April  2021, 14(4): 1607-1629. doi: 10.3934/dcdss.2020279

Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations

1. 

Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, Anhui, China

2. 

School of Mathematical Sciences, Huaqiao University, Quanzhou 362000, China

3. 

Department of Mathematics, Zhejiang Normal University, 321004, Jinhua, Zhejiang, China

* Corresponding author: Yonghui Xia. Email: xiadoc@163.com; yhxia@zjnu.cn. ORCID: 0000-0001-8918-3509. Address: Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

Received  August 2019 Revised  September 2019 Published  April 2021 Early access  February 2020

Fund Project: This work was supported in part by the National Natural Science Foundation of China under Grant (No. 11931016, No. 11671176, No. 11871251), Natural Science Foundation of Zhejiang Province under Grant (No. LY20A010016), the Project for Young and Middle-aged Teacher in Education and Science Research of Fujian Province of China under Grant JAT170028, start-up fund of Huaqiao University (Z16J0039)

This paper concerns the synchronization of a kind of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations. Multi-layer networks are a kind of complex networks with different layers, which consist of different kinds of interactions or multiple subnetworks. Additive couplings are designed to capture the different layered connections. In this paper, two pinning controllers are designed to guarantee the synchronization of the stochastic multi-layer network. One is the state-feedback pinning controller with constant control gains. The other one is the adaptive pinning controller with adaptive control gains. It is worthwhile to mention that our assumptions on the activation functions satisfy a generalized Lipschitzian condition which are weaker than those in the previous works. Moreover, as we prove, only selected part of the nodes to be controlled are enough to guarantee that the drive system and response network can be stochastically synchronized. Finally, an example and its simulations are presented to show the feasibility effectiveness of our control schemes.

Citation: Jinsen Zhuang, Yan Zhou, Yonghui Xia. Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1607-1629. doi: 10.3934/dcdss.2020279
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S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, 1994. doi: 10.1137/1.9781611970777.

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G. ChenJ. Zhou and Z. Liu, Global synchronization of coupled delayed neural networks and applications to chaotic cnn models,, International Journal of Bifurcation and Chaos, 14 (2004), 2229-2240.  doi: 10.1142/S0218127404010655.

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K. ChoromańskiM. Matuszak and J. Miȩkisz, Scale-free graph with preferential attachment and evolving internal vertex structure,, Journal of Statistical Physics, 151 (2013), 1175-1183.  doi: 10.1007/s10955-013-0749-1.

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Z.-H. GuanZ.-W. LiuG. Feng and Y.-W. Wang, Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control, IEEE Transactions on Circuits and Systems I: Regular Papers, 57 (2010), 2182-2195. 

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W. HeG. ChenQ.-L. HanW. DuJ. Cao and F. Qian, Multiagent systems on multilayer networks: Synchronization analysis and network design,, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 1655-1667.  doi: 10.1109/TSMC.2017.2659759.

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Y. LiX. WuJ. Lu and J. Lü, Synchronizability of duplex networks,, IEEE Transactions on Circuits and Systems II-Brief Papers, 63 (2016), 206-210.  doi: 10.1109/TCSII.2015.2468924.

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X. LiuH. Su and M. Z. Chen, A switching approach to designing finite-time synchronization controllers of coupled neural networks,, IEEE Transactions on Neural Networks and Learning Systems, 27 (2016), 471-482.  doi: 10.1109/TNNLS.2015.2448549.

[23]

X. LiuD. W. HoQ. Song and W. Xu, Finite/fixed-time pinning synchronization of complex networks with stochastic disturbances,, IEEE Transactions on Cybernetics, 49 (2019), 2398-2403.  doi: 10.1109/TCYB.2018.2821119.

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X. Liu, D. W. Ho and C. Xie, Prespecified-time cluster synchronization of complex networks via a smooth control approach,, IEEE Transactions on Cybernetics, 2018, 1–5. doi: 10.1109/TCYB.2018.2882519.

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J. Lu, J. Cao, D. W. C. Ho and J. Kurths, Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay,, International Journal of Bifurcation and Chaos, 22 (2012), 1250176. doi: 10.1142/S0218127412501763.

