Neutral-type, leakage, and mixed delays in fractional-order neural networks: asymptotic synchronization analysis

Călin-Adrian Popa; Călin-Adrian Popa

doi:10.3934/math.2023815

AIMS Mathematics

2023, Volume 8, Issue 7: 15969-15992. doi: 10.3934/math.2023815

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Research article

Neutral-type, leakage, and mixed delays in fractional-order neural networks: asymptotic synchronization analysis

Călin-Adrian Popa ^{1,2
,
,}

1.
Department of Mathematics, West University of Timişoara, Blvd. V. Pârvan, No. 4, Timişoara 300223, Romania
2.
Department of Computer and Software Engineering, Polytechnic University Timişoara, Blvd. V. Pârvan, No. 2, Timişoara 300223, Romania

Received: 22 December 2022 Revised: 11 April 2023 Accepted: 12 April 2023 Published: 04 May 2023
MSC : 93C10, 93C43, 93D23

The dynamics of fractional-order neural networks (FONNs) are challenging to study, since the traditional Lyapunov theory does not apply to them. Instead, Halanay-type lemmas are used to create sufficient criteria for specific dynamic properties of FONNs. The application of these lemmas, however, typically leads to conservative criteria. The Halanay-type lemma is used in a novel way in this study to develop less conservative sufficient conditions in terms of linear matrix inequalities (LMIs) for extremely general FONNs, with different types of delays, such as neutral-type, leakage, time-varying, and distributed delays. The formulation of such a general model for the fractional-order scenario is done here for the first time. In addition, a new Lyapunov-like function is established, resulting in algebraic conditions that are less conservative. Three theorems are put forward that build sufficient criteria for the asymptotic synchronization, employing state feedback control, of the proposed networks, each based on a different Lyapunov-like function. For the first time in the context of FONNs, the free weighting matrix technique is also used to greatly decrease the conservatism of the obtained sufficient conditions. One numerical simulation illustrates each of the three theorems.
- fractional-order neural networks (FONNs),
- asymptotic synchronization,
- leakage delay,
- distributed delay,
- mixed delays,
- neutral-type delay
Citation: Călin-Adrian Popa. Neutral-type, leakage, and mixed delays in fractional-order neural networks: asymptotic synchronization analysis[J]. AIMS Mathematics, 2023, 8(7): 15969-15992. doi: 10.3934/math.2023815

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Abstract

The dynamics of fractional-order neural networks (FONNs) are challenging to study, since the traditional Lyapunov theory does not apply to them. Instead, Halanay-type lemmas are used to create sufficient criteria for specific dynamic properties of FONNs. The application of these lemmas, however, typically leads to conservative criteria. The Halanay-type lemma is used in a novel way in this study to develop less conservative sufficient conditions in terms of linear matrix inequalities (LMIs) for extremely general FONNs, with different types of delays, such as neutral-type, leakage, time-varying, and distributed delays. The formulation of such a general model for the fractional-order scenario is done here for the first time. In addition, a new Lyapunov-like function is established, resulting in algebraic conditions that are less conservative. Three theorems are put forward that build sufficient criteria for the asymptotic synchronization, employing state feedback control, of the proposed networks, each based on a different Lyapunov-like function. For the first time in the context of FONNs, the free weighting matrix technique is also used to greatly decrease the conservatism of the obtained sufficient conditions. One numerical simulation illustrates each of the three theorems.

