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doi: 10.3934/dcdsb.2022005
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Asymptotic and finite-time cluster synchronization of neural networks via two different controllers

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, MO 210016, China

2. 

School of Mechatronics Engineering and Automation, Shanghai University, Shanghai, MO 200444, China

* Corresponding author: Fengli Ren

Received  June 2021 Revised  September 2021 Early access January 2022

In this paper, by using a pinning impulse controller and a hybrid controller respectively, the research difficulties of asymptotic synchronization and finite time cluster synchronization of time-varying delayed neural networks are studied. On the ground of Lyapunov stability theorem and Lyapunov-Razumikhin method, a novel sufficient criterion on asymptotic cluster synchronization of time-varying delayed neural networks is obtained. Utilizing Finite time stability theorem and hybrid control technology, a sufficient criterion on finite-time cluster synchronization is also obtained. In order to deal with time-varying delay and save control cost, pinning pulse control is introduced to promote the realization of asymptotic cluster synchronization. Following the idea of pinning control scheme, we design a progressive hybrid control to promote the realization of finite time cluster synchronization. Finally, an example is given to illustrate the theoretical results.

Citation: Juan Cao, Fengli Ren, Dacheng Zhou. Asymptotic and finite-time cluster synchronization of neural networks via two different controllers. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022005
References:
[1]

A. Abdurahman and H. Jiang, Improved control schemes for projective synchronization of delayed neural networks with unmatched coefficients, International Journal of Pattern Recognition and Artificial Intelligence, 34, (2020), 2051005. doi: 10.1142/S0218001420510052.

[2]

C. AouitiM. Bessifi and X. Li, Finite-time and fixed-time synchronization of complex-valued recurrent neural networks with discontinuous activations and time-varying delays, Circuits Systems and Signal Processing, 39 (2020), 5406-5428. 

[3]

J. CaoG. Chen and P. Li, Generalized analytical solutions and experimental confirmation of complete synchronization in a class of mutually-coupled simple nonlinear electronic circuits, Nonlinear Sciences, 113 (2017), 294-307. 

[4]

S. Ding and Z. Wang, Synchronization of coupled neural networks via an event-dependent intermittent pinning control, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 6 (2020), 1-7.  doi: 10.1109/TSMC.2020.3035173.

[5]

P. He, S.-H. Ma and T. Fan, Finite-time mixed outer synchronization of complex networks with coupling time-varying delay, Chaos, 22 (2012), 043151, 11 pp. doi: 10.1063/1.4773005.

[6]

W. HeF. Qian and J. Cao, Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control, Neural Networks, 85 (2017), 1-9.  doi: 10.1016/j.neunet.2016.09.002.

[7]

Z. Hou, J. P., Z. Liu, J. Zou and J. Luo, Theory of functional differential equations, Journal of Changsha Railway University.

[8]

A. HuJ. CaoM. Hu and L. Guo, Cluster synchronization of complex networks via event-triggered strategy under stochastic sampling, Phys. A, 434 (2015), 99-110.  doi: 10.1016/j.physa.2015.03.065.

[9]

X. JinZ. WangH. YangQ. Song and M. Xiao, Synchronization of multiplex networks with stochastic perturbations via pinning adaptive control, J. Franklin Inst., 358 (2021), 3994-4012.  doi: 10.1016/j.jfranklin.2021.03.004.

[10]

A. A. KoronovskiiO. I. MoskalenkoV. I. PonomarenkoM. D. Prokhorov and A. E. Hramov, Binary generalized synchronization, Chaos Solitons Fractals, 83 (2016), 133-139.  doi: 10.1016/j.chaos.2015.11.045.

[11]

R. KumarS. SarkarS. Das and J. Cao, Projective synchronization of delayed neural networks with mismatched parameters and impulsive effects, IEEE Trans. Neural Netw. Learn. Syst., 31 (2020), 1211-1221.  doi: 10.1109/TNNLS.2019.2919560.

[12]

K. LiJ. ZhaoH. Zhang and X. Li, On successive lag synchronization of a dynamical network with delayed couplings, IEEE Trans. Control Netw. Syst., 8 (2021), 1151-1162.  doi: 10.1109/TCNS.2021.3059218.

