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Asymptotic stability of nonlinear fractional delay differential equations with \(\alpha\in(1, 2)\): an application to fractional delay neural networks. (English) Zbl 07864167

MSC:

34K37 Functional-differential equations with fractional derivatives
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
34K20 Stability theory of functional-differential equations
68T05 Learning and adaptive systems in artificial intelligence
68T07 Artificial neural networks and deep learning
Full Text: DOI

References:

[1] Liu, P.; Wang, J.; Zeng, Z. G., Event-triggered synchronization of multiple fractional-order recurrent neural networks with time-varying delays, IEEE Trans. Neural Networks Learn. Syst., 34, 8, 4620-4630, 2023 · doi:10.1109/TNNLS.2021.3116382
[2] Wu, X.; Liu, S. T.; Wang, H. Y., Pinning synchronization of fractional memristor-based neural networks with neutral delays and reaction-diffusion terms, Commun. Nonlinear Sci. Numer. Simul., 118, 107039, 2023 · Zbl 1507.93188 · doi:10.1016/j.cnsns.2022.107039
[3] Zhang, H.; Wang, C.; Ye, R. Y.; Stamova, I.; Cao, J. D., Novel order-dependent passivity conditions of fractional generalized Cohen-Grossberg neural networks with proportional delays, Commun. Nonlinear Sci. Numer. Simul., 120, 107155, 2023 · Zbl 1509.34056 · doi:10.1016/j.cnsns.2023.107155
[4] Čermák, J.; Hornícěk, J.; Kisela, T., Stability regions for fractional differential systems with a time delay, Commun. Nonlinear Sci. Numer. Simul., 31, 108-123, 2016 · Zbl 1467.34082 · doi:10.1016/j.cnsns.2015.07.008
[5] Čermák, J.; Kisela, T., Oscillatory and asymptotic properties of fractional delay differential equations, Electron. J. Differ. Equ., 2019, 33, 2019 · Zbl 1411.34108
[6] Berkowitz, B.; Klafter, J.; Metzler, R.; Scher, H., Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk, and fractional derivative formulations, Water Resources Res., 38, 1-12, 2002 · doi:10.1029/2001WR001030
[7] Kou, S., Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins, Ann. Appl. Stat., 2, 501-535, 2008 · Zbl 1400.62272 · doi:10.1214/07-AOAS149
[8] Sakamoto, K.; Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 426-447, 2011 · Zbl 1219.35367 · doi:10.1016/j.jmaa.2011.04.058
[9] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339, 1, 77, 2000 · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[10] Achar, B. N. N.; Hanneken, J. W.; Clarke, T., Response characteristics of a fractional oscillator, Physica A, 309, 275-288, 2002 · Zbl 0995.70017 · doi:10.1016/S0378-4371(02)00609-X
[11] Čermák, J.; Kisela, T., Stabilization and destabilization of fractional oscillators via a delayed feedback control, Commun. Nonlinear Sci. Numer. Simul., 117, 106960, 2023 · Zbl 1502.34086 · doi:10.1016/j.cnsns.2022.106960
[12] Kang, Y. G.; Zhang, X. E., Some comparison of two fractional oscillators, Physica B, 405, 369-373, 2010 · doi:10.1016/j.physb.2010.04.036
[13] Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons Fractals, 7, 9, 1461-1477, 1996 · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5
[14] Tofighi, A., The intrinsic damping of the fractional oscillator, Phys. A, 329, 29-34, 2003 · doi:10.1016/S0378-4371(03)00598-3
[15] Li, Y.; Chen, Y. Q.; Podlubny, I., Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59, 1810-1821, 2010 · Zbl 1189.34015 · doi:10.1016/j.camwa.2009.08.019
[16] Tuan, H. T.; Trinh, H., A linearized stability theorem for nonlinear delay fractional differential equations, IEEE Trans. Automat. Contr., 63, 9, 3180-3186, 2018 · Zbl 1423.34092 · doi:10.1109/TAC.2018.2791485
[17] Baleanu, D.; Ranjbar, A.; Sadati, S. J.; Delavari, H.; Abdeljawad, T.; Gejji, V., Lyapunov-Krasovskii stability theorem for fractional systems with delay, Rom. J. Phys., 56, 636-643, 2011 · Zbl 1231.34005
