Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions.

Abstract

In this paper, we investigate the neural networks with a class of nondecreasing piecewise linear activation functions with 2r corner points. It is proposed that the n-neuron dynamical systems can have and only have (2r+1)(n) equilibria under some conditions, of which (r+1)(n) are locally exponentially stable and others are unstable. Furthermore, the attraction basins of these stationary equilibria are estimated. In the case of n=2, the precise attraction basin of each stable equilibrium point can be figured out, and their boundaries are composed of the stable manifolds of unstable equilibrium points. Simulations are also provided to illustrate the effectiveness of our results.