Synchronization of \(N\) coupled chaotic systems with ring connection based on special antisymmetric structure. (English) Zbl 1421.93130

Chen, Xiangyong; Qiu, Jianlong; Song, Qiang; Zhang, Ancai

Abstr. Appl. Anal. 2013, Article ID 680604, 7 p. (2013).

Summary: This paper considers the complete synchronization problem for \(N\) coupled chaotic systems with ring connections. First, we use a direct design method to design a synchronization controller. It transforms the error system into a stable system with special antisymmetric structure. And then, we get some simple stability criteria of achieving the complete synchronization. These criteria are not only easily verified but also improve and generalize previous known results. Finally, numerical examples are provided to demonstrate the effectiveness of the theoretical analysis.

MSC:

93D99	Stability of control systems
34H10	Chaos control for problems involving ordinary differential equations
34D06	Synchronization of solutions to ordinary differential equations

Keywords:

synchronization; coupled chaotic systems; antisymmetric structure

References:

[1]	Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Physical Review Letters, 64, 8, 821-824 (1990) · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821
[2]	Nana, B.; Woafo, P.; Domngang, S., Chaotic synchronization with experimental application to secure communications, Communications in Nonlinear Science and Numerical Simulation, 14, 5, 2266-2276 (2009) · doi:10.1016/j.cnsns.2008.06.028
[3]	Aguirre, C.; Campos, D.; Pascual, P.; Serrano, E., Synchronization effects using a piecewise linear map-based spiking-bursting neuron model, Neurocomputing, 69, 10-12, 1116-1119 (2006) · doi:10.1016/j.neucom.2005.12.056
[4]	Tang, S.; Liu, J. M., Chaos synchronization in semiconductor lasers with optoelectronic feedback, IEEE Journal of Quantum Electronics, 39, 6, 708-715 (2003) · doi:10.1109/JQE.2003.810775
[5]	Yan, J.-J.; Yang, Y.-S.; Chiang, T.-Y.; Chen, C.-Y., Robust synchronization of unified chaotic systems via sliding mode control, Chaos, Solitons and Fractals, 34, 3, 947-954 (2007) · Zbl 1129.93489 · doi:10.1016/j.chaos.2006.04.003
[6]	Guan, S.; Lai, C.-H.; Wei, G. W., Phase synchronization between two essentially different chaotic systems, Physical Review E, 72, 1 (2005) · doi:10.1103/PhysRevE.72.016205
[7]	Cai, N.; Jing, Y. W.; Zhang, S. Y., Adaptive synchronization and anti-synchronization of two different chaotic systems, Acta Physica Sinica, 58, 2, 802-813 (2009) · Zbl 1199.37048
[8]	Cai, N.; Jing, Y.; Zhang, S., Generalized projective synchronization of different chaotic systems based on antisymmetric structure, Chaos, Solitons and Fractals, 42, 2, 1190-1196 (2009) · Zbl 1198.93106 · doi:10.1016/j.chaos.2009.03.015
[9]	Cao, J.; Ho, D. W. C.; Yang, Y., Projective synchronization of a class of delayed chaotic systems via impulsive control, Physics Letters A, 373, 35, 3128-3133 (2009) · Zbl 1233.34017 · doi:10.1016/j.physleta.2009.06.056
[10]	He, W.; Cao, J., Generalized synchronization of chaotic systems: an auxiliary system approach via matrix measure, Chaos, 19, 1 (2009) · Zbl 1311.34113 · doi:10.1063/1.3076397
[11]	Gan, Q.; Xu, R.; Yang, P., Synchronization of non-identical chaotic delayed fuzzy cellular neural networks based on sliding mode control, Communications in Nonlinear Science and Numerical Simulation, 17, 1, 433-443 (2012) · Zbl 1239.93021 · doi:10.1016/j.cnsns.2011.05.014
[12]	Lu, J.; Cao, J.; Ho, D. W. C., Adaptive stabilization and synchronization for chaotic Lur’e systems with time-varying delay, IEEE Transactions on Circuits and Systems I, 55, 5, 1347-1356 (2008) · doi:10.1109/TCSI.2008.916462
[13]	Yu, W.; Chen, G.; Lü, J.; Kurths, J., Synchronization via pinning control on general complex networks, SIAM Journal on Control and Optimization, 51, 2, 1395-1416 (2013) · Zbl 1266.93071 · doi:10.1137/100781699
[14]	Gan, Q.; Liang, Y., Synchronization of chaotic neural networks with time delay in the leakage term and parametric uncertainties based on sampled-data control, Journal of the Franklin Institute, 349, 6, 1955-1971 (2012) · Zbl 1300.93113 · doi:10.1016/j.jfranklin.2012.05.001
[15]	Zhang, Q.-L., Synchronization of multi-chaotic systems via ring impulsive control, Journal of Control Theory and Applications, 27, 2, 226-232 (2010)
[16]	Yu, Y.; Zhang, S., Global synchronization of three coupled chaotic systems with ring connection, Chaos, Solitons and Fractals, 24, 5, 1233-1242 (2005) · Zbl 1081.37020 · doi:10.1016/j.chaos.2004.09.112
[17]	Lu, J.-A.; Han, X.-P.; Li, Y.-T.; Yu, M.-H., Adaptive coupled synchronization among multi-Lorenz systems family, Chaos, Solitons and Fractals, 31, 4, 866-878 (2007) · Zbl 1143.93333 · doi:10.1016/j.chaos.2005.10.034
[18]	Grassi, G., Propagation of projective synchronization in a series connection of chaotic systems, Journal of the Franklin Institute, 347, 2, 438-451 (2010) · Zbl 1185.93075 · doi:10.1016/j.jfranklin.2009.10.004
[19]	Liu, Y.; Lü, L., Synchronization of \(N\) different coupled chaotic systems with ring and chain connections, Applied Mathematics and Mechanics, 29, 10, 1299-1308 (2008) · Zbl 1163.93025 · doi:10.1007/s10483-008-1005-y
[20]	Tang, Y.; Fang, J.-A., Synchronization of \(N\)-coupled fractional-order chaotic systems with ring connection, Communications in Nonlinear Science and Numerical Simulation, 15, 2, 401-412 (2010) · Zbl 1221.34103 · doi:10.1016/j.cnsns.2009.03.024
[21]	Chen, X.; Qiu, J., Synchronization of \(N\) different chaotic systems based on antisymmetric structure, Mathematical Problems in Engineering, 2013 (2013) · Zbl 1299.93241 · doi:10.1155/2013/742765
[22]	Yang, L.-X.; Zhang, J., Synchronization of three identical systems and its application for secure communication with noise perturbation, Proceedings of the International Conference on Information Engineering and Computer Science (ICIECS ’09) · doi:10.1109/ICIECS.2009.5363764