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Finite-Time Mittag-Leffler Stability of Fractional-Order Quaternion-Valued Memristive Neural Networks with Impulses

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Abstract

The finite-time Mittag-Leffler stability for fractional-order quaternion-valued memristive neural networks (FQMNNs) with impulsive effect is studied here. A new mathematical expression of the quaternion-value memductance (memristance) is proposed according to the feature of the quaternion-valued memristive and a new class of FQMNNs is designed. In quaternion field, by using the framework of Filippov solutions as well as differential inclusion theoretical analysis, suitable Lyapunov-functional and some fractional inequality techniques, the existence of unique equilibrium point and Mittag-Leffler stability in finite time analysis for considered impulsive FQMNNs have been established with the order \(0<\beta <1\). Then, for the fractional order \(\beta \) satisfying \(1<\beta <2\) and by ignoring the impulsive effects, a new sufficient criterion are given to ensure the finite time stability of considered new FQMNNs system by the employment of Laplace transform, Mittag-Leffler function and generalized Gronwall inequality. Furthermore, the asymptotic stability of such system with order \(1<\beta <2\) have been investigated. Ultimately, the accuracy and validity of obtained finite time stability criteria are supported by two numerical examples.

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References

  1. Song C, Fei S, Cao J, Huang C (2019) Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control. Mathematics 7(7):599. https://doi.org/10.3390/math7070599

    Article  Google Scholar 

  2. Huang C, Zhao X, Wang X, Wang Z, Xiao M, Cao J (2019) Disparate delays-induced bifurcations in a fractional-order neural network. J Franklin Inst 356(5):2825–2846

    MathSciNet  MATH  Google Scholar 

  3. Huang C, Cao J (2018) Impact of leakage delay on bifurcation in high-order fractional BAM neural networks. Neural Netw 98:223–235

    Google Scholar 

  4. Rajchakit G, Pratap A, Raja R, Cao J, Alzabut J, Huang C (2019) Hybrid control scheme for projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks with mixed delays. Mathematics 7(8):759. https://doi.org/10.3390/math7080759

    Article  Google Scholar 

  5. Chen B, Chen J (2016) Global \(O(t^{-\alpha })\) stability and global asymptotical periodicity for a non-autonomous fractional-order neural networks with time-varying delays. Neural Netw 73:47–57

    MATH  Google Scholar 

  6. Huang C, Long X, Huang L, Fu S (2019) Stability of almost periodic Nicholson’s blowflies model involving patch structure and mortality terms. Can Math Bull. https://doi.org/10.4153/S0008439519000511

    Article  Google Scholar 

  7. Long X, Gong S (2019) New results on stability of Nicholson’s blowflies equation with multiple pairs of time-varying delays. Appl Math Lett. https://doi.org/10.1016/j.aml.2019.106027

    Article  MATH  Google Scholar 

  8. Liu Y, Tong L, Lou J, Lu J, Cao J (2017) Sampled-data control for the synchronization of Boolean control networks. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2017.2779781

    Article  Google Scholar 

  9. Huang C, Qiao Y, Huang L, Agarwal RP (2018) Dynamical behaviors of a food-chain model with stage structure and time delays. Adv Differ Equ 2018:186. https://doi.org/10.1186/s13662-018-1589-8

    Article  MathSciNet  MATH  Google Scholar 

  10. Huang C, Zhang H, Huang L (2019) Almost periodicity analysis for a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality term. Commun Pure Appl Anal 18(6):3337–3349

    MathSciNet  Google Scholar 

  11. Huang C, Zhang H, Cao J, Hu H (2019) Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator. Int J Bifurc Chaos 29(7):1950091 23 Pages

    MathSciNet  MATH  Google Scholar 

  12. Huang C, Yang Z, Yi T, Zou X (2014) On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J Differ Equ 256(7):2101–2114

    MathSciNet  MATH  Google Scholar 

  13. Rakkiyappan R, Sivaranjani R, Velmurugan G, Cao J (2016) Analysis of global \(O(t^{-\alpha })\) stability and global asymptotical periodicity for a class of fractional-order complex-valued neural networks with time varying delays. Neural Netw 77:51–69

    MATH  Google Scholar 

  14. Chen J, Li C, Yang X (2018) Asymptotic stability of delayed fractional-order fuzzy neural networks with impulse effects. J Franklin Inst 355(15):7595–7608

