American Institute of Mathematical Sciences

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April  2021, 14(4): 1465-1477. doi: 10.3934/dcdss.2020395

Finite-time exponential synchronization of reaction-diffusion delayed complex-dynamical networks

M. Syed Ali 1,, , L. Palanisamy 1, , Nallappan Gunasekaran 2, , Ahmed Alsaedi 3, and  Bashir Ahmad 3,

1. 

Department of Mathematics, Thiruvalluvar University, Vellore-632115, Tamil Nadu, India
2. 

Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama 337-8570, Japan
3. 

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics Faculty of Science, King Abdulaziz University, P.O. Box 80257, Jeddah 21589, Saudi Arabia

* Corresponding author: M. Syed Ali

Received  October 2019 Revised  March 2020 Published  April 2021 Early access  June 2020

Fund Project: This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (RG-39-130-38). The authors, therefore, acknowledge with thanks DSR technical and financial support.

This investigation looks at the issue of finite time exponential synchronization of complex dynamical systems with reaaction diffusion term. This reort studies complex networks consisting of $ N $ straightly and diffusively coupled networks. By building a new Lyapunov krasovskii functional (LKF), using Jensens inequality and convex algorithms approach stability conditions frameworks are determined. At last, a numerical precedent is given to demonstrate the practicality of the theoretical results.

Keywords: Complex dynamical systems, finite-time, reaction-diffusion, synchronization.
Mathematics Subject Classification: 34K20, 34K50, 92B20, 94D05.
Citation: M. Syed Ali, L. Palanisamy, Nallappan Gunasekaran, Ahmed Alsaedi, Bashir Ahmad. Finite-time exponential synchronization of reaction-diffusion delayed complex-dynamical networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1465-1477. doi: 10.3934/dcdss.2020395
References:
[1]

M. S. Ali, N. Gunasekaran and R. Saravanakumar, Design of passivity and passification for delayed neural networks with markovian jump parameters via non-uniform sampled-data control, Neural Computing and Applications, 30 (2018), 595-605.  doi: 10.1007/s00521-016-2682-0.

[2]

M. S. Ali, N. Gunasekaran and Q. X. Zhu, State estimation of T–S fuzzy delayed neural networks with Markovian jumping parameters using sampled-data control, Fuzzy Sets and Systems, 306 (2017), 87-104.  doi: 10.1016/j.fss.2016.03.012.

[3]

M. S. Ali, N. Gunasekaran and M. E. Rani, Robust stability of Hopfield delayed neural networks via an augmented LK functional, Neurocomputing, 234 (2017), 198-204. 

[4]

M. Syed Ali, K. Meenakshi, N. Gunasekaran and M. Usha, Finite-time passivity of discrete-time TS fuzzy neural networks with time-varying delays, Iranian Journal of Fuzzy Systems, 15 (2018), 93-107. 

[5]

M. Syed Ali, Q. X. Zhu, S. Pavithra and N. Gunasekaran, A study on $(Q, S, R)-\gamma$-dissipative synchronisation of coupled reaction-diffusion neural networks with time-varying delays, International Journal of Systems Science, 49 (2018), 755-765.  doi: 10.1080/00207721.2017.1422814.

[6]

P. Balasubramaniam and L. J. Banu, Synchronization criteria of discrete-time complex networks with time-varying delays and parameter uncertainties, Cognitive Neurodynamics, 8 (2014), 199-215.  doi: 10.1007/s11571-013-9272-y.

[7]

M. Fang, Synchronization for complex dynamical networks with time delay and discrete-time information, Applied Mathematics and Computation, 258 (2015), 1-11.  doi: 10.1016/j.amc.2015.01.106.

[8]

D. W. Gong, H. G. Zhang, Z. S. Wang and J. H. Liu, Synchronization analysis for complex networks with coupling delay based on T–S fuzzy theory, Applied Mathematical Modelling, 36 (2012), 6215-6224.  doi: 10.1016/j.apm.2012.01.041.

