On the global attractivity of non-autonomous neural networks with a distributed delay,Nonlinearity

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On the global attractivity of non-autonomous neural networks with a distributed delay
Nonlinearity ( IF 1.7 ) Pub Date : 2021-04-22 , DOI: 10.1088/1361-6544/abbc61
Leonid Berezansky ₁ , Elena Braverman ₂

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We consider a system of s nonlinear differential equations with a distributed delay $\frac{\mathrm{d}{x}_{i}}{\mathrm{d}t}={g}_{i}\left(t\right)\left[{\int }_{{h}_{i1}\left(t\right)}^{t}{\mathrm{d}}_{{\tau }_{1}}{r}_{i1}\left(t,{\tau }_{1}\right)\dots {\int }_{{h}_{is}\left(t\right)}^{t}{f}_{i}\left({x}_{1}\left({\tau }_{1}\right),{x}_{2}\left({\tau }_{2}\right),\dots ,{x}_{s}\left({\tau }_{s}\right)\right){\mathrm{d}}_{{\tau }_{s}}{r}_{is}\left(t,{\tau }_{s}\right)-{x}_{i}\left(t\right)\right]$ and obtain global asymptotic stability conditions, which are independent of delays. The ideas of the proofs are based on the notion of a strong attractor of a vector difference equation associated with a nonlinear vector differential equation. The results are applied to Hopfield neural networks and to compartment-type models of population dynamics with Nicholson’s blowflies growth law.

中文翻译：

具有分布式延迟的非自治神经网络的全局吸引力

我们考虑具有分布式延迟的s 个非线性微分方程组， $\frac{\mathrm{d}{x}_{i}}{\mathrm{d}t}={g}_{i}\left(t\right)\left[{\int }_{{ h}_{i1}\left(t\right)}^{t}{\mathrm{d}}_{{\tau }_{1}}{r}_{i1}\left(t,{\ tau }_{1}\right)\dots {\int }_{{h}_{is}\left(t\right)}^{t}{f}_{i}\left({x}_ {1}\left({\tau }_{1}\right),{x}_{2}\left({\tau }_{2}\right),\dots ,{x}_{s} \left({\tau }_{s}\right)\right){\mathrm{d}}_{{\tau }_{s}}{r}_{is}\left(t,{\tau }_{s}\right)-{x}_{i}\left(t\right)\right]$ 并获得与延迟无关的全局渐近稳定条件。证明的思想基于与非线性矢量微分方程相关联的矢量差分方程的强吸引子的概念。将结果应用于 Hopfield 神经网络和具有 Nicholson 的苍蝇生长规律的种群动态的隔室类型模型。

更新日期：2021-04-22

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