References

[1]	E. Kaslik, S. Sivasundaram, Nonlinear dynamics and chaos in fractional-order neural networks, Neural Networks, 32 (2012), 245–256. https://doi.org/10.1016/j.neunet.2012.02.030 doi: 10.1016/j.neunet.2012.02.030
[2]	L. Chen, T. Huang, J. A. T. Machado, A. M. Lopes, Y. Chai, R. Wu, Delay-dependent criterion for asymptotic stability of a class of fractional-order memristive neural networks with time-varying delays, Neural Networks, 118 (2019), 289–299. https://doi.org/10.1016/j.neunet.2019.07.006 doi: 10.1016/j.neunet.2019.07.006
[3]	M. S. Ali, M. Hymavathi, S. Senan, V. Shekher, S. Arik, Global asymptotic synchronization of impulsive fractional-order complex-valued memristor-based neural networks with time varying delays, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104869. https://doi.org/10.1016/j.cnsns.2019.104869 doi: 10.1016/j.cnsns.2019.104869
[4]	X. You, Q. Song, Z. Zhao, Global mittag-leffler stability and synchronization of discrete-time fractional-order complex-valued neural networks with time delay, Neural Networks, 122 (2020), 382–394. https://doi.org/10.1016/j.neunet.2019.11.004 doi: 10.1016/j.neunet.2019.11.004
[5]	M. S. Ali, G. Narayanan, V. Shekher, A. Alsaedi, B. Ahmad, Global mittag-leffler stability analysis of impulsive fractional-order complex-valued BAM neural networks with time varying delays, Commun. Nonlinear Sci. Numer. Simul., 83 (2020), 105088. https://doi.org/10.1016/j.cnsns.2019.105088 doi: 10.1016/j.cnsns.2019.105088
[6]	C. A. Popa, E. Kaslik, Finite-time mittag-leffler synchronization of neutral-type fractional-order neural networks with leakage delay and time-varying delays, Mathematics, 8 (2020), 1146. https://doi.org/10.3390/math8071146 doi: 10.3390/math8071146
[7]	D. Ding, Z. You, Y. Hu, Z. Yang, L. Ding, Finite-time synchronization for fractional-order memristor-based neural networks with discontinuous activations and multiple delays, Mod. Phys. Lett. B, 34 (2020), 2050162. https://doi.org/10.1142/S0217984920501626 doi: 10.1142/S0217984920501626
[8]	G. Velmurugan, R. Rakkiyappan, V. Vembarasan, J. Cao, A. Alsaedi, Dissipativity and stability analysis of fractional-order complex-valued neural networks with time delay, Neural Networks, 86 (2017), 42–53. https://doi.org/10.1016/j.neunet.2016.10.010 doi: 10.1016/j.neunet.2016.10.010
[9]	Y. Fan, X. Huang, Z. Wang, J. Xia, Y. Li. Global mittag-leffler synchronization of delayed fractional-order memristive neural networks, Adv. Differ. Equations, 2018 (2018), 338. https://doi.org/10.1186/s13662-018-1800-y doi: 10.1186/s13662-018-1800-y
[10]	C. Huang, J. Tang, Y. Niu, J. Cao, Enhanced bifurcation results for a delayed fractional neural network with heterogeneous orders, Phys. A, 526 (2019), 121014. https://doi.org/10.1016/j.physa.2019.04.250 doi: 10.1016/j.physa.2019.04.250
[11]	C. Huang, H. Liu, X. Shi, X. Chen, M. Xiao, Z. Wang, et al., Bifurcations in a fractional-order neural network with multiple leakage delays, Neural Networks, 131 (2020), 115–126. https://doi.org/10.1016/j.neunet.2020.07.015 doi: 10.1016/j.neunet.2020.07.015
[12]	C. Huang, J. Wang, X. Chen, J. Cao, Bifurcations in a fractional-order BAM neural network with four different delays, Neural Networks, 141 (2021), 344–354. https://doi.org/10.1016/j.neunet.2021.04.005 doi: 10.1016/j.neunet.2021.04.005
[13]	C. Xu, M. Liao, P. Li, L. Yao, Q. Qin, Y. Shang, Chaos control for a fractional-order jerk system via time delay feedback controller and mixed controller, Fract. Fractional, 5 (2021), 257. https://doi.org/10.3390/fractalfract5040257 doi: 10.3390/fractalfract5040257
[14]	C. Xu, Z. Liu, P. Li, J. Yan, L. Yao, Bifurcation mechanism for fractional-order three-triangle multi-delayed neural networks, Neural Process. Lett., 2022. https://doi.org/10.1007/s11063-022-11130-y doi: 10.1007/s11063-022-11130-y
