Periodic Solution for $$\nabla $$ ∇ -Stochastic High-Order Hopfield Neural Networks with Time Delays on Time Scales

Abstract

In this paper, the comparison theorem and Gronwall’s inequality with $$\nabla $$ -derivative on time scales are constructed. Based on $$\nabla $$ -stochastic integration, the $$\nabla $$ -stochastic high-order Hopfield neural networks with time delays on time scales is introduced and studied. By using contraction mapping principal and differential inequality technique on time scales, some sufficient conditions for the existence and exponential stability of periodic solutions for a class of $$\nabla $$ -stochastic high-order Hopfield neural networks with time delays on time scales are established. Our results show that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors for periodicity. Finally, a numerical example is provided to illustrate the feasibility of our results. The results of this paper are completely new even if the time scale $$\mathbb {T}=\mathbb {R}$$ or $$\mathbb {Z}$$ .