[9]

N. Gunasekaran, M. Syed Ali and S. Pavithra, Finite-time $L_\infty$ performance state estimation of recurrent neural networks with sampled-data signals, Neural Processing Letters, 51 (2019), 1379-1392.  doi: 10.1007/s11063-019-10114-9.

[10]

W. He and J. Cao, Exponential synchronization of hybrid coupled networks with delayed coupling, IEEE Transactions on Neural Networks, 21 (2010), 571-583. 

[11]

B. N. Huang, H. G. Zhang, D. W. Gong and J. Y. Wang, Synchronization analysis for static neural networks with hybrid couplings and time delays, Neurocomputing, 148 (2015), 288-293.  doi: 10.1016/j.neucom.2013.11.053.

[12]

D. H. Ji, J. H. Park, W. J. Yoo, S. C. Won and S. M. Lee, Synchronization criterion for Lur'e type complex dynamical networks with time-varying delay, Physics Letters A, 374 (2010), 1218-1227.  doi: 10.1016/j.physleta.2010.01.005.

[13]

Y.-G. Kao, J.-F. Guo, C.-H. Wang and X.-Q. Sun, Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen–Grossberg neural networks with mixed delays, Journal of the Franklin Institute, 349 (2012), 1972-1988.  doi: 10.1016/j.jfranklin.2012.04.005.

[14]

T. H. Lee, J. H. Park, H. Y. Jung, S. M. Lee and O. M. Kwon, Synchronization of a delayed complex dynamical network with free coupling matrix, Nonlinear Dynamics, 69 (2012), 1081-1090.  doi: 10.1007/s11071-012-0328-z.

[15]

J. G. Lu, Global exponential stability and periodicity of reaction–diffusion delayed recurrent neural networks with dirichlet boundary conditions, Chaos Solitons Fractals, 35 (2008), 116-125.  doi: 10.1016/j.chaos.2007.05.002.

[16]

N. N. Ma, Z. B. Liu and L. Chen, Finite-time $H_\infty$ synchronization for complex dynamical networks with markovian jump parameter, Journal of Control, Automation and Electrical Systems, 30 (2019), 75-84.  doi: 10.1007/s40313-018-00428-9.

[17]

F. Z. Nian and X. Y. Wang, Chaotic synchronization of hybrid state on complex networks, International Journal of Modern Physics C, 21 (2010), 457-469.  doi: 10.1142/S0129183110015221.

[18]

M.-J. Park, O. M. Kwon, J. H. Park, S.-M. Lee and E.-J. Cha, Synchronization criteria of fuzzy complex dynamical networks with interval time-varying delays, Applied Mathematics and Computation, 218 (2012), 11634-11647.  doi: 10.1016/j.amc.2012.05.046.

[19]

L. Scheeffer, Ueber die bedeutung der begriffe maximum und minimum iin der variationsrechnung, Mathematische Annalen, 26 (1886), 197-208.  doi: 10.1007/BF01444332.

[20]

J. Shen and J. Cao, Finite-time synchronization of coupled neural networks via discontinuous controllers, Cognitive Neurodynamics, 5 (2011), 373-385.  doi: 10.1007/s11571-011-9163-z.

[21]

M. Shirkavand, M. Pourgholi and A. Yazdizadeh, Robust fixed-time synchronisation of non-identical nodes in complex networks under input non-linearities, IET Control Theory & Applications, 13 (2019), 2095-2103. 

[22]

A. N. Langville and W. J. Stewart, The Kronecker product and stochastic automata networks, Journal of Computational and Applied Mathematics, 167 (2004), 429-447.  doi: 10.1016/j.cam.2003.10.010.

[23]

P. Park, J. W. Ko and C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47 (2011), 235-238.  doi: 10.1016/j.automatica.2010.10.014.