[18] Wang, H.; Yu, Y. G.; Wen, G. G.; Zhang, S.; Yu, J. Z., Global stability analysis of fractional-order Hopfield neural networks with time delay, Neurocomputing, 154, 15-23, 2015 · doi:10.1016/j.neucom.2014.12.031
[19] Yao, Z. C.; Yang, Z. W.; Fu, Y. Q.; Liu, S. M., Chinese J. Phys. · doi:10.1016/j.cjph.2023.03.014
[20] Yang, Z. W.; Li, Q.; Yao, Z. C., A stability analysis for multi-term fractional delay differential equations with higher order, Chaos, Solitons Fractals, 167, 112997, 2023 · doi:10.1016/j.chaos.2022.112997
[21] Xu, Y.; Yu, J. T.; Li, W. X.; Feng, J. Q., Global asymptotic stability of fractional-order competitive neural networks with multiple time-varying-delay links, Appl. Math. Comput., 389, 125498, 2021 · Zbl 1459.65052 · doi:10.1016/j.cam.2020.113361
[22] Podlubny, I., Fractional Differential Equations, 1999, Academic Press: Academic Press, San Diego · Zbl 0918.34010
[23] Zhou, Y., Basic Theory of Fractional Differential Equations, 2014, World Scientific Publishing: World Scientific Publishing, Hackensack · Zbl 1336.34001
[24] Cong, N. D.; Doan, T. S.; Siegmund, S.; Tuan, H. T., Linearized asymptotic stability for fractional differential equations, Electron. J. Qual. Theory Differ. Equ., 39, 1-13, 2016 · Zbl 1363.34004 · doi:10.14232/ejqtde.2016.1.39
[25] Breda, D.; Maset, S.; Vermiglio, R., Stability of Linear Delay Differential Equations, 2015, Springer: Springer, New York · Zbl 1269.35012
[26] Diekmann, O.; van Gils, S. A.; Verduyn Lunel, S. M.; Walther, H. O., Delay Equations, 1995, Springer: Springer, New York · Zbl 0826.34002
[27] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, 2006, Elsevier: Elsevier, Amsterdam · Zbl 1092.45003
[28] Yao, Z. C.; Yang, Z. W.; Zhang, Y. S., A stability criterion for fractional-order complex-valued differential equations with distributed delays, Chaos, Solitons Fractals, 152, 111277, 2021 · Zbl 1505.34120 · doi:10.1016/j.chaos.2021.111277
[29] Čermák, J.; Kisela, T.; Doělá, Z., Fractional differential equations with a constant delay: Stability and asymptotics of solutions, Appl. Math. Comput., 298, 336-350, 2017 · Zbl 1411.34099 · doi:10.1016/j.amc.2016.11.016
[30] Guglielmi, N.; Hairer, E., Order stars and stability for delay differential equations, Numer. Math., 83, 3, 371-383, 1999 · Zbl 0937.65079 · doi:10.1007/s002110050454
[31] Hale, J., Theory of Functional Differential Equations, 1977, Springer-Verlag: Springer-Verlag, New York · Zbl 0352.34001
[32] Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products, 2007, Elsevier/Academic Press: Elsevier/Academic Press, Amsterdam · Zbl 1208.65001
[33] Lundstrom, B.; Higgs, M.; Spain, W., Fractional differentiation by neocortical pyramidal neurons, Nat. Neurosci., 11, 11, 1335-1342, 2008 · doi:10.1038/nn.2212
[34] Li, C. P.; Zeng, F. H., Numerical Methods for Fractional Calculus, 2015, CRC Press: CRC Press, Boca Raton, FL · Zbl 1326.65033
[35] Abbaszadeh, M.; Dehghan, M., Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method, Appl. Numer. Math., 145, 488-506, 2019 · Zbl 1428.65073 · doi:10.1016/j.apnum.2019.05.005
[36] Yao, Z. C.; Yang, Z. W., Stability and asymptotics for fractional delay diffusion-wave equations, Math. Methods Appl. Sci., 46, 14, 15208-15225, 2023 · Zbl 1531.35379 · doi:10.1002/mma.9372
[37] Zhang, G. F.; Liu, L. L.; Zhang, C. J., Compact scheme for fractional diffusion-wave equation with spatial variable coefficient and delays, Appl. Anal., 101, 6, 1911-1932, 2022 · Zbl 1490.65163 · doi:10.1080/00036811.2020.1789600
[38] Orsingher, E.; Beghin, L., Time-fractional telegraph equations and telegraph processes with Brownian time, Probab. Theory Relat. Fields, 128, 1, 141-160, 2004 · Zbl 1049.60062 · doi:10.1007/s00440-003-0309-8
[39] Zhao, Z. G.; Li, C. P., Fractional difference/finite element approximations for the time-space fractional telegraph equation, Appl. Math. Comput., 219, 6, 2975-2988, 2012 · Zbl 1309.65101 · doi:10.1016/j.amc.2012.09.022
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