[27]

J. LuC. DingJ. Lou and J. Cao, Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers,, Journal of the Franklin Institute, 352 (2015), 5024-5041.  doi: 10.1016/j.jfranklin.2015.08.016.

[28]

J. LuX. GuoT. Huang and Z. Wang, Consensus of signed networked multi-agent systems with nonlinear coupling and communication delays,, Applied Mathematics and Computation, 350 (2019), 153-162.  doi: 10.1016/j.amc.2019.01.006.

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[35]

Y. Song and J. Xu, Inphase and antiphase synchronization in a delay-coupled system with applications to a delay-coupled fitzhugh-nagumo system, IEEE Transactions on Neural Networks, 23 (2012), 1659-1670. 

[36]

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[39]

L. Tang, X. Wu, J. Lü and R. M. D'Souza, Master stability functions for complete, intra-layer and inter-layer synchronization in multiplex networks, arXiv: 1611.09110, 2017.

[40]

Z. TangJ. H. Park and W.-X. Zheng, Distributed impulsive synchronization of lur'e dynamical networks via parameter variation methods,, International Journal of Robust and Nonlinear Control, 28 (2018), 1001-1015.  doi: 10.1002/rnc.3916.

[41]

Z. TangJ. H. ParkY. Wang and J. Feng, Distributed impulsive quasi-synchronization of Lur'e networks with proportional delay,, IEEE Transactions on Cybernetics, 49 (2019), 3105-3115.  doi: 10.1109/TCYB.2018.2839178.

[42]

J. WangJ. FengC. Xu and Y. Zhao, Exponential synchronization of stochastic perturbed complex networks with time-varying delays via periodically intermittent pinning,, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3146-3157.  doi: 10.1016/j.cnsns.2013.03.021.

[43]

T. Wang, S. Zhao, W. Zhou and W. Yu, Almost sure synchronization control for stochastic delayed complex networks based on pinning adaptive method,, Advances in Difference Equations, 2016 (2016), Paper No. 306, 16 pp. doi: 10.1186/s13662-016-1013-1.

[44]

Y. Wang, F. Wu, X. Mao and E. Zhu, Advances in the lasalle-type theorems for stochastic functional differential equations with infinite delay, Discrete & Continuous Dynamical Systems-B, (2019), 57–65.

[45]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440-442.  doi: 10.1038/30918.

[46]

X. WuX. ZhaoJ. LüL. Tang and J. Lu, Identifying topologies of complex dynamical networks with stochastic perturbations,, IEEE Transactions on Control of Network Systems, 3 (2016), 379-389.  doi: 10.1109/TCNS.2015.2482178.

[47]

Y. XiaZ. Yang and M. Han, Lag synchronization of unknown chaotic delayed yang-yang-type fuzzy neural networks with noise perturbation based on adaptive control and parameter identification, IEEE Transactions on Neural Networks, 20 (2009), 1165-1180. 

[48]

L. YanW. Pei and Z. Zhang, Exponential stability of sdes driven by fbm with markovian switching,, Discrete & Continuous Dynamical Systems-A, 39 (2019), 6467-6483.  doi: 10.3934/dcds.2019280.

[49]

X. YangJ. LamD. W. C. Ho and Z. Feng, Fixed-time synchronization of complex networks with impulsive effects via nonchattering control,, IEEE Transactions on Automatic Control, 62 (2017), 5511-5521.  doi: 10.1109/TAC.2017.2691303.

[50]

L. Yu and Z. Yu, Synchronization of stochastic impulsive discrete-time delayed networks via pinning control, Neurocomputing, 286, pp. 31 – 40, 2018.

[51]

W. Yu and J. Cao, Synchronization control of stochastic delayed neural networks,, Physica A: Statistical Mechanics and its Applications, 373 (2007), 252-260.  doi: 10.1016/j.physa.2006.04.105.

[52]

W. YuJ. Cao and J. Lü, Global synchronization of linearly hybrid coupled networks with time-varying delay,, SIAM Journal on Applied Dynamical Systems, 7 (2008), 108-133.  doi: 10.1137/070679090.

