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Global Exponential Stability and Synchronization of Discrete-Time Fuzzy Bidirectional Associative Memory Neural Networks via Mittag-Leffler Difference Approach

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Abstract

In recent years, exponential Euler difference methods have been widely used to discuss the neural network models depicted by integer order differential equations. However, it is rare to study fractional-order neural network models using such methods. Stimulated by this point, this paper firstly establishes Mittag-Leffler Euler difference for fractional-order fuzzy bidirectional associative memory neural networks, which includes exponential Euler difference. Secondly, global exponential stability and synchronization of the derived difference model are investigated. Compared with the classical fractional-order Euler difference method, the fuzzy Mittag-Leffler discrete-time bidirectional associative memory 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 builds a set of theory and methods to study Mittag-Leffler discrete neural networks.

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References

  1. Kosko, B.: Bidirectional associative memories. IEEE Trans. Syst. Man Cybern. 18(49–60), 49–60 (1988)

    MathSciNet  Google Scholar 

  2. Kosko, B.: Adaptive bidirectional associative memories. Appl. Opt. 26, 4947–4960 (1987)

    Google Scholar 

  3. Wu, W.Q., Yang, L.: Impulsive stochastic BAM neural networks on an invariant under a translation time scale. Acta Appl. Math. 169, 647–665 (2020)

    MathSciNet  MATH  Google Scholar 

  4. Arbi, A., Guo, Y.X., Cao, J.D.: Convergence analysis on time scales for HOBAM neural networks in the Stepanov-like weighted pseudo almost automorphic space. Neural Comput. Appl. 33, 3567–3581 (2020)

    Google Scholar 

  5. Sowmiya, C., Raja, R., Zhu, Q., Rajchakit, G.: Further mean-square asymptotic stability of impulsive discrete-time stochastic BAM neural networks with Markovian jumping and multiple time-varying delays. J. Franklin Inst. 356, 561–591 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Han, S.F., Zhang, T.W., Liu, G.X.: Stochastic dynamics of discrete-time fuzzy random BAM neural networks with time delays. Math. Probl. Eng. 2019, 9416234 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Zhang, T.W., Li, Y.K.: Global exponential stability of discrete-time almost automorphic Caputo-Fabrizio BAM fuzzy neural networks via exponential Euler technique. Knowl. Based Syst. 246, Article number: 108675 (2022)

    Google Scholar 

  8. Zhao, Y., Ren, S.S., Kurths, J.G.: Synchronization of coupled memristive competitive BAM neural networks with different time scales. Neurocomputing 427, 110–117 (2021)

    Google Scholar 

  9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Boston (2006)

    MATH  Google Scholar 

  10. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  11. Zhang, T.W., Li, Y.K., Zhou, J.W.: Almost automorphic strong oscillation in time-fractional parabolic equations. Fractal Fract. 7, 88 (2023)

    Google Scholar 

  12. Zhang, T.W., Xiong, L.L.: Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative. Appl. Math. Lett. 101, Article number: 106072 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Kumar, D., Singh, J., Tanwar, K., Baleanu, D.: A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws. Int. J. Heat Mass Transfer 138, 1222–1227 (2019)

    Google Scholar 

  14. Zhang, T.W., Zhou, J.W., Liao, Y.Z.: Exponentially stable periodic oscillation and Mittag-Leffler stabilization for fractional-order impulsive control neural networks with piecewise Caputo derivatives. IEEE Trans. Cybern. 52, 9670–9683 (2022)

    Google Scholar 

  15. El-Saka, H.A.A., Lee, S., Jang, B.: Dynamic analysis of fractional-order predator-prey biological economic system with Holling type II functional response. Nonlinear Dyn. 96, 407–416 (2019)

    MATH  Google Scholar 

  16. Sharma, R., Bhasin, S., Gaur, P., Joshi, D.: A switching-based collaborative fractional order fuzzy logic controllers for robotic manipulators. Appl. Math. Model. 73, 228–246 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Ali, M.S., Narayanan, G., Shekher, V., Alsaedi, A., Ahmad, B.: Global Mittag-Leffler stability analysis of impulsive fractional-order complex-valued BAM neural networks with time varying delays. Commun. Nonlinear Sci. Numer. Simul. 83, Article number: 105088 (2020)

    MathSciNet  MATH  Google Scholar 

  18. Ali, M.S., Hymavathi, M., Rajchakit, G., Saroha, S., Palanisamy, L., Hammachukiattikul, P.: Synchronization of fractional order fuzzy BAM neural networks with time varying delays and reaction diffusion terms. IEEE Access 8, 186551–186571 (2020)

