Abstract
In this paper, we study the quasi-synchronization control for a class of fractional-order quaternion-valued neural networks (QVNNs). Unlike previous studies, the introduce of the reaction–diffusion terms makes the model more comprehensive. It is worth mentioning that in earlier literature, there are rare results based on linear matrix inequality method when using the decomposition method (dividing the system into four real-valued ones) to study the synchronization for QVNNs, which is due to the great challenge posed by the cross-product terms. In this work, we overcome this problem by processing the cross-product term as some free variables and then obtain the quasi-synchronization criterion of the system by linear matrix inequalities (LMI) method, which extends some previous works. Finally, a numerical example is given to support the results of this paper.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Data Availability
The authors confirm that the data supporting the findings of this study are available within the article.
References
Hartley TT, Lorenzo CF (2003) Fractional-order system identification based on continuous order-distributions. Signal Process 83(11):2287–2300
Liu J, Zhai R, Liu Y, Li W, Wang B, Huang L (2021) A quasi fractional order gradient descent method with adaptive stepsize and its application in system identification. Appl Math Comput 393:125797
Aguilar CZ, Gómez-Aguilar J, Alvarado-Martínez V, Romero-Ugalde H (2020) Fractional order neural networks for system identification. Chaos Solitons Fractals 130:109444
Ali MS, Hymavathi M (2021) Synchronization of fractional order neutral type fuzzy cellular neural networks with discrete and distributed delays via state feedback control. Neural Process Lett 53(2):929–957
Chanthorn P, Rajchakit G, Ramalingam S, Lim CP, Ramachandran R (2020) Robust dissipativity analysis of Hopfield-type complex-valued neural networks with time-varying delays and linear fractional uncertainties. Mathematics 8(4):595
Fang W, Yao X, Zhao X, Yin J, Xiong N (2016) A stochastic control approach to maximize profit on service provisioning for mobile cloudlet platforms. IEEE Trans Syst Man Cybern Syst 48(4):522–534
Arena P, Caponetto R, Fortuna L, Porto D (1998) Bifurcation and chaos in noninteger order cellular neural networks. Int J Bifurc Chaos 8(7):1527–1539
Arena P, Fortuna L, Porto D (2000) Chaotic behavior in noninteger-order cellular neural networks. Phys Rev E 61(1):776
Chen J, Zeng Z, Jiang P (2014) Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw 51:1–8
Chen L, Liu C, Wu R, He Y, Chai Y (2016) Finite-time stability criteria for a class of fractional-order neural networks with delay. Neural Comput Appl 27(3):549–556
Huang C, Cao J (2020) Bifurcation mechanisation of a fractional-order neural network with unequal delays. Neural Process Lett 52(2):1171–1187
Lu J, Jiang B, Zheng WX (2022) Potential impacts of delay on stability of impulsive control systems. IEEE Trans Autom Control. https://doi.org/10.1109/TAC.2021.3120672
Niamsup P, Rajchakit M, Rajchakit G (2013) Guaranteed cost control for switched recurrent neural networks with interval time-varying delay. J Inequal Appl 2013(1):1–12
Lin B, Zhu F, Zhang J, Chen J, Chen X, Xiong NN, Mauri JL (2019) A time-driven data placement strategy for a scientific workflow combining edge computing and cloud computing. IEEE Trans Ind Inform 15(7):4254–4265
Bao H, Park JH, Cao J (2016) Synchronization of fractional-order complex-valued neural networks with time delay. Neural Netw 81:16–28
Zhang L, Song Q, Zhao Z (2017) Stability analysis of fractional-order complex-valued neural networks with both leakage and discrete delays. Appl Math Comput 298:296–309
Rakkiyappan R, Velmurugan G, Cao J (2015) Stability analysis of fractional-order complex-valued neural networks with time delays. Chaos Solitons Fractals 78:297–316
