Effect of time-varying delays' dynamic characteristics on the stability of Hopfield neural networks

This manuscript investigated the stability of Hopfield neural networks with time-varying transmission delays and leakage delays, and specially discussed the impact of the time-varying delays' dynamic characteristics on network stability. First, to characterize the dynamic features of time-varying delays transitioning continuously between short and large delays, two key parameters were introduced: a critical threshold for distinguishing whether the time-varying delays are short or large delays, and the ratio of the measure of the union of time periods, in which the time-varying delays appear as short delays, to the measure of the whole time interval. Then, by utilizing the Lyapunov stability method, some sufficient conditions for the global exponential stability of the neural networks were derived. It is pointed out that when the measure of time periods in which the time-varying delays appear as short delays is large enough, the above two parameters related to the time-varying delays' dynamic characteristics will have an important impact on the system stability, and the upper bound that the time-varying delays can achieve in the whole time interval will no longer be the dominant factor influencing stability. Additionally, the relationship between the leakage delays and the stabilization ability of the negative feedback terms was explored. Two admissible upper bounds were presented, below which the leakage delays do not completely undermine the capacity of negative feedback terms to stabilize the neural networks. Finally, some simulation experiments were conducted to validate our theoretical findings.