Isaac Scientific Publishing

Journal of Advances in Applied Mathematics

Fixed-Time Synchronization of Drive-Response Coupled Systems with Impulsive Effects

Download PDF (438.7 KB) PP. 157 - 170 Pub. Date: October 15, 2019

DOI: 10.22606/jaam.2019.44004


  • Wenjing Yang*
    College of Science, University of Shanghai for Science and Technology, Shanghai, P. R. China


In this paper, the fixed-time synchronization for drive-response coupled system(DRCS) with impulsive effects is investigated. More general controllers are designed to synchronize driveresponse coupled system(DRCS) within fixed-time. By using graph theory and Lyapunov method, strongly and non-strongly connected of topological structure of DRCS are studied deriving different criteria. What’s more, some control strategies are provided for special cases of DRCS. Furthermore, some numerical simulations are offered to demonstrate the validity of theoretical results.


Fixed-time synchronization, Drive-response coupled system, Impulsive effects, Graph theory.


[1] M. Y. Li and Z. Shuai, “Global-stability problem for coupled systems of differential equations on networks,” J. Differential Equations, vol. 248, no. 1, pp. 1–20, JAN 1 2010.

[2] D. Ganji and A. Sadighi, “Application of He’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations,” Int. J. Nonlinear Sci. Numer, vol. 7, no. 4, pp. 411–418, 2006.

[3] M. Liu and C. Bai, “Optimal harvesting of a stochastic delay competitive model,” Discrete Contin. Dyn. Syst. Ser. B, vol. 22, no. 4, pp. 1493–1508, JUN 2017.

[4] E. G. Lakatta, V. A. Maltsev, and T. M. Vinogradova, “A Coupled SYSTEM of Intracellular Ca2+ Clocks and Surface Membrane Voltage Clocks Controls the Timekeeping Mechanism of the Heart’s Pacemaker,” Circ. Res., vol. 106, no. 4, pp. 659–673, MAR 5 2010.

[5] M. Liu and C. Bai, “Analysis of a stochastic tri-trophic food-chain model with harvesting,” J. Math. Biol., vol. 73, no. 3, pp. 597–625, SEP 2016.

[6] J. Shi, P. Zhang, D. Xiao, and Q. Niu, “Proper definition of spin current in spin-orbit coupled systems,” Phys. Rev. Lett., vol. 96, no. 7, FEB 24 2006.

[7] Z. N. Low, R. A. Chinga, R. Tseng, and J. Lin, “Design and Test of a High-Power High-Efficiency Loosely Coupled Planar Wireless Power Transfer System,” IEEE Trans. Ind. Electron., vol. 56, no. 5, pp. 1801–1812, MAY 2009.

[8] D. Ye, X. Yang, and L. Su, “Fault-tolerant synchronization control for complex dynamical networks with semi-Markov jump topology,” Appl. Math. Comput., vol. 312, pp. 36–48, NOV 1 2017.

[9] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, “Epidemic processes in complex networks,” Rev. Mod. Phys., vol. 87, no. 3, pp. 925–979, AUG 31 2015.

[10] R. Guo, Z. Zhang, X. Liu, and C. Lin, “Existence, uniqueness, and exponential stability analysis for complexvalued memristor-based BAM neural networks with time delays,” Appl. Math. Comput., vol. 311, pp. 100–117, OCT 15 2017.

[11] S. Boccaletti, J. Kurths, G. Osipov, D. Valladares, and C. Zhou, “The synchronization of chaotic systems,” Phys. Rep., vol. 366, no. 1, pp. 1–101, 2002.

[12] L. Wang, Z. Wang, Q.-L. Han, and G. Wei, “Synchronization Control for a Class of Discrete-Time Dynamical Networks With Packet Dropouts: A Coding-Decoding-Based Approach,” IEEE Trans. Cybern., vol. 48, no. 8, pp. 2437–2448, AUG 2018.

[13] Z. Wu and H. Leng, “Complex hybrid projective synchronization of complex-variable dynamical networks via open-plus-closed-loop control,” J. Franklin Inst, vol. 354, no. 2, pp. 689–701, JAN 2017.

[14] M. Liu, H. Jiang, and C. Hu, “New Results for Exponential Synchronization of Memristive Cohen-Grossberg Neural Networks with Time-Varying Delays,” Neural Processing Letters, vol. 49, no. 1, pp. 79–102, FEB 2019.

[15] Y. Wu, B. Chen, and W. Li, “Synchronization of stochastic coupled systems via feedback control based on discrete-time state observations,” Nonlinear Anal-Hybri, vol. 26, pp. 68–85, NOV 2017.

[16] W. Wu and T. Chen, “Global synchronization criteria of linearly coupled neural network systems with timevarying coupling,” IEEE Trans. Neural Netw., vol. 19, no. 2, pp. 319–332, FEB 2008.

[17] A. Chandrasekar, R. Rakkiyappan, J. Cao, and S. Lakshmanan, “Synchronization of memristor-based recurrent neural networks with two delay components based on second-order reciprocally convex approach,” Neural Netw, vol. 57, pp. 79–93, SEP 2014.

[18] B. Guo, Y. Wu, Y. Xiao, and C. Zhang, “Graph-theoretic approach to synchronizing stochastic coupled systems with time-varying delays on networks via periodically intermittent control,” Appl. Math. Comput., vol. 331, pp. 341–357, AUG 15 2018.

[19] A. Chandrasekar, R. Rakkiyappan, and J. Cao, “Impulsive synchronization of Markovian jumping randomly coupled neural networks with partly unknown transition probabilities via multiple integral approach,” Neural Netw, vol. 70, pp. 27–38, OCT 2015.