[13]

W. LiJ. ZhouJ. LiT. Xie and J.-A. Lu, Cluster synchronization of two-layer networks via aperiodically intermittent pinning control, IEEE Transactions on Circuits and Systems II: Express Briefs, 68 (2020), 1338-1342.  doi: 10.1109/TCSII.2020.3027592.

[14]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.

[15]

X. Li, X. Yang and J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica J. IFAC, 117 (2020), 108981, 7 pp. doi: 10.1016/j.automatica.2020.108981.

[16]

L. Liu, K. Liu, H. Xiang and Q. Liu, Cluster synchronization for directed complex dynamical networks via pinning control, Physica A: Statistical Mechanics and its Applications, 545 (2020), 123580, 9 pp. doi: 10.1016/j.physa.2019.123580.

[17]

L. Liu and Q. Liu, Cluster synchronization in complex dynamical network of nonidentical nodes with delayed and non-delayed coupling via pinning control, Physica Scripta, 94.

[18]

P. LiuZ. Zeng and J. Wang, Asymptotic and finite-time cluster synchronization of coupled fractional-order neural networks with time delay, IEEE Trans. Neural Netw. Learn. Syst., 31 (2020), 4956-4967.  doi: 10.1109/TNNLS.2019.2962006.

[19]

S. LiuN. JiangA. ZhaoY. Zhang and K. Qiu, Secure optical communication based on cluster chaos synchronization in semiconductor lasers network, IEEE Access, 8 (2020), 11872-11879.  doi: 10.1109/ACCESS.2020.2965960.

[20]

S. LiuH. Wu and J. Cao, Fixed-time synchronization for discontinuous delayed complex-valued networks with semi-Markovian switching and hybrid couplings via adaptive control, Internat. J. Adapt. Control Signal Process., 34 (2020), 1359-1382.  doi: 10.1002/acs.3153.

[21]

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.

[22]

J. MeiM. JiangW. Xu and B. Wang, Finite-time synchronization control of complex dynamical networks with time delay, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2462-2478.  doi: 10.1016/j.cnsns.2012.11.009.

[23]

A. Ouannas, X. Wang, V.-T. Pham, G. Grassi and T. Ziar, Coexistence of identical synchronization, antiphase synchronization and inverse full state hybrid projective synchronization in different dimensional fractional-order chaotic systems, Adv. Difference Equ., 2018 (2018), Paper No. 35, 16 pp. doi: 10.1186/s13662-018-1485-2.

[24]

L. PanJ. CaoU. Al-Juboori and M. Abdel-Aty, Cluster synchronization of stochastic neural networks with delay via pinning impulsive control, Neurocomputing, 366 (2019), 109-117. 

[25]

F. RenF. Cao and J. Cao, Mittag-Leffler stability and generalized Mittag-Leffler stability of fractional-order gene regulatory networks, Neurocomputing, 160 (2015), 185-190.  doi: 10.1016/j.neucom.2015.02.049.

[26]

D. SenthilkumarM. Lakshmanan and J. Kurths, Phase synchronization in unidirectionally coupled ikeda time-delay systems, Nonlinear Sciences, 164 (2008), 35-44. 

[27]

H. WangZ.-Z. HanQ.-Y. Xie and W. Zhang, Finite-time chaos control via nonsingular terminal sliding mode control, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2728-2733.  doi: 10.1016/j.cnsns.2008.08.013.

[28]

Y. WangZ. MaJ. CaoA. Alsaedi and F. E. Alsaadi, Adaptive cluster synchronization in directed networks with nonidentical nonlinear dynamics, Complexity, 21 (2016), 380-387.  doi: 10.1002/cplx.21816.

[29]

Q. WuJ. Zhou and L. Xiang, Global exponential stability of impulsive differential equations with any time delays, Appl. Math. Lett., 23 (2010), 143-147.  doi: 10.1016/j.aml.2009.09.001.

[30]

W. WuW. Zhou and T. Chen, Cluster synchronization of linearly coupled complex networks under pinning control, IEEE Trans. Circuits Syst. I. Regul. Pap., 56 (2009), 829-839.  doi: 10.1109/TCSI.2008.2003373.

