In recent years, exponential Euler difference methods for first-order semi-linear differential equations have been developed rapidly, and various exponential Euler difference methods have become a very important research topic. For fractional-order shunting inhibitory cellular neural networks with time lags, this article first establishes a new difference method named Mittag-Leffler Euler difference, which includes exponential Euler difference. Second, the existence of unique global bounded solution and equilibrium point, exponential stability and synchronization of the derived difference model are investigated. Compared with the classical fractional-order Euler difference method, the Mittag-Leffler discrete shunting inhibitory cellular neural networks can better describe and maintain the dynamic properties of the corresponding continuous-time models. More importantly, it opens up a new way to study discrete-time fractional-order systems and establishes a set of theory and methods to study Mittag-Leffler discrete neural networks.
BouzerdoumAPinterRB. Shunting inhibitory cellular neural networks: derivation and stability analysis. IEEE T Circuits-i1993; 40(3): 215–221.
2.
HuangCXYangHDCaoJD. Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with operator. Discrete Cont Dyn-s2021; 14(4): 1259–1272.
3.
AkhmetMSeilovaRTleubergenovaM, et al. Shunting inhibitory cellular neural networks with strongly unpredictable oscillations. Commun Nonlinear Sci2020; 89: 105287.
4.
YuXXWangQR. Weighted pseudo-almost periodic solutions for shunting inhibitory cellular neural networks on time scales. Bull Malays Math Sci So2019; 42: 2055–2074.
5.
MiraouiMYaakoubiN. Measure pseudo almost periodic solutions of shunting inhibitory cellular neural networks with mixed delays. Numer Func Anal Opt2019; 40(5): 571–585.
6.
ChenZBZhangAP. Weighted pseudo almost periodic shunting inhibitory cellular neural networks with multi-proportional delays. Neural Process Lett2019; 50: 1831–1843.
7.
ZhangTWXuLJ. Mean almost periodicity and moment exponential stability of discrete-time stochastic shunting inhibitory cellular neural networks with time delays. Kybernetika2019; 55(4): 690–713.
8.
XuCJLiPL. On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator. Neurocomputing2017; 275: 377–382.
9.
ŞaylıMıYlmazE. Periodic solution for state-dependent impulsive shunting inhibitory CNNs with time-varying delays. Neural Networks2015; 68: 1–11.
10.
KilbasAASrivastavaHMTrujilloJJ. Theory and applications of fractional differential equations. Boston, MA: Elsevier, 2006.
11.
PodlubnyI. Fractional differential equations. San Diego, CA: Academic Press, 1999.
12.
ZhangTWXiongLL. Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative. Appl Math Lett2020; 101: 106072.
13.
KumarDSinghJTanwarK, et al. A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws. Int J Heat Mass Tran2019; 138: 1222–1227.
14.
ZhangTWZhouJWLiaoYZ. Exponentially stable periodic oscillation and Mittag-Leffler stabilization for fractional-order impulsive control neural networks with piecewise Caputo derivatives. IEEE T Cybernetics. Epub ahead of print 4March2021. DOI: 10.1109/TCYB.2021.3054946.
15.
El-SakaHAALeeSJangB.Dynamic analysis of fractional-order predator-prey biological economic system with Holling type II functional response. Nonlinear Dynam2019; 96(2): 407–416.
16.
SharmaRBhasinSGaurP, et al. A switching-based collaborative fractional order fuzzy logic controllers for robotic manipulators. Appl Math Model2019; 73(3): 228–246.
17.
ZhangTWLiYK. S-asymptotically periodic fractional functional differential equations with off-diagonal matrix Mittag-Leffler function kernels. Math Comput Simulat2022; 193: 331–347.
18.
NietoJJ. Solution of a fractional logistic ordinary differential equation. Appl Math Lett2022; 123: 107568.
19.
ZhangTWLiYK. Exponential Euler scheme of multi-delay Caputo–Fabrizio fractional-order differential equations. Appl Math Lett2022; 124: 107709.
20.
PecoraLMCarrollTL. Synchronization in chaotic system. Phys Rev Lett1990; 64(8): 821–824.
21.
ZhaoYRenSSKurthsJG. Synchronization of coupled memristive competitive BAM neural networks with different time scales. Neurocomputing2021; 427: 110–117.
22.
SaderMWangFYLiuZG, et al. Projective synchronization analysis for BAM neural networks with time-varying delay via novel control. Nonlinear Anal-model2021; 26(1): 41–56.
23.
MuhammadhajiATengZD. Synchronization stability on the BAM neural networks with mixed time delays. Int J Nonlin Sci Num2021; 22(1): 99–109.
24.
KongFCRajanR. Finite-time and fixed-time synchronization control of discontinuous fuzzy Cohen-Grossberg neural networks with uncertain external perturbations and mixed time delays. Fuzzy Set Syst2020; 411(4): 105–135.
25.
ZhangGDZengZGNingD. Novel results on synchronization for a class of switched inertial neural networks with distributed delays. Inform Sciences2020; 511: 114–126.