[24]

J.-A. Wang, New synchronization stability criteria for general complex dynamical networks with interval time-varying delays, Neural Computing and Applications, 28 (2017), 805-815.  doi: 10.1007/s00521-015-2108-4.

[25]

J. Y. Wang, J. W. Feng, C. Xu, Y. Zhao and J. Q. Feng, Pinning synchronization of nonlinearly coupled complex networks with time-varying delays using M-matrix strategies, Neurocomputing, 177 (2016), 89-97.  doi: 10.1016/j.neucom.2015.11.011.

[26]

L. Wang and Q.-G. Wang, Synchronization in complex networks with switching topology, Physics Letters A, 375 (2011), 3070-3074.  doi: 10.1016/j.physleta.2011.06.054.

[27]

M.-G. Wang, X.-Y. Wang and Z.-Z. Liu, A new complex network model with hierarchical and modular structures, Chinese Journal of Physics, 48 (2010), 805-813. 

[28]

Y. Xu, W. Zhou, C. Xie and D. Tong, Finite-time synchronization of the complex dynamical network with non-derivative and derivative coupling, Neurocomputing, 173 (2016), 1356-1361. 

[29]

X. S. Yang and J. D. Cao, Finite-time stochastic synchronization of complex networks, Applied Mathematical Modelling, 34 (2010), 3631-3641.  doi: 10.1016/j.apm.2010.03.012.

[30]

X. S. Yang, J. D. Cao and J. Q. Lu, Synchronization of delayed complex dynamical networks with impulsive and stochastic effects, Nonlinear Analysis: Real World Applications, 12 (2011), 2252-2266.  doi: 10.1016/j.nonrwa.2011.01.007.

[31]

X. S. Yang, J. D. Cao and Z. C. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller, SIAM Journal on Control and Optimization, 51 (2013), 3486-3510.  doi: 10.1137/120897341.

[32]

J. F. Zeng and J. D. Cao, Synchronisation in singular hybrid complex networks with delayed coupling, International Journal of Systems, Control and Communications, 3 (2011), 144-157.  doi: 10.1504/IJSCC.2011.039865.

[33]

J. Zhang and Y. B. Gao, Synchronization of coupled neural networks with time-varying delay, Neurocomputing, 219 (2017), 154-162.  doi: 10.1016/j.neucom.2016.09.004.

[34]

Y.-J. Zhang, S. Liu, R. Yang, Y.-Y. Tan and X. Y. Li, Global synchronization of fractional coupled networks with discrete and distributed delays, Physica A: Statistical Mechanics and its Applications, 514 (2019), 830-837.  doi: 10.1016/j.physa.2018.09.129.

[35]

J. Zhou, Q. J. Wu and L. Xiang, Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization, Nonlinear Dynamics, 69 (2012), 1393-1403.  doi: 10.1007/s11071-012-0355-9.

show all references

References:
[1]

M. S. Ali, N. Gunasekaran and R. Saravanakumar, Design of passivity and passification for delayed neural networks with markovian jump parameters via non-uniform sampled-data control, Neural Computing and Applications, 30 (2018), 595-605.  doi: 10.1007/s00521-016-2682-0.

[2]

M. S. Ali, N. Gunasekaran and Q. X. Zhu, State estimation of T–S fuzzy delayed neural networks with Markovian jumping parameters using sampled-data control, Fuzzy Sets and Systems, 306 (2017), 87-104.  doi: 10.1016/j.fss.2016.03.012.

[3]

M. S. Ali, N. Gunasekaran and M. E. Rani, Robust stability of Hopfield delayed neural networks via an augmented LK functional, Neurocomputing, 234 (2017), 198-204. 

[4]

M. Syed Ali, K. Meenakshi, N. Gunasekaran and M. Usha, Finite-time passivity of discrete-time TS fuzzy neural networks with time-varying delays, Iranian Journal of Fuzzy Systems, 15 (2018), 93-107. 

