acta Phys. Sin. 57 4766 (in Chinese) [敬晓丹, 吕翎 2008 物理学报 57 4766]
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[26] Wang Z L, Shi X R 2010 Nonlinear Dynam 59 559
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[36] Vincent U E, Guo R 2011 Phys. Lett. A 375 2322
[37] [38] Aghababa M P, Khanmohammadi S, Alizadeh G 2011 Appl. Math. Mode. 35 3080
[39] [40] [41] Yang Y Q, Wu X F 2012 Nonlinear Dynam. 70 197
[42] Wang H, Han Z Z, Xie Q Y Zhang W 2009 Commun. Nonlinear Sci Numer. Simulat. 14 2239
[43] [44] Hou Y Y, Wan Z L, Liao T L 2012 Nonlinear Dynam. 70 315
[45] [46] [47] Chen Z, Yang Y, Qi Q, Yuan Z 2007 Phys. Lett. A 360 696
[48] [49] Li Y X, Tang W K S, Chen G R 2005 Int. J. Bifurcat Chaos 15 3367
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[1] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[2] Zhang R X, Tian G, Li P, Yang S P 2008 Acta. Phys. Sin. 57 2073 (in Chinese) [张若洵, 田钢, 栗苹, 杨世平 2008 物理学报 57 2073]
[3] [4] [5] Li J F, Li N, Liu Y P, Gan Y 2009 Acta. Phys. Sin. 58 0779 (in Chinese) [李建芬, 李农, 刘宇平, 甘轶 2009 物理学报 58 0779]
[6] [7] Wang B, Guan Z H 2010 Nonlinear Anal. RWA 11 1925
[8] [9] Zhou P, Wei L J, Cheng X F 2009 Acta. Phys. Sin. 58 5201 (in Chinese) [周平, 危丽佳, 程雪峰 2009 物理学报 58 5201]
[10] [11] Meng J, Wang X Y 2009 Acta. Phys. Sin. 58 0819 (in Chinese) [孟娟, 王兴元 2009 物理学报 58 0819]
[12] [13] Pourmahmood M, Khanmohammadi S, Alizadeh G 2011 Commun Nonlinear Sci Numer. Simulat 16 2853
[14] Al-Sawalha M M, Noorani M 2010 Commun Nonlinear Sci. Numer Simulat. 15 1036
[15] [16] Li X F, Leung A C S, Han X P, Liu X J, Chu Y D 2011 Nonlinear Dynam 63 263
[17] [18] Jing X D, Lv L 2008 Acta Phys. Sin. 57 4766 (in Chinese) [敬晓丹, 吕翎 2008 物理学报 57 4766]
[19] [20] Shahverdiev E, Sivaprakasam S, Shore K 2002 Phys. Lett. A 292 320
[21] [22] [23] Guan Z H, Liu Z W, Feng G, Wu Y 2010 IEEE T CircuitsI 57 2182
[24] [25] Akcakaya M, Nehorai A 2010 IEEE Trans. Signal Process 58 4994
[26] Wang Z L, Shi X R 2010 Nonlinear Dynam 59 559
[27] [28] Li J F, Li N 2011 Acta. Phys. Sin. 60 080507 (in Chinese) [李建芬, 李农 2011 物理学报 60 87]
[29] [30] Yu Y G, Li H X 2010 Nonlinear Anal. RWA 11 2456
[31] [32] Yang W, Sun J T 2010 Phys. Lett. A 374 557
[33] [34] [35] Li S H, Tian Y P 2003 Chaos Soliton. Fract. 15 303
[36] Vincent U E, Guo R 2011 Phys. Lett. A 375 2322
[37] [38] Aghababa M P, Khanmohammadi S, Alizadeh G 2011 Appl. Math. Mode. 35 3080
[39] [40] [41] Yang Y Q, Wu X F 2012 Nonlinear Dynam. 70 197
[42] Wang H, Han Z Z, Xie Q Y Zhang W 2009 Commun. Nonlinear Sci Numer. Simulat. 14 2239
[43] [44] Hou Y Y, Wan Z L, Liao T L 2012 Nonlinear Dynam. 70 315
[45] [46] [47] Chen Z, Yang Y, Qi Q, Yuan Z 2007 Phys. Lett. A 360 696
[48] [49] Li Y X, Tang W K S, Chen G R 2005 Int. J. Bifurcat Chaos 15 3367
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