Multiple μμ-stability of neural networks with unbounded time-varying delays

In this paper, we are concerned with a class of recurrent neural networks with unbounded time-varying delays. Based on the geometrical configuration of activation functions, the phase space RnRn can be divided into several ΦηΦη-type subsets. Accordingly, a new set of regions ΩηΩη are proposed, and rigorous mathematical analysis is provided to derive the existence of equilibrium point and its local μμ-stability in each ΩηΩη. It concludes that the nn-dimensional neural networks can exhibit at least 3n3n equilibrium points and 2n2n of them are μμ-stable. Furthermore, due to the compatible property, a set of new conditions are presented to address the dynamics in the remaining 3n−2n3n−2n subset regions. As direct applications of these results, we can get some criteria on the multiple exponential stability, multiple power stability, multiple log-stability, multiple log–log-stability and so on. In addition, the approach and results can also be extended to the neural networks with KK-level nonlinear activation functions and unbounded time-varying delays, in which there can store (2K+1)n(2K+1)n equilibrium points, (K+1)n(K+1)n of them are locally μμ-stable. Numerical examples are given to illustrate the effectiveness of our results.