[31]

Y. Wu and L. Guo, Enhancement of intercellular electrical synchronization by conductive materials in cardiac tissue engineering, IEEE Transactions on Biomedical Engineering, 65 (2018), 264-272.  doi: 10.1109/TBME.2017.2764000.

[32]

Z. WuQ. Ye and D. Liu, Finite-time synchronization of dynamical networks coupled with complex-variable chaotic systems, International Journal of Modern Physics C, 24 (2013), 1350058.  doi: 10.1142/S0129183113500587.

[33]

X. Xiong, X. Yang, J. Cao and R. Tang, Finite-time control for a class of hybrid systems via quantized intermittent control, Sci. China Inf. Sci., 63 (2020), Paper No. 192201, 16 pp. doi: 10.1007/s11432-018-2727-5.

[34]

Z. XuP. ShiH. SuZ.-G. Wu and T. Huang, Global $H_\infty$ pinning synchronization of complex networks with sampled-data communications, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 1467-1476.  doi: 10.1109/TNNLS.2017.2673960.

[35]

Q. YangH. Wu and J. Cao, Global cluster synchronization in finite time for complex dynamical networks with hybrid couplings via aperiodically intermittent control, Optimal Control Appl. Methods, 41 (2020), 1097-1117.  doi: 10.1002/oca.2589.

[36]

S. YangC. HuJ. Yu and H. Jiang, Finite-time cluster synchronization in complex-variable networks with fractional-order and nonlinear coupling, Neural networks: The official journal of the International Neural Network Society, 135 (2021), 212-224.  doi: 10.1016/j.neunet.2020.12.015.

[37]

J. Zha and C. Li, Synchronization of complex network based on the theory of gravitational field, Acta Phys. Polon. B, 50 (2019), 87-114.  doi: 10.5506/APhysPolB.50.87.

[38]

J. ZhouJ. ChenJ. Lu and J. Lü, On applicability of auxiliary system approach to detect generalized synchronization in complex network, IEEE Trans. Automat. Control, 62 (2017), 3468-3473.  doi: 10.1109/TAC.2016.2615679.

[39]

S. ZhuJ. ZhouX. Yu and J. Lu, Bounded synchronization of heterogeneous complex dynamical networks: A unified approach, IEEE Trans. Automat. Control, 66 (2021), 1756-1762.  doi: 10.1109/TAC.2020.2995822.

show all references

References:
[1]

A. Abdurahman and H. Jiang, Improved control schemes for projective synchronization of delayed neural networks with unmatched coefficients, International Journal of Pattern Recognition and Artificial Intelligence, 34, (2020), 2051005. doi: 10.1142/S0218001420510052.

[2]

C. AouitiM. Bessifi and X. Li, Finite-time and fixed-time synchronization of complex-valued recurrent neural networks with discontinuous activations and time-varying delays, Circuits Systems and Signal Processing, 39 (2020), 5406-5428. 

[3]

J. CaoG. Chen and P. Li, Generalized analytical solutions and experimental confirmation of complete synchronization in a class of mutually-coupled simple nonlinear electronic circuits, Nonlinear Sciences, 113 (2017), 294-307. 

[4]

S. Ding and Z. Wang, Synchronization of coupled neural networks via an event-dependent intermittent pinning control, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 6 (2020), 1-7.  doi: 10.1109/TSMC.2020.3035173.

[5]

P. He, S.-H. Ma and T. Fan, Finite-time mixed outer synchronization of complex networks with coupling time-varying delay, Chaos, 22 (2012), 043151, 11 pp. doi: 10.1063/1.4773005.

[6]

W. HeF. Qian and J. Cao, Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control, Neural Networks, 85 (2017), 1-9.  doi: 10.1016/j.neunet.2016.09.002.

[7]

Z. Hou, J. P., Z. Liu, J. Zou and J. Luo, Theory of functional differential equations, Journal of Changsha Railway University.

[8]

A. HuJ. CaoM. Hu and L. Guo, Cluster synchronization of complex networks via event-triggered strategy under stochastic sampling, Phys. A, 434 (2015), 99-110.  doi: 10.1016/j.physa.2015.03.065.