[5]

M. Syed Ali, Q. X. Zhu, S. Pavithra and N. Gunasekaran, A study on $(Q, S, R)-\gamma$-dissipative synchronisation of coupled reaction-diffusion neural networks with time-varying delays, International Journal of Systems Science, 49 (2018), 755-765.  doi: 10.1080/00207721.2017.1422814.

[6]

P. Balasubramaniam and L. J. Banu, Synchronization criteria of discrete-time complex networks with time-varying delays and parameter uncertainties, Cognitive Neurodynamics, 8 (2014), 199-215.  doi: 10.1007/s11571-013-9272-y.

[7]

M. Fang, Synchronization for complex dynamical networks with time delay and discrete-time information, Applied Mathematics and Computation, 258 (2015), 1-11.  doi: 10.1016/j.amc.2015.01.106.

[8]

D. W. Gong, H. G. Zhang, Z. S. Wang and J. H. Liu, Synchronization analysis for complex networks with coupling delay based on T–S fuzzy theory, Applied Mathematical Modelling, 36 (2012), 6215-6224.  doi: 10.1016/j.apm.2012.01.041.

[9]

N. Gunasekaran, M. Syed Ali and S. Pavithra, Finite-time $L_\infty$ performance state estimation of recurrent neural networks with sampled-data signals, Neural Processing Letters, 51 (2019), 1379-1392.  doi: 10.1007/s11063-019-10114-9.

[10]

W. He and J. Cao, Exponential synchronization of hybrid coupled networks with delayed coupling, IEEE Transactions on Neural Networks, 21 (2010), 571-583. 

[11]

B. N. Huang, H. G. Zhang, D. W. Gong and J. Y. Wang, Synchronization analysis for static neural networks with hybrid couplings and time delays, Neurocomputing, 148 (2015), 288-293.  doi: 10.1016/j.neucom.2013.11.053.

[12]

D. H. Ji, J. H. Park, W. J. Yoo, S. C. Won and S. M. Lee, Synchronization criterion for Lur'e type complex dynamical networks with time-varying delay, Physics Letters A, 374 (2010), 1218-1227.  doi: 10.1016/j.physleta.2010.01.005.

[13]

Y.-G. Kao, J.-F. Guo, C.-H. Wang and X.-Q. Sun, Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen–Grossberg neural networks with mixed delays, Journal of the Franklin Institute, 349 (2012), 1972-1988.  doi: 10.1016/j.jfranklin.2012.04.005.

[14]

T. H. Lee, J. H. Park, H. Y. Jung, S. M. Lee and O. M. Kwon, Synchronization of a delayed complex dynamical network with free coupling matrix, Nonlinear Dynamics, 69 (2012), 1081-1090.  doi: 10.1007/s11071-012-0328-z.

[15]

J. G. Lu, Global exponential stability and periodicity of reaction–diffusion delayed recurrent neural networks with dirichlet boundary conditions, Chaos Solitons Fractals, 35 (2008), 116-125.  doi: 10.1016/j.chaos.2007.05.002.

[16]

N. N. Ma, Z. B. Liu and L. Chen, Finite-time $H_\infty$ synchronization for complex dynamical networks with markovian jump parameter, Journal of Control, Automation and Electrical Systems, 30 (2019), 75-84.  doi: 10.1007/s40313-018-00428-9.

[17]

F. Z. Nian and X. Y. Wang, Chaotic synchronization of hybrid state on complex networks, International Journal of Modern Physics C, 21 (2010), 457-469.  doi: 10.1142/S0129183110015221.

[18]

M.-J. Park, O. M. Kwon, J. H. Park, S.-M. Lee and E.-J. Cha, Synchronization criteria of fuzzy complex dynamical networks with interval time-varying delays, Applied Mathematics and Computation, 218 (2012), 11634-11647.  doi: 10.1016/j.amc.2012.05.046.

