Delay-coupled fractional order complex Cohen-Grossberg neural networks under parameter uncertainty: Synchronization stability criteria

Pratap Anbalagan; Evren Hincal; Raja Ramachandran; Dumitru Baleanu; Jinde Cao; Chuangxia Huang; Michal Niezabitowski; Pratap Anbalagan; Evren Hincal; Raja Ramachandran; Dumitru Baleanu; Jinde Cao; Chuangxia Huang; Michal Niezabitowski

doi:10.3934/math.2021172

AIMS Mathematics

2021, Volume 6, Issue 3: 2844-2873. doi: 10.3934/math.2021172

Previous Article Next Article

Research article

Delay-coupled fractional order complex Cohen-Grossberg neural networks under parameter uncertainty: Synchronization stability criteria

1.
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
2.
Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi-630 004, India
3.
Department of Mathematics, Cankaya University, Ankara 06530, Turkey
4.
School of Mathematics, Southeast University, Nanjing 210096, China, and Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea
5.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha, 410114, China
6.
Faculty of Automatic Control, Electronics and Computer Science, Department of Automatic Control, and Robotics, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland

Received: 27 October 2020 Accepted: 31 December 2020 Published: 06 January 2021
MSC : 26A33, 34K37

This paper inspects the issues of synchronization stability and robust synchronization stability for fractional order coupled complex interconnected Cohen-Grossberg neural networks under linear coupling delays. For investigation of synchronization stability results, the comparison theorem for multiple delayed fractional order linear system is derived at first. Then, by means of given fractional comparison principle, some inequality methods, Kronecker product technique and classical Lyapunov-functional, several asymptotical synchronization stability criteria are addressed in the voice of linear matrix inequality (LMI) for the proposed model. Moreover, when parameter uncertainty exists, we also the investigate on the robust synchronization stability criteria for complex structure on linear coupling delayed Cohen-Grossberg type neural networks. At last, the validity of the proposed analytical results are performed by two computer simulations.
- synchronization stability,
- fractional order,
- complex coupled Cohen-Grossberg neural networks,
- Kronecker product,
- linear coupling delay
Citation: Pratap Anbalagan, Evren Hincal, Raja Ramachandran, Dumitru Baleanu, Jinde Cao, Chuangxia Huang, Michal Niezabitowski. Delay-coupled fractional order complex Cohen-Grossberg neural networks under parameter uncertainty: Synchronization stability criteria[J]. AIMS Mathematics, 2021, 6(3): 2844-2873. doi: 10.3934/math.2021172

Related Papers:

Abstract

This paper inspects the issues of synchronization stability and robust synchronization stability for fractional order coupled complex interconnected Cohen-Grossberg neural networks under linear coupling delays. For investigation of synchronization stability results, the comparison theorem for multiple delayed fractional order linear system is derived at first. Then, by means of given fractional comparison principle, some inequality methods, Kronecker product technique and classical Lyapunov-functional, several asymptotical synchronization stability criteria are addressed in the voice of linear matrix inequality (LMI) for the proposed model. Moreover, when parameter uncertainty exists, we also the investigate on the robust synchronization stability criteria for complex structure on linear coupling delayed Cohen-Grossberg type neural networks. At last, the validity of the proposed analytical results are performed by two computer simulations.

