Random Attractors of a Stochastic Hopfield Neural Network Model with Delays

Wenjie Hu ^[1] ; Quanxin Zhu ^[1] ; Peter E. Kloeden ^[3] ; Yueliang Duan ^[2]
1. [1] Hunan Normal University
  
  Hunan Normal University
  
  China
2. [2] Shantou University
  
  Shantou University
  
  China
3. [3] Universität Tübingen
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Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 5, 2024
Idioma: inglés
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Resumen
- The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is firstly proved that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous version, which is compact for (where is the delay) by a construction based on the random semiflow generated by the diffusion term due to Mohammed ( Stoch. Stoch. Rep. 29: 89–131, 1990). Then, this version is shown to generate a random dynamical system (RDS) by a Wong–Zakai approximation, after which the existence of a random absorbing set is obtained via uniform apriori estimate of the solutions. Subsequently, the pullback asymptotic compactness of the RDS generated by the stochastic delayed HNNM is established and hence the existence of random attractors is obtained. Sufficient conditions under which the attractors turn out to be an exponential attracting stationary solution are also given. Finally, some numerical simulations illustrate the results.