[19]

L. Scheeffer, Ueber die bedeutung der begriffe maximum und minimum iin der variationsrechnung, Mathematische Annalen, 26 (1886), 197-208.  doi: 10.1007/BF01444332.

[20]

J. Shen and J. Cao, Finite-time synchronization of coupled neural networks via discontinuous controllers, Cognitive Neurodynamics, 5 (2011), 373-385.  doi: 10.1007/s11571-011-9163-z.

[21]

M. Shirkavand, M. Pourgholi and A. Yazdizadeh, Robust fixed-time synchronisation of non-identical nodes in complex networks under input non-linearities, IET Control Theory & Applications, 13 (2019), 2095-2103. 

[22]

A. N. Langville and W. J. Stewart, The Kronecker product and stochastic automata networks, Journal of Computational and Applied Mathematics, 167 (2004), 429-447.  doi: 10.1016/j.cam.2003.10.010.

[23]

P. Park, J. W. Ko and C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47 (2011), 235-238.  doi: 10.1016/j.automatica.2010.10.014.

[24]

J.-A. Wang, New synchronization stability criteria for general complex dynamical networks with interval time-varying delays, Neural Computing and Applications, 28 (2017), 805-815.  doi: 10.1007/s00521-015-2108-4.

[25]

J. Y. Wang, J. W. Feng, C. Xu, Y. Zhao and J. Q. Feng, Pinning synchronization of nonlinearly coupled complex networks with time-varying delays using M-matrix strategies, Neurocomputing, 177 (2016), 89-97.  doi: 10.1016/j.neucom.2015.11.011.

[26]

L. Wang and Q.-G. Wang, Synchronization in complex networks with switching topology, Physics Letters A, 375 (2011), 3070-3074.  doi: 10.1016/j.physleta.2011.06.054.

[27]

M.-G. Wang, X.-Y. Wang and Z.-Z. Liu, A new complex network model with hierarchical and modular structures, Chinese Journal of Physics, 48 (2010), 805-813. 

[28]

Y. Xu, W. Zhou, C. Xie and D. Tong, Finite-time synchronization of the complex dynamical network with non-derivative and derivative coupling, Neurocomputing, 173 (2016), 1356-1361. 

[29]

X. S. Yang and J. D. Cao, Finite-time stochastic synchronization of complex networks, Applied Mathematical Modelling, 34 (2010), 3631-3641.  doi: 10.1016/j.apm.2010.03.012.

[30]

X. S. Yang, J. D. Cao and J. Q. Lu, Synchronization of delayed complex dynamical networks with impulsive and stochastic effects, Nonlinear Analysis: Real World Applications, 12 (2011), 2252-2266.  doi: 10.1016/j.nonrwa.2011.01.007.

[31]

X. S. Yang, J. D. Cao and Z. C. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller, SIAM Journal on Control and Optimization, 51 (2013), 3486-3510.  doi: 10.1137/120897341.

[32]

J. F. Zeng and J. D. Cao, Synchronisation in singular hybrid complex networks with delayed coupling, International Journal of Systems, Control and Communications, 3 (2011), 144-157.  doi: 10.1504/IJSCC.2011.039865.

[33]

J. Zhang and Y. B. Gao, Synchronization of coupled neural networks with time-varying delay, Neurocomputing, 219 (2017), 154-162.  doi: 10.1016/j.neucom.2016.09.004.

[34]

Y.-J. Zhang, S. Liu, R. Yang, Y.-Y. Tan and X. Y. Li, Global synchronization of fractional coupled networks with discrete and distributed delays, Physica A: Statistical Mechanics and its Applications, 514 (2019), 830-837.  doi: 10.1016/j.physa.2018.09.129.

[35]

J. Zhou, Q. J. Wu and L. Xiang, Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization, Nonlinear Dynamics, 69 (2012), 1393-1403.  doi: 10.1007/s11071-012-0355-9.

Figure 1.  Error trajectories of the system in Example 1 with node 6
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