title: Multiple μ-stability of neural networks with unbounded time-varying delays.
abstract: 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.
authors.affiliation: Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, PR China. Electronic address: wang.lili@mail.shufe.edu.cn.
authors.forename: Lili
authors.initials: L
authors.lastname: Wang
authors.affiliation: School of Computer Sciences, Key Laboratory of Nonlinear Mathematics Science, School of Mathematical Sciences, Fudan University, Shanghai, 200433, PR China. Electronic address: tchen@fudan.edu.cn.
authors.forename: Tianping
authors.initials: T
authors.lastname: Chen
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D016571:Neural Networks, Computer; D013997:Time Factors
keywords: -stability; Multistability; Neural networks; Nonlinear activation function; Unbounded delays