[53]

B. Zhang, J. Zhuang, H. Liu, J. Cao and Y. Xia, Master-slave synchronization of a class of fractional-order takagi-sugeno fuzzy neural networks,, Advance in Difference Equations, 2018 (2018), Paper No. 473, 11 pp. doi: 10.1186/s13662-018-1918-y.

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Y. ZhangJ. ZhuangY. XiaY. BaiJ. Cao and L. Gu, Fixed-time synchronization of the impulsive memristor-based neural networks,, Communications in Nonlinear Science and Numerical Simulation, 77 (2019), 40-53.  doi: 10.1016/j.cnsns.2019.04.021.

[55]

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[56]

J. Zhuang, J. Cao, L. Tang, Y. Xia and M. Perc, Synchronization analysis for stochastic delayed multi-layer network with additive couplings, IEEE Transactions on Systems, Man, and Cybernetics: Systems. doi: 10.1109/TSMC.2018.2866704.

show all references

References:
[1]

A. Barabasi and R. Albert, Emergence of scaling in random networks,, Science, 286 (1999), 509-512.  doi: 10.1126/science.286.5439.509.

[2]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, 1994. doi: 10.1137/1.9781611970777.

[3]

J. CaoG. Chen and P. Li, Global synchronization in an array of delayed neural networks with hybrid coupling, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38 (2008), 488-498. 

[4]

J. Cao and J. Lu, Adaptive synchronization of neural networks with or without time-varying delay,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 16 (2006), 013133, 6 pp. doi: 10.1063/1.2178448.

[5]

J. CaoZ. Wang and Y. Sun, Synchronization in an array of linearly stochastically coupled networks with time delays,, Physica A: Statistical Mechanics and its Applications, 385 (2007), 718-728.  doi: 10.1016/j.physa.2007.06.043.

[6]

G. ChenJ. Zhou and Z. Liu, Global synchronization of coupled delayed neural networks and applications to chaotic cnn models,, International Journal of Bifurcation and Chaos, 14 (2004), 2229-2240.  doi: 10.1142/S0218127404010655.

[7]

K. ChoromańskiM. Matuszak and J. Miȩkisz, Scale-free graph with preferential attachment and evolving internal vertex structure,, Journal of Statistical Physics, 151 (2013), 1175-1183.  doi: 10.1007/s10955-013-0749-1.

[8]

Z.-H. GuanZ.-W. LiuG. Feng and Y.-W. Wang, Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control, IEEE Transactions on Circuits and Systems I: Regular Papers, 57 (2010), 2182-2195. 

[9]

B. GuoY. WuY. Xiao and C. Zhang, Graph-theoretic approach to synchronizing stochastic coupled systems with time-varying delays on networks via periodically intermittent control,, Applied Mathematics and Computation, 331 (2018), 341-357.  doi: 10.1016/j.amc.2018.03.020.

[10]

W. HeG. ChenQ.-L. HanW. DuJ. Cao and F. Qian, Multiagent systems on multilayer networks: Synchronization analysis and network design,, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 1655-1667.  doi: 10.1109/TSMC.2017.2659759.

[11]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, Siam Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.

[12] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2013. 
[13]

J. Hu and C. Yuan, Strong convergence of neutral stochastic functional differential equations with two time-scales, Discrete & Continuous Dynamical Systems-B, 24 (2019), 5831-5848. 

[14]

Y. LiX. WuJ. Lu and J. Lü, Synchronizability of duplex networks,, IEEE Transactions on Circuits and Systems II-Brief Papers, 63 (2016), 206-210.  doi: 10.1109/TCSII.2015.2468924.

[15]

X. Li and J. Cao, Adaptive synchronization for delayed neural networks with stochastic perturbation,, Journal of the Franklin Institute, 345 (2008), 779-791.  doi: 10.1016/j.jfranklin.2008.04.012.

[16]

X. LiD. O'Regan and H. Akca, Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays,, IMA Journal of Applied Mathematics, 80 (2015), 85-99.  doi: 10.1093/imamat/hxt027.