[15]	C. Xu, D. Mu, Z. Liu, Y. Pang, M. Liao, P. Li, et al., Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks, Nonlinear Anal., 27 (2022), 1–24. https://doi.org/10.15388/namc.2022.27.28491 doi: 10.15388/namc.2022.27.28491
[16]	C. Xu, D. Mu, Z. Liu, Y. Pang, M. Liao, C. Aouiti, New insight into bifurcation of fractional-order 4d neural networks incorporating two different time delays, Commun. Nonlinear Sci. Numer. Simul., 118 (2023), 107043. https://doi.org/10.1016/j.cnsns.2022.107043 doi: 10.1016/j.cnsns.2022.107043
[17]	C. Xu, W. Zhang, C. Aouiti, Z. Liu, L. Yao, Bifurcation insight for a fractional-order stage-structured predator-prey system incorporating mixed time delays, Math. Methods Appl. Sci., 2023. https://doi.org/10.1002/mma.9041 doi: 10.1002/mma.9041
[18]	W. Zhang, H. Zhang, J. Cao, H. Zhang, D. Chen, Synchronization of delayed fractional-order complex-valued neural networks with leakage delay, Phys. A, 556 (2020), 124710. https://doi.org/10.1016/j.physa.2020.124710 doi: 10.1016/j.physa.2020.124710
[19]	S. Yang, H. Jiang, C. Hu, J. Yu, Synchronization for fractional-order reaction-diffusion competitive neural networks with leakage and discrete delays, Neurocomputing, 436 (2021), 47–57. https://doi.org/10.1016/j.neucom.2021.01.009 doi: 10.1016/j.neucom.2021.01.009
[20]	Z. Wu, Multiple asymptotic stability of fractional-order quaternion-valued neural networks with time-varying delays, Neurocomputing, 448 (2021), 301–312. https://doi.org/10.1016/j.neucom.2021.03.079 doi: 10.1016/j.neucom.2021.03.079
[21]	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 (2021), 125498. https://doi.org/10.1016/j.amc.2020.125498 doi: 10.1016/j.amc.2020.125498
[22]	A. Singh, J. N. Rai, Stability of fractional order fuzzy cellular neural networks with distributed delays via hybrid feedback controllers, Neural Process. Lett., 53 (2021), 1469–1499. https://doi.org/10.1007/s11063-021-10460-7 doi: 10.1007/s11063-021-10460-7
[23]	I. Stamova, G. Stamov, Impulsive control strategy for the mittag-leffler synchronization of fractional-order neural networks with mixed bounded and unbounded delays, AIMS Math., 6 (2021), 2287–2303. https://doi.org/10.3934/math.2021138 doi: 10.3934/math.2021138
[24]	S. M. A. Pahnehkolaei, A. Alfi, J. A. T. Machado, Delay-dependent stability analysis of the QUAD vector field fractional order quaternion-valued memristive uncertain neutral type leaky integrator echo state neural networks, Neural Networks, 117 (2019), 307–327. https://doi.org/10.1016/j.neunet.2019.05.015 doi: 10.1016/j.neunet.2019.05.015
[25]	I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, 1998.
[26]	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. Numer. Simul., 22 (2015), 650–659. https://doi.org/10.1016/j.cnsns.2014.10.008 doi: 10.1016/j.cnsns.2014.10.008
[27]	P. Liu, M. Kong, Z. Zeng, Projective synchronization analysis of fractional-order neural networks with mixed time delays, IEEE Trans. Cybern., 52 (2022), 6798–6808. https://doi.org/10.1109/TCYB.2020.3027755 doi: 10.1109/TCYB.2020.3027755
[28]	J. Cao, D. W. C. Ho, X. Huang, LMI-based criteria for global robust stability of bidirectional associative memory networks with time delay, Nonlinear Anal., 66 (2027), 1558–1572. https://doi.org/10.1016/j.na.2006.02.009 doi: 10.1016/j.na.2006.02.009
[29]	J. Jia, X. Huang, Y. Li, J. Cao, A. Alsaedi, Global stabilization of fractional-order memristor-based neural networks with time delay, IEEE Trans. Neural Networks Learn. Syst., 31 (2020), 997–1009. https://doi.org/10.1109/TNNLS.2019.2915353 doi: 10.1109/TNNLS.2019.2915353
[30]	S. Yang, J. Yu, C. Hu, H. Jiang, Finite-time synchronization of memristive neural networks with fractional-order, IEEE Trans. Syst. Man Cybern., 51 (2021), 3739–3750. https://doi.org/10.1109/TSMC.2019.2931046 doi: 10.1109/TSMC.2019.2931046

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