[9]

X. JinZ. WangH. YangQ. Song and M. Xiao, Synchronization of multiplex networks with stochastic perturbations via pinning adaptive control, J. Franklin Inst., 358 (2021), 3994-4012.  doi: 10.1016/j.jfranklin.2021.03.004.

[10]

A. A. KoronovskiiO. I. MoskalenkoV. I. PonomarenkoM. D. Prokhorov and A. E. Hramov, Binary generalized synchronization, Chaos Solitons Fractals, 83 (2016), 133-139.  doi: 10.1016/j.chaos.2015.11.045.

[11]

R. KumarS. SarkarS. Das and J. Cao, Projective synchronization of delayed neural networks with mismatched parameters and impulsive effects, IEEE Trans. Neural Netw. Learn. Syst., 31 (2020), 1211-1221.  doi: 10.1109/TNNLS.2019.2919560.

[12]

K. LiJ. ZhaoH. Zhang and X. Li, On successive lag synchronization of a dynamical network with delayed couplings, IEEE Trans. Control Netw. Syst., 8 (2021), 1151-1162.  doi: 10.1109/TCNS.2021.3059218.

[13]

W. LiJ. ZhouJ. LiT. Xie and J.-A. Lu, Cluster synchronization of two-layer networks via aperiodically intermittent pinning control, IEEE Transactions on Circuits and Systems II: Express Briefs, 68 (2020), 1338-1342.  doi: 10.1109/TCSII.2020.3027592.

[14]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.

[15]

X. Li, X. Yang and J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica J. IFAC, 117 (2020), 108981, 7 pp. doi: 10.1016/j.automatica.2020.108981.

[16]

L. Liu, K. Liu, H. Xiang and Q. Liu, Cluster synchronization for directed complex dynamical networks via pinning control, Physica A: Statistical Mechanics and its Applications, 545 (2020), 123580, 9 pp. doi: 10.1016/j.physa.2019.123580.

[17]

L. Liu and Q. Liu, Cluster synchronization in complex dynamical network of nonidentical nodes with delayed and non-delayed coupling via pinning control, Physica Scripta, 94.

[18]

P. LiuZ. Zeng and J. Wang, Asymptotic and finite-time cluster synchronization of coupled fractional-order neural networks with time delay, IEEE Trans. Neural Netw. Learn. Syst., 31 (2020), 4956-4967.  doi: 10.1109/TNNLS.2019.2962006.

[19]

S. LiuN. JiangA. ZhaoY. Zhang and K. Qiu, Secure optical communication based on cluster chaos synchronization in semiconductor lasers network, IEEE Access, 8 (2020), 11872-11879.  doi: 10.1109/ACCESS.2020.2965960.

[20]

S. LiuH. Wu and J. Cao, Fixed-time synchronization for discontinuous delayed complex-valued networks with semi-Markovian switching and hybrid couplings via adaptive control, Internat. J. Adapt. Control Signal Process., 34 (2020), 1359-1382.  doi: 10.1002/acs.3153.

[21]

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.

[22]

J. MeiM. JiangW. Xu and B. Wang, Finite-time synchronization control of complex dynamical networks with time delay, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2462-2478.  doi: 10.1016/j.cnsns.2012.11.009.

[23]

A. Ouannas, X. Wang, V.-T. Pham, G. Grassi and T. Ziar, Coexistence of identical synchronization, antiphase synchronization and inverse full state hybrid projective synchronization in different dimensional fractional-order chaotic systems, Adv. Difference Equ., 2018 (2018), Paper No. 35, 16 pp. doi: 10.1186/s13662-018-1485-2.

[24]

L. PanJ. CaoU. Al-Juboori and M. Abdel-Aty, Cluster synchronization of stochastic neural networks with delay via pinning impulsive control, Neurocomputing, 366 (2019), 109-117. 

[25]

F. RenF. Cao and J. Cao, Mittag-Leffler stability and generalized Mittag-Leffler stability of fractional-order gene regulatory networks, Neurocomputing, 160 (2015), 185-190.  doi: 10.1016/j.neucom.2015.02.049.

