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,Phi(n)-type subsets. Accordingly, a new set of regions ohm(n) are proposed, and rigorous mathematical analysis is provided to derive the existence of equilibrium point and its local mu-stability in each Q. It concludes that the n-dimensional neural networks can exhibit at least 3(n) equilibrium points and 2 of them are A-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 timevarying delays, in which there can store (2K+1)(n) equilibrium points, (K +1)(n) of them are locally.mu-stable. Numerical examples are given to illustrate the effectiveness of our results. (C) 2014 Elsevier Ltd. All rights reserved.