26.
LiuXYuYG. Synchronization analysis for discrete fractional-order complex-valued neural networks with time delays. Neural Comput Appl2021; 33(16): 10503–10514.
27.
YuJHuCJiangH, et al. Exponential and adaptive synchronization of inertial complex-valued neural networks: a non-reduced order and non-separation approach. Neural Netw2020; 124: 50–59.
28.
YouXXSongQKZhaoZJ. Global Mittag-Leffler stability and synchronization of discrete-time fractional-order complex-valued neural networks with time delay. Neural Netw2020; 122: 382–394.
29.
ButcherJC. Numerical methods for ordinary differential equations. Oxford: John Wiley & Sons, 2016.
30.
DaiXJXiaoAGBuWP. Stochastic fractional integro-differential equations with weakly singular kernels: well-posedness and Euler–Maruyama approximation. Discrete Cont Dyn-B2021; 27: 4231–4253.
31.
HuangZKMohamadSGaoF. Multi-almost periodicity in semi-discretizations of a general class of neural networks. Math Comput Simulat2014; 101: 43–60.
32.
HuPHuangCM. Delay dependent asymptotic mean square stability analysis of the stochastic exponential Euler method. J Comput Appl Math2021; 382: 113068.
33.
ZhangTWHanSFZhouJW. Dynamic behaviours for semi-discrete stochastic Cohen-Grossberg neural networks with time delays. J Franklin I2020; 357(17): 13006–13040.
34.
HuJWangJ. Global exponential periodicity and stability of discrete-time complex-valued recurrent neural networks with time-delays. Neural Networks2015; 66: 119–130.
35.
CalvoMPPalenciaC. A class of explicit multistep exponential integrators for semilinear problems. Numer Math2006; 102: 367–381.
36.
OstermannAThalhammerMWrightWM. A class of exponential general linear methods. BIT Number Math2006; 46(2): 409–431.
37.
HochbruckMOstermannASchweitzerJ. Exponential Rosenbrock-type methods. SIAM J Numer Anal2009; 47(1): 786–803.
38.
WangXMDongXG. A concise tutorial on computational methods. Beijing, China: Science Press, 2010.
39.
GuoBLPuXKHuangFH. Frational partial differential equations and their numerical solutions. Beijing, China: Science Press, 2011.
40.
WuGCDengZGBaleanuD, et al. New variable-order fractional chaotic systems for fast image encryption. Chaos2019; 29(8): 083103.
WuGCZengDQBaleanuD. Fractional impulsive differential equations: exact solutions, integral equations and short memory case. Fract Calc Appl Anal2019; 22(1): 180–192.
43.
LiuPZengZGWangJ. Global synchronization of coupled fractional-order recurrent neural networks. IEEE Trans Neural Netw Learn Syst2019; 30(8): 2358–2368.
44.
DaiXJXiaoAG. A note on Euler method for the overdamped generalized Langevin equation with fractional noise. Appl Math Lett2021; 111: 106669.
45.
YangZWZhengXCZhangZQ, et al. Strong convergence of a Euler-Maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noise. Chaos Soliton Fract2021; 142: 110392.
46.
SharmaRKXiongXKamalS, et al. Discrete-time super-twisting fractional-order differentiator with implicit Euler method. IEEE T Circuits-ii2021; 68(4): 1238–1242.
47.
WangJRFečkanMZhouY. Presentation of solutions of impulsive fractional Langevin equations and existence results. Eur Phys J Special Topics2013; 222(8): 1857–1874.
48.
AliMSNarayananGShekherV, et al. Global Mittag-Leffler stability analysis of impulsive fractional-order complex-valued BAM neural networks with time varying delays. Commun Nonlinear Sci2020; 83: 105088.
49.
AliMSNarayananGSevgenS, et al. Global stability analysis of fractional-order fuzzy BAM neural networks with time delay and impulsive effects. Commun Nonlinear Sci2019; 78: 104853.
50.
StamovaIStamovGSimeonovS, et al. Mittag-Leffler stability of impulsive fractional-order bi-directional associative memory neural networks with time-varying delays. T I Meas Control2018; 40(10): 3068–3077.
51.
WangFYangYQXuXY, et al. Global asymptotic stability of impulsive fractional-order BAM neural networks with time delay. Neural Comput Appl2017; 28: 345–352.
52.
ZhengXCWangH. A hidden-memory variable-order time-fractional optimal control model: analysis and approximation. SIAM J Control Optim2021; 59(3): 1851–1880.
53.
ZhengXCWangH. An error estimate of a numerical approximation to a hidden-memory variable-order space-time fractional diffusion equation. SIAM J Numer Anal2020; 58(5): 2492–2514.
54.
SamkoSGRossB. Integration and differentiation to a variable fractional order. Integr Transf Spec F1993; 1(4): 277–300.
55.
ZhengXCWangH. An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J Numer Anal2020; 58(1): 330–352.
56.
ZhengXCWangH. Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions. IMA J Numer Anal2021; 41(2): 1522–1545.
