Spatial-temporal dynamics of a non-monotone reaction-diffusion Hopfield’s neural network model with delays

In this paper, the spatial-temporal dynamics of a delayed reaction-diffusion Hopfield’s neural network model (HNNM) under a Neumann boundary condition is considered. Our main concern is the absolute, also called the delay-independent, global or local stability of the trivial and nontrivial steady states of the HNNM with non-monotone activation functions. The dissipativity of the semiflow generated by HNNM is first proved and then the absolute attractivity of both the trivial and the nontrivial steady states is obtained by adopting the idea of connecting the spatial-temporal dynamics of the HNNM with the asymptotic behaviors of a finite-dimensional discrete dynamical system. Specifically, it is shown that the strong attractors of the finite-dimensional discrete dynamical system generated by the nonlinear activation function of the HNNM are indeed the attractors of the corresponding delayed reaction-diffusion HNNM. Numerical simulations are also conducted at last to illustrate the effectiveness of our established results.