Double almost periodicity for high-order Hopfield neural networks with slight vibration in time variables

In this paper, based on almost periodic time scales, we propose the concept of double almost periodicity of neural networks and analyze almost periodic hybrid dynamics for a new type of Hopfield neural networks that is with slight vibration in time variables. Not only can this new type of neural networks unify discrete, continuous and q-difference analysis, but also it initiates an original idea of slight vibration in time variables of neural networks to precisely describe the double almost periodicity for the status of neural networks. Moreover, a new timescale functional is established and some new Lyapunov functional on time scales are constructed to obtain some sufficient conditions for the existence and ψ-exponential stability of almost periodic solutions for this new type of high-order Hopfield neural networks with variable delays. Finally, several numerical examples with simulation are provided to illustrate the effectiveness and feasibility of our results.