    Google Scholar 

  19. Ali, M.S., Narayanan, G., Sevgen, S., Shekher, V., Arik, S.: Global stability analysis of fractional-order fuzzy BAM neural networks with time delay and impulsive effects. Commun. Nonlinear Sci. Numer. Simul. 78, Article number: 104853 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Stamova, I., Stamov, G., Simeonov, S., Ivanov, A.: Mittag-Leffler stability of impulsive fractional-order bi-directional associative memory neural networks with time-varying delays. Trans. Inst. Meas. Control 40, 3068–3077 (2018)

    Google Scholar 

  21. Wang, F., Yang, Y.Q., Xu, X.Y., Li, L.: Global asymptotic stability of impulsive fractional-order BAM neural networks with time delay. Neural Comput. Appl. 28, 345–352 (2017)

    Google Scholar 

  22. Wu, A.L., Zeng, Z.G., Song, X.G.: Global Mittag-Leffler stabilization of fractional-order bidirectional associative memory neural networks. Neurocomputing 177, 489–496 (2016)

    Google Scholar 

  23. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    MATH  Google Scholar 

  24. Yang, T., Yang, L.B.: The global stability of fuzzy cellular neural networks. IEEE Trans. Cric. Syst. I(43), 880–883 (1996)

    MathSciNet  Google Scholar 

  25. Zhang, T.W., Li, Y.K.: S-asymptotically periodic fractional functional differential equations with off-diagonal matrix Mittag-Leffler function kernels. Math. Comput. Simul. 193, 331–347 (2022)

    MathSciNet  MATH  Google Scholar 

  26. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic system. Phys. Rev. Lett. 64, 821–824 (1990)

    MathSciNet  MATH  Google Scholar 

  27. Lakshmanan, S., Prakash, M., Lim, C.P., Rakkiyappan, R., Balasubramaniam, P., Nahavandi, S.: Synchronization of an inertial neural network with time-varying delays and its application to secure communication. IEEE Trans. Neural Netw. Learn. Syst. 29, 195–207 (2018)

    MathSciNet  Google Scholar 

  28. Cheng, S., Ji, J.C., Zhou, J.: Fast synchronization of directionally coupled chaotic systems. Appl. Math. Model. 37, 127–136 (2013)

    MathSciNet  MATH  Google Scholar 

  29. Chen, L., Yin, H., Huang, T., Yuan, L., Zheng, S., Yin, L.: Chaos in fractional-order discrete neural networks with application to image encryption. Neural Netw. 125, 174–184 (2020)

    Google Scholar 

  30. Xiao, J.Y., Zhong, S.M., Wen, S.P.: Improved approach to the problem of the global Mittag-Leffler synchronization for fractional-order multidimension-valued BAM neural networks based on new inequalities. Neural Netw. 133, 87–100 (2021)

    MATH  Google Scholar 

  31. Muhammadhaji, A., Teng, Z.D.: Synchronization stability on the BAM neural networks with mixed time delays. Int. J. Nonlinear Sci. Numer. Simul. 22, 99–109 (2021)

    MathSciNet  MATH  Google Scholar 

  32. Pratap, A., Raja, R., Cao, J.D., Rihan, F.A., Seadawy, A.R.: Quasi-pinning synchronization and stabilization of fractional order BAM neural networks with delays and discontinuous neuron activations. Chaos Solitons & Fractals 131, Article number: 109491 (2020)

    MathSciNet  MATH  Google Scholar 

  33. Lin, F., Zhang, Z.Q.: Global asymptotic synchronization of a class of BAM neural networks with time delays via integrating inequality techniques. J. Syst. Sci. Complex. 33, 366–382 (2020)

    MathSciNet  MATH  Google Scholar 

  34. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, United Kingdom (2016)

    MATH  Google Scholar 

  35. Huang, Z.K., Mohamad, S., Gao, F.: Multi-almost periodicity in semi-discretizations of a general class of neural networks. Math. Comput. Simul. 101, 43–60 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Hu, P., Huang, C.M.: Delay dependent asymptotic mean square stability analysis of the stochastic exponential Euler method. J. Comput. Appl. Math. 382, Article number: 113068 (2021)

    MathSciNet  MATH  Google Scholar 

  37. Zhang, T.W., Han, S.F., Zhou, J.W.: Dynamic behaviours for semi-discrete stochastic Cohen-Grossberg neural networks with time delays. J. Franklin Inst. 357, 13006–13040 (2020)

    MathSciNet  MATH  Google Scholar 

  38. Hu, J., Wang, J.: Global exponential periodicity and stability of discrete-time complex-valued recurrent neural networks with time-delays. Neural Netw. 66, 119–130 (2015)

    MATH  Google Scholar 

  39. Calvo, M.P., Palencia, C.: A class of explicit multistep exponential integrators for semilinear problems. Numer. Math. 102, 367–381 (2006)

