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Strong convergence and almost sure exponential stability of balanced numerical approximations to stochastic delay Hopfield neural networks

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  • Rathinasamy, Anandaraman
  • Mayavel, Pichamuthu

Abstract

This paper deals with the balanced numerical schemes of the stochastic delay Hopfield neural networks. The balanced methods have a strong convergence rate of at least 12 and the balanced schemes have almost sure exponential stability under certain conditions. Under the Lipchitz and linear growth conditions, the balanced Euler methods are proved to have a strong convergence of order 12 in mean-square sense. Using the Lipchitz conditions on the various parameters of the model, based on the semimartingale convergence theorem and some reasonable assumptions, the balanced Euler methods of the stochastic delay Hopfield neural networks are proved to be almost sure exponentially stable. Numerical experiments are provided to illustrate the theoretical results which are derived in this paper. The computational efficiency of the balanced methods is demonstrated by numerical tests and compared to the Euler–Maruyama approximation scheme of the stochastic delay Hopfield neural networks. Furthermore, the obtained numerical results show that the balanced numerical methods of stochastic delay Hopfield neural networks are very efficient with the least error and have the best step size region for almost sure mean square stable.

Suggested Citation

  • Rathinasamy, Anandaraman & Mayavel, Pichamuthu, 2023. "Strong convergence and almost sure exponential stability of balanced numerical approximations to stochastic delay Hopfield neural networks," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    • Handle: RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006476
    DOI: 10.1016/j.amc.2022.127573
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    References listed on IDEAS

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    1. Qian Guo & Wenwen Xie & Taketomo Mitsui, 2013. "Convergence and Stability of the Split-Step -Milstein Method for Stochastic Delay Hopfield Neural Networks," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, April.
    2. Rathinasamy, A. & Narayanasamy, J., 2019. "Mean square stability and almost sure exponential stability of two step Maruyama methods of stochastic delay Hopfield neural networks," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 126-152.
    3. Tan, Jianguo & Tan, Yahua & Guo, Yongfeng & Feng, Jianfeng, 2020. "Almost sure exponential stability of numerical solutions for stochastic delay Hopfield neural networks with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    4. Liu, Linna & Zhu, Quanxin, 2015. "Almost sure exponential stability of numerical solutions to stochastic delay Hopfield neural networks," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 698-712.
    5. G. N. Milstein & Eckhard Platen & H. Schurz, 1998. "Balanced Implicit Methods for Stiff Stochastic Systems," Published Paper Series 1998-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
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    1. Rathinasamy, Anandaraman & Mayavel, Pichamuthu, 2023. "The balanced split step theta approximations of stochastic neutral Hopfield neural networks with time delay and Poisson jumps," Applied Mathematics and Computation, Elsevier, vol. 455(C).
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    4. Xu, Hongjie & Luo, Huantian & Fan, Xu-Qian, 2024. "Stability of stochastic delay Hopfield neural network with Poisson jumps," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).

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