Periodic solutions and its exponential stability of reaction-diffusion recurrent neural networks with continuously distributed delays.

*(English)*Zbl 1094.35128Summary: Both exponential stability and periodic oscillatory solutions are considered for reaction-diffusion recurrent neural networks with continuously distributed delays. By constructing suitable Lyapunov functional, using \(M\)-matrix theory and some analysis techniques, some simple sufficient conditions are given ensuring the global exponential stability and the existence of periodic oscillatory solutions for reaction-diffusion recurrent neural networks with continuously distributed delays. Moreover, the exponential convergence rate is estimated. These results have leading significance in the design and applications of globally exponentially stable and periodic oscillatory neural circuits for reaction-diffusion recurrent neural networks with continuously distributed delays. Two examples are given to illustrate the correctness of the obtained results.

##### MSC:

35Q80 | Applications of PDE in areas other than physics (MSC2000) |

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

35K57 | Reaction-diffusion equations |

35B35 | Stability in context of PDEs |

##### Keywords:

Recurrent neural networks; Reaction-diffusion; Distributed delays; Global exponential stability; Periodic oscillatory solutions; Lyapunov functional; Periodic oscillatory neural circuits
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\textit{Q. Song} et al., Nonlinear Anal., Real World Appl. 7, No. 1, 65--80 (2006; Zbl 1094.35128)

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