Multiple μ-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 R(n) can be divided into several Φη-type subsets. Accordingly, a new set of regions Ωη are proposed, and rigorous mathematical analysis is provided to derive the existence of equilibrium point and its local μ-stability in each Ωη. It concludes that the n-dimensional neural networks can exhibit at least 3(n) equilibrium points and 2(n) of them are μ-stable. Furthermore, due to the compatible property, a set of new conditions are presented to address the dynamics in the remaining 3(n)-2(n) 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 K-level nonlinear activation functions and unbounded time-varying delays, in which there can store (2K+1)(n) equilibrium points, (K+1)(n) of them are locally μ-stable. Numerical examples are given to illustrate the effectiveness of our results.