[26]

D. SenthilkumarM. Lakshmanan and J. Kurths, Phase synchronization in unidirectionally coupled ikeda time-delay systems, Nonlinear Sciences, 164 (2008), 35-44. 

[27]

H. WangZ.-Z. HanQ.-Y. Xie and W. Zhang, Finite-time chaos control via nonsingular terminal sliding mode control, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2728-2733.  doi: 10.1016/j.cnsns.2008.08.013.

[28]

Y. WangZ. MaJ. CaoA. Alsaedi and F. E. Alsaadi, Adaptive cluster synchronization in directed networks with nonidentical nonlinear dynamics, Complexity, 21 (2016), 380-387.  doi: 10.1002/cplx.21816.

[29]

Q. WuJ. Zhou and L. Xiang, Global exponential stability of impulsive differential equations with any time delays, Appl. Math. Lett., 23 (2010), 143-147.  doi: 10.1016/j.aml.2009.09.001.

[30]

W. WuW. Zhou and T. Chen, Cluster synchronization of linearly coupled complex networks under pinning control, IEEE Trans. Circuits Syst. I. Regul. Pap., 56 (2009), 829-839.  doi: 10.1109/TCSI.2008.2003373.

[31]

Y. Wu and L. Guo, Enhancement of intercellular electrical synchronization by conductive materials in cardiac tissue engineering, IEEE Transactions on Biomedical Engineering, 65 (2018), 264-272.  doi: 10.1109/TBME.2017.2764000.

[32]

Z. WuQ. Ye and D. Liu, Finite-time synchronization of dynamical networks coupled with complex-variable chaotic systems, International Journal of Modern Physics C, 24 (2013), 1350058.  doi: 10.1142/S0129183113500587.

[33]

X. Xiong, X. Yang, J. Cao and R. Tang, Finite-time control for a class of hybrid systems via quantized intermittent control, Sci. China Inf. Sci., 63 (2020), Paper No. 192201, 16 pp. doi: 10.1007/s11432-018-2727-5.

[34]

Z. XuP. ShiH. SuZ.-G. Wu and T. Huang, Global $H_\infty$ pinning synchronization of complex networks with sampled-data communications, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 1467-1476.  doi: 10.1109/TNNLS.2017.2673960.

[35]

Q. YangH. Wu and J. Cao, Global cluster synchronization in finite time for complex dynamical networks with hybrid couplings via aperiodically intermittent control, Optimal Control Appl. Methods, 41 (2020), 1097-1117.  doi: 10.1002/oca.2589.

[36]

S. YangC. HuJ. Yu and H. Jiang, Finite-time cluster synchronization in complex-variable networks with fractional-order and nonlinear coupling, Neural networks: The official journal of the International Neural Network Society, 135 (2021), 212-224.  doi: 10.1016/j.neunet.2020.12.015.

[37]

J. Zha and C. Li, Synchronization of complex network based on the theory of gravitational field, Acta Phys. Polon. B, 50 (2019), 87-114.  doi: 10.5506/APhysPolB.50.87.

[38]

J. ZhouJ. ChenJ. Lu and J. Lü, On applicability of auxiliary system approach to detect generalized synchronization in complex network, IEEE Trans. Automat. Control, 62 (2017), 3468-3473.  doi: 10.1109/TAC.2016.2615679.

[39]

S. ZhuJ. ZhouX. Yu and J. Lu, Bounded synchronization of heterogeneous complex dynamical networks: A unified approach, IEEE Trans. Automat. Control, 66 (2021), 1756-1762.  doi: 10.1109/TAC.2020.2995822.

Figure 1.  The error dynamic trajectory when $ i = 1, 2 $ without controller
Figure 2.  The error dynamic trajectory when $ i = 3, 4, 5 $ without controller
Figure 3.  The error dynamic trajectory when $ i = 1, 2 $ with the pinning impulsive controllers (6)
Figure 4.  The error dynamic trajectory when $ i = 3, 4, 5 $ with the pinning impulsive controllers (6)
Figure 5.  The error dynamic trajectory $ i = 1, 2 $ with the hybrid controllers (23)
Figure 6.  The error dynamic trajectory when $ i = 3, 4, 5 $ with the hybrid controllers (23)
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