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November  2011, 16(4): 1071-1082. doi: 10.3934/dcdsb.2011.16.1071

Synchronization of chaotic systems with time-varying coupling delays

Tingwen Huang 1, , Guanrong Chen 2, and  Juergen Kurths 3,

1. 

Texas A&M University at Qatar, Doha, P.O.Box 23874, Qatar
2. 

Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China
3. 

Institute for Physics, University of Potsdam, Am Neuen Palais, Gebude 19, D-14415 Potsdam, Germany

Received  October 2010 Revised  May 2011 Published  August 2011

In this paper, we study the complete synchronization of a class of time-varying delayed coupled chaotic systems using feedback control. In terms of Linear Matrix Inequalities, a sufficient condition is obtained through using a Lyapunov-Krasovskii functional and differential equation inequalities. The conditions can be easily verified and implemented. We present two simulation examples to illustrate the effectiveness of the proposed method.
Keywords: Chaotic System, Synchronization, Time-varying Delay..
Mathematics Subject Classification: Primary: 34A20, 34G2.
Citation: Tingwen Huang, Guanrong Chen, Juergen Kurths. Synchronization of chaotic systems with time-varying coupling delays. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1071-1082. doi: 10.3934/dcdsb.2011.16.1071
References:
[1]

S. Boccaletti, J. Kurths, G. Osipov, D. L. Vallares and C. S. Zhou, The synchronization of chaotic systems, Phys. Rep., 366 (2002), 1-101. doi: 10.1016/S0370-1573(02)00137-0.

[2]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities In Systems And Control Theory," SIAM Studies in Applied Mathematics, 15, SIAM, Philadephia, PA, 1994.

[3]

J. Cao and J. Lu, Adaptive synchronization of neural networks with or without time-varying delay, Chaos, 16 (2006), 013133, 6 pp.

[4]

P. Colet and R. Roy, Digital communication with synchronized chaotic lasers, Opt. Lett., 19 (1994), 2056. doi: 10.1364/OL.19.002056.

[5]

K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992.

[6]

H. G. Schuster, ed., "Handbook of Chaos Control: Foundations and Applications," Wiley-VCH, Weinheim, 1999.

[7]

H. Huang, G. Feng and Y. Sun, Robust synchronization of chaotic systems subject to parameter uncertainties, Chaos, 19 (2009), 033128. doi: 10.1063/1.3212940.

[8]

T. Huang, C. Li and X. Liu, Synchronization of chaotic systems with delay using intermittent linear state feedback, Chaos, 18 (2008), 033122, 8 pp.

[9]

C. Li, G. Feng and X. Liao, Stabilization of nonlinear systems via periodically intermittent control, IEEE Trans. Circuits and Systems II, 54 (2006), 1019-1023.

[10]

C. Li, X. Liao and K. Wong, Chaotic lag synchronization of coupled time-delayed systems and its application in secure communication, Physica D, 194 (2004), 187-202. doi: 10.1016/j.physd.2004.02.005.

[11]

X. Liu, T. Chen and W. Lu, Cluster synchronization for linearly coupled complex networks, Journal of Industrial and Management Optimization (JIMO), 7 (2011), 87-101. doi: 10.3934/jimo.2011.7.87.

[12]

J. Lu, J. Cao and D. Ho, Adaptive stabilization and synchronization for chaotic lur’e systems with time-varying delay, IEEE Transactions on Circuits and Systems I: Regular Papers, 55 (2008), 1347-1356. doi: 10.1109/TCSI.2008.916462.

[13]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824. doi: 10.1103/PhysRevLett.64.821.

[14]

J. Qing, Projective synchronization of a new hyperchaotic Lorenz system, Physics Letters A, 370 (2007), 40-45. doi: 10.1016/j.physleta.2007.05.028.

[15]

F. Rogister, D. Pieroux, M. Sciamanna, P. Megret and M. Blondel, Anticipating synchronization of two chaotic laser diodes by incoherent optical coupling and its application to secure communications, Optics Communications, 207 (2002), 295-306. doi: 10.1016/S0030-4018(02)01494-3.

[16]

M. Rosenblum and A. Pikovsky, Phase synchronization of chaotic oscillators, Phys. Rev. Lett., 76 (1996), 1804-1807. doi: 10.1103/PhysRevLett.76.1804.

[17]

N. Rulkov and M. Sushchik, Generalized synchronization of chaos in directionally coupled chaotic systems, Phys. Rev. E, 51 (1995), 980-994. doi: 10.1103/PhysRevE.51.980.

[18]

S. Sivaprakasam and P. Spencer, Regimes of chaotic synchronization in external-cavity laser diodes, IEEE Journal of Quantum Electronics, 38 (2002), 1155-1161. doi: 10.1109/JQE.2002.801949.

[19]

Q. Song and J. Cao, Global dissipativity analysis on uncertain neural networks with mixed time-varying delays, Chaos, 18 (2008), 043126, 10 pp.

[20]

K. Thornburg, M. Moller, R. Roy and T. Carr, Chaos and coherence in coupled lasers, Phys. Rev. E, 55 (1997), 3865. doi: 10.1103/PhysRevE.55.3865.

