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**Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions.**
*(English)*
Zbl 1134.35066

Summary: 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 zero 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 result 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 are given. These conditions are easy to check and have important leading significance in the design and application. Finally, two numerical examples are given to show the effectiveness of the obtained results.

### MSC:

35K57 | Reaction-diffusion equations |

35B35 | Stability in context of PDEs |

35R10 | Partial functional-differential equations |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

### Keywords:

Lyapunov functionals
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\textit{J. G. Lu}, Chaos Solitons Fractals 35, No. 1, 116--125 (2008; Zbl 1134.35066)

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