[17]

X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay,, Applied Mathematics and Computation, 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[18]

X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method,, Applied Mathematics and Computation, 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

[19]

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

[20]

Y. LiJ. LouZ. Wang and F. Alsaadi, Synchronization of nonlinearly coupled dynamical networks under hybrid pinning impulsive controllers,, Journal of the Franklin Institute, 355 (2018), 6520-6530.  doi: 10.1016/j.jfranklin.2018.06.021.

[21]

X. LiuD. W. HoW. Yu and J. Cao, A new switching design to finite-time stabilization of nonlinear systems with applications to neural networks,, Neural Networks, 57 (2014), 94-102.  doi: 10.1016/j.neunet.2014.05.025.

[22]

X. LiuH. Su and M. Z. Chen, A switching approach to designing finite-time synchronization controllers of coupled neural networks,, IEEE Transactions on Neural Networks and Learning Systems, 27 (2016), 471-482.  doi: 10.1109/TNNLS.2015.2448549.

[23]

X. LiuD. W. HoQ. Song and W. Xu, Finite/fixed-time pinning synchronization of complex networks with stochastic disturbances,, IEEE Transactions on Cybernetics, 49 (2019), 2398-2403.  doi: 10.1109/TCYB.2018.2821119.

[24]

X. Liu, D. W. Ho and C. Xie, Prespecified-time cluster synchronization of complex networks via a smooth control approach,, IEEE Transactions on Cybernetics, 2018, 1–5. doi: 10.1109/TCYB.2018.2882519.

[25]

J. Lu and D. W. C. Ho, Globally exponential synchronization and synchronizability for general dynamical networks, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 40 (2010), 350-361. 

[26]

J. Lu, J. Cao, D. W. C. Ho and J. Kurths, Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay,, International Journal of Bifurcation and Chaos, 22 (2012), 1250176. doi: 10.1142/S0218127412501763.

[27]

J. LuC. DingJ. Lou and J. Cao, Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers,, Journal of the Franklin Institute, 352 (2015), 5024-5041.  doi: 10.1016/j.jfranklin.2015.08.016.

[28]

J. LuX. GuoT. Huang and Z. Wang, Consensus of signed networked multi-agent systems with nonlinear coupling and communication delays,, Applied Mathematics and Computation, 350 (2019), 153-162.  doi: 10.1016/j.amc.2019.01.006.

[29]

X. Mao, A note on the lasalle-type theorems for stochastic differential delay equations,, Journal of Mathematical Analysis and Applications, 268 (2002), 125-142.  doi: 10.1006/jmaa.2001.7803.

[30]

X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[31] P. V. Mieghem, Graph Spectra for Complex Networks, Cambridge University Press, Cambridge, 2011. 
[32]

T. PanD. JiangT. Hayat and A. Alsaedi, Extinction and periodic solutions for an impulsive sir model with incidence rate stochastically perturbed,, Physica A: Statistical Mechanics and its Applications, 505 (2018), 385-397.  doi: 10.1016/j.physa.2018.03.012.

[33]

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

[34]

L. ShiX. YangY. Li and Z. Feng, Finite-time synchronization of nonidentical chaotic systems with multiple time-varying delays and bounded perturbations,, Nonlinear Dynamics, 83 (2016), 75-87.  doi: 10.1007/s11071-015-2310-z.

[35]

Y. Song and J. Xu, Inphase and antiphase synchronization in a delay-coupled system with applications to a delay-coupled fitzhugh-nagumo system, IEEE Transactions on Neural Networks, 23 (2012), 1659-1670. 

[36]

G. Stamov and I. Stamova, Impulsive fractional functional differential systems and lyapunov method for the existence of almost periodic solutions,, Reports on Mathematical Physics, 75 (2015), 73-84.  doi: 10.1016/S0034-4877(15)60025-8.

[37]

I. Stamova, Global mittag-leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays,, Nonlinear Dynamics, 77 (2014), 1251-1260.  doi: 10.1007/s11071-014-1375-4.