[21]

J. Wang, Z. Yang, T. Huang and M. Xiao, Local and global exponential synchronization of complex delayed dynamical networks with general topology, Discrete and Continuous Dynamical Systems-Series B, 16 (2011), 393-408. doi: 10.3934/dcdsb.2011.16.393.

[22]

R. Zhen, X. Wu and J. Zhang, "Sliding Model Synchronization Controller Design for Chaotic Neural Network with Time-Varying Delay," Proceedings of the 8th World Congress on Intelligent Control and Automation, China, (2010), 3914-3919. doi: 10.1109/WCICA.2010.5554977.

show all references

References:
[1]

S. Boccaletti, J. Kurths, G. Osipov, D. L. Vallares and C. S. Zhou, The synchronization of chaotic systems, Phys. Rep., 366 (2002), 1-101. doi: 10.1016/S0370-1573(02)00137-0.

[2]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities In Systems And Control Theory," SIAM Studies in Applied Mathematics, 15, SIAM, Philadephia, PA, 1994.

[3]

J. Cao and J. Lu, Adaptive synchronization of neural networks with or without time-varying delay, Chaos, 16 (2006), 013133, 6 pp.

[4]

P. Colet and R. Roy, Digital communication with synchronized chaotic lasers, Opt. Lett., 19 (1994), 2056. doi: 10.1364/OL.19.002056.

[5]

K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992.

[6]

H. G. Schuster, ed., "Handbook of Chaos Control: Foundations and Applications," Wiley-VCH, Weinheim, 1999.

[7]

H. Huang, G. Feng and Y. Sun, Robust synchronization of chaotic systems subject to parameter uncertainties, Chaos, 19 (2009), 033128. doi: 10.1063/1.3212940.

[8]

T. Huang, C. Li and X. Liu, Synchronization of chaotic systems with delay using intermittent linear state feedback, Chaos, 18 (2008), 033122, 8 pp.

[9]

C. Li, G. Feng and X. Liao, Stabilization of nonlinear systems via periodically intermittent control, IEEE Trans. Circuits and Systems II, 54 (2006), 1019-1023.

[10]

C. Li, X. Liao and K. Wong, Chaotic lag synchronization of coupled time-delayed systems and its application in secure communication, Physica D, 194 (2004), 187-202. doi: 10.1016/j.physd.2004.02.005.

[11]

X. Liu, T. Chen and W. Lu, Cluster synchronization for linearly coupled complex networks, Journal of Industrial and Management Optimization (JIMO), 7 (2011), 87-101. doi: 10.3934/jimo.2011.7.87.

[12]

J. Lu, J. Cao and D. Ho, Adaptive stabilization and synchronization for chaotic lur’e systems with time-varying delay, IEEE Transactions on Circuits and Systems I: Regular Papers, 55 (2008), 1347-1356. doi: 10.1109/TCSI.2008.916462.

[13]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824. doi: 10.1103/PhysRevLett.64.821.

[14]

J. Qing, Projective synchronization of a new hyperchaotic Lorenz system, Physics Letters A, 370 (2007), 40-45. doi: 10.1016/j.physleta.2007.05.028.

[15]

F. Rogister, D. Pieroux, M. Sciamanna, P. Megret and M. Blondel, Anticipating synchronization of two chaotic laser diodes by incoherent optical coupling and its application to secure communications, Optics Communications, 207 (2002), 295-306. doi: 10.1016/S0030-4018(02)01494-3.

[16]

M. Rosenblum and A. Pikovsky, Phase synchronization of chaotic oscillators, Phys. Rev. Lett., 76 (1996), 1804-1807. doi: 10.1103/PhysRevLett.76.1804.

[17]

N. Rulkov and M. Sushchik, Generalized synchronization of chaos in directionally coupled chaotic systems, Phys. Rev. E, 51 (1995), 980-994. doi: 10.1103/PhysRevE.51.980.

[18]

S. Sivaprakasam and P. Spencer, Regimes of chaotic synchronization in external-cavity laser diodes, IEEE Journal of Quantum Electronics, 38 (2002), 1155-1161. doi: 10.1109/JQE.2002.801949.

[19]

Q. Song and J. Cao, Global dissipativity analysis on uncertain neural networks with mixed time-varying delays, Chaos, 18 (2008), 043126, 10 pp.

[20]

K. Thornburg, M. Moller, R. Roy and T. Carr, Chaos and coherence in coupled lasers, Phys. Rev. E, 55 (1997), 3865. doi: 10.1103/PhysRevE.55.3865.

[21]

J. Wang, Z. Yang, T. Huang and M. Xiao, Local and global exponential synchronization of complex delayed dynamical networks with general topology, Discrete and Continuous Dynamical Systems-Series B, 16 (2011), 393-408. doi: 10.3934/dcdsb.2011.16.393.

[22]

R. Zhen, X. Wu and J. Zhang, "Sliding Model Synchronization Controller Design for Chaotic Neural Network with Time-Varying Delay," Proceedings of the 8th World Congress on Intelligent Control and Automation, China, (2010), 3914-3919. doi: 10.1109/WCICA.2010.5554977.

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