Chanthorn P, Rajchakit G, Thipcha J, Emharuethai C, Sriraman R, Lim CP, Ramachandran R (2020) Robust stability of complex-valued stochastic neural networks with time-varying delays and parameter uncertainties. Mathematics 8(5):742
Chanthorn P, Rajchakit G, Humphries U, Kaewmesri P, Sriraman R, Lim CP (2020) A delay-dividing approach to robust stability of uncertain stochastic complex-valued hopfield delayed neural networks. Symmetry 12(5):683
Syed Ali M, Narayanan G, Orman Z, Shekher V, Arik S (2020) Finite time stability analysis of fractional-order complex-valued memristive neural networks with proportional delays. Neural Process Lett 51(1):407–426
Chen X, Song Q, Li Z (2017) Design and analysis of quaternion-valued neural networks for associative memories. IEEE Trans Syst Man Cybern Syst 48(12):2305–2314
Humphries U, Rajchakit G, Kaewmesri P, Chanthorn P, Sriraman R, Samidurai R, Lim CP (2020) Stochastic memristive quaternion-valued neural networks with time delays: an analysis on mean square exponential input-to-state stability. Mathematics 8(5):815
Aouiti C, Bessifi M (2021) Periodically intermittent control for finite-time synchronization of delayed quaternion-valued neural networks. Neural Comput Appl 33(12):6527–6547
Chen X, Song Q (2017) State estimation for quaternion-valued neural networks with multiple time delays. IEEE Trans Syst Man Cybern Syst 49(11):2278–2287
Liu Y, Zhang D, Lou J, Lu J, Cao J (2017) Stability analysis of quaternion-valued neural networks: decomposition and direct approaches. IEEE Trans Neural Netw Learn Syst 29(9):4201–4211
Pratap A, Raja R, Alzabut J, Cao J, Rajchakit G, Huang C (2020) Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field. Math Methods Appl Sci 43(10):6223–6253
Ali MS, Narayanan G, Nahavandi S, Wang J, Cao J (2021) Global dissipativity analysis and stability analysis for fractional-order quaternion-valued neural networks with time delays. IEEE Trans Syst Man Cybern Syst.(https://doi.org/10.1109/TSMC.2021.3065114)
Jian J, Wu K, Wang B (2020) Global Mittag-Leffler boundedness of fractional-order fuzzy quaternion-valued neural networks with linear threshold neurons. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2020.3014659
Humphries U, Rajchakit G, Kaewmesri P, Chanthorn P, Sriraman R, Samidurai R, Lim CP (2020) Global stability analysis of fractional-order quaternion-valued bidirectional associative memory neural networks. Mathematics 8(5):801
Song X, Man J, Song S, Ahn CK (2020) Finite/fixed-time anti-synchronization of inconsistent Markovian quaternion-valued memristive neural networks with reaction–diffusion terms. IEEE Trans Circuits Syst I Regul Pap 68(1):363–375
Zhang R, Zeng D, Park JH, Lam H-K, Xie X (2020) Fuzzy sampled-data control for synchronization of T-S fuzzy reaction-diffusion neural networks with additive time-varying delays. IEEE Trans Cybern 51(5):2384–2397
Song X, Man J, Fu Z, Wang M, Lu J (2019) Memory-based state estimation of T-S fuzzy Markov jump delayed neural networks with reaction-diffusion terms. Neural Process Lett 50(3):2529–2546
Narayanan G, Ali MS, Alam MI, Rajchakit G, Boonsatit N, Kumar P, Hammachukiattikul P (2021) Adaptive fuzzy feedback controller design for finite-time Mittag-Leffler synchronization of fractional-order quaternion-valued reaction–diffusion fuzzy molecular modeling of delayed neural networks. IEEE Access 9:130862–130883
Wang G, Shen Y (2014) Exponential synchronization of coupled memristive neural networks with time delays. Neural Comput Appl 24(6):1421–1430
Zhang W, Li C, Huang T, Tan J (2015) Exponential stability of inertial BAM neural networks with time-varying delay via periodically intermittent control. Neural Comput Appl 26(7):1781–1787
Li L, Sun Y, Lu J, Cao J (2022) Dynamic quantization driven synchronization of networked systems under event-triggered mechanism. IEEE Trans Circuits Syst I Regul Pap 69(4):1728–1740
Sun Y, Li L, Ho DW (2021) Quantized synchronization control of networked nonlinear systems: dynamic quantizer design with event-triggered mechanism. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2021.3090999