[38]

X. TanJ. CaoX. Li and A. Alsaedi, Leader-following mean square consensus of stochastic multi-agent systems with input delay via event-triggered control,, IET Control Theory & Applications, 12 (2018), 299-309.  doi: 10.1049/iet-cta.2017.0462.

[39]

L. Tang, X. Wu, J. Lü and R. M. D'Souza, Master stability functions for complete, intra-layer and inter-layer synchronization in multiplex networks, arXiv: 1611.09110, 2017.

[40]

Z. TangJ. H. Park and W.-X. Zheng, Distributed impulsive synchronization of lur'e dynamical networks via parameter variation methods,, International Journal of Robust and Nonlinear Control, 28 (2018), 1001-1015.  doi: 10.1002/rnc.3916.

[41]

Z. TangJ. H. ParkY. Wang and J. Feng, Distributed impulsive quasi-synchronization of Lur'e networks with proportional delay,, IEEE Transactions on Cybernetics, 49 (2019), 3105-3115.  doi: 10.1109/TCYB.2018.2839178.

[42]

J. WangJ. FengC. Xu and Y. Zhao, Exponential synchronization of stochastic perturbed complex networks with time-varying delays via periodically intermittent pinning,, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3146-3157.  doi: 10.1016/j.cnsns.2013.03.021.

[43]

T. Wang, S. Zhao, W. Zhou and W. Yu, Almost sure synchronization control for stochastic delayed complex networks based on pinning adaptive method,, Advances in Difference Equations, 2016 (2016), Paper No. 306, 16 pp. doi: 10.1186/s13662-016-1013-1.

[44]

Y. Wang, F. Wu, X. Mao and E. Zhu, Advances in the lasalle-type theorems for stochastic functional differential equations with infinite delay, Discrete & Continuous Dynamical Systems-B, (2019), 57–65.

[45]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440-442.  doi: 10.1038/30918.

[46]

X. WuX. ZhaoJ. LüL. Tang and J. Lu, Identifying topologies of complex dynamical networks with stochastic perturbations,, IEEE Transactions on Control of Network Systems, 3 (2016), 379-389.  doi: 10.1109/TCNS.2015.2482178.

[47]

Y. XiaZ. Yang and M. Han, Lag synchronization of unknown chaotic delayed yang-yang-type fuzzy neural networks with noise perturbation based on adaptive control and parameter identification, IEEE Transactions on Neural Networks, 20 (2009), 1165-1180. 

[48]

L. YanW. Pei and Z. Zhang, Exponential stability of sdes driven by fbm with markovian switching,, Discrete & Continuous Dynamical Systems-A, 39 (2019), 6467-6483.  doi: 10.3934/dcds.2019280.

[49]

X. YangJ. LamD. W. C. Ho and Z. Feng, Fixed-time synchronization of complex networks with impulsive effects via nonchattering control,, IEEE Transactions on Automatic Control, 62 (2017), 5511-5521.  doi: 10.1109/TAC.2017.2691303.

[50]

L. Yu and Z. Yu, Synchronization of stochastic impulsive discrete-time delayed networks via pinning control, Neurocomputing, 286, pp. 31 – 40, 2018.

[51]

W. Yu and J. Cao, Synchronization control of stochastic delayed neural networks,, Physica A: Statistical Mechanics and its Applications, 373 (2007), 252-260.  doi: 10.1016/j.physa.2006.04.105.

[52]

W. YuJ. Cao and J. Lü, Global synchronization of linearly hybrid coupled networks with time-varying delay,, SIAM Journal on Applied Dynamical Systems, 7 (2008), 108-133.  doi: 10.1137/070679090.

[53]

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Figure 1.  an example of multi-layer network with 2 layers
Figure 2.  Multi-layer network with two layers and 100 nodes. (a) First layer: a Watts-Strogatz small-world graph. (b) Second layer: a scale-free graph
Figure 3.  The dynamic behavior of the systems (2) and (3) without control input
Figure 4.  The dynamic behavior of system (3) and the total error under controller (8)
Figure 5.  The dynamic behavior of system (3) and the total error under controller (20)
Figure 6.  The control gains under the state-feedback pinning controller versus the average controller gains under the adaptive pinning controller
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