Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions

In this paper, we investigate the exact existence and dynamical behaviors of multiple equilibrium points for delayed competitive neural networks (DCNNs) with a class of nondecreasing piecewise linear activation functions with 2r(r≥1)2r(r≥1) corner points. It is shown that under some conditions, the N  -neuron DCNNs can have and only have (2r+1)N(2r+1)N equilibrium points, (r+1)N(r+1)N of which are locally exponentially stable, based on decomposition of state space, fixed point theorem and matrix theory. In addition, for the activation function with two corner points, the dynamical behaviors of all equilibrium points for 2-neuron delayed Hopfield neural networks(DHNNs) are completely analyzed, and a sufficient criterion derived for ensuring the networks have exactly nine equilibrium points, four of which are stable and others are unstable, by discussing the distribution of roots of the corresponding characteristic equation of the linearized delayed system. Finally, two examples with their simulations are presented to verify the theoretical analysis.