    MathSciNet  MATH  Google Scholar 

  40. Ostermann, A., Thalhammer, M., Wright, W.M.: A class of exponential general linear methods. BIT Numer. Math. 46, 409–431 (2006)

    MathSciNet  MATH  Google Scholar 

  41. Hochbruck, M., Ostermann, A., Schweitzer, J.: Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47, 786–803 (2009)

    MathSciNet  MATH  Google Scholar 

  42. Wang, X.M., Dong, X.G.: A Concise Tutorial on Computational Methods. Science Press, Beijing (2010)

    Google Scholar 

  43. Zhang, T.W., Li, Y.K., Zhou, J.W.: Mittag-Leffler Euler \(\nabla\)-differences for Caputo fractional-order systems. Results Phys. 37, Article number: 105482 (2022)

    Google Scholar 

  44. Zhang, T.W., Li, Y.K.: Exponential Euler scheme of multi-delay Caputo-Fabrizio fractional-order differential equations. Appl. Math. Lett. 124, Article number: 107709 (2022)

    MathSciNet  MATH  Google Scholar 

  45. Guo, B.L., Pu, X.K., Huang, F.H.: Frational Patial Differential Equations and their Numerical Solutions. Science Press, Beijing (2011)

    Google Scholar 

  46. Fěckan, M., Wang, J.R.: Periodic impulsive fractional differential equations. Adv. Nonlinear Anal. 8, 482–496 (2019)

    MathSciNet  MATH  Google Scholar 

  47. Wu, G.C., Zeng, D.Q., Baleanu, D.: Fractional impulsive differential equations: exact solutions, integral equations and short memory case. Fract. Calculus Appl. Anal. 22, 180–192 (2019)

    MathSciNet  MATH  Google Scholar 

  48. Wu, G.C., Deng, Z.G., Baleanu, D., Zeng, D.Q.: New variable-order fractional chaotic systems for fast image encryption. Chaos 29, Article number: 083103 (2019)

    MathSciNet  MATH  Google Scholar 

  49. Liu, P., Zeng, Z.G., Wang, J.: Global synchronization of coupled fractional-order recurrent neural networks. IEEE Trans. Neural Netw. Learn. Syst. 30, 2358–2368 (2019)

    MathSciNet  Google Scholar 

  50. Dai, X.J., Xiao, A.G.: A note on Euler method for the overdamped generalized Langevin equation with fractional noise. Appl. Math. Lett. 111, Article number: 106669 (2021)

    MathSciNet  MATH  Google Scholar 

  51. Yang, Z.W., Zheng, X.C., Zhang, Z.Q., Wang, H.: Strong convergence of a Euler-Maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noise. Chaos Solitons Fractals 142, Article number: 110392 (2021)

    MathSciNet  MATH  Google Scholar 

  52. Sharma, R.K., Xiong, X., Kamal, S., Ghosh, S.: Discrete-time super-twisting fractional-order differentiator with implicit euler method. IEEE Tran. Circ. Syst. II 68, 1238–1242 (2021)

    Google Scholar 

  53. Wang, J.R., Fečkan, M., Zhou, Y.: Presentation of solutions of impulsive fractional Langevin equations and existence results. Eur. Phys. J. Spec. Top. 222, 1857–1874 (2013)

  54. Garrappa, R., Popolizio, M.: Generalized exponential time differencing methods for fractional order problems. Comput. Math. Appl. 62, 876–890 (2011)

    MathSciNet  MATH  Google Scholar 

  55. Zhao, L.H., Wen, S.P., Xu, M., Shi, K.B., Zhu, S., Huang, T.W.: PID control for output synchronization of multiple output coupled complex networks. IEEE Trans. Netw. Sci. Eng. 9, 1553–1566 (2022)

    MathSciNet  Google Scholar 

  56. Wen, S.P., Ni, X.Z., Wang, H.M., Zhu, S., Shi, K.B., Huang, T.W.: Observer-based adaptive synchronization of multiagent systems with unknown parameters under attacks. IEEE Trans. Neural Netw. Learn. Syst. 33, 3109–3119 (2022)

    MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by Scientific Research Fund Project of Education Department of Yunnan Province under Grant No. 2021J0835.

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  1. Oxbridge College, Kunming University of Science and Technology, Kunming, 650106, China

    Yuntao Liu

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Liu, Y. Global Exponential Stability and Synchronization of Discrete-Time Fuzzy Bidirectional Associative Memory Neural Networks via Mittag-Leffler Difference Approach. Int. J. Fuzzy Syst. 25, 1922–1934 (2023). https://doi.org/10.1007/s40815-023-01482-5

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