References

[1]	L. Chen, J. Cao, R. Wu, J. A. Tenreiro Machado, A. M. Lopes, H. Yang, Stability and synchronization of fractional-order memristive neural networks with multiple delays, Neural Networks, 94 (2017), 76–85. doi: 10.1016/j.neunet.2017.06.012
[2]	L. O. Chua, L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits Syst., 35 (1988), 1273–1290. doi: 10.1109/31.7601
[3]	M. A. Cohen, S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst., Man, Cybernetics, 5 (1983), 815–826.
[4]	J. Cheng, J. H. Park, J. Cao, W. Qi, A hidden mode observation approach to finite-time SOFC of Markovian switching with quantization, Nonlinear Dynam., 100 (2020), 509–521. doi: 10.1007/s11071-020-05501-0
[5]	C. Huang, X. Long, L. Huang, S. Fu, Stability of almost periodic Nicholson's blowflies model involving patch structure and mortality terms, Can. Math. Bull., 63 (2020), 405–422. doi: 10.4153/S0008439519000511
[6]	J. Cheng, J. H. Park, X. Zhao, H. Karimi, J. Cao, Quantized nonstationary filtering of network-based Markov switching RSNSs: A multiple hierarchical structure strategy, IEEE Transactions on Automatic Control, 65 (2020), 4816–4823. doi: 10.1109/TAC.2019.2958824
[7]	L. Duan, M. Shi, C. Huang, X. Fang, Synchronization in finite-/fixed-time of delayed diffusive complex-valued neural networks with discontinuous activations, Chaos, Solitons Fractals, (2020), DOI: 10.1016/j.chaos.2020.110386.
[8]	M. A. Duarte-Mermoud, N. Aguila-Camacho, J. A. Gallegos, R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci., 22 (2015), 650–659. doi: 10.1016/j.cnsns.2014.10.008
[9]	A. S. Elwakil, Fractional-order circuits and systems: An emerging interdisciplinary research area, IEEE Circ Syst. Mag, 10 (2010), 40–50.
[10]	R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, 2014.
[11]	C. M. Gray, Synchronous oscillations in neuronal systems: Mechanisms and functions, J. Comput. Neurosci., 1 (1994), 11–38. doi: 10.1007/BF00962716
[12]	B. B. He, H. C. Zhou, Y. Q. Chen, C. H. Kou, Asymptotical stability of fractional order systems with time delay via an integral inequality, IET Control Theory Appl., 12 (2018), 1748–1754. doi: 10.1049/iet-cta.2017.1144
[13]	H. Wang, Y. Yu, G. Wen, S. Zhang, J. Yu, Global stability analysis of fractional-order Hopfield neural networks with time delay, Neurocomputing, 154 (2015), 15–23. doi: 10.1016/j.neucom.2014.12.031
[14]	C. Song, S. Fei, J. Cao, C. Huang, Robust Synchronization of Fractional-Order Uncertain Chaotic Systems Based on Output Feedback Sliding Mode Control. Mathematics, 7 (2019), 599. Available from: https://doi.org/10.3390/math7070599.
[15]	C. Huang, X. Zhao, J. Cao, F. Alsaadi, Global dynamics of neoclassical growth model with multiple pairs of variable delays, Nonlinearity, 33(12) (2020), 6819–6834.
[16]	C. Huang, L. Yang, J. Cao, Asymptotic behavior for a class of population dynamics, AIMS Math., 5 (2020), 3378–3390. doi: 10.3934/math.2020218
[17]	H. Yang, Weighted pseudo almost periodicity on neutral type CNNs involving multi-proportional delays and D operator, AIMS Math., 6 (2020), 1865–1879.
[18]	C. Huang, J. Wang, L. Huang, New results on asymptotically almost periodicity of delayed Nicholson-type system involving patch structure, Electron. J. Diff. Eq., 2020 (2020), 1–17. doi: 10.1186/s13662-019-2438-0
[19]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier B.V, 2006.
[20]	A. Langville, W. Stewart, The Kronecker product and stochastic automata networks, J. Comput. Appl. Math., 167 (2004), 429–447. doi: 10.1016/j.cam.2003.10.010
[21]	N. Laskin, Fractional market dynamics, Physica A, 287 (2000), 482–492.
[22]	S. Liang, R. Wu, L. Chen, Adaptive pinning synchronization in fractional-order uncertain complex dynamical networks with delay, Physica A, 444 (2015), 49–62.
[23]	B. Li, N. Wang, X. Ruan, Q. Pan, Pinning and adaptive synchronization of fractional-order complex dynamical networks with and without time-varying delay, Adv. Differ. Eq., (2018). Available from: https://doi.org/10.1186/s13662-017-1454-1.
[24]	H. Li, C. Hu, Y. Jiang, Z. Wang, Z. Teng, Pinning adaptive and impulsive synchronization of fractional-order complex dynamical networks, Chaos, Solitons Fractals, 92 (2016), 142–149. doi: 10.1016/j.chaos.2016.09.023
[25]	Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810–1821. doi: 10.1016/j.camwa.2009.08.019
[26]	R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59 (2010), 1586–1593. doi: 10.1016/j.camwa.2009.08.039
[27]	M. L. Morgado, N. J. Ford, P. M. Lima, Analysis and numerical methods for fractional differential equations with delay, J. Comput. Appl. Math., 252 (2013), 159–168.