    MathSciNet  MATH  Google Scholar 

  15. 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

    MathSciNet  MATH  Google Scholar 

  16. Liu P, Nie X, Liang J, Cao J (2018) Multiple Mittag-Leffler stability of fractional-order competitive neural networks with Gaussian activation functions. Neural Netw 108:452–465

    Google Scholar 

  17. Yang X, Song Q, Liu Y, Zhao Z (2015) Finite-time stability analysis of fractional-order neural networks with delay. Neurocomputing 152:19–26

    Google Scholar 

  18. Wu R, Hei X, Chen L (2013) Finite-Time stability of fractional-order neural networks with delay. Commun Theor Phys 60:189–193

    MathSciNet  MATH  Google Scholar 

  19. Chua L (1971) Memristor: the missing circuit element. IEEE Trans Circuits Theory 18:507–519

    Google Scholar 

  20. Bernard W (1960) An adaptive adaline neuron using chemical memistors, Stanford Electronics Laboratories Technical Report 1553-2

  21. Strukov D, Snider G, Stewart D, Williams R (2008) The missing memristor found. Nature 453:80–83

    Google Scholar 

  22. Kim H, Sah M, Yang C, Roska T, Chua L (2012) Memristor bridge synapses. Proc IEEE 100(6):2061–2070. https://doi.org/10.1109/JPROC.2011.2166749 [6074916]

    Article  Google Scholar 

  23. Miller K, Nalwa K, Bergerud A, Neihart N, Chaudhary S (2010) Memristive behavior in thin anodic titania. IEEE Electron Device Lett 31(7):737–739

    Google Scholar 

  24. Sun J, Shen Y, Yin Q, Xu C (2012) Compound synchronization of four memristor chaotic oscillator systems and secure communication. Chaos 22(4):1–10

    MATH  Google Scholar 

  25. Corinto F, Ascoli A, Gilli M (2011) Nonlinear dynamics of memristor oscillators. IEEE Trans Circuits Syst I 58(6):1323–1336

    MathSciNet  Google Scholar 

  26. Wang L, Song Q, Liu R, Zhao Z, Alsaadi F (2017) Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with both leakage and time-varying delays. Neurocomputing 245:86–101

    Google Scholar 

  27. Rakkiyappan R, Velmurugan G, Cao J (2014) Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays. Nonlinear Dyn 78:2823–2836

    MathSciNet  MATH  Google Scholar 

  28. Sudbery A (1979) Quaternionic analysis, Math. Proc. Camb. Phil. Soc

  29. Hu J, Zeng C, Tan J (2017) Boundedness and periodicity for linear threshold discrete-time quaternion-valued neural network with time-delays. Neurocomputing 267:417–425

    Google Scholar 

  30. Tu Z, Cao J, Ahmed A, Hayat T (2017) Global dissipativity analysis for delayed quaternion-valued neural networks. Neural Netw 89:97–104

    Google Scholar 

  31. Tu Z, Zhao Y, Ding N, Feng N, Wei Z (2019) Stability analysis of quaternion-valued neural networks with both discrete and distributed delays. Appl Math Comput 343:342–353

    MathSciNet  MATH  Google Scholar 

  32. Tan M, Liu Y, Xu D (2019) Multistability analysis of delayed quaternion-valued neural networks with nonmonotonic piecewise nonlinear activation functions. Appl Math Comput 341:229–255

    MathSciNet  MATH  Google Scholar 

  33. You X, Song Q, Liang J, Liu Y, Alsaadi F (2018) Global \(\mu \)-stability of quaternion-valued neural networks with mixed time-varying delays. Neurocomputing 290:12–25

    Google Scholar 

  34. Samidurai R, Sriraman R, Zhu S (2019) Leakage delay-dependent stability analysis for complex-valued neural networks with discrete and distributed time-varying delays. Neurocomputing 338:262–273

    Google Scholar 

  35. Samidurai R, Sriraman R, Cao J, Tu Z (2018) Effects of leakage delay on global asymptotic stability of complex-valued neural networks with interval time-varying delays via new complex-valued Jensens inequality. Int J Adapt Control Signal Process 32:1294–1312

    MathSciNet  MATH  Google Scholar 

  36. Tan M, Liu Y, Xu D (2019) Multistability analysis of delayed quaternion valued neural networks with nonmonotonic piecewise nonlinear activation functions. Appl Math Comput 341:229–255

