American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020248

Finite-time cluster synchronization of coupled dynamical systems with impulsive effects

Tianhu Yu 1,2, , Jinde Cao 1,, and  Chuangxia Huang 3,

1. 

School of Mathematics, Southeast University, Nanjing 210096, China
2. 

Department of Mathematics, Luoyang Normal University, Luoyang 471934, China
3. 

Department of Applied Mathematics, Changsha University of Science and Technology, Changsha 410114, China

* Corresponding author: Jinde Cao

Received  December 2019 Revised  June 2020 Published  August 2020

In our paper, the finite-time cluster synchronization problem is investigated for the coupled dynamical systems in networks. Based on impulsive differential equation theory and differential inequality method, two novel Lyapunov-based finite-time stability results are proposed and be used to obtain the finite-time cluster synchronization criteria for the coupled dynamical systems with synchronization and desynchronization impulsive effects, respectively. The settling time with respect to the average impulsive interval is estimated according to the sufficient synchronization conditions. It is illustrated that the introduced settling time is not only dependent on the initial conditions, but also dependent on the impulsive effects. Compared with the results without stabilizing impulses, the attractive domain of the finite-time stability can be enlarged by adding impulsive control input. Conversely, the smaller attractive domain can be obtained when the original system is subject to the destabilizing impulses. By using our criteria, the continuous feedback control can always be designed to finite-time stabilize the unstable impulsive system. Several existed results are extended and improved in the literature. Finally, typical numerical examples involving the large-scale complex network are outlined to exemplify the availability of the impulsive control and continuous feedback control, respectively.

Keywords: Cluster synchronization, coupled dynamical systems, finite-time stability, impulsive effects, differential inequality.
Mathematics Subject Classification: Primary: 34D06, 34H15; Secondary: 49N25.
Citation: Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020248
References:
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show all references

References:
[1]

F. Amato, M. Ariola and C. Cosentino, Finite-time stability of linear time-varying systems: analysis and controller design, IEEE Trans. Automat. Control, 55 (2010), 1003-1008.  doi: 10.1109/TAC.2010.2041680.  Google Scholar

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K. L. Babcock and R. M. Westervelt, Dynamics of simple electronic neural networks, Physica D, 28 (1987), 305-316.  doi: 10.1016/0167-2789(87)90021-2.  Google Scholar

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[14]

L. V. Gambuzza and M. Frasca, A criterion for stability of cluster synchronization in networks with external equitable partitions, Automatica J. IFAC, 100 (2019), 212-218.  doi: 10.1016/j.automatica.2018.11.026.  Google Scholar

[15]

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[16]

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[17]

Y. Hong, Z.-P. Jiang ZP and G. Feng, Finite-time input-to-state stability and applications to finite-time control design, SIAM J. Control Optim., 48 (2010), 4395-4418.  doi: 10.1137/070712043.  Google Scholar

[18]

Y. Hong, J. Wang and D. Cheng, Adaptive finite-time control of nonlinear systems with parametric uncertainty, IEEE Trans. Automat. Control, 51 (2006), 858-862.  doi: 10.1109/TAC.2006.875006.  Google Scholar

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B. Hu, Z.-H. Guan, G. Chen and et. al, Multistability of delayed hybrid impulsive neural networks with application to associative memories, IEEE Trans. Neural Netw. Learn. Syst., 30 (2019), 1537-1551.  doi: 10.1109/TNNLS.2018.2870553.  Google Scholar

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C. Hu, J. Yu, Z. Chen and et. al, Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks, Neural Networks, 89 (2017), 74-83.  doi: 10.1016/j.neunet.2017.02.001.  Google Scholar

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[22]

J. Huang, C. Wen, W. Wang and Y.-D. Song, Adaptive finite-time consensus control of a group of uncertain nonlinear mechanical systems, Automatica J. IFAC, 51 (2015), 292-301.  doi: 10.1016/j.automatica.2014.10.093.  Google Scholar

[23]

S. Jalan and R. E. Amritkar, Self-organized and driven phase synchronization in coupled maps, Physical Review Letters, 90 (2003), 014101. Google Scholar

[24]

S. Jalan, R. E. Amritkar and C. K. Hu, Synchronized clusters in coupled map networks. I. Numerical studies, Physical Review E, 72 (2005), 016212. Google Scholar

[25]

H. K. Khalil and J. W. Grizzle, Nonlinear Systems, Prentice Hall, Upper Saddle River, 2002. Google Scholar

[26]

M. Kumar, D. P. Garg and V. Kumar, Segregation of heterogeneous units in a swarm of robotic agents, IEEE Trans. Automat. Control, 55 (2010), 743-748.  doi: 10.1109/TAC.2010.2040494.  Google Scholar

[27]

Z. Li, Z. Duan, G. Chen and et. al, Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint, IEEE Trans. Circuits Syst. I. Regul. Pap., 57 (2010), 213-224.  doi: 10.1109/TCSI.2009.2023937.  Google Scholar

[28]

X. Li, D. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.  Google Scholar

[29]