[28]	I. Petras, Fractional-order nonlinear systems: Modeling, analysis and simulation, Springer Berlin, 2011.
[29]	I. Podlubny, Fractional differential equations, San Diego California: Academic Press, 1999.
[30]	C. Rajivganthi, F. A. Rihan, S. Lakshmanan, P. Muthukumar, Finite-time stability analysis for fractional-order Cohen–Grossberg BAM neural networks with time delays, Neural Comput. Appl., 29 (2018), 1309–1320. doi: 10.1007/s00521-016-2641-9
[31]	G. Ren, Y. Yu, Pinning synchronization of fractional general complex dynamical networks with time Delay, IFAC Papers Online, 50-1 (2017), 8058–8065.
[32]	X. Ruan, A. Wu, Multi-quasi-synchronization of coupled fractional-order neural networks with delays via pinning impulsive control, Adv. Diff. Eq., 359 (2017). Available from: https://doi.org/10.1186/s13662-017-1417-6.
[33]	N. Sene, Stability analysis of electrical RLC circuit described by the Caputo-Liouville generalized fractional derivative, Alex. Eng. J., 59(2020), 2083–2090. doi: 10.1016/j.aej.2020.01.008
[34]	W. Shuxue, H. Yanli, R. Shunyan, Synchronization and robust synchronization for fractional-order coupled neural networks, IEEE Access, 5 (2017), 12439–12448. doi: 10.1109/ACCESS.2017.2721950
[35]	Q. Shuihan, H. Yanli, R. Shunyan, Finite-time synchronization of coupled Cohen-Grossberg neural networks with and without coupling delays, J. Franklin Inst., 355 (2018), 4379–4403. doi: 10.1016/j.jfranklin.2018.04.023
[36]	P. Thiran, K. R. Crounse, L. O. Chua, M. Hasler, Pattern formation properties of autonomous cellular neural networks, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., 42 (1995), 757–774. doi: 10.1109/81.473585
[37]	G. Velmurugan, R. Rakkiyappan, J. Cao, Finite-time synchronization of fractional-order memristor-based neural networks with time delays, Neural Networks, 73 (2016), 36–46. doi: 10.1016/j.neunet.2015.09.012
[38]	L. Wan, A. Wu, Mittag-Leffler stability analysis of fractional-order fuzzy Cohen-Grossberg neural networks with deviating argument, Advances Diff. Eq., (2017). Available from: https//doi.org/10.1186/s13662-017-1368-y.
[39]	F. Wang, Y. Q. Yang, M. F. Hu, Asymptotic stability of delayed fractional-order neural networks with impulsive effects, Neurocomputing, 154 (2015), 239–244. doi: 10.1016/j.neucom.2014.11.068
[40]	F. Wang, Y. Yang, A. Hu, X. Xu, Exponential synchronization of fractional-order complex networks via pinning impulsive control, Nonlinear Dyn., 82 (2015), 1979–1987. doi: 10.1007/s11071-015-2292-x
[41]	L. Wang, H. Wu, T. Huang, S. Ren, J. Wu, Pinning control for synchronization of coupled reaction-diffusion neural networks with directed topologies, IEEE Trans. Syst., Man, Cybernetics: Syst., 46 (2016), 1109–1120. doi: 10.1109/TSMC.2015.2476491
[42]	Y. Xu, J. Yu, W. Li, J. Feng, Global asymptotic stability of fractional-order competitive neural networks with multiple time-varying-delay links, Appl. Math. Comput., 389 (2020), 125498.
[43]	X. Yang, C. Li, T. Huang, Q. Song, J. Huang, Synchronization of fractional-order memristor-based complex-valued neural networks with uncertain parameters and time delays, Chaos, Solitons Fractals, 110 (2018), 105–123. doi: 10.1016/j.chaos.2018.03.016
[44]	H. Yanli, C. Weizhong, R. Shunyan, Z. Zewei, Analysis and pinning control for generalized synchronization of delayed coupled neural networks with different dimensional nodes, J. Franklin Inst., 355 (2018), 5968–5997. doi: 10.1016/j.jfranklin.2018.05.055
[45]	H. Yanli, Q. Shuihan, R. Shunyan, Z. Zewei, Fixed-time synchronization of coupled Cohen-Grossberg neural networks with and without parameter uncertainties, Neurocomputing, 315 (2018), 157–168. doi: 10.1016/j.neucom.2018.07.013
[46]	A. Pratap, R. Raja, J. Cao, C. Huang, M. Niezabitowski, O. Bagdasar, Stability of discrete-time fractional-order time-delayed neural networks in complex field, Math. Methods Appl. Sci., (2020). Available from: https://doi.org/10.1002/mma.6745.
[47]	X. M. Zhang, S. Y. Sheng, G. P. Lu, Y. F. Zheng, Synchronization for arrays of coupled jumping delayed neural networks and its application to image encryption, Proceeding of the $56$th Annual Conference on Decision and Control, 2017.
[48]	J. Zhou, T. P. Chen, L. Xiang, Chaotic lag synchronization of coupled delayed neural networks and its applications in secure communication, Circuits Syst. Signal Process., 24 (2005), 599–613. doi: 10.1007/s00034-005-2410-y

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)