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Global exponential stability and periodicity of reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions

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  • Lu, Jun Guo

Abstract

In this paper, the global exponential stability and periodicity for a class of reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions are addressed by constructing suitable Lyapunov functionals and utilizing some inequality techniques. We first prove global exponential converge to 0 of the difference between any two solutions of the original reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions, the existence and uniqueness of equilibrium is the direct results of this procedure. This approach is different from the usually used one where the existence, uniqueness of equilibrium and stability are proved in two separate steps. Furthermore, we prove periodicity of the reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions. Sufficient conditions ensuring the global exponential stability and the existence of periodic oscillatory solutions for the reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions are given. These conditions are easy to check and have important leading significance in the design and application of reaction–diffusion recurrent neural networks with delays. Finally, two numerical examples are given to show the effectiveness of the obtained results.

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  • Lu, Jun Guo, 2008. "Global exponential stability and periodicity of reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 116-125.
    • Handle: RePEc:eee:chsofr:v:35:y:2008:i:1:p:116-125
    DOI: 10.1016/j.chaos.2007.05.002
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    3. Gani Stamov & Stefania Tomasiello & Ivanka Stamova & Cvetelina Spirova, 2019. "Stability of Sets Criteria for Impulsive Cohen-Grossberg Delayed Neural Networks with Reaction-Diffusion Terms," Mathematics, MDPI, vol. 8(1), pages 1-20, December.
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    10. Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.
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    13. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay," Mathematics, MDPI, vol. 11(14), pages 1-19, July.
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    15. Stamov, Gani & Stamova, Ivanka & Martynyuk, Anatoliy & Stamov, Trayan, 2021. "Almost periodic dynamics in a new class of impulsive reaction–diffusion neural networks with fractional-like derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    16. Lu, Jun Guo & Lu, Lin Ji, 2009. "Global exponential stability and periodicity of reaction–diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1538-1549.
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