Rajchakit M, Niamsup P, Rajchakit G (2013) A switching rule for exponential stability of switched recurrent neural networks with interval time-varying delay. Adv Differ Equ 2013(1):1–10
Aouiti C, Assali EA, Cherif F, Zeglaoui A (2020) Fixed-time synchronization of competitive neural networks with proportional delays and impulsive effect. Neural Comput Appl 32(17):13245–13254
Xiao J, Cao J, Cheng J, Zhong S, Wen S (2020) Novel methods to finite-time Mittag-Leffler synchronization problem of fractional-order quaternion-valued neural networks. Inf Sci 526:221–244
Li H, Zhang L, Hu C, Jiang H, Cao J (2020) Global Mittag-Leffler synchronization of fractional-order delayed quaternion-valued neural networks: direct quaternion approach. Appl Math Comput 373:125020
Chen J, Chen B, Zeng Z (2018) Global asymptotic stability and adaptive ultimate Mittag-Leffler synchronization for a fractional-order complex-valued memristive neural networks with delays. IEEE Trans Syst Man Cybern Syst 49(12):2519–2535
Fan Y, Huang X, Li Y, Xia J, Chen G (2018) Aperiodically intermittent control for quasi-synchronization of delayed memristive neural networks: an interval matrix and matrix measure combined method. IEEE Trans Syst Man Cybern Syst 49(11):2254–2265
Song X, Li X, Song S, Zhang Y, Ning Z (2021) Quasi-synchronization of coupled neural networks with reaction-diffusion terms driven by fractional Brownian motion. J Franklin Inst 358(4):2482–2499
Li R, Gao X, Cao J (2019) Quasi-state estimation and quasi-synchronization control of quaternion-valued fractional-order fuzzy memristive neural networks: vector ordering approach. Appl Math Comput 362:124572
Yang X, Li C, Song Q, Chen J, Huang J (2018) Global Mittag-Leffler stability and synchronization analysis of fractional-order quaternion-valued neural networks with linear threshold neurons. Neural Netw 105:88–103
Li H, Kao Y, Hu C, Jiang H, Jiang Y (2021) Robust exponential stability of fractional-order coupled quaternion-valued neural networks with parametric uncertainties and impulsive effects. Chaos Solitons Fractals 143:110598
Podlubny I (1999) Fractional Differential Equations. Academic Press, New York
Nishio Y, Ushida A (1996) Quasi-synchronization phenomena in chaotic circuits coupled by one resistor. IEEE Trans Circuits Syst I Fundam Theory Appl 43(6):491–496
Kilbas AA, Saigo M, Saxena RK (2004) Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transform Spec Funct 15(1):31–49
Yang S, Yu J, Hu C, Jiang H (2018) Quasi-projective synchronization of fractional-order complex-valued recurrent neural networks. Neural Netw 104:104–113
Lu JG (2008) Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions. Chaos Solitons Fractals 35(1):116–125
Xie L (1996) Output feedback \({H}_\infty \) control of systems with parameter uncertainty. Int J Control 63(4):741–750
Li X, Zhang W, Fang J, Li H (2019) Event-triggered exponential synchronization for complex-valued memristive neural networks with time-varying delays. IEEE Trans Neural Netw Learn Syst 31(10):4104–4116
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grants 61976081 and 62203153, in part by the Natural Science Fund for Excellent Young Scholars of Henan Province under Grant 202300410127, in part by Key Scientific Research Projects of Higher Education Institutions in Henan Province under Grant 22A413001, in part by Top Young Talents in Central Plains under Grant Yuzutong (2021)44, in part by Technology Innovative Teams in University of Henan Province under Grant 23IRTSTHN012, and in part by the Natural Science Fund for Young Scholars of Henan Province under Grant 222300420151.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sun, X., Song, X., Man, J. et al. Quasi-Synchronization for Fractional-Order Reaction–Diffusion Quaternion-Valued Neural Networks: An LMI Approach. Neural Process Lett 55, 4499–4517 (2023). https://doi.org/10.1007/s11063-022-11054-7
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11063-022-11054-7