    MathSciNet  MATH  Google Scholar 

  37. Zhou Y, Li C, Chen L, Huang T (2018) Global exponential stability of memristive Cohen–Grossberg neural networks with mixed delays and impulse time window. Neurocomputing 275:2384–2391

    Google Scholar 

  38. Ning L, Cao J (2018) Global dissipativity analysis of quaternion-valued memristor-based neural networks with proportional delay. Neurocomputing 321:103–113

    Google Scholar 

  39. Liu Y, Zheng Y, Lu J, Cao J, Rutkowski L (2019) Constrained quaternion-variable convex optimization: a quaternion-valued recurrent neural network approach. IEEE Trans Neural Netw Learn Syst. https://doi.org/10.1109/TNNLS.2019.2916597

    Article  Google Scholar 

  40. 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

    Google Scholar 

  41. Xiao J, Zhong S (2019) Synchronization and stability of delayed fractional-order memristive quaternion-valued neural networks with parameter uncertainties. Neurocomputing 363:321–338

    Google Scholar 

  42. Li X, Ding Y (2017) Razumikhin-type theorems for time-delay systems with persistent impulses. Syst Control Lett 107:22–27

    MathSciNet  MATH  Google Scholar 

  43. Yang X, Li X, Xi Q, Duan P (2018) Review of stability and stabilization for impulsive delayed systems. Math Biosci Eng 15(6):1495–1515

    MathSciNet  MATH  Google Scholar 

  44. Li X, Ho DWC, Cao J (2019) Finite-time stability and settling-time estimation of nonlinear impulsive systems. Automatica 99:361–368

    MathSciNet  MATH  Google Scholar 

  45. Wu H, Zhang X, Xue S, Wang L, Wang Y (2016) LMI conditions to global Mittag-Leffler stability of fractional-order neural networks with impulses. Neurocomputing 193:148–154

    Google Scholar 

  46. Zhang X, Niu P, Ma Y, Wei Y, Li Guoqiang (2017) Global Mittag-Leffler stability analysis of fractional-order impulsive neural networks with one-side Lipschitz condition. Neural Netw 94:67–75

    MATH  Google Scholar 

  47. Wang L, Song Q, Liu Y, Zhao Z, Alsaadi F (2017) Global asymptotic stability of impulsive fractional-order complex-valued neural networks with time delay. Neurocomputing 243:49–59

    Google Scholar 

  48. Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  49. Ye H, Gao J, Ding Y (2007) A generalized Gronwall inequality and its application to a fractional differential equation. J Math Anal Appl 328(2):1075–1081

    MathSciNet  MATH  Google Scholar 

  50. Zhang S, Yu Y, Wang H (2015) Mittag-Leffler stability of fractional-order Hopfield neural networks. Nonlinear Anal Hybrid Syst 16:104–121

    MathSciNet  MATH  Google Scholar 

  51. De la Sen M (2011) About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory and Applications (1), Article ID: 867932, 1-19

  52. Chen B, Chen J (2015) Global asymptotical \(\omega \)-periodicity of a fractional-order non-autonomous neural networks. Neural Netw 68:78–88

    MATH  Google Scholar 

  53. Filippov AF (1988) Differential equations with discontinuous right-hand sides. Kluwer, Dordrecht

    MATH  Google Scholar 

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Authors and Affiliations

  1. Vel Tech High Tech Dr Rangarajan Dr Sakunthala Engineering College, Chennai, 600 062, India

    A. Pratap & J. Dianavinnarasi

  2. Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi, 630 004, India

    R. Raja

  3. Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, 12435, Saudi Arabia

    J. Alzabut

  4. School of Mathematics, Southeast University, Nanjing, 211189, China

    J. Cao

  5. Department of Mathematics, Faculty of Science, Maejo University, Chiang Mai, Thailand

    G. Rajchakit

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  1. A. Pratap
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  2. R. Raja
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  4. J. Dianavinnarasi
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  5. J. Cao
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  6. G. Rajchakit
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Correspondence to J. Cao.

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Pratap, A., Raja, R., Alzabut, J. et al. Finite-Time Mittag-Leffler Stability of Fractional-Order Quaternion-Valued Memristive Neural Networks with Impulses. Neural Process Lett 51, 1485–1526 (2020). https://doi.org/10.1007/s11063-019-10154-1

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