[20] C. Li, W. Sun, and J. Kurths, “Synchronization between two coupled complex networks,” PHYSICAL REVIEW E, 2007.

[21] L.M.Pecora and T.L.carroll, “Synchronization chaotic system,” Phys.Lett, vol. 64, no. 64, pp. 821–824, 1990.

[22] Y. Liu, C. Zhu, D. Chu, and W. Li, “Synchronization of stochastic coupled systems with time-varying coupling structure on networks via discrete-time state feedback control,” Neurocomputing, vol. 285, pp. 104–116, APR 12 2018.

[23] V. Haimo, “Finite-time controllers,” SIAM J. Control Optim, vol. 24, no. 4, pp. 760–770, 1986.

[24] S. Bowong and F. Kakmeni, “Chaos control and duration time of a class of uncertain chaotic systems,” Phys. Lett. A, vol. 316, no. 3-4, pp. 206–217, SEP 22 2003.

[25] Z.-H. Guan, F.-L. Sun, Y.-W. Wang, and T. Li, “Finite-Time Consensus for Leader-Following Second-Order Multi-Agent Networks,” IEEE Trans. Circuits Syst. I, vol. 59, no. 11, pp. 2646–2654, NOV 2012.

[26] J. Yang, Xinsong; Lu, “Finite-time synchronization of coupled networks with markovian topology and impulsive effects,” IEEE Transactions On Automatic Control, vol. 61, no. 8, pp. 2256–2261, Aug. 2016.

[27] W. Zhang, X. Yang, C. Xu, J. Feng, and C. Li, “Finite-Time Synchronization of Discontinuous Neural Networks With Delays and Mismatched Parameters,” IEEE Trans. Neural Netw. Learn. Syst., vol. 29, no. 8, pp. 3761–3771, AUG 2018.

[28] J. Li, G. Wei, D. Ding, and Y. Li, “Finite-time control in probability for time-varying systems with measurement censoring,” J. Franklin Inst, vol. 356, no. 4, pp. 1677–1694, MAR 2019.

[29] A. Polyakov, “Nonlinear Feedback Design for Fixed-Time Stabilization of Linear Control Systems,” IEEE Trans. Autom. Control, vol. 57, no. 8, SI, pp. 2106–U1, AUG 2012.

[30] W. Zhang, C. Li, T. Huang, and J. Huang, “Fixed-time synchronization of complex networks with nonidentical nodes and stochastic noise perturbations,” Physica A, vol. 492, pp. 1531–1542, FEB 15 2018.

[31] J. H. D. W. C. Yang, Xinsong; Lam, “Fixed-time synchronization of complex networks with impulsive effects via nonchattering control,” IEEE Transactions On Automatic Control, vol. 62, no. 11, pp. 5511–5521, Nov. 2017.

[32] Y. Wan, J. Cao, G. Wen, and W. Yu, “Robust fixed-time synchronization of delayed cohen-grossberg neural networks,” Neural Networks, vol. 73, pp. 86–94, JAN 2016.

[33] J. Liu, Y. Yu, Q. Wang, and C. Sun, “Fixed-time event-triggered consensus control for multi-agent systems with nonlinear uncertainties,” Neurocomputing, vol. 260, pp. 497–504, OCT 18 2017.

[34] J. C. J. Lu, D. W. C. Ho, “Synchronization of delayed complex dynamical networks with impulsive and stochastic effects,” Automatica, vol. 46, pp. 1215–1221, 2010.

[35] X. S. A. Khadra, X. Liu, “Application of impulsive synchronization to communication security vol.” IEEE Trans. Circuits Syst. I,, vol. 50, no. 3, pp. 341–351, 2003.

[36] C. W. X. Li, M. Bohner, “Impulsive differential equations: Periodic solutions and applications,” Automatica, vol. 52, pp. 173–178, 2015.

[37] T. W. H.Li, C.Li, “Fixed-time stabilization of impulsive cohen-grossberg bam neural networks,” Neural Netw, vol. 98, pp. 203–211, 2018.

[38] Y. W. X. Zhang, Wenbing; Tang, “Synchronization of nonlinear dynamical networks with heterogeneous impulses,” IEEE Transactions on circuits and systems1-regular papers, vol. 61, no. 4, pp. 1220–1228, Apr. 2014.

[39] F. L. J. He, Wangli; Qian, “Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control: Error estimation, optimization and design,” Automatica, vol. 63, pp. 249–262, Dec. 2015.

[40] J. Li, Xiaodi; Wu, “Stability of nonlinear differential systems with state-dependent delayed impulses,” Automatica, vol. 64, pp. 63–69, Feb. 2016.

[41] X. Liu, T. Chen, and W. Lu, “Consensus problem in directed networks of multi-agents via nonlinear protocols,” Phys. Lett. A, vol. 373, no. 35, pp. 3122–3127, AUG 24 2009.

[42] M.Goldberg, “Equivalence constants constants for i norms of matrices,” Linear Multilinear Algebra, vol. 21, no. 2, pp. 173–179, 1987.

[43] Y. Sun, W. Li, and D. Zhao, “Finite-time stochastic outer synchronization between two complex dynamical networks with different topologies,” Chaos, vol. 22, no. 2, JUN 2012.

[44] Q. Yi and Z. H. Shenming Hua, “Impulsive synchronization of rossller chaotic system with applications to communications,” JOURNAL OF AIR FORCE ENGINEERING UNIVERSITY, vol. 2, no. 6, pp. 63–65, 2001.