Multistability and complete convergence analysis on high-order neural networks with a class of nonsmooth activation functions

In this paper, we are concerned with a class of high-order neural networks (HONNs) with nonsmooth activation functions. A set of new sufficient conditions ensuring the coexistence of 3n3n equilibrium points and the local stability of 2n2n equilibrium points are proposed, which reveal that the high-order interactions between neurons also play an important role on the multistability of HONNs. Besides, every solution is shown to converge to a certain equilibrium point, that is, the systems are also completely stable. Furthermore, for the 2-neuron neural networks, we can get that the stable manifolds of unstable equilibrium points constitute the boundaries of attraction basins of stable equilibrium points, despite the nonlinearity of high-order items of HONNs. Several numerical examples are presented to illustrate the effectiveness of our criteria.