X. Liu, Adaptive finite time stability of delayed systems with applications to network synchronization, (2020), arXiv: 2002.00145. Google Scholar

[30]

X. Liu and T. Chen, Finite-time and fixed-time cluster synchronization with or without pinning control, IEEE Transactions on Cybernetics, 48 (2018), 240-252.  doi: 10.1109/TCYB.2016.2630703.  Google Scholar

[31]

Z. Liu, W. S. Wong and H. Cheng, Cluster synchronization of coupled systems with nonidentical linear dynamics, Internat. J. Robust Nonlinear Control, 27 (2017), 1462-1479.   Google Scholar

[32]

J. Lu, D. W. C. Ho and J. Cao, A unified synchronization criterion for impulsive dynamical networks, Automatica J. IFAC, 46 (2010), 1215-1221.  doi: 10.1016/j.automatica.2010.04.005.  Google Scholar

[33]

W. Lu, B. Liu and T. Chen, Cluster synchronization in networks of coupled nonidentical dynamical systems, phChaos, 20 (2010), 013120, 12 pp. doi: 10.1063/1.3329367.  Google Scholar

[34]

E. Moulay and W. Perruquetti, Finite time stability and stabilization of a class of continuous systems, J. Math. Anal. Appl., 323 (2006), 1430-1443.  doi: 10.1016/j.jmaa.2005.11.046.  Google Scholar

[35]

S. G. Nersesov and W. M. Haddad, Finite-time stabilization of nonlinear impulsive dynamical systems, Nonlinear Analysis: Hybrid Systems, 2 (2008), 832-845.  doi: 10.1016/j.nahs.2007.12.001.  Google Scholar

[36]

A. Pratap, R. Raja, J. Alzabut and et. al, Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field, Mathematical Methods in the Applied Sciences, 43 (2020), 6223-6253.  doi: 10.1002/mma.6367.  Google Scholar

[37]

M. T. Schaub, N. O'Clery N, Y. N. Billeh, et. al, Graph partitions and cluster synchronization in networks of oscillators, Chaos, 26 (2016), 094821, 14 pp. doi: 10.1063/1.4961065.  Google Scholar

[38]

Y. Shen and X. Xia, Semi-global finite-time observers for nonlinear systems, Automatica J. IFAC, 44 (2008), 3152-3156.  doi: 10.1016/j.automatica.2008.05.015.  Google Scholar

[39]

C. Song, S. Fei, Jinde Cao, et. al, Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control, Mathematics 7 (2019), 599. doi: 10.3390/math7070599.  Google Scholar

[40]

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, et. al, Complete characterization of the stability of cluster synchronization in complex dynamical networks, Science Advances, 2 (2016), e1501737. arXiv: 1507.04381v2. doi: 10.1126/sciadv.1501737.  Google Scholar

[41]

I. Stamova, Stability Analysis of Impulsive Functional Differential Equations, Walter de Gruyter GmbH & Co. KG, Berlin, 2009. doi: 10.1515/9783110221824.  Google Scholar

[42]

Y.-Z. Sun, S.-Y. Leng, Y.-C. Lai, et al, Closed-loop control of complex networks: a trade-off between time and energy, Phys. Rev. Lett., 119 (2017), 198301, 6 pp. doi: 10.1103/PhysRevLett.119.198301.  Google Scholar

[43]

Z.-Y. Sun, M.-M. Yun and T. Li, A new approach to fast global finite-time stabilization of high-order nonlinear system, Automatica J. IFAC, 81 (2017), 455-463.  doi: 10.1016/j.automatica.2017.04.024.  Google Scholar

[44]

Z. Tang, J. H. Park and H. Shen, Finite-time cluster synchronization of Lur'e networks: A nonsmooth approach, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 1213-1224.  doi: 10.1109/TSMC.2017.2657779.  Google Scholar

[45]

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Figure 1.  Phase plots of (a). the system (61) and (b). the system (62) in Example 1
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Figure 2.  Time histories of (a). the coupled system without control input, (b-d). the variables $ x_{i1} $, $ x_{i2} $ and $ x_{i3} $ of the coupled system with synchronization impulsive effect in Example 1
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Figure 3.  Under control input, time histories of (a). the error function $ E(t) $ in Eq. (63), (b-d). the variables $ e_{i1} $, $ e_{i2} $ and $ e_{i3} $ of the synchronization error system in Example 1
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Figure 4.  With desynchronization impulses, time histories of (a-c). the variables $ x_{i1} $, $ x_{i2} $ and $ x_{i3} $ of the coupled system with nonidentical nodes (61) and (62), (d-f). the variables $ e_{i1} $, $ e_{i2} $ and $ e_{i3} $ of the synchronization error system in Example 1
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Figure 5.  Time histories of (a-c). the variables $ x_{i1} $, $ x_{i2} $ and $ x_{i3} $ of the complex networks, (d-f). the variables $ e_{i1} $, $ e_{i2} $ and $ e_{i3} $ of the synchronization error system in Example 2
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