Random recurrent neural networks with delays
type: Item
instance of: scholarly article
title: Random recurrent neural networks with delays
zbMATH Open document ID: 1453.34105
author: Meiyu Sui
author: Yejuan Wang
author: Xiaoying Han
author: Peter E. Kloeden
published in: Journal of Differential Equations
publication date: 2020-07-02T00:00:00Z
review text: This paper is dedicated to study the asymptotic behavior of some class of random functional differential equations of the form \[ \frac{dx_{i}(t,\omega)}{dt}=-x_{i}(t,\omega)+\sum\limits_{j=i-N}^{i+N}\lambda_{ij}(\theta_{t}\omega)F_{i}(t,x_{jt}(\omega))+J_{i}(t),\ i\in\mathbb Z \] with the initial condition \[ x_{i}(t)=\phi_{i}(t-\tau),\ \ t\in [\tau -h,r],\ \ i\in\mathbb Z \] or in the vector form \[ \frac{dx(t,\omega)}{dt}=G(t,\theta_{t}\omega,x_{t})\ \ \mbox{for all}\ \ t\ge \tau,\ \ x(t)=\phi(t-\tau) \ \mbox{for all}\ t\in [\tau-h,\tau],\tag{3} \] where \[ \phi \in l^{2}:=\{x=(x_{i})_{i\in\mathbb Z}:\ \sum\limits_{i\in\mathbb Z}x_{i}^{2}<\infty \},\tag{4} \] \((\Omega,\mathcal F,\mathbb R)\) is a probability space, \((\Omega,\mathbb R,\theta))\) is a random dynamical system on the event space \(\Omega\) and \((Q,\mathbb R,\xi)\) is a deterministic dynamical system on the metric space \(Q\). The authors study the problem \begin{itemize} \item[--] of existence of a solution (at least one) of the Cauchy problem (3) and they give sufficient conditions under which equation (3) generates a non-autonomous random cocycle dynamical system; \item[--] of existence of random pullback attractors for equation (3); \item[--] of periodicity of attractors for periodic equation (3); \item[--] of periodicity of trajectories in the random pullback attractors. \end{itemize} I consider that that these results are new and interesting for the experts in the nonautonomous functional differential equations and random dynamical systems.
Mathematics Subject Classification ID: 34K50
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34K09
Mathematics Subject Classification ID: 34D45
zbMATH DE Number: 7216762
zbMATH Keywords: random recurrent neural network
zbMATH Keywords: random attractor
zbMATH Keywords: multi-valued non compact random dynamical system
zbMATH Keywords: extremal random complete trajectory
zbMATH Keywords: variable delay
zbMATH Keywords: monotone
reviewed by: David Cheban
MaRDI profile type: Publication
full work available at URL: j.jde.2020.06.008
OpenAlex ID: W3036571702
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DOI: 10.1016/J.JDE.2020.06.008
description: scientific article; zbMATH DE number 7216762